Programming Assignment 3: Paths in Graphs Solution

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Description

Introduction

Welcome to your third programming assignment of the Algorithms on Graphs class! In this and the next pro-gramming 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.

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;

  1. check whether a given graph is bipartite.

Passing Criteria: 1 out of 2

Passing this programming assignment requires passing at least 1 out of 2 programming challenges from this assignment. In turn, passing a programming challenge 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.

Contents

1

Computing the Minimum Number of Flight Segments

4

2

Checking whether a Graph is Bipartite

6

3

Appendix

8

3.1

Compiler Flags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

3.2

Frequently Asked Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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 and — the number of vertices and the number of edges respectively. The vertices are always numbered from 1 to . Each of the following lines defines an edge in the format u v where 1 ≤ , ≤ are endpoints of the edge. If the problem deals with an undirected graph this defines an undirected edge between and . In case of a directed graph this defines a directed edge from to . If the problem deals with a weighted graph then each edge is given as u v w where and are vertices and 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:

  • 5

2 1

  • 3

1 4

2 4

3 2

lu57324e4us tmp 77a295f2f816da3d

  • 3

    • 2

  • A directed graph with five vertices and eight edges.

  • 8

4 3

1 2

3 1

3 4

2 5

  • 1

  • 4

  • 3

lu57324e4us tmp 4f9586afed74f3ce

2 lu57324e4us tmp e63c4177ed6359fb 5 lu57324e4us tmp e63c4177ed6359fb 4

1 lu57324e4us tmp e4b4e7f1f87ae4e0 3

  • A directed graph with five vertices and one edge.

  • 1

4 3

lu57324e4us tmp aae7bff941bc0aab

2 5 4

  • 3

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

2

  • A weighted directed graph with three vertices and three edges.

3 3

2 3 9

1 3 5

12-2

lu57324e4us tmp f37476502a01ec9d

3

5 lu57324e4us tmp c39125480c62a5ca lu57324e4us tmp 5f37b868e76ef55a 9

1 lu57324e4us tmp 9514b103a9b0243 2

2

3

  • 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 vertices and edges and two vertices and , compute the length of a shortest path between and (that is, the minimum number of edges in a path from to ).

Input Format. A graph is given in the standard format. The next line contains two vertices and .

Constraints. 2 ≤ ≤ 105, 0 ≤ ≤ 105, ̸= , 1 ≤ , ≤ .

Output Format. Output the minimum number of edges in a path from to , or −1 if there is no path.

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 4

  • 2

4 1

2 3

3 1

2 4

Output:

2

lu57324e4us tmp b0c4d73a8b8c28a

  • 3

12

There is a unique shortest path between vertices 2 and 4 in this graph: 2 − 1 − 4.

4

Sample 2.

Input:

  • 4

  • 2

1 3

3 4

1 4

3 5

Output:

-1

lu57324e4us tmp 373e40db8c31beb3

3 4 5

1 2

There is no path between vertices 3 and 5 in this graph.

5

  • 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 vertices and edges, check whether it is bipartite.

Input Format. A graph is given in the standard format.

Constraints. 1 ≤ ≤ 105, 0 ≤ ≤ 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 (sec)

2

2

3

10

3

4

10

10

6

Memory Limit. 512MB.

Sample 1.

Input:

4 4

  • 2

4 1

2 3

3 1

Output:

0

lu57324e4us tmp b0c4d73a8b8c28a

  • 3

12

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.

6

Sample 2.

Input:

  • 4

  • 2

4 2

3 4

1 4

Output:

1

lu57324e4us tmp faa46158faf46229

3 4 5

  • 2

This graph is bipartite: assign the vertices 4 and 5 the white color, assign all the remaining vertices the black color.

What To Do

Adapt the breadth-first search to solve this problem.

7

  • Appendix

3.1 Compiler Flags

  • (gcc 7.4.0). File extensions: .c. Flags:

gcc – pipe – O2 – std = c11 < filename > – lm

C++ (g++ 7.4.0). 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 4.6.2). File extensions: .cs. Flags:

mcs

Go (golang 1.13.4). File extensions: .go. Flags

go

Haskell (ghc 8.0.2). File extensions: .hs. Flags:

ghc – O2

Java (OpenJDK 1.8.0_232). File extensions: .java. Flags:

javac – encoding UTF -8

java – Xmx1024m

JavaScript (NodeJS 12.14.0). File extensions: .js. No flags:

nodejs

Kotlin (Kotlin 1.3.50). File extensions: .kt. Flags:

kotlinc

java – Xmx1024m

Python (CPython 3.6.9). File extensions: .py. No flags:

python3

Ruby (Ruby 2.5.1p57). File extensions: .rb.

ruby

Rust (Rust 1.37.0). File extensions: .rs.

rustc

Scala (Scala 2.12.10). File extensions: .scala.

scalac

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3.2 Frequently Asked Questions

Why My Submission Is Not Graded?

