# Programming Assignment 2: Decomposition of Graphs solution

Introduction Welcome to your second programming assignment of the Algorithms on Graphs class! In this assignment, we focus on directed graphs and their parts. In this programming assignment, the grader will show you the input and output data if your solution fails on any of the tests. This is done to help you to get used to the algorithmic problems in general and get some experience debugging your programs while knowing exactly on which tests they fail. However, for all the following programming assignments, the grader will show the input data only in case your solution fails on one of the first few tests (please review the questions 6.4 and 6.5 in the FAQ section for a more detailed explanation of this behavior of the grader).

Learning Outcomes Upon completing this programming assignment you will be able to: 1. check consistency of Computer Science curriculum; 2. find an order of courses that is consistent with prerequisite dependencies; 3. check whether any intersection of a city is reachable from any other intersection.

Passing Criteria: 2 out of 3 Passing thisprogramming assignmentrequires passingat least2out of3code problemsfrom thisassignment. In turn, passing a code problem requires implementing a solution that passes all the tests for this problem in the grader and does so under the time and memory limits specified in the problem statement.

1

Contents 1 Graph Representation in Programming Assignments 3

2 Problem: Checking Consistency of CS Curriculum 5

3 Problem: Determining an Order of Courses 7

4 Advanced Problem: Checking Whether Any Intersection in a City is Reachable from Any Other 9

5 General Instructions and Recommendations on Solving Algorithmic Problems 11 5.1 Reading the Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2 Designing an Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.3 Implementing Your Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.4 Compiling Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.5 Testing Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.6 Submitting Your Program to the Grading System . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.7 Debugging and Stress Testing Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Frequently Asked Questions 14 6.1 I submit the program, but nothing happens. Why? . . . . . . . . . . . . . . . . . . . . . . . . 14 6.2 I submit the solution only for one problem, but all the problems in the assignment are graded. Why? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.3 What are the possible grading outcomes, and how to read them? . . . . . . . . . . . . . . . . 14 6.4 How to understand why my program fails and to fix it? . . . . . . . . . . . . . . . . . . . . . 15 6.5 Why do you hide the test on which my program fails? . . . . . . . . . . . . . . . . . . . . . . 15 6.6 My solution does not pass the tests? May I post it in the forum and ask for a help? . . . . . . 16 6.7 My implementation always fails in the grader, though I already tested and stress tested it a lot. Would not it be better if you give me a solution to this problem or at least the test cases that you use? I will then be able to fix my code and will learn how to avoid making mistakes. Otherwise, I do not feel that I learn anything from solving this problem. I am just stuck. . . . 16

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1 Graph Representation in Programming Assignments In programming assignments, graphs are given as follows. The first line contains non-negative integers n and m — the number of vertices and the number of edges respectively. The vertices are always numbered from 1 to n. Each of the following m lines defines an edge in the format u v where 1 ≤ u,v ≤ n are endpoints of the edge. If the problem deals with an undirected graph this defines an undirected edge between u and v. In case of a directed graph this defines a directed edge from u to v. If the problem deals with a weighted graph then each edge is given as u v w where u and v are vertices and w is a weight. It is guaranteed that a given graph is simple. That is, it does not contain self-loops (edges going from a vertex to itself) and parallel edges. Examples: ∙ An undirected graph with four vertices and five edges: 4 5 2 1 4 3 1 4 2 4 3 2

1 2

34

∙ A directed graph with five vertices and eight edges. 5 8 4 3 1 2 3 1 3 4 2 5 5 1 5 4 5 3

1 3

2 5 4

∙ A directed graph with five vertices and one edge. 5 1 4 3

1 3

2 5 4

Note that the vertices 1, 2, and 5 are isolated (have no adjacent edges), but they are still present in the graph.

