AN INTRODUCTION TO THE

USA COMPUTING OLYMPIAD

Darren Yao
2020

R (cid:140) (cid:135)

Java Edition

Foreword

This book was written as a comprehensive and up-to-date training resource for the USA
Computing Olympiad. The goal was to create an “Art of Problem Solving” of sorts for the
USACO: a one-stop-shop guide to prepare competitive programmers for the Bronze and Silver
divisions of the USACO contests.

My primary motivation for writing this book was the struggle to find the right resources
when I first started doing USACO contests. When I eventually reached the Platinum division,
new competitors often asked me for help in structuring their competitive programming
practice. Since I always found myself explaining that the USACO lacked comprehensive
training resources, I decided to write this book.

I would like to thank a number of people for their contributions to this book. In particular,
Michael Cao for writing sections 10.6 and 10.7 and helping with content revisions, Jason Chen
for writing section 14.2 and extensive help with both content and LaTeX formatting, and
Aaryan Prakash, Rishab Parthasarathy, Kevin Wang, and Stephanie Wu for their valuable
and constructive feedback on early draft versions of the book.

I’d also like to thank the USACO discord community for supporting me through my
competitive programming journey; it was because of them that my competitive programming
successes, and this book, are possible.

Cover design by Dylan Yu.

Author’s Profile

Darren Yao is a USACO Platinum competitor. You can find his website at https:

//darrenyao.com/.

Copyright c(cid:13)2020 by Darren Yao

All rights reserved. No part of this book may be reproduced or used in any manner without
the prior written permission from the copyright owner.

i

Contents

I Basic Techniques

1 The Beginning

1.1 Competitive Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Contests and Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Competitive Programming Practice . . . . . . . . . . . . . . . . . . . . . . .
1.4 About This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Elementary Techniques

2.1
Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Time/Space Complexity and Algorithm Analysis

3.1 Big O Notation and Complexity Calculations . . . . . . . . . . . . . . . . . .
3.2 Common Complexities and Constraints . . . . . . . . . . . . . . . . . . . . .

4 Built-in Data Structures

4.1 Dynamic Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Stacks and the Various Types of Queues
. . . . . . . . . . . . . . . . . . . .
4.3 Sets and Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

II Bronze

5 Simulation

5.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Complete Search

6.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Generating Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTS

7 Additional Bronze Topics

7.1 Square and Rectangle Geometry . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Ad-hoc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

III Silver

8 Sorting and comparators

8.1 Comparators
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Sorting by Multiple Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 Greedy Algorithms

Introductory Example: Studying Algorithms . . . . . . . . . . . . . . . . . .
9.1
9.2 Example: The Scheduling Problem . . . . . . . . . . . . . . . . . . . . . . .
9.3 Failure Cases of Greedy Algorithms . . . . . . . . . . . . . . . . . . . . . . .
9.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Graph Theory

10.1 Graph Basics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Graph Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4 Graph Traversal Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5 Floodfill
. . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6 Disjoint-Set Data Structure
10.7 Bipartite Graphs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 Prefix Sums

11.1 Prefix Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Two Dimensional Prefix Sums . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 Binary Search

12.1 Binary Search on the Answer
. . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 Elementary Number Theory

13.1 Prime Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 GCD and LCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTS

14 Additional Silver Topics

14.1 Two Pointers
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 Line Sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3 Bitwise Operations and Subsets . . . . . . . . . . . . . . . . . . . . . . . . .
14.4 Ad-hoc Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IV Problem Set

15 Parting Shots

iv

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Part I

Basic Techniques

1

Chapter 1

The Beginning

1.1 Competitive Programming

Welcome to the world of competitive programming! If you’ve had some basic programming
experience with Java (perhaps at the level of an introductory course like AP Computer Science
A), and are interested in competitive programming, then this book is for you. (If your primary
language is C++, we also have a C++ edition of this book; please refer to that instead). If
you currently do not know how to code, there are numerous resources available online to help
you learn.

This book aims to guide you through your competitive programming journey by providing
a framework in which to learn the important contest topics. From competitive programming,
not only do you improve at programming, but you improve your problem-solving skills which
will help you in other areas. If at any point you have questions, feedback, or notice any
mistakes, please contact me at darren.yao@gmail.com. Best of luck, and enjoy the ride!

The goal of competitive programming is to write code to solve given problems quickly.
These problems are not open problems; they are problems that are designed to be solved in
the short timeframe of a contest, and have already been solved by the problem writer and
testers. In general, each problem in competitive programming is solved by a two-step process:
coming up with the algorithm, and then implementing it into working code. The degree of
mathematics knowledge varies from contest to contest, but generally the level of mathematics
required is relatively elementary, and we will review important topics in this book.

A contest generally lasts for several hours, and consists of a set of problems. For each
problem, when you complete your code, you submit it to a grader, which checks the answers
calculated by the your program against a set of predetermined test cases. For each problem,
you are given a time limit and a memory limit that your program must satisfy. Grading
varies between contests; sometimes there is partial credit for passing some cases, while other
times grading is all-or-nothing. For those of you with experience in software development,
note that competitive programming is quite different, as the goal is to write programs that
compute the correct answer, run quickly, and can be implemented quickly. Note that nowhere
was maintainability of code mentioned. This means that you should throw away everything
you know about traditional code writing; you don’t need to bother documenting your code,
because it only needs to be readable to you, during the contest.

2

CHAPTER 1. THE BEGINNING

3

1.2 Contests and Resources

The USA Computing Olympiad is a national programming competition that occurs four
times a year, with December, January, February, and US Open contests. The regular contests
are four hours long, and the US Open is five hours long. Each contest contains three problems.
Solutions are evaluated and scored against a set of predetermined test cases that are not
visible to the student. Scoring is out of 1000 points, with each problem being weighted
equally. There are four divisions of contests: Bronze, Silver, Gold, and Platinum. After each
contest, students who meet the contest-dependent cutoff for promotion will compete in the
next division for future contests.

While this book is primarily focused on the USACO, CodeForces is another contest
programming platform that many students use for practice. CodeForces holds 2-hour contests
very frequently, which are more focused on fast solving compared to USACO. However, we
do think CodeForces is a valuable training platform, so many exercises and problems will
come from there. We encourage you to create a CodeForces account and solve the provided
problems there. CodeForces submissions are all-or-nothing; unlike USACO, there is no partial
credit and you only receive credit for a problem if you pass all of the test cases.

We will also include some exercises from Antti Laaksonen’s website CSES. It contains a
selection of standard problems that you can use to learn and practice well-known algorithms
and techniques. You should note that CSES’s grader is very slow, so don’t worry if you
encounter a Time Limit Exceeded verdict; as long as you pass the majority of test cases
within the time limit, and your time complexity is reasonable, you can consider the problem
solved, and move on.

1.3 Competitive Programming Practice

Reaching a high level in competitive programming requires dedication and motivation.
For many people, their practice is inefficient because they do problems that are too easy, too
hard, or simply of the wrong type. This book aims to correct that by providing comprehensive
problem sets for each topic covered on the USA Computing Olympiad, as well as an extensive
selection of problems across all topics in the final chapter.

In the lower divisions, most problems use relatively elementary algorithms; the main
challenge is deciding which algorithm to use, and implementing it correctly. In a contest,
you should spend the bulk of your time thinking about the problem and coming up with the
algorithm, rather than typing code. Thus, you should practice your implementation skills,
so that during the contest, you can implement the algorithm quickly and correctly, without
resorting to debugging.

On Exercises and Practice Problems

You improve at competitive programming by solving problems, so we strongly recommend
that you make use of the included exercises in each section before moving on. Some of the
problems will be easy, and some of them will be hard. This is because problems that you
practice with should be of the appropriate difficulty. You don’t necessarily need to complete
all the exercises at the end of each chapter, just do what you think is right for you. A

CHAPTER 1. THE BEGINNING

4

problem at the right level of difficulty should be one of two types: either you struggle with
the problem for a while before coming up with a working solution, or you miss it slightly and
need to consult the solution for some small part. If you instantly come up with the solution,
a problem is likely too easy, and if you’re missing multiple steps, it might be too hard.

In general, especially on harder problems, I think it’s fine to read the solution relatively
early on, as long as you’re made several different attempts at it and you can learn effectively
from the solution.

• On a bronze problem, read the solution after 15-20 minutes of no meaningful progress,

after you’ve exhausted every idea you can think of.

• On a silver problem, read the solution after 30-40 minutes of no meaningful progress.

When you get stuck and consult the solution, you should not read the entire solution at
once, and you certainly shouldn’t look at the solution code. Instead, it’s better to read the
solution step by step until you get unstuck, at which point you should go back and finish the
problem, and implement it yourself. Reading the full solution or its code should be seen as a
last resort.

IDEs and Text Editors

Here’s some IDEs and text editors often used by competitive programmers:

• Java: Visual Studio Code or IntelliJ/Eclipse

• C++: Visual Studio Code, CodeBlocks, vim/gvim, Sublime Text.

• Do not use online IDEs that display your code publicly, like the free version of ideone.

This allows other users to copy your code, and you may get flagged for cheating.

1.4 About This Book

This book aims to prepare students for the Bronze and Silver division of the USACO, with
the goal of qualifying for Gold. We will do this by covering all the necessary algorithms, data
structures, and skills to pass the Bronze and Silver contests. Many examples and practice
problems have been provided; these are the most important part of studying competitive
programming, so make sure you pay careful attention to the examples and solve the practice
problems, which usually come from previous USACO contests. This book is intended for
those who have some programming experience – Basic knowledge of Java at the level of
an introductory class like AP Computer Science is expected. This book begins with some
necessary background knowledge, which is then followed by lessons on common topics that
appear on the Bronze and Silver divisions of USACO, and then examples. At the end of
each chapter will be a set of problems from USACO, CodeForces, and CSES, where you can
practice what you’ve learned in the chapter.

The primary purpose of this book is to compile all of the topics needed for a beginner in
one book, and provide all the resources needed, to make the process of studying for contests
easier.

Chapter 2

Elementary Techniques

2.1

Input and Output

In your CS classes, you’ve probably implemented input and output using standard input
and standard output, or using Scanner to read input and System.out.print to print output.
In CodeForces and CSES, input and output are standard, and the above methods work.
However, Scanner and System.out.print are slow when we have to handle inputting and
outputting tens of thousands of lines. Thus, we use BufferedReader and PrintWriter
instead, which are faster because they buffer the input and output and handle it all at once
as opposed to parsing each line individually.

However, in USACO, input is read from a file called problemname.in, and printing
output to a file called problemname.out. Note that you’ll have to rename the .in and .out
files. Essentially, replace every instance of the word template in the word below with the
input/output file name, which should be given in the problem.

In order to test a program, create a file called problemname.in, and then run the program.

The output will be printed to problemname.out.

Below, we have included Java example code for input and output in USACO. We import
the entire util and io libraries for ease of use. The template is intentionally kept short so
you can type it out, since use of prewritten code is not allowed in USACO as of the 2020-2021
season.

import java.io.*;
import java.util.*;

public class template {

public static void main(String[] args) throws IOException {

BufferedReader r = new BufferedReader(new

FileReader("template.in"));

(cid:44)→
PrintWriter pw = new PrintWriter(new BufferedWriter(new

(cid:44)→

FileWriter("template.out")));

StringTokenizer st = new StringTokenizer(r.readLine());
int n = Integer.parseInt(st.nextToken());

5

CHAPTER 2. ELEMENTARY TECHNIQUES

6

r.close();

pw.close();

}

}

We have several important functions that are used in reading input and printing output:

Method

Description

r.readLine()
st.nextToken()
Integer.parseInt

Reads the next line of the input
Reads the next token (up to a whitespace) and returns as a ‘String‘.
Converts the ‘String‘ returned by the ‘StringTokenizer‘ to an ‘int‘.

Double.parseDouble Converts the ‘String‘ returned by the ‘StringTokenizer‘ to a ‘double‘.

Long.parseLong
pw.println()
pw.print()

Converts the ‘String‘ returned by the ‘StringTokenizer‘ to a ‘long‘
Prints the argument to designated output stream and adds newline
Prints the argument to designated output stream

For example, if we’re reading the following input,

1 2 3

our code (inside the main method) will look like this:

BufferedReader r = new BufferedReader(new FileReader("template.in"));
PrintWriter pw = new PrintWriter(new BufferedWriter(new

(cid:44)→

FileWriter("template.out")));

StringTokenizer st = new StringTokenizer(r.readLine());
int a = Integer.parseInt(st.nextToken());
int b = Integer.parseInt(st.nextToken());
int c = Integer.parseInt(st.nextToken());

r.close();
pw.close();

Now, let’s suppose we wanted to read this input, which is presented on different lines,

with different data types:

100000000000
SFDFSDFSDFD
3

Then our code would be

CHAPTER 2. ELEMENTARY TECHNIQUES

7

BufferedReader r = new BufferedReader(new FileReader("template.in"));
PrintWriter pw = new PrintWriter(new BufferedWriter(new

(cid:44)→

FileWriter("template.out")));

StringTokenizer st = new StringTokenizer(r.readLine());
long a = Long.parseLong(st.nextToken());
st = new StringTokenizer(r.readLine());
String b = st.nextToken();
st = new StringTokenizer(r.readLine());
int c = Integer.parseInt(st.nextToken());

r.close();
pw.close();

Note how we have to re-declare the ‘StringTokenizer‘ every time we read in a new line.

For CodeForces, CSES, and other contests that use standard input and output, here is a

nicer template, which essentially functions as a faster Scanner:

import java.io.*;
import java.util.*;

public class template {

static class InputReader {
BufferedReader reader;
StringTokenizer tokenizer;

public InputReader(InputStream stream) {

reader = new BufferedReader(new InputStreamReader(stream), 32768);
tokenizer = null;

}

String next() { // reads in the next string

while (tokenizer == null || !tokenizer.hasMoreTokens()) {

try {

tokenizer = new StringTokenizer(reader.readLine());

} catch (IOException e) {

throw new RuntimeException(e);

}

}
return tokenizer.nextToken();

}

public int nextInt() { // reads in the next int

return Integer.parseInt(next());

CHAPTER 2. ELEMENTARY TECHNIQUES

8

}

public long nextLong() { // reads in the next long

return Long.parseLong(next());

}

public double nextDouble() { // reads in the next double

return Double.parseDouble(next());

}

}

static InputReader r = new InputReader(System.in);
static PrintWriter pw = new PrintWriter(System.out);

public static void main(String[] args) {

// YOUR CODE HERE

pw.close(); // flushes the output once printing is done

}

}

Here’s a brief description of the methods in our InputReader class, with an instance r,

and PrintWriter with an instance pw.

