Description
Introduction
Welcome to your fourth programming assignment of the Algorithms on Graphs class! In this assignments we focus on shortest paths in weighted graphs.
Learning Outcomes
Upon completing this programming assignment you will be able to:
1. compute the minimum cost of a flight from one city to another one;
2. detect anomalies in currency exchange rates;
3. compute optimal way of exchanging the given currency into all other currencies.
Passing Criteria: 2 out of 3
Passing this programming assignment requires passing at least 2 out of 3 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 Cost of a Flight 4
3 Problem: Detecting Anomalies in Currency Exchange Rates 7
4 Advanced Problem: Exchanging Money Optimally 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 Are you going to support my favorite language in programming assignments? . . . . . . . . . 16 6.8 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
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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 Cost of a Flight
Problem Introduction
Now, you are interested in minimizing not the number of segments, but the total cost of a flight. For this you construct a weighted graph: the weight of an edge from one city to another one is the cost of the corresponding flight.
Problem Description
Task. Given an directed graph with positive edge weights and with n vertices and m edges as well as two vertices u and v, compute the weight of a shortest path between u and v (that is, the minimum total weight of 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. 1 ≤ n ≤ 103, 0 ≤ m ≤ 105, u 6= v, 1 ≤ u,v ≤ n, edge weights are non-negative integers not exceeding 103. Output Format. Output the minimum weight of a path from u to v, or −1 if there is no path. Time Limits. C: 2 sec, C++: 2 sec, Java: 3 sec, Python: 10 sec. C#: 3 sec, Haskell: 4 sec, JavaScript: 10 sec, Ruby: 10 sec, Scala: 6 sec.
Memory Limit. 512Mb.
Sample 1. Input: 4 4 1 2 1 4 1 2 2 3 2 1 3 5 1 3 Output: 3 Explanation:
1 2
34
1
2 2 5
There is a unique shortest path from vertex 1 to vertex 3 in this graph (1 → 2 → 3), and it has weight 3.
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Sample 2. Input: 5 9 1 2 4 1 3 2 2 3 2 3 2 1 2 4 2 3 5 4 5 4 1 2 5 3 3 4 4 1 5 Output: 6 Explanation:
1
2
3
4
5
4
2
2
4
131 4 3
There are two paths from 1 to 5 of total weight 6: 1 → 3 → 5 and 1 → 3 → 2 → 5. Sample 3. Input: 3 3 1 2 7 1 3 5 2 3 2 3 2 Output: -1 Explanation:
1 2
3
7
5 2
There is no path from 3 to 2.
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: dijkstra
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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: Detecting Anomalies in Currency Exchange Rates
Problem Introduction You are given a list of currencies c1,c2,…,cn together with a list of exchange rates: rij is the number of units of currency cj that one gets for one unit of ci. You would like to check whether it is possible to start with one unit of some currency, perform a sequence of exchanges, and get more than one unit of the same currency. In other words, you would like to find currencies ci1,ci2,…,cik such that ri1,i2 ·ri2,i3 ·rik−1,ik,rik,i1 1. For this, you construct the following graph: vertices are currencies c1,c2,…,cn, the weight of an edge from ci to cj is equal to −logrij. There it suffices to check whether is a negative cycle in this graph. Indeed, assume that a cycle ci → cj → ck → ci has negative weight. This means that −(logcij + logcjk + logcki) < 0 and hence logcij + logcjk + logcki 0. This, in turn, means that
rijrjkrki = 2log cij2log cjk2log cki = 2log cij+log cjk+log cki 1.
Problem Description
Task. Given an directed graph with possibly negative edge weights and with n vertices and m edges, check whether it contains a cycle of negative weight.
Input Format. A graph is given in the standard format. Constraints. 1 ≤ n ≤ 103, 0 ≤ m ≤ 104, edge weights are integers of absolute value at most 103. Output Format. Output 1 if the graph contains a cycle of negative weight and 0 otherwise.
Time Limits. C: 2 sec, C++: 2 sec, Java: 3 sec, Python: 10 sec. C#: 3 sec, Haskell: 4 sec, JavaScript: 10 sec, Ruby: 10 sec, Scala: 6 sec.
Memory Limit. 512Mb.
Sample 1. Input: 4 4 1 2 -5 4 1 2 2 3 2 3 1 1 Output: 1 Explanation:
1 2
34
−5
2 2 1
The weight of the cycle 1 → 2 → 3 is equal to −2, that is, negative.
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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: negative cycle
What To Do
To solve this problem, it is enough to implement carefully the corresponding algorithm covered in the lectures.
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4 Advanced Problem: Exchanging Money Optimally
(Recall that advanced problems are not covered in the video lectures and require additional ideas to be solved. We therefore strongly recommend you start solving these problems only when you are done with the basic problems.)
Problem Introduction
Now, you would like to compute an optimal way of exchanging the given currency ci into all other currencies. For this, you find shortest paths from the vertex ci to all the other vertices.
Problem Description
Task. Given an directed graph with possibly negative edge weights and with n vertices and m edges as well as its vertex s, compute the length of shortest paths from s to all other vertices of the graph.
Input Format. A graph is given in the standard format. Constraints. 1 ≤ n ≤ 103, 0 ≤ m ≤ 104, 1 ≤ s ≤ n, edge weights are integers of absolute value at most 109.
Output Format. For all vertices i from 1 to n output the following on a separate line: • “*”, if there is no path from s to u; • “-”, if there is a path from s to u, but there is no shortest path from s to u (that is, the distance from s to u is −∞); • the length of a shortest path otherwise. Time Limits. C: 2 sec, C++: 2 sec, Java: 3 sec, Python: 10 sec. C#: 3 sec, Haskell: 4 sec, JavaScript: 10 sec, Ruby: 10 sec, Scala: 6 sec.
Memory Limit. 512Mb.
Sample 1. Input: 6 7 1 2 10 2 3 5 1 3 100 3 5 7 5 4 10 4 3 -18 6 1 -1 1 Output: 0 10 * Explanation:
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1
2 3
4
56
10
5
−18 10− 1 100 7
The first line of the output states that the distance from 1 to 1 is equal to 0. The second one shows that the distance from 1 to 2 is 10 (the corresponding path is 1 → 2). The next three lines indicate that the distance from 1 to vertices 3, 4, and 5 is equal to −∞: indeed, one first reaches the vertex 3 through edges 1 → 2 → 3 and then makes the length of a path arbitrary small by making sufficiently many walks through the cycle 3 → 5 → 4 of negative weight. The last line of the output shows that there is no path from 1 to 6 in this graph.
Sample 2. Input: 5 4 1 2 1 4 1 2 2 3 2 3 1 -5 4 Output: 0 * Explanation:
1
2 3
4 51
2
−5
2
In this case, the distance from 4 to vertices 1, 2, and 3 is −∞ since there is a negative cycle 1 → 2 → 3 that is reachable from 4. The distance from 4 to 4 is zero. There is no path from 4 to 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. Filename: shortest paths
What To Do
To solve this problem, it is enough to implement carefully the corresponding algorithm covered in the lectures.
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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
• C++ (g++ 5.2.1). File extensions: .cc, .cpp. Flags: g++ -pipe -O2 -std=c++11
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.js 0.10.25). 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 compiles your program (detecting the programming language based on your file extension, see Subsection 5.4) 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