3

∙ A weighted directed graph with three vertices and three edges. 3 3 2 3 9 1 3 5 1 2 -2

1 2

3

−2

5 9

4

2 Problem: Checking Consistency of CS Curriculum Problem Introduction A Computer Science curriculum specifies the prerequisites for each course as a list of courses that should be taken before taking this course. You would like to perform a consistency check of the curriculum, that is, to check that there are no cyclic dependencies. For this, you construct the following directed graph: vertices correspond to courses, there is a directed edge (u,v) is the course u should be taken before the course v. Then, it is enough to check whether the resulting graph contains a cycle.

Problem Description Task. Check whether a given directed graph with n vertices and m edges contains a cycle. Input Format. A graph is given in the standard format. Constraints. 1 ≤ n ≤ 103, 0 ≤ m ≤ 103. Output Format. Output 1 if the graph contains a cycle and 0 otherwise. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time (sec) 1 1 1.5 5 1.5 2 5 5 3 Memory Limit. 512MB. Sample 1. Input: 4 4 1 2 4 1 2 3 3 1 Output: 1 Explanation:

1 2

34

This graph contains a cycle: 3 → 1 → 2 → 3. Sample 2. Input: 5 7 1 2 2 3 1 3 3 4 1 4 2 5 3 5

5

Output: 0 Explanation:

1 2

34 5

There is no cycle in this graph. This can be seen, for example, by noting that all edges in this graph go from a vertex with a smaller number to a vertex with a larger number.

Starter Files The starter solutions for this problem read the input data from the standard input, pass it to a blank procedure, and then write the result to the standard output. You are supposed to implement your algorithm in this blank procedure if you are using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch. Filename: acyclicity

What To Do Tosolvethisproblem,itisenoughtoimplementcarefullythecorrespondingalgorithmcoveredinthelectures.

6

3 Problem: Determining an Order of Courses Problem Introduction Now, when you are sure that there are no cyclic dependencies in the given CS curriculum, you would like to find an order of all courses that is consistent with all dependencies. For this, you find a topological ordering of the corresponding directed graph.

Problem Description Task. Compute a topological ordering of a given directed acyclic graph (DAG) with n vertices and m edges. Input Format. A graph is given in the standard format. Constraints. 1 ≤ n ≤ 105, 0 ≤ m ≤ 105. The given graph is guaranteed to be acyclic. Output Format. Output any topological ordering of its vertices. (Many DAGs have more than just one topological ordering. You may output any of them.) Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time (sec) 2 2 3 10 3 4 10 10 6

Memory Limit. 512MB. Sample 1. Input: 4 3 1 2 4 1 3 1 Output: 4 3 1 2 Explanation:

1 2

34

4 3 1 2

Sample 2. Input: 4 1 3 1 Output: 2 3 1 4 Explanation:

1 2

34

2 3 1 4

7

Sample 3. Input: 5 7 2 1 3 2 3 1 4 3 4 1 5 2 5 3 Output: 5 4 3 2 1 Explanation:

1 2

34 5

5 4 3 2 1

Starter Files The starter solutions for this problem read the input data from the standard input, pass it to a blank procedure, and then write the result to the standard output. You are supposed to implement your algorithm in this blank procedure if you are using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch. Filename: toposort

What To Do Tosolvethisproblem,itisenoughtoimplementcarefullythecorrespondingalgorithmcoveredinthelectures.

8

4 Advanced Problem: Checking Whether Any Intersection in a City is Reachable from Any Other Westronglyrecommendyoustartsolvingadvancedproblemsonlywhenyouaredonewiththebasicproblems (for some advanced problems, algorithms are not covered in the video lectures and require additional ideas to be solved; for some other advanced problems, algorithms are covered in the lectures, but implementing them is a more challenging task than for other problems).

Problem Introduction The police department of a city has made all streets one-way. You would like to check whether it is still possible to drive legally from any intersection to any other intersection. For this, you construct a directed graph: vertices are intersections, there is an edge (u,v) whenever there is a (one-way) street from u to v in the city. Then, it suffices to check whether all the vertices in the graph lie in the same strongly connected component.