Method

r.next()
r.nextInt()
r.nextLong()

Description

Reads the next token (up to a whitespace) and returns a String
Reads the next token (up to a whitespace) and returns as an int
Reads the next token (up to a whitespace) and returns as a long

r.nextDouble() Reads the next token (up to a whitespace) and returns as a double
Prints the argument to designated output stream and adds newline
Prints the argument to designated output stream

pw.println()
pw.print()

Here’s an example to show how input/output works. Let’s say we want to write a program

that takes three numbers as input and prints their sum.

// InputReader template code above
static InputReader r = new InputReader(System.in);
static PrintWriter pw = new PrintWriter(System.out);

public static void main(String[] args) {

int a = r.nextInt();
int b = r.nextInt();
int c = r.nextInt()
pw.println(a + b + c);

CHAPTER 2. ELEMENTARY TECHNIQUES

9

pw.close();

}

2.2 Data Types

There are several main data types that are used in contests: 32-bit and 64-bit integers,

floating point numbers, booleans, characters, and strings.

The 32-bit integer supports values between −2 147 483 648 and 2 147 483 647, which is
roughly equal to ± 2×109. If the input, output, or any intermediate values used in calculations
exceed the range of a 32-bit integer, then a 64-bit integer must be used. The range of the
64-bit integer is between −9 223 372 036 854 775 808 and 9 223 372 036 854 775 807 which is
roughly equal to ± 9 × 1018. Contest problems are usually set such that the 64-bit integer is
sufficient. If it’s not, the problem will ask for the answer modulo m, instead of the answer
itself, where m is a prime. In this case, make sure to use 64-bit integers, and take the
remainder of x modulo m after every step using x %= m;.

Floating point numbers are used to store decimal values. It is important to know that
floating point numbers are not exact, because the binary architecture of computers can only
store decimals to a certain precision. Hence, we should always expect that floating point
numbers are slightly off. Contest problems will accommodate this by either asking for the
greatest integer less than 10k times the value, or will mark as correct any output that is
within a certain (cid:15) of the judge’s answer.

Boolean variables have two possible states: true and false. We’ll usually use booleans to
mark whether a certain process is done, and arrays of booleans to mark which components of
an algorithm have finished.

Character variables represent a single Unicode character. They are returned when you
access the character at a certain index within a string. Characters are represented using the
ASCII standard, which assigns each character to a corresponding integer; this allows us to do
arithmetic with them, for example, System.out.println('f' - 'a'); will print 5.

Strings are effectively arrays of characters. You can easily access the character at a certain
index and take substrings of the string. String problems on USACO are generally very easy
and don’t involve any special data structures.

Chapter 3

Time/Space Complexity and
Algorithm Analysis

In programming contests, there is a strict limit on program runtime. This means that
in order to pass, your program needs to finish running within a certain timeframe. For
USACO, this limit is 4 seconds for Java submissions. A conservative estimate for the number
of operations the grading server can handle per second is 108.

3.1 Big O Notation and Complexity Calculations

We want a method of characterizing how many operations it takes to run each algorithm,
in terms of the input size n. Fortunately, this can be done relatively easily using Big O
notation, which expresses worst-case complexity as a function of n, as n gets arbitrarily large.
Complexity is an upper bound for the number of steps an algorithm requires, as a function of
the input size. In Big O notation, we denote the complexity of a function as O(f (n)), where
f (n) is a function without constant factors or lower-order terms. We’ll see some examples of
how this works, as follows.

The following code is O(1), because it executes a constant number of operations.

int a = 5;
int b = 7;
int c = 4;
int d = a + b + c + 153;

Input and output operations are also assumed to be O(1).

In the following examples, we assume that the code inside the loops is O(1).
The time complexity of loops is the number of iterations that the loop runs multiplied by

the amount of operations per iteration. The following code examples are both O(n).

for(int i = 1; i <= n; i++){

// constant time code here

}

10

CHAPTER 3. TIME/SPACE COMPLEXITY AND ALGORITHM ANALYSIS

11

int i = 0;
while(i < n){

// constant time node here
i++;

}

Because we ignore constant factors and lower order terms, for loops where we loop up to

5n + 17 or n + 457737 would also be O(n):

We can find the time complexity of multiple loops by multiplying together the time
complexities of each loop. The following example is O(nm), because the outer loop runs O(n)
iterations and the inner loop O(m).

for(int i = 1; i <= n; i++){

for(int j = 1; j <= m; j++){

// constant time code here

}

}

If an algorithm contains multiple blocks, then its time complexity is the worst time
complexity out of any block. For example, if an algorithm has an O(n) block and an O(n2)
block, the overall time complexity is O(n2).

Functions of different variables generally are not considered lower-order terms with respect
to each other, so we must include both terms. For example, if an algorithm has an O(n2)
block and an O(nm) block, the overall time complexity would be O(n2 + nm).

3.2 Common Complexities and Constraints

Complexity factors that come from some common algorithms and data structures are as

follows:

• Mathematical formulas that just calculate an answer: O(1)

• Unordered set/map: O(1) per operation

• Binary search: O(log n)

• Ordered set/map or priority queue: O(log n) per operation

• Prime factorization of an integer, or checking primality or compositeness of an integer:

√

O(

n)

• Reading in n items of input: O(n)

• Iterating through an array or a list of n elements: O(n)

CHAPTER 3. TIME/SPACE COMPLEXITY AND ALGORITHM ANALYSIS

12

• Sorting: usually O(n log n) for default sorting algorithms (mergesort, for example

Collections.sort or Arrays.sort on objects)

• Java Quicksort (Arrays.sort function on primitives on pathological worst-case data

sets, don’t use this in CodeForces rounds. . . ): O(n2).

• Iterating through all subsets of size k of the input elements: O(nk). For example,

iterating through all triplets is O(n3).

• Iterating through all subsets: O(2n)

• Iterating through all permutations: O(n!)

Here are conservative upper bounds on the value of n for each time complexity. You can
probably get away with more than this, but this should allow you to quickly check whether
an algorithm is viable.

n

Possible complexities

n ≤ 10
n ≤ 20
n ≤ 80
n ≤ 400
n ≤ 7500
n ≤ 7 · 104
n ≤ 5 · 105
n ≤ 5 · 106
n ≤ 1012
n ≤ 1018 O(log2 n), O(log n), O(1)

O(n!), O(n7), O(n6)
O(2n · n), O(n5)
O(n4)
O(n3)
O(n2)
√
n)
O(n
O(n log n)
O(n)
n log n), O(

O(

n)

√

√

Chapter 4

Built-in Data Structures

A data structure determines how data is stored. (is it sorted? indexed? what operations
does it support?) Each data structure supports some operations efficiently, while other
operations are either inefficient or not supported at all. This chapter introduces the data
structures in the Java standard library that are frequently used in competitive programming.
Java default Collections data structures are designed to store any type of object. However,
we usually don’t want this; instead, we want our data structures to only store one type of
data, like integers, or strings. We do this by putting the desired data type within the <>
brackets when declaring the data structure, as follows:

ArrayList<String> list = new ArrayList<String>();

This creates an ArrayList structure that only stores objects of type String.
For our examples below, we will primarily use the Integer data type, but note that you
can have Collections of any object type, including Strings, other Collections, or user-defined
objects.

Collections data types always contain an add method for adding an element to the
collection, and a remove method which removes and returns a certain element from the
collection. They also support the size() method, which returns the number of elements in
the data structure, and the isEmpty() method, which returns true if the data structure is
empty, and false otherwise.

4.1 Dynamic Arrays

You’re probably already familiar with regular (static) arrays. Now, there are also dynamic
arrays (ArrayList in Java) that support all the functions that a normal array does, and can
resize itself to accommodate more elements. In a dynamic array, we can also add and delete
elements at the end in O(1) time.

However, we need to be careful that we only add elements to the end of the ArrayList;

insertion and deletion in the middle of the ArrayList is O(n).

13

CHAPTER 4. BUILT-IN DATA STRUCTURES

14

ArrayList<Integer> list = new ArrayList<Integer>(); // declare the dynamic array
list.add(2); // [2]
list.add(3); // [2, 3]
list.add(7); // [2, 3, 7]
list.add(5); // [2, 3, 7, 5]
list.set(1, 4); // sets element at index 1 to 4 -> [2, 4, 7, 5]
list.remove(1); // removes element at index 1 -> [2, 7, 5]
// this remove method is O(n); to be avoided
list.add(8); // [2, 7, 5, 8]
list.remove(list.size()-1); // [2, 7, 5]
// here, we remove the element from the end of the list; this is O(1).
System.out.println(list.get(2)); // 5

To iterate through a static or dynamic array, we can use either the regular for loop or the

for-each loop.

Arrays.sort(arr) is used to sort a static array, and Collections.sort(list) a dynamic

array. The default sort function sorts the array in ascending order.

In array-based contest problems, we’ll use one-, two-, and three-dimensional static arrays
most of the time. However, we can also have static arrays of dynamic arrays, dynamic arrays
of static arrays, and so on. Usually, the choice between a static array and a dynamic array is
just personal preference.

4.2 Stacks and the Various Types of Queues

Stacks

A stack is a Last In First Out (LIFO) data structure that supports three operations:
push, which adds an element to the top of the stack, pop, which removes an element from
the top of the stack, and peek, which retrieves the element at the top without removing it,
all in O(1) time. Think of it like a real-world stack of papers.

Stack<Integer> s = new Stack<Integer>();
s.push(1); // [1]
s.push(13); // [1, 13]
System.out.println(s.size()); // 2
s.pop(); // [1]
System.out.println(s.peek()); // 1
s.pop(); // []
System.out.println(s.size()); // 0

Queues

A queue is a First In First Out (FIFO) data structure that supports three operations of
add, insertion at the back of the queue, poll, deletion from the front of the queue, and peek,

CHAPTER 4. BUILT-IN DATA STRUCTURES

15

which retrieves the element at the front without removing it, all in O(1) time. Java doesn’t
actually have a Queue class; it’s only an interface. The most commonly used implementation
is the LinkedList, declared as follows: Queue q = new LinkedList();.

Queue<Integer> q = new LinkedList<Integer>();
q.add(1); // [1]
q.add(3); // [3, 1]
q.poll(); // [3]
q.add(4); // [4, 3]
System.out.println(q.size()); // 2
System.out.println(q.peek()); // 4

Deques

A deque (usually pronounced “deck”) stands for double ended queue and is a combination
of a stack and a queue, in that it supports O(1) insertions and deletions from both the
front and the back of the deque. In Java, the deque class is called ArrayDeque. The four
methods for adding and removing are addFirst, removeFirst, addLast, and removeLast.
The methods for retrieving the first and last elements without removing are peekFirst and
peekLast.

ArrayDeque<Integer> deque = new ArrayDeque<Integer>();
deque.addFirst(1); // [1]
deque.addLast(2); // [1, 2]
deque.addFirst(3); // [3, 1, 2]
deque.addFirst(4); // [3, 1, 2, 4]
deque.removeLast(); // [3, 1, 2]
deque.removeFirst(); // [1, 2]

Priority Queues

A priority queue supports the following operations:

insertion of elements, deletion of
the element considered highest priority, and retrieval of the highest priority element, all in
O(log n) time according to the number of elements in the priority queue. Priority is based on
a comparator function, but by default the lowest element is at the front of the priority queue.
The priority queue is one of the most important data structures in competitive programming,
so make sure you understand how and when to use it. By default, the Priority Queue puts
the lowest element at the front of the queue.

PriorityQueue<Integer> pq = new PriorityQueue<Integer>();
pq.add(4); // [4]
pq.add(2); // [4, 2]
pq.add(1); // [4, 2, 1]

CHAPTER 4. BUILT-IN DATA STRUCTURES

16

pq.add(3); // [4, 3, 2, 1]
System.out.println(pq.peek()); // 1
pq.poll(); // [4, 3, 2]
pq.poll(); // [4, 3]
pq.add(5); // [5, 4, 3]

4.3 Sets and Maps

A set is a collection of objects that contains no duplicates. There are two types of sets:

unordered sets (HashSet in Java), and ordered set (TreeSet in Java).

Unordered Sets

The unordered set works by hashing, which is assigning a usually-unique code to every
variable/object which allows insertions, deletions, and searches in O(1) time, albeit with a
high constant factor, as hashing requires a large constant number of operations. However,
as the name implies, elements are not ordered in any meaningful way, so traversals of an
unordered set will return elements in some arbitrary order. The operations on an unordered
set are add, which adds an element to the set if not already present, remove, which deletes
an element if it exists, and contains, which checks whether the set contains that element.

HashSet<Integer> set = new HashSet<Integer>();
set.add(1); // [1]
set.add(4); // [1, 4] in arbitrary order
set.add(2); // [1, 4, 2] in arbitrary order
set.add(1); // [1, 4, 2] in arbitrary order
// the add method did nothing because 1 was already in the set
System.out.println(set.contains(1)); // true
set.remove(1); // [2, 4] in arbitrary order
System.out.println(set.contains(5)); // false
set.remove(0); // [2, 4] in arbitrary order
// if the element to be removed does not exist, nothing happens

for(int element : set){

System.out.println(element);

}
// You can iterate through an unordered set, but it will do so in arbitrary

(cid:44)→

order

Ordered Sets

The second type of set data structure is the ordered or sorted set. Insertions, deletions,
and searches on the ordered set require O(log n) time, based on the number of elements

CHAPTER 4. BUILT-IN DATA STRUCTURES

17

in the set. As well as those supported by the unordered set, the ordered set also allows
four additional operations: first, which returns the lowest element in the set, last, which
returns the highest element in the set, lower, which returns the greatest element strictly less
than some element, and higher, which returns the least element strictly greater than it.