You need to create a submission and upload the source file(rather than the executable file) of your solution. Make sure that after uploading the file with your solution you press the blue “Submit” button at the bottom. After that, the grading starts, and the submission being graded is enclosed in an orange rectangle. After the testing is finished, the rectangle disappears, and the results of the testing of all problems are shown.

What Are the Possible Grading Outcomes?

There are only two outcomes: “pass” or “no pass.” To pass, your program must return a correct answer on all the test cases we prepared for you, and do so under the time and memory constraints specified in the problem statement. If your solution passes, you get the corresponding feedback “Good job!” and get a point for the problem. Your solution fails if it either crashes, returns an incorrect answer, works for too long, or uses too much memory for some test case. The feedback will contain the index of the first test case on which your solution failed and the total number of test cases in the system. The tests for the problem are numbered from 1 to the total number of test cases for the problem, and the program is always tested on all the tests in the order from the first test to the test with the largest number.

Here are the possible outcomes:

  • Good job! Hurrah! Your solution passed, and you get a point!

  • Wrong answer. Your solution outputs incorrect answer for some test case. Check that you consider all the cases correctly, avoid integer overflow, output the required white spaces, output the floating point numbers with the required precision, don’t output anything in addition to what you are asked to output in the output specification of the problem statement.

  • Time limit exceeded. Your solution worked longer than the allowed time limit for some test case. Check again the running time of your implementation. Test your program locally on the test of max-imum size specified in the problem statement and check how long it works. Check that your program doesn’t wait for some input from the user which makes it to wait forever.

  • Memory limit exceeded. Your solution used more than the allowed memory limit for some test case. Estimate the amount of memory that your program is going to use in the worst case and check that it does not exceed the memory limit. Check that your data structures fit into the memory limit. Check that you don’t create large arrays or lists or vectors consisting of empty arrays or empty strings, since those in some cases still eat up memory. Test your program locally on the tests of maximum size specified in the problem statement and look at its memory consumption in the system.

  • Cannot check answer. Perhaps the output format is wrong. This happens when you output something different than expected. For example, when you are required to output either “Yes” or “No”, but instead output 1 or 0. Or your program has empty output. Or your program outputs not only the correct answer, but also some additional information (please follow the exact output format specified in the problem statement). Maybe your program doesn’t output anything, because it crashes.

  • Unknown signal 6 (or 7, or 8, or 11, or some other). This happens when your program crashes. It can be because of a division by zero, accessing memory outside of the array bounds, using uninitialized variables, overly deep recursion that triggers a stack overflow, sorting with a contradictory comparator, removing elements from an empty data structure, trying to allocate too much memory, and many other reasons. Look at your code and think about all those possibilities. Make sure that you use the same compiler and the same compiler flags as we do.

Internal error: exception… Most probably, you submitted a compiled program instead of a source code.

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  • Grading failed. Something wrong happened with the system. Report this through Coursera or edX Help Center.

May I Post My Solution at the Forum?

Please do not post any solutions at the forum or anywhere on the web, even if a solution does not pass the tests (as in this case you are still revealing parts of a correct solution). Our students follow the Honor Code: “I will not make solutions to homework, quizzes, exams, projects, and other assignments available to anyone else (except to the extent an assignment explicitly permits sharing solutions).”

Do I Learn by Trying to Fix My Solution?

My implementation always fails in the grader, though I already tested and stress tested it a lot. Would not it be better if you gave 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.

First of all, learning from your mistakes is one of the best ways to learn.

The process of trying to invent new test cases that might fail your program is difficult but is often enlightening. Thinking about properties of your program makes you understand what happens inside your program and in the general algorithm you’re studying much more.

Also, it is important to be able to find a bug in your implementation without knowing a test case and without having a reference solution, just like in real life. Assume that you designed an application and an annoyed user reports that it crashed. Most probably, the user will not tell you the exact sequence of operations that led to a crash. Moreover, there will be no reference application. Hence, it is important to learn how to find a bug in your implementation yourself, without a magic oracle giving you either a test case that your program fails or a reference solution. We encourage you to use programming assignments in this class as a way of practicing this important skill.

If you have already tested your program on all corner cases you can imagine, constructed a set of manual test cases, applied stress testing, etc, but your program still fails, try to ask for help on the forum. We encourage you to do this by first explaining what kind of corner cases you have already considered (it may happen that by writing such a post you will realize that you missed some corner cases!), and only afterwards asking other learners to give you more ideas for tests cases.

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