Problem Description Task. Compute the number of strongly connected components of a given directed graph with n vertices and m edges. Input Format. A graph is given in the standard format. Constraints. 1 ≤ n ≤ 104, 0 ≤ m ≤ 104. Output Format. Output the number of strongly connected components. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time (sec) 1 1 1.5 5 1.5 2 5 5 3

Memory Limit. 512MB. Sample 1. Input: 4 4 1 2 4 1 2 3 3 1 Output: 2 Explanation:

1 2

34

This graph has two strongly connected components: {1,3,2}, {4}.

9

Sample 2. Input: 5 7 2 1 3 2 3 1 4 3 4 1 5 2 5 3 Output: 5 Explanation:

1 2

34 5

This graph has five strongly connected components: {1}, {2}, {3}, {4}, {5}. Starter Files The starter solutions for this problem read the input data from the standard input, pass it to a blank procedure, and then write the result to the standard output. You are supposed to implement your algorithm in this blank procedure if you are using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch.

What To Do Tosolvethisproblem,itisenoughtoimplementcarefullythecorrespondingalgorithmcoveredinthelectures.

10

5 General Instructions and Recommendations on Solving Algorithmic Problems Your main goal in an algorithmic problem is to implement a program that solves a given computational problem in just few seconds even on massive datasets. Your program should read a dataset from the standard input and write an answer to the standard output. Below we provide general instructions and recommendations on solving such problems. Before reading them, go through readings and screencasts in the first module that show a step by step process of solving two algorithmic problems: link.

5.1 Reading the Problem Statement You start by reading the problem statement that contains the description of a particular computational task as well as time and memory limits your solution should fit in, and one or two sample tests. In some problems your goal is just to implement carefully an algorithm covered in the lectures, while in some other problems you first need to come up with an algorithm yourself.

5.2 Designing an Algorithm If your goal is to design an algorithm yourself, one of the things it is important to realize is the expected running time of your algorithm. Usually, you can guess it from the problem statement (specifically, from the subsection called constraints) as follows. Modern computers perform roughly 108–109 operations per second. So, if the maximum size of a dataset in the problem description is n = 105, then most probably an algorithm with quadratic running time is not going to fit into time limit (since for n = 105, n2 = 1010) while a solution with running time O(nlogn) will fit. However, an O(n2) solution will fit if n is up to 103 = 1000, and if n is at most 100, even O(n3) solutions will fit. In some cases, the problem is so hard that we do not know a polynomial solution. But for n up to 18, a solution with O(2nn2) running time will probably fit into the time limit. To design an algorithm with the expected running time, you will of course need to use the ideas covered in the lectures. Also, make sure to carefully go through sample tests in the problem description.

5.3 Implementing Your Algorithm When you have an algorithm in mind, you start implementing it. Currently, you can use the following programming languages to implement a solution to a problem: C, C++, C#, Haskell, Java, JavaScript, Python2, Python3, Ruby, Scala. For all problems, we will be providing starter solutions for C++, Java, and Python3. If you are going to use one of these programming languages, use these starter files. For other programming languages, you need to implement a solution from scratch.

5.4 Compiling Your Program For solving programming assignments, you can use any of the following programming languages: C, C++, C#, Haskell, Java, JavaScript, Python2, Python3, Ruby, and Scala. However, we will only be providing starter solution files for C++, Java, and Python3. The programming language of your submission is detected automatically, based on the extension of your submission. We have reference solutions in C++, Java and Python3 which solve the problem correctly under the given restrictions, and in most cases spend at most 1/3 of the time limit and at most 1/2 of the memory limit. You can also use other languages, and we’ve estimated the time limit multipliers for them, however, we have no guarantee that a correct solution for a particular problem running under the given time and memory constraints exists in any of those other languages. Your solution will be compiled as follows. We recommend that when testing your solution locally, you use the same compiler flags for compiling. This will increase the chances that your program behaves in the