TreeSet<Integer> set = new TreeSet<Integer>();
set.add(1); // [1]
set.add(14); // [1, 14]
set.add(9); // [1, 9, 14]
set.add(2); // [1, 2, 9, 14]
System.out.println(set.higher(7)); // 9
System.out.println(set.higher(9)); // 14
System.out.println(set.lower(5)); // 2
System.out.println(set.first()); // 1
System.out.println(set.last()); // 14
set.remove(set.higher(6)); // [1, 2, 14]
System.out.println(set.higher(23); // ERROR, no such element exists

The primary limitation of the ordered set is that we can’t efficiently access the kth largest
element in the set, or find the number of elements in the set greater than some arbitrary x.
These operations can be handled using a data structure called an order statistic tree, but
that is beyond the scope of this book.

Maps

A map is a set of ordered pairs, each containing a key and a value. In a map, all keys
are required to be unique, but values can be repeated. Maps have three primary methods:
one to add a specified key-value pairing, one to retrieve the value for a given key, and one
to remove a key-value pairing from the map. Like sets, maps can be unordered (HashSet
in Java) or ordered (TreeSet in Java). In an unordered map, hashing is used to support
O(1) operations. In an ordered map, the entries are sorted in order of key. Operations are
O(log n), but accessing or removing the next key higher or lower than some input k is also
supported.

Unordered Maps

In the unordered map, the put(key, value) method assigns a value to a key and places
the key and value pair into the map. The get(key) method returns the value associated with
the key. The containsKey(key) method checks whether a key exists in the map. Lastly,
remove(key) removes the map entry associated with the specified key. All of these operations
are O(1), but again, due to the hashing, this has a high constant factor.

HashMap<Integer, Integer> map = new HashMap<Integer, Integer>();
map.put(1, 5); // [(1, 5)]
map.put(3, 14); // [(1, 5); (3, 14)]

CHAPTER 4. BUILT-IN DATA STRUCTURES

18

map.put(2, 7); // [(1, 5); (3, 14); (2, 7)]
map.remove(2); // [(1, 5); (3, 14)]
System.out.println(map.get(1)); // 5
System.out.println(map.containsKey(7)); // false
System.out.println(map.containsKey(1)); // true

Ordered Maps

The ordered map supports all of the operations that an unordered map supports, and
additionally supports firstKey/firstEntry and lastKey/lastEntry, returning the lowest
key/entry and the highest key/entry, as well as higherKey/higherEntry and lowerKey/
lowerEntry, returning the lowest key/entry strictly higher than the specified key, or the
highest key/entry strictly lower than the specified key.

TreeMap<Integer, Integer> map = new TreeMap<Integer, Integer>();
map.put(3, 5); // [(3, 5)]
map.put(11, 4); // [(3, 5); (11, 4)]
map.put(10, 491); // [(3, 5); (10, 491); (11, 4)]
System.out.println(map.firstKey()); // 3
System.out.println(map.firstEntry()); // (3, 5)
System.out.println(map.lastEntry()); // (11, 4)
System.out.println(map.higherEntry(4)); // (10, 491)
map.remove(11); // [(3, 5); (10, 491)]
System.out.println(map.lowerKey(4)); // 3
System.out.println(map.lowerKey(3)); // ERROR

A note on unordered sets and maps: In USACO contests, they’re generally fine, but in
CodeForces contests, you should always use sorted sets and maps. This is because the built-in
hashing algorithm is vulnerable to pathological data sets causing abnormally slow runtimes,
in turn causing failures on some test cases.

Multisets

Lastly, there is the multiset, which is essentially a sorted set that allows multiple copies
of the same element. While there is no Multiset in Java, we can implement one using the
TreeMap from values to their respective frequencies. We declare the TreeMap implementation
globally so that we can write functions for adding and removing elements from it.

static TreeMap<Integer, Integer> multiset = new TreeMap<Integer, Integer>();

public static void main(String[] args){

...

}

CHAPTER 4. BUILT-IN DATA STRUCTURES

19

static void add(int x){

if(multiset.containsKey(x)){

multiset.put(x, multiset.get(x) + 1);

} else {

multiset.put(x, 1);

}

}

static void remove(int x){

multiset.put(x, multiset.get(x) - 1);
if(multiset.get(x) == 0){

multiset.remove(x);

}

}

The first, last, higher, and lower operations still function as intended; just use firstKey,

lastKey, higherKey, and lowerKey respectively.

4.4 Problems

Again, note that CSES’s grader is very slow, so don’t worry if you encounter a Time
Limit Exceeded verdict; as long as you pass the majority of test cases within the time limit,
you can consider the problem solved, and move on.

1. CSES Problem Set Task 1621: Distinct Numbers

https://cses.fi/problemset/task/1621

2. CSES Problem Set Task 1084: Apartments
https://cses.fi/problemset/task/1084

3. CSES Problem Set Task 1091: Concert Tickets
https://cses.fi/problemset/task/1091

4. CSES Problem Set Task 1163: Traffic Lights
https://cses.fi/problemset/task/1163

5. CSES Problem Set Task 1164: Room Allocation
https://cses.fi/problemset/task/1164

Part II

Bronze

20

Chapter 5

Simulation

In many problems, we can simply simulate what we’re told to do by the problem statement.
Since there’s no formal algorithm involved, the intent of the problem is to assess competence
with one’s programming language of choice and knowledge of built-in data structures. At
least in USACO Bronze, when a problem statement says to find the end result of some
process, or to find when something occurs, it’s usually sufficient to simulate the process.

5.1 Example 1

Alice and Bob are standing on a 2D plane. Alice starts at the point (0, 0), and Bob
starts at the point (R, S) (1 ≤ R, S ≤ 1000). Every second, Alice moves M units to the
right, and N units up. Every second, Bob moves P units to the left, and Q units down.
(1 ≤ M, N, P, Q ≤ 10). Determine if Alice and Bob will ever meet (be at the same point at
the same time), and if so, when.
INPUT FORMAT:
The first line of the input contains R and S.
The second line of the input contains M , N , P , and Q.
OUTPUT FORMAT:
Please output a single integer containing the number of seconds after the start at which Alice
and Bob meet. If they never meet, please output −1.
Solution
We can simulate the process. After inputting the values of R, S, M , N , P , and Q, we can
keep track of Alice’s and Bob’s x- and y-coordinates. To start, we initialize variables for their
respective positions. Alice’s coordinates are initially (0, 0), and Bob’s coordinates are (R, S)
respectively. Every second, we increase Alice’s x-coordinate by M and her y-coordinate by
N , and decrease Bob’s x-coordinate by P and his y-coordinate by Q.

Now, when do we stop? First, if Alice and Bob ever have the same coordinates, then we
are done. Also, since Alice strictly moves up and to the right and Bob strictly moves down
and to the left, if Alice’s x- or y-coordinates are ever greater than Bob’s, then it is impossible
for them to meet. Example code will be displayed below (Here, as in other examples, input
processing will be omitted):

21

CHAPTER 5. SIMULATION

22

int ax = 0; int ay = 0; // alice's x and y coordinates
int bx = r; int by = s; // bob's x and y coordinates
int t = 0; // keep track of the current time
while(ax < bx && ay < by){

// every second, update alice's and bob's coordinates and the time
ax += m; ay += n;
bx -= p; by -= q;
t++;

}
if(ax == bx && ay == by){ // if they are in the same location

out.println(t); // they meet at time t

} else {

out.println(-1); // they never meet

}
out.close(); // flush the output

5.2 Example 2

There are N buckets (5 ≤ N ≤ 105), each with a certain capacity Ci (1 ≤ Ci ≤ 100). One
day, after a rainstorm, each bucket is filled with Ai units of water (1 ≤ Ai ≤ Ci). Charlie
then performs the following process: he pours bucket 1 into bucket 2, then bucket 2 into
bucket 3, and so on, up until pouring bucket N − 1 into bucket N . When Charlie pours
bucket B into bucket B + 1, he pours as much as possible until bucket B is empty or bucket
B + 1 is full. Find out how much water is in each bucket once Charlie is done pouring.
INPUT FORMAT:
The first line of the input contains N .
The second line of the input contains the capacities of the buckets, C1, C2, . . . , Cn.
The third line of the input contains the amount of water in each bucket A1, A2, . . . , An.
OUTPUT FORMAT:
Please print one line of output, containing N space-separated integers: the final amount of
water in each bucket once Charlie is done pouring.
Solution:
Once again, we can simulate the process of pouring one bucket into the next. The amount of
water poured from bucket B to bucket B + 1 is the smaller of the amount of water in bucket
B (after all previous operations have been completed) and the remaining space in bucket
B + 1, which is CB+1 − AB+1. We can just handle all of these operations in order, using an
array C to store the maximum capacities of each bucket, and an array A to store the current
water level in each bucket, which we update during the process. Example code is below (note
that arrays are zero-indexed, so the indices of our buckets go from 0 to N − 1 rather than
from 1 to N ).

CHAPTER 5. SIMULATION

23

for(int i = 0; i < n-1; i++){

int amt = Math.min(A[i], C[i+1]-A[i+1]);
// the amount of water to be poured is the lesser of
// the amount of water in the current bucket and
// the amount of additional water that the next bucket can hold
A[i] -= amt; // remove the amount from the current bucket
A[i+1] += amt; // add it to the next bucket

}

for(int i = 0; i < n; i++){

pw.print(A[i] + " ");
// print the amount of water in each bucket at the end

}
pw.println(); // print newline
pw.close(); // flush the output

5.3 Problems

1. USACO December 2018 Bronze Problem 1: Mixing Milk

http://www.usaco.org/index.php?page=viewproblem2&cpid=855

2. USACO December 2017 Bronze Problem 3: Milk Measurement

http://www.usaco.org/index.php?page=viewproblem2&cpid=761

3. USACO US Open 2017 Bronze Problem 1: The Lost Cow

http://www.usaco.org/index.php?page=viewproblem2&cpid=735

4. USACO February 2017 Bronze Problem 3: Why Did the Cow Cross the Road III

http://www.usaco.org/index.php?page=viewproblem2&cpid=713

5. USACO January 2016 Bronze Problem 3: Mowing the Field

http://www.usaco.org/index.php?page=viewproblem2&cpid=593

6. USACO December 2017 Bronze Problem 2: The Bovine Shuffle
http://usaco.org/index.php?page=viewproblem2&cpid=760

7. USACO February 2016 Bronze Problem 2: Circular Barn

http://usaco.org/index.php?page=viewproblem2&cpid=616

Chapter 6

Complete Search

In many problems (especially in Bronze), it’s sufficient to check all possible cases in
the solution space, whether it be all elements, all pairs of elements, or all subsets, or all
permutations. Unsurprisingly, this is called complete search (or brute force), because it
completely searches the entire solution space.

6.1 Example 1

You are given N (3 ≤ N ≤ 5000) integer points on the coordinate plane. Find the square
of the maximum Euclidean distance (aka length of the straight line) between any two of the
points.
INPUT FORMAT:
The first line contains an integer N .
The second line contains N integers, the x-coordinates of the points: x1, x2, . . . , xn (−1000 ≤
xi ≤ 1000).
The third line contains N integers, the y-coordinates of the points: y1, y2, . . . , yn (−1000 ≤
yi ≤ 1000).
OUTPUT FORMAT:
Print one integer, the square of the maximum Euclidean distance between any two of the
points.
Solution:
We can brute-force every pair of points and find the square of the distance between them,
by squaring the formula for Euclidean distance: distance2 = (x2 − x1)2 + (y2 − y1)2. Thus,
we store the coordinates in arrays X[] and Y[], such that X[i] and Y[i] are the x- and
y-coordinates of the ith point, respectively. Then, we iterate through all possible pairs of
points, using a variable max to store the maximum square of distance between any pair seen
so far, and if the square of the distance between a pair is greater than our current maximum,

24

CHAPTER 6. COMPLETE SEARCH

25

we set our current maximum to it.

Algorithm: Finds the maximum Euclidean distance between any two of the given
points

Function maxDist

: points an array of n ordered pairs

Input
Output : the maximum Euclidean distance between any two of the points
max ← 0
for i ← 1 to n do

for j ← i + 1 to n do

if dist(points[i], points[j])2 > max then
max ← dist(points[i], points[j])2

end

end

end
return max

int max = 0; // storing the current maximum
for(int i = 0; i < n; i++){ // for each first point

for(int j = i+1; j < n; j++){ // for each second point

int dx = x[i] - x[j];
int dy = y[i] - y[j];
max = Math.max(max, dx*dx + dy*dy);
// if the square of the distance between the two points is greater than
// our current maximum, then update the maximum

}

}
pw.println(max);

A couple notes: first, since we’re iterating through all pairs of points, we start the j loop
from j = i + 1 so that point i and point j are never the same point. Furthermore, it makes
it so that each pair is only counted once. In this problem, it doesn’t matter whether we
double-count pairs or whether we allow i and j to be the same point, but in other problems
where we’re counting something rather than looking at the maximum, it’s important to be
careful that we don’t overcount. Secondly, the problem asks for the square of the maximum
Euclidean distance between any two points. Some students may be tempted to maintain the
maximum distance in a variable, and then square it at the end when outputting. However,
the problem here is that while the square of the distance between two integer points is always
an integer, the distance itself isn’t guaranteed to be an integer. Thus, we’ll end up shoving a
non-integer value into an integer variable, which truncates the decimal part. Using a floating
point variable isn’t likely to work either, due to precision errors (use of floating point decimals
should generally be avoided when possible).

CHAPTER 6. COMPLETE SEARCH

26

6.2 Generating Permutations

A permutation is a reordering of a list of elements. Some problems will ask for an
ordering of elements that satisfies certain conditions. In these problems, if N ≤ 10, we can
probably iterate through all permutations and check each permutation for validity. For a list
of N elements, there are N ! ways to permute them, and generally we’ll need to read through
each permutation once to check its validity, for a time complexity of O(N · N !).

In Java, we’ll have to implement this ourselves, which is called Heap’s Algorithm (no
relation to the heap data structure). What’s going to be in the check function depends on
the problem, but it should verify whether the current permutation satisfies the constraints
given in the problem.

