Description
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 this programming assignment requires passing at least 1 out of 2 code problems from this assignment.
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
4 3
• 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
4 3
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:
1 2
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
To solve this problem, it is enough to implement carefully the corresponding algorithm covered in the lectures.
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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
4 3
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
What To Do
Adapt the breadth-first search to solve this problem.
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4 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.
4.1 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.
4.2 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
, n
2 = 1010) while
a solution with running time O(n log n) will fit. However, an O(n
2
) solution will fit if n is up to 103 = 1000,
and if n is at most 100, even O(n
3
) 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(2nn
2
) 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.
4.3 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.
4.4 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 ++11 < filename > – lm
If your C/C++ compiler does not recognize -std=c++11 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 -O
• Java (Open JDK 8). File extensions: .java. Flags:
javac – encoding UTF -8
• 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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4.5 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.
4.6 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
compiles your program (detecting the programming language based on your file extension, see Subsection 4.4)
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).
4.7 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 4.5. 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.
Go ahead, and we hope you pass the assignment soon!
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5 Frequently Asked Questions
5.1 I submit the program, but nothing happens. Why?
You need to create submission and upload the file with your solution in one of the programming languages C,
C++, Java, or Python (see Subsections 4.3 and 4.4). Make sure that after uploading the file with your solution
you press on the blue “Submit” button in 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 is shown to you.
5.2 I submit the solution only for one problem, but all the problems in the
assignment are graded. Why?
Each time you submit any solution, the last uploaded solution for each problem is tested. Don’t worry: this
doesn’t affect your score even if the submissions for the other problems are wrong. As soon as you pass the
sufficient number of problems in the assignment (see in the pdf with instructions), you pass the assignment.
After that, you can improve your result if you successfully pass more problems from the assignment. We
recommend working on one problem at a time, checking whether your solution for any given problem passes
in the system as soon as you are confident in it. However, it is better to test it first, please refer to the
reading about stress testing: link.
5.3 What are the possible grading outcomes, and how to read them?
Your solution may either pass or not. To pass, it must work without crashing and return the correct answers
on all the test cases we prepared for you, and do so under the time limit and memory limit constraints
specified in the problem statement. If your solution passes, you get the corresponding feedback “Good job!”
and get a point for the problem. If your solution fails, it can be because it crashes, returns wrong answer,
works for too long or uses too much memory for some test case. The feedback will contain the number of
the test case on which your solution fails 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 test number 1 to the test with the biggest number.
Here are the possible outcomes:
Good job! Hurrah! Your solution passed, and you get a point!
Wrong answer. Your solution has output incorrect answer for some test case. If it is a sample test case from
the problem statement, or if you are solving Programming Assignment 1, you will also see the input
data, the output of your program and the correct answer. Otherwise, you won’t know the input, the
output, and the correct answer. Check that you consider all the cases correctly, avoid integer overflow,
output the required white space, 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. See this reading on testing: link.
Time limit exceeded. Your solution worked longer than the allowed time limit for some test case. If it
is a sample test case from the problem statement, or if you are solving Programming Assignment 1,
you will also see the input data and the correct answer. Otherwise, you won’t know the input and the
correct answer. Check again that your algorithm has good enough running time estimate. Test your
program locally on the test of maximum size allowed by the problem statement and see how long it
works. Check that your program doesn’t wait for some input from the user which makes it to wait
forever. See this reading on testing: link.
Memory limit exceeded. Your solution used more than the allowed memory limit for some test case. If it
is a sample test case from the problem statement, or if you are solving Programming Assignment 1,
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you will also see the input data and the correct answer. Otherwise, you won’t know the input and the
correct answer. Estimate the amount of memory that your program is going to use in the worst case
and check that it is less than the memory limit. Check that you don’t create too large arrays or data
structures. 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 test of
maximum size allowed by the problem statement and look at its memory consumption in the system.
Cannot check answer. Perhaps output format is wrong. This happens when you output something
completely different than expected. For example, you are required to output word “Yes” or “No”, but
you output number 1 or 0, or vice versa. Or your program has empty output. Or your program outputs
not only the correct answer, but also some additional information (this is not allowed, so please follow
exactly the 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 division by zero, accessing memory outside of the array bounds, using uninitialized variables, too deep recursion that triggers stack overflow, sorting with 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
compilers and the same compiler options as we do. Try different testing techniques from this reading:
link.
Internal error: exception… Most probably, you submitted a compiled program instead of a source
code.
Grading failed. Something very wrong happened with the system. Contact Coursera for help or write in
the forums to let us know.
5.4 How to understand why my program fails and to fix it?
If your program works incorrectly, it gets a feedback from the grader. For the Programming Assignment 1,
when your solution fails, you will see the input data, the correct answer and the output of your program in
case it didn’t crash, finished under the time limit and memory limit constraints. If the program crashed,
worked too long or used too much memory, the system stops it, so you won’t see the output of your program
or will see just part of the whole output. We show you all this information so that you get used to the
algorithmic problems in general and get some experience debugging your programs while knowing exactly
on which tests they fail.
However, in the following Programming Assignments throughout the Specialization you will only get so
much information for the test cases from the problem statement. For the next tests you will only get the
result: passed, time limit exceeded, memory limit exceeded, wrong answer, wrong output format or some
form of crash. We hide the test cases, because it is crucial for you to learn to test and fix your program
even without knowing exactly the test on which it fails. In the real life, often there will be no or only partial
information about the failure of your program or service. You will need to find the failing test case yourself.
Stress testing is one powerful technique that allows you to do that. You should apply it after using the other
testing techniques covered in this reading.
5.5 Why do you hide the test on which my program fails?
Often beginner programmers think by default that their programs work. Experienced programmers know,
however, that their programs almost never work initially. Everyone who wants to become a better programmer needs to go through this realization.
When you are sure that your program works by default, you just throw a few random test cases against
it, and if the answers look reasonable, you consider your work done. However, mostly this is not enough. To
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make one’s programs work, one must test them really well. Sometimes, the programs still don’t work although
you tried really hard to test them, and you need to be both skilled and creative to fix your bugs. Solutions
to algorithmic problems are one of the hardest to implement correctly. That’s why in this Specialization you
will gain this important experience which will be invaluable in the future when you write programs which
you really need to get right.
It is crucial for you to learn to test and fix your programs yourself. In the real life, often there will be
no or only partial information about the failure of your program or service. Still, you will have to reproduce
the failure to fix it (or just guess what it is, but that’s rare, and you will still need to reproduce the failure
to make sure you have really fixed it). When you solve algorithmic problems, it is very frequent to make
subtle mistakes. That’s why you should apply the testing techniques described in this reading to find the
failing test case and fix your program.
5.6 My solution does not pass the tests? May I post it in the forum and ask
for a help?
No, please do not post any solutions in 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). Recall the third item
of the Coursera 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).
This includes both solutions written by me, as well as any solutions provided by the course staff or others”
(link).
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.
First of all, you always learn from your mistakes.
The process of trying to invent new test cases that might fail your program and proving them wrong is
often enlightening. This thinking about the invariants which you expect your loops, ifs, etc. to keep and
proving them wrong (or right) 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. 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, once again, it is important to be able to
locate 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 a lot (considered all corner cases that you can imagine, constructed a set of
manual test cases, applied stress testing), but your program still fails and you are stuck, 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 when writing such a post you will realize that you missed some corner cases!)
and only then asking other learners to give you more ideas for tests cases.
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