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same way on your machine and on the testing machine (note that a buggy program may behave differently when compiled by different compilers, or even by the same compiler with different flags). ∙ C (gcc 5.2.1). File extensions: .c. Flags: gcc -pipe -O2 -std=c11 <filename -lm ∙ C++ (g++ 5.2.1). File extensions: .cc, .cpp. Flags: g++ -pipe -O2 -std=c++14 <filename -lm If your C/C++ compiler does not recognize -std=c++14 flag, try replacing it with -std=c++0x flag or compiling without this flag at all (all starter solutions can be compiled without it). On Linux and MacOS, you most probably have the required compiler. On Windows, you may use your favorite compiler or install, e.g., cygwin. ∙ C# (mono 3.2.8). File extensions: .cs. Flags: mcs ∙ Haskell (ghc 7.8.4). File extensions: .hs. Flags: ghc -O2 ∙ Java (Open JDK 8). File extensions: .java. Flags: javac -encoding UTF-8 java -Xmx1024m ∙ JavaScript (Node v6.3.0). File extensions: .js. Flags: nodejs ∙ Python 2 (CPython 2.7). File extensions: .py2 or .py (a file ending in .py needs to have a first line which is a comment containing “python2”). No flags: python2 ∙ Python 3 (CPython 3.4). File extensions: .py3 or .py (a file ending in .py needs to have a first line which is a comment containing “python3”). No flags: python3 ∙ Ruby (Ruby 2.1.5). File extensions: .rb. ruby ∙ Scala (Scala 2.11.6). File extensions: .scala. scalac

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5.5 Testing Your Program When your program is ready, you start testing it. It makes sense to start with small datasets (for example, sample tests provided in the problem description). Ensure that your program produces a correct result. You then proceed to checking how long does it take your program to process a massive dataset. For this, it makes sense to implement your algorithm as a function like solve(dataset) and then implement an additional procedure generate() that produces a large dataset. For example, if an input to a problem is a sequence of integers of length 1 ≤ n ≤ 105, then generate a sequence of length exactly 105, pass it to your solve() function, and ensure that the program outputs the result quickly. Also, check the boundary values. Ensure that your program processes correctly sequences of size n = 1,2,105. If a sequence of integers from 0 to, say, 106 is given as an input, check how your program behaves when it is given a sequence 0,0,...,0 or a sequence 106,106,...,106. Check also on randomly generated data. For each such test check that you program produces a correct result (or at least a reasonably looking result). In the end, we encourage you to stress test your program to make sure it passes in the system at the first attempt. See the readings and screencasts from the first week to learn about testing and stress testing: link.

5.6 Submitting Your Program to the Grading System When you are done with testing, you submit your program to the grading system. For this, you go the submission page, create a new submission, and upload a file with your program. The grading system then compilesyourprogram(detectingtheprogramminglanguagebasedonyourfileextension,seeSubsection5.4) and runs it on a set of carefully constructed tests to check that your program always outputs a correct result and that it always fits into the given time and memory limits. The grading usually takes no more than a minute, but in rare cases when the servers are overloaded it might take longer. Please be patient. You can safely leave the page when your solution is uploaded. As a result, you get a feedback message from the grading system. The feedback message that you will love to see is: Good job! This means that your program has passed all the tests. On the other hand, the three messages Wrong answer, Time limit exceeded, Memory limit exceeded notify you that your program failed due to one these three reasons. Note that the grader will not show you the actual test you program have failed on (though it does show you the test if your program have failed on one of the first few tests; this is done to help you to get the input/output format right).