As an example, here are the permutations generated by Heap’s Algorithm for [1, 2, 3]:

[1, 2, 3], [2, 1, 3], [3, 1, 2], [1, 3, 2], [2, 3, 1], [3, 2, 1]

Algorithm: Iterate over all permutations of a given input array, performing some
action on each permutation

Function generatePermutations

Input : An array arr, and its length k
if k = 1 then

process the current permutation

else

generatePermutations (arr, k − 1)
for i ← 0 to k − 1 do
if k is even then

swap indices i and k − 1 of arr

else

swap indices 0 and k − 1 of arr

end
generatePermutations (arr, k − 1)

end

end

Code for iterating over all permutations is as follows:

// this method is called with k equal to the length of arr
static void generate(int[] arr, int k){

if(k == 1){

check(arr); // check the current permutation for validity

} else {

generate(arr, k-1);
for(int i = 0; i < k-1; i++){

if(k % 2 == 0){

swap(arr, i, k-1);
// swap indices i and k-1 of arr

CHAPTER 6. COMPLETE SEARCH

27

} else {

swap(arr, 0, k-1);
// swap indices 0 and k-1 of arr

}
generate(arr, k-1);

}

}

}

6.3 Problems

1. USACO February 2020 Bronze Problem 1: Triangles

http://usaco.org/index.php?page=viewproblem2&cpid=1011

2. USACO January 2020 Bronze Problem 2: Photoshoot

http://www.usaco.org/index.php?page=viewproblem2&cpid=988
(Hint: Figure out what exactly you’re complete searching)

3. USACO December 2019 Bronze Problem 1: Cow Gymnastics

http://usaco.org/index.php?page=viewproblem2&cpid=963
(Hint: Brute force over all possible pairs)

4. USACO February 2016 Bronze Problem 1: Milk Pails

http://usaco.org/index.php?page=viewproblem2&cpid=615

5. USACO January 2018 Bronze Problem 2: Lifeguards

http://usaco.org/index.php?page=viewproblem2&cpid=784
(Hint: Try removing each lifeguard one at a time).

6. USACO December 2019 Bronze Problem 2: Where Am I?

http://usaco.org/index.php?page=viewproblem2&cpid=964
(Hint: Brute force over all possible substrings)

7. (Permutations) USACO December 2019 Bronze Problem 3: Livestock Lineup

http://usaco.org/index.php?page=viewproblem2&cpid=965

8. (Permutations) CSES Problem Set Task 1624: Chessboard and Queens

https://cses.fi/problemset/task/1624

9. USACO US Open 2016 Bronze Problem 3: Field Reduction

http://www.usaco.org/index.php?page=viewproblem2&cpid=641
(Hint: For this problem, you can’t do a full complete search; you have to do a reduced
search)

10. USACO December 2018 Bronze Problem 3: Back and Forth

http://www.usaco.org/index.php?page=viewproblem2&cpid=857
(This problem is relatively hard)

Chapter 7

Additional Bronze Topics

7.1 Square and Rectangle Geometry

The extent of “geometry” problems on USACO Bronze are usually quite simple and
limited to intersections and unions of squares and rectangles. These usually only include two
or three squares or rectangles, in which case you can simply draw out cases on paper, which
should logically lead to a solution.

In Java, the Rectangle class may be of use for finding intersections and unions of

rectangles.

Here are some of the functions that the Rectangle class has:

• Find whether a certain point or rectangle is contained within another rectangle

• Find the intersection or union of two rectangles

• Translate, scale, or shrink rectangles

For exact details and documentation, refer to the Rectangle class page on the official

javadoc: https://docs.oracle.com/javase/7/docs/api/java/awt/Rectangle.html

The problems given at the end of the chapter should encompass all the techniques you

need to know for geometry problems in the Bronze division.

7.2 Ad-hoc

Ad-hoc problems are problems that don’t fall into any standard algorithmic category
with well known solutions. They are usually unique problems intended to be solved with
unconventional techniques. In ad-hoc problems, it’s helpful to look at the constraints given in
the problem and devise potential time complexities of solutions; this, combined with details
in the problem statement itself, may give an outline of the solution.

Unfortunately, since ad-hoc problems don’t have solutions consisting of well known
algorithms, we can’t systematically teach you how to do them. The best way of learning how
to do ad-hoc is to practice. Of course, the problem solving intuition from math contests (if
you did them) is quite helpful, but otherwise, you can develop this intuition from practicing
ad-hoc problems.

28

CHAPTER 7. ADDITIONAL BRONZE TOPICS

29

While solving these problems, make sure to utilize what you’ve learned about the built-in
data structures and algorithmic complexity analysis, from chapters 2, 3, and 4. Since ad-hoc
problems comprise a significant portion of bronze problems, we’ve included a large selection
of them below for your practice.

7.3 Problems

Square and Rectangle Geometry

1. USACO December 2017 Bronze Problem 1: Blocked Billboard

http://usaco.org/index.php?page=viewproblem2&cpid=759

2. USACO December 2018 Bronze Problem 1: Blocked Billboard II
http://usaco.org/index.php?page=viewproblem2&cpid=783

3. CodeForces Round 587 (Div. 3) Problem C: White Sheet
https://codeforces.com/contest/1216/problem/C

4. USACO December 2016 Bronze Problem 1: Square Pasture

http://usaco.org/index.php?page=viewproblem2&cpid=663

Ad-hoc problems

5. USACO January 2016 Bronze Problem 1: Promotion Counting
http://usaco.org/index.php?page=viewproblem2&cpid=591

6. USACO January 2020 Bronze Problem 1: Word Processor

http://usaco.org/index.php?page=viewproblem2&cpid=987

7. USACO US Open 2019 Bronze Problem 1: Bucket Brigade

http://usaco.org/index.php?page=viewproblem2&cpid=939

8. USACO January 2018 Bronze Problem 3: Out of Place

http://usaco.org/index.php?page=viewproblem2&cpid=785

9. USACO December 2016 Bronze Problem 2: Block Game

http://usaco.org/index.php?page=viewproblem2&cpid=664

10. USACO February 2020 Bronze Problem 3: Swapity Swap

http://usaco.org/index.php?page=viewproblem2&cpid=1013
(This problem is quite hard for bronze.)

11. USACO February 2018 Bronze Problem 1: Teleportation

http://usaco.org/index.php?page=viewproblem2&cpid=807

12. USACO February 2018 Bronze Problem 2: Hoofball

http://usaco.org/index.php?page=viewproblem2&cpid=808

CHAPTER 7. ADDITIONAL BRONZE TOPICS

30

13. USACO US Open 2019 Bronze Problem 3: Cow Evolution

http://usaco.org/index.php?page=viewproblem2&cpid=941
(Warning: This problem is extremely difficult for bronze.)

Part III

Silver

31

Chapter 8

Sorting and comparators

8.1 Comparators

Java has built-in functions for sorting: Arrays.sort(arr) for arrays, and Collections.
sort(list) for ArrayLists. However, if we use custom objects, or if we want to sort elements
in a different order, then we’ll need to use a Comparator.

Normally, sorting functions rely on moving objects with a lower value ahead of objects
with a higher value if sorting in ascending order, and vice versa if in descending order. This
is done through comparing two objects at a time. What a Comparator does is compare two
objects as follows, based on our comparison criteria:

• If object x is less than object y, return a negative number

• If object x is greater than object y, return a positive number

• If object x is equal to object y, return 0.

In addition to returning the correct number, comparators should also satisfy the following

conditions:

• The function must be consistent with respect to reversing the order of the arguments:
if compare(x, y) is positive, then compare(y, x) should be negative and vice versa

• The function must be transitive. If compare(x, y) > 0 and compare(y, z) > 0, then
compare(x, z) > 0. Same applies if the compare functions return negative numbers.

• Equality must be consistent.

If compare(x, y) = 0, then compare(x, z) and
compare(y, z) must both be positive, both negative, or both zero. Note that they
don’t have to be equal, they just need to have the same sign.

Java has default functions for comparing ints, longs, and doubles. The Integer.compare(),
Long.compare(), and Double.compare() functions take two arguments x and y and compare
them as described above.

Now, there are two ways of implementing this in Java: Comparable, and Comparator.
They essentially serve the same purpose, but Comparable is generally easier and shorter to

32

CHAPTER 8. SORTING AND COMPARATORS

33

code. Comparable is a function implemented within the class containing the custom object,
while Comparator is its own class. For our example, we’ll use a Person class that contains a
person’s height and weight, and sort in ascending order by height.

If we use Comparable, we’ll need to put implements Comparable<Person> into the
heading of the class. Furthermore, we’ll need to implement the compareTo method. Essentially,
compareTo(x) is the compare function that we described above, with the object itself as the
first argument: compare(self, x).

static class Person implements Comparable<Person>{

int height, weight;
public Person(int h, int w){

height = h; weight = w;

}
public int compareTo(Person p){

return Integer.compare(height, p.height);

}

}

When using Comparable, we can just call Arrays.sort(arr) or Collections.sort(list)

on the array or list as usual.

If instead we choose to use Comparator, we’ll need to declare a second Comparator class,

and then implement that:

static class Person{

int height, weight;
public Person(int h, int w){

height = h; weight = w;

}

}
static class Comp implements Comparator<Person>{
public int compare(Person a, Person b){

return Integer.compare(a.height, b.height);

}

}

When using Comparator, the syntax for using the built-in sorting function requires
a second argument: Arrays.sort(arr, new Comp()), or Collections.sort(list, new
Comp()).

If we instead wanted to sort in descending order, this is also very simple. Instead of the
comparing function returning Integer.compare(x, y) of the arguments, it should instead
return -Integer.compare(x, y).

CHAPTER 8. SORTING AND COMPARATORS

34

8.2 Sorting by Multiple Criteria

Now, suppose we wanted to sort a list of Persons in ascending order, primarily by height
and secondarily by weight. We can do this quite similarly to how we handled sorting by one
criterion earlier. What the compareTo function needs to do is to compare the weights if the
heights are equal, and otherwise compare heights, as that’s the primary sorting criterion.

static class Person implements Comparable<Person>{

int height, weight;
public Person(int h, int w){

height = h; weight = w;

}
public int compareTo(Person p){
if(height == p.height){

return Integer.compare(weight, p.weight);

} else {

return Integer.compare(height, p.height);

}

}

}

Sorting with more criteria is done similarly.
An alternative way of representing custom objects is with arrays. Instead of using a custom
object to store data about each person, we can simply use int[], where each int array is of
size 2, and stores pairs of height, weight, probably in the form of a list like ArrayList<int[]>.
Since arrays aren’t objects in the usual sense, we need to use Comparator. Example for
sorting by the same two criteria as above:

static class Comp implements Comparator<int[]>{

public int compare(int[] a, int[] b){

if(a[0] == b[0]){

return Integer.compare(a[1], b[1]);

} else {

return Integer.compare(a[0], b[0]);

}

}

}

I don’t recommend using arrays to represent objects, mostly because it’s confusing, but

it’s worth noting that some competitors do this.

8.3 Problems

1. USACO US Open 2018 Silver Problem 2: Lemonade Line

http://www.usaco.org/index.php?page=viewproblem2&cpid=835

CHAPTER 8. SORTING AND COMPARATORS

35

2. CodeForces Round 633 (Div. 2) Problem B: Sorted Adjacent Differences

https://codeforces.com/problemset/problem/1339/B

3. CodeForces Round 579 (Div. 3) Problem E: Boxers

https://codeforces.com/problemset/problem/1203/E

4. USACO January 2019 Silver Problem 3: Mountain View

http://www.usaco.org/index.php?page=viewproblem2&cpid=896

5. USACO US Open 2016 Silver Problem 1: Field Reduction
http://www.usaco.org/index.php?page=open16results

Chapter 9

Greedy Algorithms

Greedy algorithms are algorithms that select the most optimal choice at each step, instead
of looking at the solution space as a whole. This reduces the problem to a smaller problem at
each step. However, as greedy algorithms never recheck previous steps, they sometimes lead
to incorrect answers. Moreover, in a certain problem, there may be more than one possible
greedy algorithm; usually only one of them is correct. This means that we must be extremely
careful when using the greedy method. However, when they are correct, greedy algorithms
are extremely efficient.

Greedy is not a single algorithm, but rather a way of thinking that is applied to problems.
There’s no one way to do greedy algorithms. Hence, we use a selection of well-known examples
to help you understand the greedy paradigm.

Usually, when using a greedy algorithm, there is a heuristic or value function that

determines which choice is considered most optimal.

9.1

Introductory Example: Studying Algorithms

Steph wants to improve her knowledge of algorithms over winter break. She has a total of
X (1 ≤ X ≤ 104) minutes to dedicate to learning algorithms. There are N (1 ≤ N ≤ 100)
algorithms, and each one of them requires ai (1 ≤ ai ≤ 100) minutes to learn. Find the
maximum number of algorithms she can learn.

The solution is quite simple. The first observation we make is that Steph should prioritize
learning algorithms from easiest to hardest; in other words, start with learning the algorithm
that requires the least amount of time, and then choose further algorithms in increasing order
of time required. Let’s look at the following example:

X = 15,

N = 6,

ai = {4, 3, 8, 4, 7, 3}

After sorting the array, we have {3, 3, 4, 4, 7, 8}. Within the maximum of 15 minutes, Steph
can learn four algorithms in a total of 3 + 3 + 4 + 4 = 14 minutes. The implementation
of this algorithm is very simple. We sort the array, and then take as many elements as
possible while the sum of times of algorithms chosen so far is less than X. Sorting the array
takes O(N log N ) time, and iterating through the array takes O(N ) time, for a total time
complexity of O(N log N ).

36

CHAPTER 9. GREEDY ALGORITHMS

37

// read in the input, store the algorithms in int[] algorithms
Arrays.sort(algorithms);
int minutes = 0; // number of minutes used so far
int i = 0;
while(minutes + algorithms[i] <= x){

// while there is enough time, learn more algorithms
minutes += algorithms[i];
i++;

}
pw.println(i); // print the ans
pw.close();

9.2 Example: The Scheduling Problem

There are N events, each described by their starting and ending times. Jason would like
to attend as many events as possible, but he can only attend one event at a time, and if he
chooses to attend an event, he must attend the entire event. Traveling between events is
instantaneous.

Earliest Ending Next Event (Correct)

The correct approach to this problem is to always select the next possible event that ends

as soon as possible.

A brief explanation of correctness is as follows. If we have two events E1 and E2, with
E2 ending later than E1, then it is always optimal to select E1. This is because selecting E1
gives us more choices for future events. If we can select an event to go after E2, then that
event can also go after E1, because E1 ends first. Thus, the set of events that can go after E2
is a subset of the events that can go after E1, making E1 the optimal choice.

