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Module: Course: Specialization: Paths in Graphs (Week 4 out of 5) Algorithms on Graphs (Course 3 out of 6) Data Structures and Algorithms Programming Assignment 4: Paths in Graphs Revision: June 2, 2016 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. 1 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 5.1 Reading the Problem Statement . . . . . . . . . . . . . . 5.2 Designing an Algorithm . . . . . . . . . . . . . . . . . . . 5.3 Implementing Your Algorithm . . . . . . . . . . . . . . . . 5.4 Compiling Your Program . . . . . . . . . . . . . . . . . . 5.5 Testing Your Program . . . . . . . . . . . . . . . . . . . . 5.6 Submitting Your Program to the Grading System . . . . . 5.7 Debugging and Stress Testing Your Program . . . . . . . Algorithmic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 11 11 11 13 13 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 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 2 4 1 2 3 5 1 3 4 4 2 4 3 1 2 • A directed graph with five vertices and eight edges. 5 4 1 3 3 2 5 5 5 8 3 2 1 4 5 1 4 3 2 5 1 4 3 • A weighted directed graph with three vertices and three edges. 3 2 1 1 3 39 35 2 -2 5 1 3 −2 3 9 2 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. 10 sec, Ruby: 10 sec, Scala: 6 sec. C#: 3 sec, Haskell: 4 sec, JavaScript: Memory Limit. 512Mb. Sample 1. Input: 44 121 412 232 135 13 Output: 3 Explanation: 4 2 1 5 1 3 2 2 There is a unique shortest path from vertex 1 to vertex 3 in this graph (1 → 2 → 3), and it has weight 3. 4 Sample 2. Input: 59 124 132 232 321 242 354 541 253 344 15 Output: 6 Explanation: 1 2 2 4 1 2 4 4 3 1 3 3 4 5 There are two paths from 1 to 5 of total weight 6: 1 → 3 → 5 and 1 → 3 → 2 → 5. Sample 3. Input: 33 127 135 232 32 Output: -1 Explanation: 3 5 1 7 2 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 5 What To Do To solve this problem, it is enough to implement carefully the corresponding algorithm covered in the lectures. 6 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 − log rij . 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 −(log cij + log cjk + log cki ) < 0 and hence log cij + log cjk + log cki > 0. This, in turn, means that rij rjk rki = 2log cij 2log cjk 2log 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. 10 sec, Ruby: 10 sec, Scala: 6 sec. C#: 3 sec, Haskell: 4 sec, JavaScript: Memory Limit. 512Mb. Sample 1. Input: 44 1 2 -5 412 232 311 Output: 1 Explanation: 4 2 1 1 −5 3 2 2 The weight of the cycle 1 → 2 → 3 is equal to −2, that is, negative. 7 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. 8 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. 10 sec, Ruby: 10 sec, Scala: 6 sec. Memory Limit. 512Mb. Sample 1. Input: 67 1 2 10 235 1 3 100 357 5 4 10 4 3 -18 6 1 -1 1 Output: 0 10 * Explanation: 9 C#: 3 sec, Haskell: 4 sec, JavaScript: 5 2 10 1 −1 100 3 −18 4 10 7 6 5 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: 54 121 412 232 3 1 -5 4 Output: 0 * Explanation: 2 2 1 1 2 3 −5 4 5 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. 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(n log n) 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(2n n2 ) 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 11 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 12 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. Go ahead, and we hope you pass the assignment soon! 13 6 6.1 Frequently Asked Questions 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 5.3 and 5.4). 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. 6.2 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. 6.3 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, the output of your program and the correct answer. Otherwise, you won’t know the input, the output 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, 14 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. 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: code. exception... Most probably, you submitted a compiled program instead of a source Grading failed. Something very wrong happened with the system. Contact Coursera for help or write in the forums to let us know. 6.4 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. 6.5 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. 15 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 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. 6.6 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). 6.7 Are you going to support my favorite language in programming assignments? Currently, we are going to support C, C++, Java, and Python only, but we may add other programming languages later if there appears a huge need. To express your interest in a particular programming language, please post its name in this thread or upvote the corresponding option if it is already there. 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. 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 16 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. 17

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