5.7 Debugging and Stress Testing Your Program If your program failed, you will need to debug it. Most probably, you didn’t follow some of our suggestions from the section 5.5. See the readings and screencasts from the first week to learn about debugging your program: link. You are almost guaranteed to find a bug in your program using stress testing, because the way these programming assignments and tests for them are prepared follows the same process: small manual tests, tests for edge cases, tests for large numbers and integer overflow, big tests for time limit and memory limit checking, random test generation. Also, implementation of wrong solutions which we expect to see and stress testing against them to add tests specifically against those wrong solutions

Learning Outcomes Upon completing this programming assignment you will be able to: 1. check consistency of Computer Science curriculum; 2. find an order of courses that is consistent with prerequisite dependencies; 3. check whether any intersection of a city is reachable from any other intersection.

Passing Criteria: 2 out of 3 Passing thisprogramming assignmentrequires passingat least2out of3code problemsfrom thisassignment. In turn, passing a code problem requires implementing a solution that passes all the tests for this problem in the grader and does so under the time and memory limits specified in the problem statement.

1

Contents 1 Graph Representation in Programming Assignments 3

2 Problem: Checking Consistency of CS Curriculum 5

3 Problem: Determining an Order of Courses 7

4 Advanced Problem: Checking Whether Any Intersection in a City is Reachable from Any Other 9

5 General Instructions and Recommendations on Solving Algorithmic Problems 11 5.1 Reading the Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2 Designing an Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.3 Implementing Your Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.4 Compiling Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.5 Testing Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.6 Submitting Your Program to the Grading System . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.7 Debugging and Stress Testing Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Frequently Asked Questions 14 6.1 I submit the program, but nothing happens. Why? . . . . . . . . . . . . . . . . . . . . . . . . 14 6.2 I submit the solution only for one problem, but all the problems in the assignment are graded. Why? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.3 What are the possible grading outcomes, and how to read them? . . . . . . . . . . . . . . . . 14 6.4 How to understand why my program fails and to fix it? . . . . . . . . . . . . . . . . . . . . . 15 6.5 Why do you hide the test on which my program fails? . . . . . . . . . . . . . . . . . . . . . . 15 6.6 My solution does not pass the tests? May I post it in the forum and ask for a help? . . . . . . 16 6.7 My implementation always fails in the grader, though I already tested and stress tested it a lot. Would not it be better if you give me a solution to this problem or at least the test cases that you use? I will then be able to fix my code and will learn how to avoid making mistakes. Otherwise, I do not feel that I learn anything from solving this problem. I am just stuck. . . . 16

2

1 Graph Representation in Programming Assignments In programming assignments, graphs are given as follows. The first line contains non-negative integers n and m — the number of vertices and the number of edges respectively. The vertices are always numbered from 1 to n. Each of the following m lines defines an edge in the format u v where 1 ≤ u,v ≤ n are endpoints of the edge. If the problem deals with an undirected graph this defines an undirected edge between u and v. In case of a directed graph this defines a directed edge from u to v. If the problem deals with a weighted graph then each edge is given as u v w where u and v are vertices and w is a weight. It is guaranteed that a given graph is simple. That is, it does not contain self-loops (edges going from a vertex to itself) and parallel edges. Examples: ∙ An undirected graph with four vertices and five edges: 4 5 2 1 4 3 1 4 2 4 3 2

1 2

34

∙ A directed graph with five vertices and eight edges. 5 8 4 3 1 2 3 1 3 4 2 5 5 1 5 4 5 3

1 3

2 5 4

∙ A directed graph with five vertices and one edge. 5 1 4 3

1 3

2 5 4

Note that the vertices 1, 2, and 5 are isolated (have no adjacent edges), but they are still present in the graph.

3

∙ A weighted directed graph with three vertices and three edges. 3 3 2 3 9 1 3 5 1 2 -2

1 2

3

−2

5 9

4

2 Problem: Checking Consistency of CS Curriculum Problem Introduction A Computer Science curriculum specifies the prerequisites for each course as a list of courses that should be taken before taking this course. You would like to perform a consistency check of the curriculum, that is, to check that there are no cyclic dependencies. For this, you construct the following directed graph: vertices correspond to courses, there is a directed edge (u,v) is the course u should be taken before the course v. Then, it is enough to check whether the resulting graph contains a cycle.