For the following code, let’s say we have the array events of events, which each contain a
start and an end point. We’ll be using the following static class to store each event (a review
of the previous chapter!)

static class Event implements Comparable<Event>{

int start; int end;
public Event(int s, int e){
start = s; end = e;

}
public int compareTo(Event e){

CHAPTER 9. GREEDY ALGORITHMS

38

return Integer.compare(this.end, e.end);

}

}

// read in the input, store the events in Event[] events.
Arrays.sort(events); // sorts by comparator we defined above
int currentEventEnd = -1; // end of event currently attending
int ans = 0; // how many events were attended?
for(int i = 0; i < n; i++){ // process events in order of end time

if(events[i].start >= currentEventEnd){ // if event can be attended
// we know that this is the earliest ending event that we can attend
// because of how the events are sorted
currentEventEnd = events[i].end;
ans++;

}

}
pw.println(ans);
pw.close();

Earliest Starting Next Event (Incorrect)

To emphasize the importance of selecting the right criteria, we review an incorrect solution
that always selects the next possible event that begins as soon as possible. Let’s look at the
following example, where the selected events are highlighted in red:

In this case, the greedy algorithm selects to attend only one event. However, the optimal

solution would be the following:

9.3 Failure Cases of Greedy Algorithms

We’ll provide a few common examples of when greedy fails, so that you can avoid falling

into obvious traps and wasting time getting wrong answers in contest.

CHAPTER 9. GREEDY ALGORITHMS

39

Coin Change

This problem gives several coin denominations, and asks for the minimum number of
coins needed to make a certain value. The greedy algorithm of taking the largest possible
coin denomination that fits in the remaining capacity can be used to solve this problem only
in very specific cases (it can be proven that it works for the American as well as the Euro
coin systems). However, it doesn’t work in the general case.

Knapsack

The knapsack problem gives a number of items, each having a weight and a value, and
we want to choose a subset of these items. We are limited to a certain weight, and we want
to maximize the value of the items that we take.

Let’s take the following example, where we have a maximum capacity of 4:

Item Weight Value Value Per Weight

A
B
C

3
2
2

18
10
10

6
5
5

If we use greedy based on highest value first, we choose item A and then we are done, as
we don’t have remaining weight to fit either of the other two. Using greedy based on value
per weight again selects item A and then quits. However, the optimal solution is to select
items B and C, as they combined have a higher value than item A alone. In fact, there is no
working greedy solution. The solution to this problem uses dynamic programming, which is
beyond the scope of this book.

9.4 Problems

1. USACO December 2015 Silver Problem 2: High Card Wins

http://usaco.org/index.php?page=viewproblem2&cpid=571

2. USACO February 2018 Silver Problem 1: Rest Stops

http://www.usaco.org/index.php?page=viewproblem2&cpid=810

3. USACO February 2017 Silver Problem 1: Why Did The Cow Cross The Road

http://www.usaco.org/index.php?page=viewproblem2&cpid=714

Chapter 10

Graph Theory

Graph theory is one of the most important topics at the Silver level and above. Graphs
can be used to represent many things, from images to wireless signals, but one of the simplest
analogies is to a map. Consider a map with several cities and highways connecting the cities.
Some of the problems relating to graphs are:

• If we have a map with some cities and roads, what’s the shortest distance I have to

travel to get from point A to point B?

• Consider a map of cities and roads. Is city A connected to city B? Consider a region to
be a group of cities such that each city in the group can reach any other city in said
group, but no other cities. How many regions are in this map, and which cities are in
which region?

10.1 Graph Basics

Graphs are made up of nodes and edges, where nodes are connected by edges. Graphs
can have either weighted edges, in which each edge has a certain length, or unweighted, in
which case all edges have the same length. Edges are either directed, which means they can
be traveled upon in one direction, or undirected, which means that they can be traveled
upon in both directions.

4

2

6

5

1

3

5

5

2

1

4

4

2

−1

2

3

3

5

6

Figure 10.1: An undirected unweighted graph (left) and a directed weighted graph (right)

40

CHAPTER 10. GRAPH THEORY

41

A connected component is a set of nodes within which any node can reach any other
node. For example, in this graph, nodes 1, 2, and 3 are a connected component, nodes 4 and
5 are a connected component, and node 6 is its own component.

3

1

2

6

4

5

Figure 10.2: Connected components in a graph

10.2 Trees

A tree is a special type of graph satisfying two constraints: it is acyclic, meaning there
are no cycles, and the number of edges is one less than the number of nodes. Trees satisfy
the property that for any two nodes A and B, there is exactly one way to travel between A
and B.

1

6

3

7

2

5

4

Figure 10.3: A tree graph

The root of a tree is the one vertex that is placed at the top, and is where we usually
start our tree traversals from. Usually, problems don’t tell us where the tree is rooted at, and
it usually doesn’t matter either; trees can be arbitrarily rooted (here, we’ll use the convention
of rooting at index 1).

Every node except the root node has a parent. The parent of a node s is defined as
follows: On the path from the root to s, the node that is one closer to the root than s is the
parent of s. Each non-root node has a unique parent.

Child nodes are the opposite. They lie one farther away from the root than their parent
node. Unlike parent nodes, these are not unique. Each node can have arbitrarily many child
nodes, and nodes can also have zero children. If a node s is the parent of a node t, then t is
the child node of s.

A leaf node is a node that has no children. Leaf nodes can be identified quite easily

because there is only one edge adjacent to them.

CHAPTER 10. GRAPH THEORY

42

In our example tree above, node 1 is the root, nodes 2 and 3 are children of node 1, nodes
4, 5, and 6 are children of 2, and node 7 is the child of 3. Nodes 4, 5, 6, and 7 are leaf nodes.

10.3 Graph Representations

Usually, in a graph with N edges and M edges, we’ll number the nodes 0 through N − 1.
If the problem gives the nodes numbered 1 through N , simply decrease the endpoint node
numbers of edges by 1 as you input them, in order to accommodate zero-indexing of arrays.
However, in problem statements, input and output, the node labels will usually be 1 through
N , so that’s what we’ll use in our examples.

Graphs will usually be given in an input format similar to the following: First, integers
N and M denoting the number of nodes and edges, respectively. Then, M lines, each with
integers a and b, representing edges; if the graph is undirected, then there is an edge between
nodes a and b, and if the graph is directed, then there is an edge from a to b.

For example, the input below would be for the following graph (without the comments):

6 7 // 6 nodes, 7 edges
// the following lines represent edges.
1 2
1 4
1 5
2 3
2 4
3 5
4 6

4

2

6

5

1

3

Figure 10.4: The graph corresponding to the above input

Graphs can be represented in three ways: Adjacency List, Adjacency Matrix, and Edge
List. Regardless of how the graph is represented, it’s important that it be stored globally
and statically, because we need to be able to access it from outside the main method, and
call the graph searching and traversal methods on it.

CHAPTER 10. GRAPH THEORY

43

Adjacency List

The adjacency list is the most commonly used method of storing graphs. When we use
DFS, BFS, Dijkstra’s, or other single-source graph traversal algorithms, we’ll want to use an
adjacency list. In an adjacency list, we maintain a length N array of lists. Each list stores
the neighbors of one node. In an undirected graph, if there is an edge between node a and
node b, we add a to the list of b’s neighbors, and b to the list of a’s neighbors. In a directed
graph, if there is an edge from node a to node b, we add b to the list of a’s neighbors, but
not vice versa.

4

4

2

2

1

3

6

5

3

4

1

3

9

5

Figure 10.5: An example of a weighted undirected graph

Adjacency list representation of the graph in fig. 10.5:

adj[0]
adj[1]
adj[2]
adj[3]
adj[4]
adj[5]

(1, 9), (3, 4), (4, 3)
(0, 9), (2, 5), (3, 2)
(1, 5), (4, 4), (5, 1)
(0, 4), (1, 2), (5, 3)
(0, 3), (2, 4)
(2, 1), (3, 3)

Adjacency lists take up O(N + M ) space, because each node corresponds to one list of
neighbors, and each edge corresponds to either one or two endpoints (directed vs undirected).
In an adjacency list, we can find (and iterate through) the neighbors of a node easily. Hence,
the adjacency list is the graph representation we should be using most of the time.

Often, we’ll want to maintain a array visited, which is a boolean array representing
whether each node has been visited. When we visit node k (0-indexed), we mark visited[k]
true, so that we know not to return to it.

Code for setting up an adjacency list is as follows:

static int n, m; // number of nodes and edges
static ArrayList<Integer>[] adj; // adjacency list

public static void main(String[] args){

n = r.nextInt(); // reads in number of nodes
m = r.nextInt(); // reads in number of edges

CHAPTER 10. GRAPH THEORY

44

adj = new ArrayList[n]; // adjacency list
// Java doesn't allow ArrayList<Integer>[n]
boolean[] visited = new boolean[n];
for(int i = 0; i < n; i++){

adj[i] = new ArrayList<Integer>(); // initializes the ArrayLists

}
for(int i = 0; i < m; i++){ // reading in each of the m edges

int a = r.nextInt()-1; // we subtract 1 because our array is

zero-indexed

(cid:44)→
int b = r.nextInt()-1;
adj[a].add(b);
adj[b].add(a); // omit this line if the graph is directed

}

}

If we’re dealing with a weighted graph, we’ll declare an Edge class or struct that stores
two variables: the second endpoint of the edge, and the weight of the edge, and we store an
array of lists of edges rather than an array of lists of integers.

static class Edge{

int to;
int weight;
public Edge(int to, int weight){

this.to = to;
this.edge = edge;

}

}

Adjacency Matrix

Another way of representing graphs is the adjacency matrix, which is an N by N 2-
dimensional array that stores for each pair of indices (a, b), stores whether there is an
edge between a and b. Start by initializing every entry in the matrix to zero (this is done
automatically in Java), and then for undirected graphs, for each edge between indices a and
b, set adj[a][b] and adj[b][a] to 1 (if unweighted) or the edge weight (if weighted). If
the graph is directed, for an edge from a to b, only set adj[a][b].

CHAPTER 10. GRAPH THEORY

45

4

4

2

2

1

3

6

5

3

4

1

3

9

5

Figure 1.5 repeated for convenience

Adjacency matrix representation of the graph in fig. 1.5:

× 0 1 2 3 4 5
0 0 9 0 4 3 0
1 9 0 5 2 0 0
2 0 5 0 0 4 1
3 4 2 0 0 0 3
4 3 0 4 0 0 0
5 0 0 1 3 0 0

At the Silver level, we generally won’t be using the adjacency matrix much, but it’s helpful
to know if it does come up. The primary use of the adjacency matrix is the Floyd-Warshall
algorithm, which is beyond the scope of this book.

Code for setting up an adjacency matrix is as follows:

static int n, m; // number of nodes and edges
static int[][] adj; // adj matrix

public static void main(String[] args){

n = r.nextInt();
m = r.nextInt();
adj = new int[n][n];
for(int i = 0; i < m; i++){ // read in each of the m edges

int a = r.nextInt()-1;
int b = r.nextInt()-1;
adj[a][b] = 1; // or set equal to w if graph is weighted
adj[b][a] = 1; // or set equal to w if graph is weighted;
// ignore above line if graph is directed

}

}

CHAPTER 10. GRAPH THEORY

46

Edge List

The last graph representation is the edge list. Usually, we use this in weighted undirected
graphs when we want to sort the edges by weight (for DSU, for example; see section 10.6).
In the edge list, we simply store a single list of all the edges, in the form (a, b, w) where a
and b are the nodes that the edge connects, and w is the edge weight. Note that in an edge
list, we do NOT add each edge twice; there is only one place for us to add the edges, so we
only do so once.

4

4

2

2

1

3

6

5

3

4

1

3

9

5

Figure 1.5 repeated for convenience

Edge list representation of the graph in fig. 1.5:

(0, 1, 9), (0, 3, 4), (0, 4, 3), (1, 3, 2), (3, 5, 3), (2, 4, 4), (2, 1, 5), (2, 5, 1)

We’ll need an edge class, such as the following:

static class Edge implements Comparable<Edge>{

int a, b, w;
public Edge(int a, int b, int w){

this.a = a;
this.b = b;
this.w = w;

}
public int compareTo(Edge e){ // sort order is ascending, by weight
// to sort in descending order, just negate the value of the compare

function.
return Integer.compare(w, e.w);

(cid:44)→

}

}

Code for the edge list is as follows, using the above edge class:

static int n, m; // number of nodes and edges
static ArrayList<Edge> edges;

CHAPTER 10. GRAPH THEORY

47

public static void main(String[] args){

n = r.nextInt();
m = r.nextInt();
edges = new ArrayList<Edge>();
for(int i = 0; i < m; i++){ // for each of the m edges

int a = r.nextInt()-1;
int b = r.nextInt()-1;
// subtract 1 to maintain zero-indexing of vertices
int w = r.nextInt();
edges.add(new Edge(a, b, w)); // add the edge to the list

}
Collections.sort(edges);

}

10.4 Graph Traversal Algorithms

Graph traversal is the process of visiting or checking each vertex in a graph. This is useful
when we want to determine which vertices can be visited, whether there exists a path from
one vertex to another, and so forth. There are two algorithms for graph traversal, namely
depth-first search (DFS) and breadth-first search (BFS).

Depth-first search

Depth-first search continues down a single path to the end, then it backtracks to check
other vertices. Depth-first search will process all nodes that are reachable (connected by
edges) to the starting node. Let’s look at an example of how this works. Depth first-search
can start at any node, but by convention we’ll start the search at node 1. We’ll use the
following color scheme: blue for nodes we have already visited, red for nodes we are currently
processing, and black for nodes that have not been visited yet.

The DFS starts from node 1 and then goes to node 2, as it’s the only neighbor of node 1:

1

2

5

3

4

1

2

5

3

4

Now, the DFS goes to node 3 and then 4, following a single path to the end until it has no
more nodes to process:

CHAPTER 10. GRAPH THEORY

48

1

2

5

3

4

1

2

5

3

4

Lastly, the DFS backtracks to visit node 5, which was skipped over previously.

