# Programming Assignment 3: Paths in Graphs solution

Introduction Welcome to your third programming assignment of the Algorithms on Graphs class! In this and the next programming assignments you will be practicing implementing algorithms for ﬁnding shortest paths in graphs. Recall that starting from this programming assignment, the grader will show you only the ﬁrst few tests (see the questions 5.4 and 5.5 in the FAQ section).
Learning Outcomes Upon completing this programming assignment you will be able to: 1. compute the minimum number of ﬂight segments to get from one city to another one; 2. check whether a given graph is bipartite.
Passing Criteria: 1 out of 2 Passing thisprogramming assignmentrequires passingat least1out of2code 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.
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Contents 1 Graph Representation in Programming Assignments 3
2 Problem: Computing the Minimum Number of Flight Segments 4
3 Problem: Checking whether a Graph is Bipartite 6
4 General Instructions and Recommendations on Solving Algorithmic Problems 8 4.1 Reading the Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2 Designing an Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.3 Implementing Your Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.4 Compiling Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.5 Testing Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.6 Submitting Your Program to the Grading System . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.7 Debugging and Stress Testing Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5 Frequently Asked Questions 11 5.1 I submit the program, but nothing happens. Why? . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2 I submit the solution only for one problem, but all the problems in the assignment are graded. Why? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.3 What are the possible grading outcomes, and how to read them? . . . . . . . . . . . . . . . . 11 5.4 How to understand why my program fails and to ﬁx it? . . . . . . . . . . . . . . . . . . . . . 12 5.5 Why do you hide the test on which my program fails? . . . . . . . . . . . . . . . . . . . . . . 12 5.6 My solution does not pass the tests? May I post it in the forum and ask for a help? . . . . . 13 5.7 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. . . 13
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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
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• 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 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
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2 Problem: Computing the Minimum Number of Flight Segments Problem Introduction You would like to compute the minimum number of ﬂight segments to get from one city to another one. For this, you construct the following undirected graph: vertices represent cities, there is an edge between two vertices whenever there is a ﬂight between the corresponding two cities. Then, it suﬃces to ﬁnd a shortest path from one of the given cities to the other one.
Problem Description Task. Given an undirected graph with n vertices and m edges and two vertices u and v, compute the length of a shortest path between u and v (that is, the minimum number of edges in a path from u to v). Input Format. A graph is given in the standard format. The next line contains two vertices u and v. Constraints. 2 ≤ n ≤ 105, 0 ≤ m ≤ 105, u 6= v, 1 ≤ u,v ≤ n. Output Format. Output the minimum number of edges in a path from u to v, or −1 if there is no path. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time in seconds 2 2 3 10 3 4 10 10 6
Memory Limit. 512Mb. Sample 1. Input: 4 4 1 2 4 1 2 3 3 1 2 4 Output: 2 Explanation:
1 2
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There is a unique shortest path between vertices 2 and 4 in this graph: 2−1−4.
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Sample 2. Input: 5 4 5 2 1 3 3 4 1 4 3 5 Output: -1 Explanation:
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3 4 5
There is no path between vertices 3 and 5 in this graph.
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: bfs
What To Do Tosolvethisproblem, itisenoughtoimplementcarefullythecorrespondingalgorithmcoveredinthelectures.
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3 Problem: Checking whether a Graph is Bipartite Problem Introduction An undirected graph is called bipartite if its vertices can be split into two parts such that each edge of the graph joins to vertices from diﬀerent parts. Bipartite graphs arise naturally in applications where a graph is used to model connections between objects of two diﬀerent types (say, boys and girls; or students and dormitories). An alternative deﬁnition is the following: a graph is bipartite if its vertices can be colored with two colors (say, black and white) such that the endpoints of each edge have diﬀerent colors.
Problem Description Task. Given an undirected graph with n vertices and m edges, check whether it is bipartite. Input Format. A graph is given in the standard format. Constraints. 1 ≤ n ≤ 105, 0 ≤ m ≤ 105. Output Format. Output 1 if the graph is bipartite and 0 otherwise. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time in seconds 2 2 3 10 3 4 10 10 6
Memory Limit. 512Mb. Sample 1. Input: 4 4 1 2 4 1 2 3 3 1 Output: 0 Explanation:
1 2
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This graph is not bipartite. To see this assume that the vertex 1 is colored white. Then the vertices 2 and 3 should be colored black since the graph contains the edges {1,2} and {1,3}. But then the edge {2,3} has both endpoints of the same color.
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Sample 2. Input: 5 4 5 2 4 2 3 4 1 4 Output: 1 Explanation:
1 2
3 4 5
This graph is bipartite: assign the vertices 4 and 5 the white color, assign all the remaining vertices the black color.
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: bipartite