Problem Description Task. Check whether a given directed graph with n vertices and m edges contains a cycle. Input Format. A graph is given in the standard format. Constraints. 1 ≤ n ≤ 103, 0 ≤ m ≤ 103. Output Format. Output 1 if the graph contains a cycle and 0 otherwise. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time (sec) 1 1 1.5 5 1.5 2 5 5 3 Memory Limit. 512MB. Sample 1. Input: 4 4 1 2 4 1 2 3 3 1 Output: 1 Explanation:

1 2

34

This graph contains a cycle: 3 → 1 → 2 → 3. Sample 2. Input: 5 7 1 2 2 3 1 3 3 4 1 4 2 5 3 5

5

Output: 0 Explanation:

1 2

34 5

There is no cycle in this graph. This can be seen, for example, by noting that all edges in this graph go from a vertex with a smaller number to a vertex with a larger number.

Starter Files The starter solutions for this problem read the input data from the standard input, pass it to a blank procedure, and then write the result to the standard output. You are supposed to implement your algorithm in this blank procedure if you are using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch. Filename: acyclicity

What To Do Tosolvethisproblem,itisenoughtoimplementcarefullythecorrespondingalgorithmcoveredinthelectures.

6

3 Problem: Determining an Order of Courses Problem Introduction Now, when you are sure that there are no cyclic dependencies in the given CS curriculum, you would like to find an order of all courses that is consistent with all dependencies. For this, you find a topological ordering of the corresponding directed graph.

Problem Description Task. Compute a topological ordering of a given directed acyclic graph (DAG) with n vertices and m edges. Input Format. A graph is given in the standard format. Constraints. 1 ≤ n ≤ 105, 0 ≤ m ≤ 105. The given graph is guaranteed to be acyclic. Output Format. Output any topological ordering of its vertices. (Many DAGs have more than just one topological ordering. You may output any of them.) Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time (sec) 2 2 3 10 3 4 10 10 6

Memory Limit. 512MB. Sample 1. Input: 4 3 1 2 4 1 3 1 Output: 4 3 1 2 Explanation:

1 2

34

4 3 1 2

Sample 2. Input: 4 1 3 1 Output: 2 3 1 4 Explanation:

1 2

34

2 3 1 4

7

Sample 3. Input: 5 7 2 1 3 2 3 1 4 3 4 1 5 2 5 3 Output: 5 4 3 2 1 Explanation:

1 2

34 5

5 4 3 2 1

Starter Files The starter solutions for this problem read the input data from the standard input, pass it to a blank procedure, and then write the result to the standard output. You are supposed to implement your algorithm in this blank procedure if you are using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch. Filename: toposort

What To Do Tosolvethisproblem,itisenoughtoimplementcarefullythecorrespondingalgorithmcoveredinthelectures.

8

4 Advanced Problem: Checking Whether Any Intersection in a City is Reachable from Any Other Westronglyrecommendyoustartsolvingadvancedproblemsonlywhenyouaredonewiththebasicproblems (for some advanced problems, algorithms are not covered in the video lectures and require additional ideas to be solved; for some other advanced problems, algorithms are covered in the lectures, but implementing them is a more challenging task than for other problems).

Problem Introduction The police department of a city has made all streets one-way. You would like to check whether it is still possible to drive legally from any intersection to any other intersection. For this, you construct a directed graph: vertices are intersections, there is an edge (u,v) whenever there is a (one-way) street from u to v in the city. Then, it suffices to check whether all the vertices in the graph lie in the same strongly connected component.