1

2

5

3

4

Depth-first search is implemented recursively because it allows for much simpler and shorter
code. The algorithm is as follows:

Algorithm: Recursive implementation for depth-first traversal of a graph

Function DFS

Input : start, the 0-indexed number of the starting vertex
visted(start) ← true
foreach vertex k adjacent to start do

if visited(k) is false then

DFS (k)

end

end

Code:

static void dfs(int node){

visited[node] = true;
for(int next : adj[node]){
if(!visited[next]){
dfs(next);

}

}

}

Breadth-first search

Breadth-first search visits nodes in order of distance away from the starting node; it first

visits all nodes that are one edge away, then all nodes that are two edges away, and so on.
Let’s use the same example graph that we used earlier: The BFS starts from node 1 and

then goes to node 2, as it’s the only neighbor of node 1:

CHAPTER 10. GRAPH THEORY

49

1

2

5

3

4

1

2

5

3

4

Now, the BFS goes to node 3, and then node 5, because both of them are two edges away
from node 1:

1

2

5

3

4

1

2

5

3

4

Lastly, the BFS visits node 4, which is farthest.

1

2

5

3

4

The breadth-first search algorithm cannot be implemented recursively, so it’s significantly
longer. Thus, when both BFS and DFS work, DFS is usually the better option.

BFS can be used for finding the distance away from a starting node for all nodes in an

unweighted graph, as we show below:

CHAPTER 10. GRAPH THEORY

50

The algorithm is as follows:

Algorithm: Breadth-first traversal of a graph

Function BFS

Input : start, the 0-indexed number of the starting vertex
foreach vertex v do
dist[v] ← −1
visited[v] ← false

end
dist[start] ← 0
Let q be a queue of integers
Add start to q
while q is not empty do

Pop the first element from q, call it v
foreach neighbor u of v do

if node u has not yet been visited then

dist[u] ← dist[v] + 1
Add u to q

end

end

end

Once the BFS finishes, the array dist contains the distances from the start node to each

node.

Example code is below. Note that the array dist[] is initially filled with −1’s to denote

that none of the nodes have been processed yet.

static void bfs(int k){

Arrays.fill(dist, -1); // fill distance array with -1's
Queue<Integer> q = new LinkedList<Integer>();
dist[k] = 0;
q.add(k);
while(!q.isEmpty()){

int v = q.poll();
for(int e : adj[v]){

if(dist[e] == -1){

dist[e] = dist[v] + 1;
q.add(e);

}

}

}

}

CHAPTER 10. GRAPH THEORY

51

Iterative DFS

If you encounter stack overflows while using recursive DFS, you can write an iterative

DFS, which is just BFS but with nodes stored on a stack rather than a queue.

10.5 Floodfill

Floodfill is an algorithm that identifies and labels the connected component that a
particular cell belongs to, in a multi-dimensional array. Essentially, it’s DFS, but on a grid,
and we want to find the connected component of all the connected cells with the same number.
For example, let’s look at the following grid and see how floodfill works, starting from the
top-left cell. The color scheme will be the same: red for the node currently being processed,
blue for nodes already visited, and uncolored for nodes not yet visited.

2
2
2

2
2
2

2
2
2

2
2
2

2
2
2

2
2
2

2
1
2

2
1
2

2
1
2

2
1
2

2
1
2

2
1
2

1
1
1

1
1
1

1
1
1

1
1
1

1
1
1

1
1
1

As opposed to an explicit graph where the edges are given, a grid is an implicit graph.
This means that the neighbors are just the nodes directly adjacent in the four cardinal
directions.

Usually, grids given in problems will be N by M , so the first line of the input contains the
numbers N and M . In this example, we will use an two-dimensional integer array to store the
grid, but depending on the problem, a two-dimensional character array or a two-dimensional
boolean array may be more appropriate. Then, there are N rows, each with M numbers
containing the contents of each square in the grid. Example input might look like the following
(varies between problems):

CHAPTER 10. GRAPH THEORY

52

3 4
1 1 2 1
2 3 2 1
1 3 3 3

And we’ll want to input the grid as follows:

static int[][] grid;
static int n, m;

public static void main(String[] args){

int n = r.nextInt();
int m = r.nextInt();
grid = new int[n][m];
for(int i = 0; i < n; i++){

for(int j = 0; j < m; j++){

grid[i][j] = r.nextInt();

}

}

}

When doing floodfill, we will maintain an N × M array of booleans to keep track of
which squares have been visited, and a global variable to maintain the size of the current
component we are visiting. Make sure to store the grid, the visited array, dimensions, and
the current size variable globally.

This means that we want to recursively call the search function from the squares above,
below, and to the left and right of our current square. The algorithm to find the size of a
connected component in a grid using floodfill is as follows (we’ll also maintain a 2d visited

CHAPTER 10. GRAPH THEORY

53

array):

Algorithm: Floodfill of a graph

Function main

// Input/output, global vars, etc hidden
for i ← 0 to n − 1 do

for j ← 0 to m − 1 do

if the square at (i, j) is not visited then

currentSize ← 0
floodfill(i, j, grid[i][j])
Process the connected component

end

end

end

Function floodfill

: r, c, color

Input
// row and column index of starting square, target color
if r or c is out of bounds then

return

end
if the cell at (r, c) is the wrong color then

return

end
if the square at (r, c) has already been visited then

return

end
visited[r][c] ← true
currentSize ← currentSize + 1
floodfill(r, c + 1, color)
floodfill(r, c − 1, color)
floodfill(r − 1, c, color)
floodfill(r + 1, c, color)

The code below shows the global/static variables we need to maintain while doing floodfill,

and the floodfill algorithm itself.

static int[][] grid; // the grid itself
static int n, m; // grid dimensions, rows and columns
static boolean[][] visited; // keeps track of which nodes have been visited
static int currentSize = 0; // reset to 0 each time we start a new component

public static void main(String[] args){

/**
* input code and other problem-specific stuff here
*/

CHAPTER 10. GRAPH THEORY

54

for(int i = 0; i < n; i++){

for(int j = 0; j < m; j++){
if(!visited[i][j]){

currentSize = 0;
floodfill(i, j, grid[i][j]);
// start a floodfill if the square hasn't already been visited,
// and then store or otherwise use the component size
// for whatever it's needed for

}

}

}

}

static void floodfill(int r, int c, int color){

if(r < 0 || r >= n || c < 0 || c >= m) return; // if outside grid
if(grid[r][c] != color) return; // wrong color
if(visited[r][c]) return; // already visited this square
visited[r][c] = true; // mark current square as visited
currentSize++; // increment the size for each square we visit
// recursively call floodfill for neighboring squares
floodfill(r, c+1, color);
floodfill(r, c-1, color);
floodfill(r-1, c, color);
floodfill(r+1, c, color);

}

10.6 Disjoint-Set Data Structure

Let’s say we want to construct a graph, one edge at a time. We also want to be able to
add additional nodes, and query whether two nodes are connected. We can naively solve
this problem by adding the edges and running a floodfill each time, before finally checking
whether two nodes have the same color. This yields a time complexity of O(nm) for a graph
of n nodes and m edges.

However, we can do better than this using a data structure known as Disjoint-Set Union,

or DSU for short. This data structure supports two operations:

• Add an edge between two nodes.

• Check if two nodes are connected.

To achieve this, we store sets as trees, with the root of the tree representing the “parent”
of the set. Initially, we store each node as its own set. Then, we combine their sets when we
add an edge between two nodes. The image below illustrates this structure.

CHAPTER 10. GRAPH THEORY

55

In this graph, 1 is the parent of the set containing 3, 2, and 4.
To implement this, let’s store the parent of each node in the tree represented by that
node’s set. Then, to merge two sets, we set the parent of one tree’s root to the other tree’s
root, like so:

The following methods demonstrate this idea:

static int[] parent; //stores the parent nodes

static void initialize(int N){

for(int i = 0; i < N; i++){

parent[i] = i; //initially, the root of each set is the node itself

}

}

static int find(int x){ //finds the root of the set of x

if(x == parent[x]){ //if x is the parent of itself, it is the root

return x;

}
else{

return find(parent[x]); //otherwise, recurse to the parent of x

}

CHAPTER 10. GRAPH THEORY

56

}

static void union(int a, int b){ //merges the sets of a and b

int c = find(a); //find the root of a
int d = find(b); //find the root of b
if(c != d){

parent[d] = c; //merge the sets by setting the parent of d to c

}

}

However, this naive implementation of a DSU isn’t much better than simply running a
floodfill. As the recursing up the tree of a set to find it’s root can be time-consuming for
trees with long chains, the runtime ultimately degrades to still being O(nm) for n nodes and
m edges.

Now that we understand the general idea of a DSU, we can improve the runtime of this
implementation using an optimization known as path compression. The general idea is to
reassign nodes in the tree as you are recursively calling the find method to prevent long
chains from forming. Here is a rewritten find method representing this idea:

static int find(int x){
if(x == parent[x]){

return x;

}
else{

// we set the direct parent to the root of the set to reduce path length
return parent[x] = find(parent[x]);

}

}

The following image demonstrates how the tree with parent 1 is compressed after find(6)
is called. All of the bolded nodes in the final tree were visited during the recursive operation,
and now point to the root.

CHAPTER 10. GRAPH THEORY

57

With this new optimization, the runtime reduces to O(n log n), far better than our naive
algorithm. Further optimizations can reduce the runtime of DSU to nearly constant. However,
those techniques and the proof of complexity for these optimizations are both unnecessary for
and out of the scope of the USACO Silver division, so they will not be included in this book.

10.7 Bipartite Graphs

A bipartite graph is a graph such that each node can be colored in one of two colors,
such that no two adjacent nodes have the same color. For example, the following graph is
bipartite:

1

2

4

3

5

A graph is bipartite if and only if there are no cycles of odd length. For example, the

following graph is not bipartite, because it contains a cycle of length 3.

1

3

2

4

The following image depicts how a bipartite graph splits vertices into two “groups”

depending on their color.

CHAPTER 10. GRAPH THEORY

58

In order to check whether a graph is bipartite, we use a modified breadth-first search.

Algorithm: Bipartiteness check

Function bipartite

: a graph

Input
Output : whether the graph is bipartite or not
Assign color 1 to the starting vertex
// Use the following modified bfs
foreach vertex v processed in bfs do

d ← dist(start, v)
if d is odd then

Assign color 2 to vertex v

else

Assign color 1 to vertex v

end
foreach vertex w adjacent to v do

if w and v are the same color then
return false // not bipartite

end

end

end
return true // bipartite

10.8 Problems

DFS/BFS Problems

1. USACO January 2018 Silver Problem 3: MooTube

http://www.usaco.org/index.php?page=viewproblem2&cpid=788

CHAPTER 10. GRAPH THEORY

59

2. USACO December 2016 Silver Problem 3: Moocast

http://www.usaco.org/index.php?page=viewproblem2&cpid=668

3. USACO US Open 2016 Silver Problem 3: Closing the Farm

http://www.usaco.org/index.php?page=viewproblem2&cpid=644

DSU Problems

Many of these problems do not require DSU. However, they become much easier to do if

you understand it.

4. USACO US Open Silver Problem 3: The Moo Particle

http://usaco.org/index.php?page=viewproblem2&cpid=1040

5. USACO January 2018 Silver Problem 3: MooTube

http://www.usaco.org/index.php?page=viewproblem2&cpid=788

6. USACO December 2019 December Problem 3: Milk Visits

http://usaco.org/index.php?page=viewproblem2&cpid=968

7. USACO US Open 2016 Gold Problem 2: Closing the Farm

http://www.usaco.org/index.php?page=viewproblem2&cpid=646

8. USACO January Contest 2020 Silver Problem 3: Wormhole Sort

http://www.usaco.org/index.php?page=viewproblem2&cpid=992

Other Graph Problems

9. (Bipartite Graphs) USACO February 2019 Silver Problem 3: The Great Revegetation

http://www.usaco.org/index.php?page=viewproblem2&cpid=920

10. CodeForces Round 595 (Div. 3) Problem B2: Books Exchange
https://codeforces.com/problemset/problem/1249/B2

Chapter 11

Prefix Sums

11.1 Prefix Sums

Let’s say we have an integer array arr with N elements, and we want to process Q queries
to find the sum of the elements between two indices a and b, inclusive, with different values
of a and b for every query. For the purposes of this chapter, we will assume that the original
array is 1-indexed, meaning arr[0] = 0 (which is a dummy index), and the actual array
elements occupy indices 1 through N (this means that the array actually has length N + 1).

Let’s use the following example 1-indexed array arr, with N = 6:

Index i
arr[i]

0
0

1
1

2
6

3
4

4
2

5
5

6
3

Naively, for every query, we can iterate through all entries from index a to index b to add
them up. Since we have Q queries and each query requires a maximum of O(N ) operations
to calculate the sum, our total time complexity is O(N Q). For most problems of this nature,
the constraints will be N, Q ≤ 105, so N Q is on the order of 1010. This is not acceptable; it
will almost always exceed the time limit.

Instead, we can use prefix sums to process these array sum queries. We designate a prefix
sum array prefix[]. First, since we’re 1-indexing the array, set prefix[0] = 0, then for
indices k such that 1 ≤ k ≤ n, define the prefix sum array as follows:

prefix[k] =

k
(cid:88)

i=1

arr[i]

Basically, what this means is that the element at index k of the prefix sum array stores the
sum of all the elements in the original array from index 1 up to k. This can be calculated
easily in O(N ) by the following formula:

prefix[k] = prefix[k-1] + arr[k]

For the example case, our prefix sum array looks like this:

Index i
prefix[i]

0
0

1
1

2
7

3
11

4
13

5
18

6
21

60

CHAPTER 11. PREFIX SUMS

61

Now, when we want to query for the sum of the elements of arr between (1-indexed)

indices a and b inclusive, we can use the following formula:

b
(cid:88)

i=a

arr[i] =

b
(cid:88)

i=1

arr[i] −

a−1
(cid:88)

i=1

arr[i]

Using our definition of the elements in the prefix sum array, we have

b
(cid:88)

i=a

arr[i] = prefix[b] − prefix[a-1]

Since we are only querying two elements in the prefix sum array, we can calculate subarray
sums in O(1) per query, which is much better than the O(N ) per query that we had before.
Now, after an O(N ) preprocessing to calculate the prefix sum array, each of the Q queries
takes O(1) time. Thus, our total time complexity is O(N + Q), which should now pass the
time limit.

Let’s do an example query and find the subarray sum between indices a = 2 and b = 5,
inclusive, in the 1-indexed arr. From looking at the original array, we see that this is
(cid:80)5

i=2 arr[i] = 6 + 4 + 2 + 5 = 17.

Index i
arr[i]

0
0

1
1

2
6

3
4

4
2

5
5

6
3

Using prefix sums: prefix[5] − prefix[1] = 18 − 1 = 17.

Index i
prefix[i]

0
0

1
1

2
7

3
11

4
13

5
18

6
21

11.2 Two Dimensional Prefix Sums

Now, what if we wanted to process Q queries for the sum over a subrectangle of a N
rows by M columns matrix in two dimensions? Let’s assume both rows and columns are
1-indexed, and we use the following matrix as an example:

0
0
0
0
0

0
1
1
4
7

0
5
7
6
5

0
6
11
1
4

0
11
9
3
2

0
8
4
2
3

Naively, each sum query would then take O(N M ) time, for a total of O(QN M ). This is

too slow.

Let’s take the following example region, which we want to sum:

CHAPTER 11. PREFIX SUMS

62

0
0
0
0
0

0
1
1
4
7

0
5
7
6
5

0
6
11
1
4

0
11
9
3
2

0
8
4
2
3

Manually summing all the cells, we have a submatrix sum of 7 + 11 + 9 + 6 + 1 + 3 = 37.
The first logical optimization would be to do one-dimensional prefix sums of each row.
Then, we’d have the following row-prefix sum matrix. The desired subarray sum of each row
in our desired region is simply the green cell minus the red cell in that respective row. We do
this for each row, to get (28 − 1) + (14 − 4) = 37.

0
0
0
0
0

0
1
1
4
7

0
6
8
10
12

0
12
19
11
16

0
23
28
14
18

0
31
32
16
21

Now, if we wanted to find a submatrix sum, we could break up the submatrix into a
subarray for each row, and then add their sums, which would be calculated using the prefix
sums method described earlier. Since the matrix has N rows, the time complexity of this is
O(QN ). This is better, but still usually not fast enough.

To do better, we can do two-dimensional prefix sums. In our two dimensional prefix sum

array, we have

prefix[a][b] =

a
(cid:88)

b
(cid:88)

i=1

j=1

arr[i][j]

This can be calculated as follows for row index 1 ≤ i ≤ n and column index 1 ≤ j ≤ m:

prefix[i][j] = prefix[i-1][j] + prefix[i][j-1] − prefix[i-1][j-1] + arr[i][j]

The submatrix sum between rows a and A and columns b and B, can thus be expressed as
follows:

A
(cid:88)

B
(cid:88)

arr[i][j] = prefix[A][B] − prefix[a-1][B]

i=a

j=b

− prefix[A][b-1] + prefix[a-1][b-1]

Summing the blue region from above using the 2d prefix sums method, we add the value
of the green square, subtract the values of the red squares, and then add the value of the
gray square. In this example, we have 65 − 23 − 6 + 1 = 37, as expected.

0
0
0
0
0

0
1
2
6
13

0
6
14
24
36

0
12
31
42
58

0
23
51
65
83

0
31
63
79
100

Since no matter the size of the submatrix we are summing, we only need to access 4 values
of the 2d prefix sum array, this runs in O(1) per query after an O(N M ) preprocessing. This
is fast enough.

CHAPTER 11. PREFIX SUMS

11.3 Problems

63

1. USACO December 2015 Silver Problem 3: Breed Counting

http://usaco.org/index.php?page=viewproblem2&cpid=572

2. USACO January 2016 Silver Problem 2: Subsequences Summing to Sevens

http://usaco.org/index.php?page=viewproblem2&cpid=595

3. USACO December 2017 Silver Problem 1: My Cow Ate My Homework
http://www.usaco.org/index.php?page=viewproblem2&cpid=762

4. USACO January 2017 Silver Problem 2: Hoof, Paper, Scissors

http://www.usaco.org/index.php?page=viewproblem2&cpid=691

5. (2D Prefix Sums) USACO February 2019 Silver Problem 2: Painting the Barn

http://www.usaco.org/index.php?page=viewproblem2&cpid=919

Chapter 12

Binary Search

12.1 Binary Search on the Answer

You’re probably already familiar with the concept of binary searching for a number in
a sorted array. However, binary search can be extended to binary searching on the answer
itself. When we binary search on the answer, we start with a search space, where we know
the answer lies in. Then, each iteration of the binary search cuts the search space in half,
so the algorithm tests O(log N ) values, which is efficient and much better than testing each
possible value in the search space.

Similarly to how binary search on an array only works on a sorted array, binary search
on the answer only works if the answer function is monotonic. Let’s say we have a function
check(x) that returns true if the answer of x is possible, and false otherwise. Usually, in
such problems, we’ll want to find the maximum or minimum value of x such that check(x)
is true.

In order for binary search to work, the search space must look like something of the

following form, using a check function as we described above.

true true true true true false false false false

Then, we find the point at which true becomes false, using binary search.
Below, we present two algorithms for binary search. The first implementation may be
more intuitive, because it’s closer to the binary search most students learned, while the

64

CHAPTER 12. BINARY SEARCH

65

second implementation is shorter.

Algorithm: Binary searching for the answer

Function binarySearch1

left ← lower bound of search space
right ← upper bound of search space
ans ← −1
while left ≤ right do

mid ← (left + right)/2
if check(mid) then
left ← mid + 1
ans ← mid

else

right ← mid − 1

end
return ans

Algorithm: Binary searching for the answer

Function binarySearch2

pos ← 0
max ← upper bound of search space
for (a = max; a ≥ 1; a /= 2) do
while check(pos + a) do

pos ← pos + a

end

end
return pos

12.2 Example

Source: Codeforces Round 577 (Div. 2) Problem C
https://codeforces.com/contest/1201/problem/C

Given an array arr of n integers, where n is odd, we can perform the following operation
on it k times: take any element of the array and increase it by 1. We want to make the
median of the array as large as possible, after k operations.
Constraints: 1 ≤ n ≤ 2 · 105, 1 ≤ k ≤ 109 and n is odd.
The solution is as follows: we first sort the array in ascending order. Then, we binary
search for the maximum possible median. We know that the number of operations required
to raise the median to x increases monotonically as x increases, so we can use binary search.
For a given median value x, the number of operations required to raise the median to x is

n
(cid:88)

i=(n+1)/2

max(0, x − arr[i])

CHAPTER 12. BINARY SEARCH

66

If this value is less than or equal to k, then x can be the median, so our check function
returns true. Otherwise, x cannot be the median, so our check function returns false.

Solution code (using the second implementation of binary search):

static int n;
static long k;
static long[] arr;
public static void main(String[] args) {

n = r.nextInt(); k = r.nextLong();
arr = new long[n];
for(int i = 0; i < n; i++){

arr[i] = r.nextLong();

}
Arrays.sort(arr);

pw.println(search());
pw.close();

}

// binary searches for the correct answer
static long search(){

long pos = 0; long max = (long)2E9;
for(long a = max; a >= 1; a /= 2){

while(check(pos+a)) pos += a;

}
return pos;

}

// checks whether the number of given operations is sufficient
// to raise the median of the array to x
static boolean check(long x){

long operationsNeeded = 0;
for(int i = (n-1)/2; i < n; i++){

operationsNeeded += Math.max(0, x-arr[i]);

}
if(operationsNeeded <= k){ return true; }
else{ return false; }

}

12.3 Problems

1. USACO December 2018 Silver Problem 1: Convention

http://www.usaco.org/index.php?page=viewproblem2&cpid=858

CHAPTER 12. BINARY SEARCH

67

2. USACO January 2016 Silver Problem 1: Angry Cows

http://usaco.org/index.php?page=viewproblem2&cpid=594

3. USACO January 2017 Silver Problem 1: Cow Dance Show

http://www.usaco.org/index.php?page=viewproblem2&cpid=690

4. Educational Codeforces Round 60 Problem C: Magic Ship

https://codeforces.com/problemset/problem/1117/C (Also uses prefix sums)

5. USACO January 2020 Silver Problem 2: Loan Repayment

http://www.usaco.org/index.php?page=viewproblem2&cpid=991
(Warning: extremely difficult for silver)

Chapter 13

Elementary Number Theory

13.1 Prime Factorization

A number a is called a divisor or a factor of a number b if b is divisible by a, which means
that there exists some integer k such that b = ka. Conventionally, 1 and n are considered
divisors of n. A number n > 1 is prime if its only divisors are 1 and n. Numbers greater
than 1 that are not prime are composite.

Every number has a unique prime factorization: a way of decomposing it into a product

of primes, as follows:

n = p1

a1p2

a2 · · · pk

ak

where the pi are distinct primes and the ai are positive integers.

Now, we will discuss how to find the prime factorization of an integer.

Algorithm: Finds the prime factorization of a number

Function factor

: n, the number to be factorized
Input
Output : v, a list of all the prime factors
v ← empty list
for i ← 2 to (cid:98)

n(cid:99) do

√

while n is divisible by i do

n ← n/i
Add i to the list v

end

end
return v ;

√

√

This algorithm runs in O(

n) time, because the for loop checks divisibility for at most
n values. Even though there is a while loop inside the for loop, dividing n by i quickly
reduces the value of n, which means that the outer for loop runs less iterations, which actually
speeds up the code.

68

CHAPTER 13. ELEMENTARY NUMBER THEORY

69

Let’s look at an example of how this algorithm works, for n = 252.

i

n

2 252
2 126
63
2
21
3
7
3

v

{}
{2}
{2, 2}
{2, 2, 3}
{2, 2, 3, 3}

At this point, the for loop terminates, because i is already 3 which is greater than (cid:98)
7(cid:99). In
the last step, we add 7 to the list of factors v, because it otherwise won’t be added, for a
final prime factorization of {2, 2, 3, 3, 7}.

√

13.2 GCD and LCM

The greatest common divisor (GCD) of two integers a and b is the largest integer
that is a factor of both a and b. In order to find the GCD of two numbers, we use the
Euclidean Algorithm, which is as follows:

gcd(a, b) =

(cid:40)

a
gcd(b, a mod b)

b = 0
b (cid:54)= 0

This algorithm is very easy to implement using a recursive function, as follows:

public int gcd(int a, int b){
if(b == 0) return a;
return gcd(b, a % b);

}

Finding the GCD of two numbers can be done in O(log n) time, where n = min(a, b).

The least common multiple (LCM) of two integers a and b is the smallest integer

divisible by both a and b.

The LCM can easily be calculated from the following property with the GCD:

lcm(a, b) =

a · b
gcd(a, b)

If we want to take the GCD or LCM of more than two elements, we can do so two at a time,
in any order. For example,

gcd(a1, a2, a3, a4) = gcd(a1, gcd(a2, gcd(a3, a4)))

CHAPTER 13. ELEMENTARY NUMBER THEORY

70

13.3 Modular Arithmetic

In modular arithmetic, instead of working with integers themselves, we work with their
remainders when divided by m. We call this taking modulo m. For example, if we take
m = 23, then instead of working with x = 247, we use x mod 23 = 17. Usually, m will be a
large prime, given in the problem; the two most common values are 109 + 7, and 998 244 353.
Modular arithmetic is used to avoid dealing with numbers that overflow built-in data types,
because we can take remainders, according to the following formulas:

(a + b) mod m = (a mod m + b mod m) mod m
(a − b) mod m = (a mod m − b mod m) mod m

(a · b)

(mod m) = ((a mod m) · (b mod m)) mod m

ab mod m = (a mod m)b mod m

Under a prime moduli, division does exist; however it’s rarely used in problems and is

beyond the scope of this book.

13.4 Problems

1. CodeForces VK Cup 2012 Wildcard Round 1

https://codeforces.com/problemset/problem/162/C

Chapter 14

Additional Silver Topics

14.1 Two Pointers

The two pointers method iterates two pointers across an array, to track the start and end
of an interval, or two values in a sorted array that we are currently checking. Both pointers
are monotonic; meaning each pointer starts at one end of the array and only move in one
direction.

2SUM Problem

Given an array of N elements (1 ≤ N ≤ 105), find two elements that sum to X. We can
solve this problem using two pointers; sort the array, then set one pointer at the beginning
and one pointer at the end of the array. Then, we consider the sum of the numbers at the
indices of the pointers. If the sum is too small, advance the left pointer towards the right,
and if the sum is too large, advance the right pointer towards the left. Repeat until either
the correct sum is found, or the pointers meet (in which case there is no solution).

Let’s take the following example array, where N = 6 and X = 15

First, we sort the array:

1

7

11

10

5

13

1

5

7

10

11

13

We then place the left pointer at the start of the array, and the right pointer at the end

of the array.

1

5

7

10

11

13

Then, run and repeat this process: If the sum of the pointer elements is less than X,
move the left pointer one step to the right. If the sum is greater than X, move the right
pointer one step to the left. The example is as follows. First, the sum 1 + 13 = 14 is too
small, so we move the left pointer one step to the right.

1

5

7

10

11

13

71

CHAPTER 14. ADDITIONAL SILVER TOPICS

72

Now, 5 + 13 = 18 overshoots the sum we want, so we move the right pointer one step to

the left.

1

5

7

10

11

13

At this point we have 5 + 11 = 16, still too big. We continue moving the right pointer to

the left.

1

5

7

10

11

13

Now, we have the correct sum, and we are done.
Code is as follows:

int left = 0; int right = n-1;
while(left < right){

if(arr[left] + arr[right] == x){

break;

} else if(arr[left] + arr[right] < x){

left++;

} else {

right--;

}

}
// if left >= right after the loop ends, no answer exists.

Subarray Sum

Given an array of N (1 ≤ N ≤ 105) positive elements, find a contiguous subarray that

sums to X.

We can do this in a similar manner to how we did the 2SUM problem: except this time we
start both pointers at the left, and the pointers mark the beginning and end of the subarray
we are currently checking. We advance the right pointer one step to the right if the total of
the current subarray is too small, advance the left pointer one step to the right if the current
total is too large, and we are done when we find the correct total.

Maximum subarray sum

Another problem that isn’t quite solved by two pointers, but is somewhat related, is the

maximum subarray sum problem.

Given an array of N integers (1 ≤ N ≤ 105), which can be positive or negative, find the

maximum sum of a contiguous subarray.