Problem Description Task. Compute the number of strongly connected components of a given directed graph with n vertices and m edges. Input Format. A graph is given in the standard format. Constraints. 1 ≤ n ≤ 104, 0 ≤ m ≤ 104. Output Format. Output the number of strongly connected components. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time (sec) 1 1 1.5 5 1.5 2 5 5 3

Memory Limit. 512MB. Sample 1. Input: 4 4 1 2 4 1 2 3 3 1 Output: 2 Explanation:

1 2

34

This graph has two strongly connected components: {1,3,2}, {4}.

9

Sample 2. Input: 5 7 2 1 3 2 3 1 4 3 4 1 5 2 5 3 Output: 5 Explanation:

1 2

34 5

This graph has five strongly connected components: {1}, {2}, {3}, {4}, {5}. Starter Files The starter solutions for this problem read the input data from the standard input, pass it to a blank procedure, and then write the result to the standard output. You are supposed to implement your algorithm in this blank procedure if you are using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch.

What To Do Tosolvethisproblem,itisenoughtoimplementcarefullythecorrespondingalgorithmcoveredinthelectures.

10

5 General Instructions and Recommendations on Solving Algorithmic Problems Your main goal in an algorithmic problem is to implement a program that solves a given computational problem in just few seconds even on massive datasets. Your program should read a dataset from the standard input and write an answer to the standard output. Below we provide general instructions and recommendations on solving such problems. Before reading them, go through readings and screencasts in the first module that show a step by step process of solving two algorithmic problems: link.

5.1 Reading the Problem Statement You start by reading the problem statement that contains the description of a particular computational task as well as time and memory limits your solution should fit in, and one or two sample tests. In some problems your goal is just to implement carefully an algorithm covered in the lectures, while in some other problems you first need to come up with an algorithm yourself.

5.2 Designing an Algorithm If your goal is to design an algorithm yourself, one of the things it is important to realize is the expected running time of your algorithm. Usually, you can guess it from the problem statement (specifically, from the subsection called constraints) as follows. Modern computers perform roughly 108–109 operations per second. So, if the maximum size of a dataset in the problem description is n = 105, then most probably an algorithm with quadratic running time is not going to fit into time limit (since for n = 105, n2 = 1010) while a solution with running time O(nlogn) will fit. However, an O(n2) solution will fit if n is up to 103 = 1000, and if n is at most 100, even O(n3) solutions will fit. In some cases, the problem is so hard that we do not know a polynomial solution. But for n up to 18, a solution with O(2nn2) running time will probably fit into the time limit. To design an algorithm with the expected running time, you will of course need to use the ideas covered in the lectures. Also, make sure to carefully go through sample tests in the problem description.

5.3 Implementing Your Algorithm When you have an algorithm in mind, you start implementing it. Currently, you can use the following programming languages to implement a solution to a problem: C, C++, C#, Haskell, Java, JavaScript, Python2, Python3, Ruby, Scala. For all problems, we will be providing starter solutions for C++, Java, and Python3. If you are going to use one of these programming languages, use these starter files. For other programming languages, you need to implement a solution from scratch.

5.4 Compiling Your Program For solving programming assignments, you can use any of the following programming languages: C, C++, C#, Haskell, Java, JavaScript, Python2, Python3, Ruby, and Scala. However, we will only be providing starter solution files for C++, Java, and Python3. The programming language of your submission is detected automatically, based on the extension of your submission. We have reference solutions in C++, Java and Python3 which solve the problem correctly under the given restrictions, and in most cases spend at most 1/3 of the time limit and at most 1/2 of the memory limit. You can also use other languages, and we’ve estimated the time limit multipliers for them, however, we have no guarantee that a correct solution for a particular problem running under the given time and memory constraints exists in any of those other languages. Your solution will be compiled as follows. We recommend that when testing your solution locally, you use the same compiler flags for compiling. This will increase the chances that your program behaves in the