We can solve this problem using Kadane’s algorithm, which works as follows: we iterate
through the elements of the array, and for each index i, we maintain the maximum subarray
sum of a subarray ending at i in the variable current, and the maximum subarray sum of a
subarray ending at or before i, in the variable best.

Example code is below.

CHAPTER 14. ADDITIONAL SILVER TOPICS

73

int best = 0; int current = 0;
for(int i = 0; i < n; i++){

current = Math.max(0, current + arr[i]);
best = Math.max(best, current);

}

14.2 Line Sweep

Line sweep is the technique of sorting a set of points or line segments and then processing
them in order (this usually means from left to right). The name line sweep comes from the
fact that we are sweeping an imaginary vertical line across the plane containing the points or
segments.

To describe this technique, we’ll be using the 2019 US Open problem, “Cow Steeplechase

II”.

http://usaco.org/index.php?page=viewproblem2&cpid=943
In this problem, we are given some line segments and asked to find one line segment and
remove it such that the resulting segments form no intersections. It is guaranteed that this is
always possible.

First of all, let’s observe it is sufficient to find any two line segments that intersect. Once
we have done this, the solution is guaranteed to be one of these two segments. Then, out of
the two, the segment with multiple intersections is the answer (because removing any other
segment decreases the number of intersections by at most 1, and only removing the segment
with multiple intersections ensures there are no intersections).

If both segments have one intersection, that means the intersect with each other, so we
should return the one with the smallest index (as per the problem statement). Now, the
problem reduces to two parts: checking if two line segments intersect, and processing the line
segments using a line sweep.

Checking If Two Segments Intersect

To check if two line segments intersect, we will use a fact from algebra: if we have the
points A = (xa, ya), B = (xb, yb), and C = (xc, yc), then the (signed) area of (cid:52)ABC, denoted
[ABC], is (xb − xa)(yc − ya) − (xc − xa)(yb − ya). This can be derived from the cross product
of the vectors

−→
AB and

−→
AC.

The part that will help us is the fact that this area is signed, which means that [ABC] is

positive if A, B, and C occur in counterclockwise order,

negative if A, B, and C occur in clockwise order, and

zero if A, B, and C are collinear.

Then, the key observation is that two segments P Q and XY intersect if the two conditions
hold:

CHAPTER 14. ADDITIONAL SILVER TOPICS

74

• [XP Q] and [Y P Q] have different signs

• [P XY ] and [QXY ] have different signs

For example, in the figure below, [X1P1Q1] and [Q1X1Y1] are positive because their vertices
occur in counterclockwise order, and [Y1P1Q1] and [P1X1Y1] are negative because their vertices
occur in clockwise order. Therefore, we know that X1Y1 and P1Q1 intersect. Similarly, on
the right, we know that [P2X2Y2] and [Q2X2Y2] have vertices both going in clockwise order,
so their signed areas are the same, and therefore P2Q2 and X2Y2 don’t intersect.

X1

P1

Q1

P2

X2

Q2

Y1

Y2

If the two conditions hold and some of the signs are zero, then this means that the segments
intersect at their endpoints. If the problem does not count these as intersecting, then consider
zero to have the same sign as both positive and negative.

However, there is a special case. If the signs of all four areas are zero, then all four points
lie on a line. To check if they intersect in this case, we just check whether one point is
between the others. In particular, we check if P or Q is on XY or if X is on P Q. We don’t
need to check if Y is on P Q because if the segments do intersect, we will have two instances
of points on the other segments.
Here’s a full implementation:

public class Point {
public int x, y;
public Point(int x, int y){
this.x = x; this.y = y;

}

}

public static int sign(Point A, Point B, Point C) {

int area = (B.x-A.x) * (C.y-A.y) - (C.x-A.x) * (B.y-A.y);
if (area > 0) return 1;
if (area < 0) return -1;
return 0;

}

public static boolean between(Point P, Point X, Point Y) {

return ((X.x <= P.x && P.x <= Y.x) || (Y.x <= P.x && P.x <= X.x))

&& ((X.y <= P.y && P.y <= Y.y) || (Y.y <= P.y && P.y <= X.y));

CHAPTER 14. ADDITIONAL SILVER TOPICS

75

}

public static boolean intersectQ(Point P, Point Q, Point X, Point Y) {

int[] signs = {sign(P, X, Y), sign(Q, X, Y), sign(X, P, Q), sign(Y, P, Q)};
if (signs[0] == 0 && signs[1] == 0 && signs[2] == 0 && signs[3] == 0)

return between(P, X, Y) || between(Q, X, Y) || between(X, P, Q);

return signs[0] != signs[1] && signs[2] != sign[3];

}

Processing Line Segments

Let’s break apart the N line segments into 2N events, one for each start and end point.
We’ll store whether some event is a start point or an end point, and which start points
correspond to each end point.

Then, we process the endpoints in order of x coordinate from left to right, maintaining a
set of currently processed segments, which is sorted by y. When we hit an endpoint, we either
add or remove a segment from the set, depending on whether we start or end a segment.
Every time we add a segment, we check it for intersection with the segment above it and the
segment below it. In addition, every time we remove a segment, we check the segment above
it and the segment below it for intersection. Once we find an intersection, we are done.

14.3 Bitwise Operations and Subsets

Binary Representations of Integers

In programming, numbers are stored as binary representations. This means that a number

x is represented as

n
(cid:88)

ai2i,

x =

i=0
where the ais are either 0 or 1 and n = (cid:98)log2 x(cid:99).

For example:

17 = 24 + 20 = 100012

Each digit in the binary representation, which is either 0 or 1, is called a bit.

Bitwise Operations

There are several binary operations on binary numbers called bitwise operations. These
operations are applied separately for each bit position. The common binary operations are
shown in table 14.1:

CHAPTER 14. ADDITIONAL SILVER TOPICS

76

Bit A Bit B A and B A or B A xor B

1
1
0
0

1
0
1
0

1
0
0
0

1
1
1
0

0
1
1
0

Table 14.1: The outputs of bitwise operations on two bits

The AND operation (&) returns 1 if and only if both bits are 1.

19 & 27

1 0 0 1 1 = 19
AN D 1 1 0 1 1 = 27
1 0 0 1 1 = 19

=

The OR operation (|) returns 1 if either bit is 1.

19 | 27

1 0 0 1 1 = 19
OR 1 1 0 1 1 = 27
= 1 1 0 1 1 = 27

The XOR operation (∧) returns 1 if and only if exactly one of the bits is 1.

19 ∧ 26

1 0 0 1 1 = 19
XOR 1 1 0 1 1 = 27
0 1 0 0 0 = 8

=

Finally, the left shift operator x << k multiplies x by 2k. Watch for overflow and use the

long data type if necessary. For example:

1 << 5 = 1 · 25 = 32
7 << 2 = 7 · 22 = 28

Exercises

Calculate by converting the numbers to binary, applying the bit operations, and then

converting back to decimal numbers:

(a) 19 & 34

(b) 14 | 29

(c) 10 ∧ 19

(d) 3 << 5

Answer: 2

Answer: 31

Answer: 25

Answer: 96

CHAPTER 14. ADDITIONAL SILVER TOPICS

77

Generating Subsets

Occasionally in a problem we’ll want to iterate through every possible subset of a given
set, either to find a subset that satisfies some condition, or to find the number of subsets that
satisfy some condition. Also, some problems might ask you to find the number of partitions
of a set into 2 groups that satisfy a certain condition. In this case, we will iterate through all
possible subsets, and check each subset for validity (first adding the non-selected elements to
the second subset if necessary).

In a set of N elements, there are 2N possible subsets, because for each of the N elements,
there are two choices: either in the subset, or not in the subset. Subset problems usually
require a time complexity of O(N · 2N ), because each subset has an average of O(N ) elements.
Now, let’s look at how we can generate the subsets. We can represent subsets as binary
numbers from 0 to 2N − 1. Then, each bit represents whether or not a certain element is in
the subset. Let’s look at an example set of a, b, c.

number binary

subset

0
1
2
3
4
5
6
7

000
001
010
011
100
101
110
111

{ }
{a}
{b}
{a, b}
{c}
{a, c}
{b, c}
{a, b, c}

Algorithm: The algorithm for generating all subsets of a given input array

Function generateSubsets

Input : An array arr, and its length n
for i ← 0 to 2n − 1 do

Declare list
for j = 0 to n-1 do

if the bit in the binary representation of i corresponding to 2j is 1 then

Add arr[j] to the list

end

end
Process the list

end

In the following code, our original set is represented by the array arr[] with length n.

int ans = 0;
for(int i = 0; i < (1<<n); i++){
// this loop iterates through the 2^n subsets, one by one.
// 1 << n is a shortcut for 2^n

CHAPTER 14. ADDITIONAL SILVER TOPICS

78

ArrayList<Integer> list = new ArrayList<Integer>();
// we create a new list for each subset and add
// the elements to it
for(int j = 0; j < n; j++){
if((i & (1 << j)) > 0){

// (1 << j) is the number where only the bit representing 2^j is 1.
list.add(arr[j]); // if the respective bit of i is 1,
// add that element to the list

}

}
if(valid(list)){

// code is not included here, but this method will vary depending on the
// problem to check if a certain subset is valid
// and increments the answer counter if so.
ans++;

}

}

14.4 Ad-hoc Problems

The silver division also often has ad hoc problems. They primarily rely on non-standard
algorithmic thinking and problem solving ability. You develop these skills by solving problems;
thus, we don’t have much content to teach you about ad hoc problems, but we provide a
selection of problems at the end of the chapter for your practice.

14.5 Problems

Two Pointers

1. CSES Problem Set Task 1640: Sum of Two Values

https://cses.fi/problemset/task/1640

2. CSES Problem Set Task 1643: Maximum Subarray Sum

https://cses.fi/problemset/task/1643

Line Sweep

3. USACO US Open 2019 Silver Problem 2: Cow Steeplechase II

http://usaco.org/index.php?page=viewproblem2&cpid=943

Subsets

4. (Subsets) CSES Problem Set Task 1623: Apple Division

https://cses.fi/problemset/task/1623

CHAPTER 14. ADDITIONAL SILVER TOPICS

79

Ad hoc problems

5. USACO February 2016 Silver Problem 1: Circular Barn

http://usaco.org/index.php?page=viewproblem2&cpid=618

6. USACO US Open 2019 Silver Problem 1: Left Out

http://www.usaco.org/index.php?page=viewproblem2&cpid=942

7. USACO February 2019 Silver Problem 1: Sleepy Cow Herding

http://www.usaco.org/index.php?page=viewproblem2&cpid=918

8. USACO January 2017 Silver Problem 3: Secret Cow Code

http://www.usaco.org/index.php?page=viewproblem2&cpid=692

9. USACO January 2020 Silver Problem 1: Berry Picking

http://www.usaco.org/index.php?page=viewproblem2&cpid=990

10. USACO December 2019 Silver Problem 2: Meetings

http://www.usaco.org/index.php?page=viewproblem2&cpid=967
(Warning: extremely difficult)

Part IV

Problem Set

80

Chapter 15

Parting Shots

You improve at competitive programming primarily by doing problems, so we leave you
with an extensive selection of CodeForces problems for your practice. This consists of five
problem sets of ten problems each, increasing in difficulty. The problems mostly use topics
covered in the book, but may require some ingenuity to find the solution. If you get stuck,
you can search for the editorial. Best of luck!

Set 1

1. https://codeforces.com/problemset/problem/1227/B

2. https://codeforces.com/problemset/problem/1196/B

3. https://codeforces.com/problemset/problem/1195/B

4. https://codeforces.com/problemset/problem/1294/B

5. https://codeforces.com/problemset/problem/1288/B

6. https://codeforces.com/problemset/problem/1293/A

7. https://codeforces.com/problemset/problem/1213/B

8. https://codeforces.com/problemset/problem/1207/B

9. https://codeforces.com/problemset/problem/1324/B

10. https://codeforces.com/problemset/problem/1327/A

Set 2

1. https://codeforces.com/problemset/problem/1182/B

2. https://codeforces.com/problemset/problem/1183/D

3. https://codeforces.com/problemset/problem/1183/C

4. https://codeforces.com/problemset/problem/1133/C

81

CHAPTER 15. PARTING SHOTS

82

5. https://codeforces.com/problemset/problem/1249/B2

6. https://codeforces.com/problemset/problem/1194/B

7. https://codeforces.com/problemset/problem/1271/C

8. https://codeforces.com/problemset/problem/1326/C

9. https://codeforces.com/problemset/problem/1294/C

10. https://codeforces.com/problemset/problem/1272/B

Set 3

1. https://codeforces.com/problemset/problem/1169/B

2. https://codeforces.com/problemset/problem/1102/D

3. https://codeforces.com/problemset/problem/978/F

4. https://codeforces.com/problemset/problem/1196/C

5. https://codeforces.com/problemset/problem/1154/D

6. https://codeforces.com/problemset/problem/1272/D

7. https://codeforces.com/problemset/problem/1304/C

8. https://codeforces.com/problemset/problem/1296/C

9. https://codeforces.com/contest/1263/problem/D

10. https://codeforces.com/contest/1339/problem/C

Set 4

1. https://codeforces.com/problemset/problem/1281/B

2. https://codeforces.com/problemset/problem/1196/D2

3. https://codeforces.com/problemset/problem/1165/D

4. https://codeforces.com/problemset/problem/1238/C

5. https://codeforces.com/problemset/problem/1234/D

6. https://codeforces.com/problemset/problem/1198/B

7. https://codeforces.com/problemset/problem/1198/A

8. https://codeforces.com/problemset/problem/1077/D

9. https://codeforces.com/problemset/problem/1303/C

10. https://codeforces.com/problemset/problem/1098/A

CHAPTER 15. PARTING SHOTS

83

Set 5

1. https://codeforces.com/problemset/problem/1185/D

2. https://codeforces.com/problemset/problem/1195/D2

3. https://codeforces.com/problemset/problem/1154/E

4. https://codeforces.com/contest/1195/problem/C

5. https://codeforces.com/problemset/problem/1196/E

6. https://codeforces.com/problemset/problem/1328/D

7. https://codeforces.com/problemset/problem/1253/D

8. https://codeforces.com/problemset/problem/1157/E

9. https://codeforces.com/problemset/problem/1185/C2

10. https://codeforces.com/problemset/problem/1209/D