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same way on your machine and on the testing machine (note that a buggy program may behave differently when compiled by different compilers, or even by the same compiler with different flags). ∙ C (gcc 5.2.1). File extensions: .c. Flags: gcc -pipe -O2 -std=c11 <filename -lm ∙ C++ (g++ 5.2.1). File extensions: .cc, .cpp. Flags: g++ -pipe -O2 -std=c++14 <filename -lm If your C/C++ compiler does not recognize -std=c++14 flag, try replacing it with -std=c++0x flag or compiling without this flag at all (all starter solutions can be compiled without it). On Linux and MacOS, you most probably have the required compiler. On Windows, you may use your favorite compiler or install, e.g., cygwin. ∙ C# (mono 3.2.8). File extensions: .cs. Flags: mcs ∙ Haskell (ghc 7.8.4). File extensions: .hs. Flags: ghc -O2 ∙ Java (Open JDK 8). File extensions: .java. Flags: javac -encoding UTF-8 java -Xmx1024m ∙ JavaScript (Node v6.3.0). File extensions: .js. Flags: nodejs ∙ Python 2 (CPython 2.7). File extensions: .py2 or .py (a file ending in .py needs to have a first line which is a comment containing “python2”). No flags: python2 ∙ Python 3 (CPython 3.4). File extensions: .py3 or .py (a file ending in .py needs to have a first line which is a comment containing “python3”). No flags: python3 ∙ Ruby (Ruby 2.1.5). File extensions: .rb. ruby ∙ Scala (Scala 2.11.6). File extensions: .scala. scalac

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5.5 Testing Your Program When your program is ready, you start testing it. It makes sense to start with small datasets (for example, sample tests provided in the problem description). Ensure that your program produces a correct result. You then proceed to checking how long does it take your program to process a massive dataset. For this, it makes sense to implement your algorithm as a function like solve(dataset) and then implement an additional procedure generate() that produces a large dataset. For example, if an input to a problem is a sequence of integers of length 1 ≤ n ≤ 105, then generate a sequence of length exactly 105, pass it to your solve() function, and ensure that the program outputs the result quickly. Also, check the boundary values. Ensure that your program processes correctly sequences of size n = 1,2,105. If a sequence of integers from 0 to, say, 106 is given as an input, check how your program behaves when it is given a sequence 0,0,...,0 or a sequence 106,106,...,106. Check also on randomly generated data. For each such test check that you program produces a correct result (or at least a reasonably looking result). In the end, we encourage you to stress test your program to make sure it passes in the system at the first attempt. See the readings and screencasts from the first week to learn about testing and stress testing: link.

5.6 Submitting Your Program to the Grading System When you are done with testing, you submit your program to the grading system. For this, you go the submission page, create a new submission, and upload a file with your program. The grading system then compilesyourprogram(detectingtheprogramminglanguagebasedonyourfileextension,seeSubsection5.4) and runs it on a set of carefully constructed tests to check that your program always outputs a correct result and that it always fits into the given time and memory limits. The grading usually takes no more than a minute, but in rare cases when the servers are overloaded it might take longer. Please be patient. You can safely leave the page when your solution is uploaded. As a result, you get a feedback message from the grading system. The feedback message that you will love to see is: Good job! This means that your program has passed all the tests. On the other hand, the three messages Wrong answer, Time limit exceeded, Memory limit exceeded notify you that your program failed due to one these three reasons. Note that the grader will not show you the actual test you program have failed on (though it does show you the test if your program have failed on one of the first few tests; this is done to help you to get the input/output format right).

5.7 Debugging and Stress Testing Your Program If your program failed, you will need to debug it. Most probably, you didn’t follow some of our suggestions from the section 5.5. See the readings and screencasts from the first week to learn about debugging your program: link. You are almost guaranteed to find a bug in your program using stress testing, because the way these programming assignments and tests for them are prepared follows the same process: small manual tests, tests for edge cases, tests for large numbers and integer overflow, big tests for time limit and memory limit checking, random test generation. Also, implementation of wrong solutions which we expect to see and stress testing against them to add tests specifically against those wrong solutions

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