Programming Assignment 2: Linear Programming

Description

Introduction
Welcome to your second programming assignment of the Advanced Algorithms and Complexity class! In
this programming assignment, you will be practicing reducing real-world problems to linear programming
and implementing algorithms to solve them.
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. Implement Gaussian Elimination, brute-force algorithm for Linear Programming and Simplex Method.
2. Design and implement efficient algorithms for the following computational problems:
(a) inferring energy values of ingredients from the menu with calorie counts;
(b) optimal diet problem;
(c) online advertisement allocation problem.
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 Problem: Infer Energy Values of Ingredients 3
2 Problem: Optimal Diet Problem 6
3 Advanced Problem: Online Advertisement Allocation 10
4 General Instructions and Recommendations on Solving Algorithmic Problems 13
4.1 Reading the Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Designing an Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 Implementing Your Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.4 Compiling Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.5 Testing Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.6 Submitting Your Program to the Grading System . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.7 Debugging and Stress Testing Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Frequently Asked Questions 16
5.1 I submit the program, but nothing happens. Why? . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 I submit the solution only for one problem, but all the problems in the assignment are graded.
Why? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.3 What are the possible grading outcomes, and how to read them? . . . . . . . . . . . . . . . . 16
5.4 How to understand why my program fails and to fix it? . . . . . . . . . . . . . . . . . . . . . 17
5.5 Why do you hide the test on which my program fails? . . . . . . . . . . . . . . . . . . . . . . 17
5.6 My solution does not pass the tests? May I post it in the forum and ask for a help? . . . . . 18
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. . . 18
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1 Problem: Infer Energy Values of Ingredients
Problem Introduction
In this problem, you will apply Gaussian Elimination to infer the energy
values of ingredients given a restaurant menu with calorie counts and
ingredient lists provided for each item.
Problem Description
Task. You’re looking into a restaurant menu which shows for each dish the list of ingredients with amounts
and the estimated total energy value in calories. You would like to find out the energy values of
individual ingredients (then you will be able to estimate the total energy values of your favorite dishes).
Input Format. The first line of the input contains an integer 𝑛 — the number of dishes in the menu, and
it happens so that the number of different ingredients is the same. Each of the next 𝑛 lines contains
description 𝑎1, 𝑎2, . . . , 𝑎𝑛, 𝐸 of a single menu item. 𝑎𝑖
is the amount of 𝑖-th ingredient in the dish,
and 𝐸 is the estimated total energy value of the dish. If the ingredient is not used in the dish, the
amount will be specified as 𝑎𝑖 = 0; beware that although the amount of any ingredient in any
real menu would be positive, we will test that your algorithm works even for negative
amounts 𝑎𝑖 < 0. Constraints. 0 ≤ 𝑛 ≤ 20; −1000 ≤ 𝑎𝑖 ≤ 1000. Output Format. Output 𝑛 real numbers — for each ingredient, what is its energy value. These numbers can be non-integer, so output them with at least 3 digits after the decimal point. Your output for a particular test input will be accepted if all the numbers in the output are considered correct. The amounts and energy values are of course approximate, and the computations in real numbers on a computer are not always precise, so each of the numbers in your output will be considered correct if either absolute or relative error is less than 10−2 . That is, if the correct number is 5.245000, and you output 5.235001, your number will be considered correct, but 5.225500 will not be accepted. Also, if the correct number is 1001, and you output 1000, your answer will be considered correct, because the relative error will be less than 10−2 , but if the correct answer is 0.1, and you output 0.05, your answer will not be accepted, because in this case both the absolute error (0.05) and the relative error (0.5) are more than 10−2 . Note that we ask you to output at least 3 digits after the decimal point, although we only require precision of 10−2 , intentionally: if you output only 2 digits after the decimal point, your answer can be rejected while being correct because of the rounding issues. The easiest way to avoid this mistake is to output at least 3 digits after the decimal point. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time in seconds 1 1 1.5 5 1.5 2 5 5 3 Memory Limit. 512MB. 3 Sample 1. Input: 0 Output: Explanation: There are no dishes in the menu — you don’t need to output anything in this case. Sample 2. Input: 4 1 0 0 0 1 0 1 0 0 5 0 0 1 0 4 0 0 0 1 3 Output: 1.000000 5.000000 4.000000 3.000000 Explanation: This is an easy test. Each dish contains just one component, and the amount used is exactly 1, so the energy value of each ingredient is just equal to the energy value of the whole dish in which it is used. Sample 3. Input: 2 1 1 3 2 3 7 Output: 2.000000 1.000000 Explanation: You can see that the numbers match: 1 · 2.0 + 1 · 1.0 = 3 and 2 · 2.0 + 3 · 1.0 = 7. If you output 1.994000 and 1.009000 instead of 2.000000 and 1.000000 respectively, your answer will still be accepted, but don’t forget to output at least 3 digits after the decimal point! Sample 4. Input: 2 5 -5 -1 -1 -2 -1 Output: 0.200000 0.400000 Explanation: Beware that there will be tests with negative amounts and negative total energy values, although this is impossible in reality! Also note that the answers can be non-integer! You can check that the numbers match: 5 · 0.2 + (−5) · 0.4 = −1 and (−1) · 0.2 + (−2) · 0.4 = −1. 4 Starter Files The starter solutions for this problem read the data from the input, pass it to a blank procedure and output the result. They also contain some convenience functions and data structures. You need to change the main procedure to implement Gaussian Elimination if you are using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch. Filename: energy_values What To Do Implement the Gaussian Elimination algorithm from the lectures. Need Help? Ask a question or see the questions asked by other learners at this forum thread. 5 2 Problem: Optimal Diet Problem Problem Introduction In this problem, you will implement an algorithm for solving linear programming with only a few inequalities and apply it to determine the optimal diet. Problem Description Task. You want to optimize your diet: that is, make sure that your diet satisfies all the recommendations of nutrition experts, but you also get maximum pleasure from your food and drinks. For each dish and drink you know all the nutrition facts, cost of one item, and an estimation of how much you like it. Your budget is limited, of course. The recommendations are of the form “total amount of calories consumed each day should be at least 1000” or “the amount of water you drink in liters should be at least twice the amount of food you eat in kilograms”, and so on. You optimize the total pleasure which is the sum of pleasure you get from consuming each particular dish or drink, and that is proportional to the amount amount𝑖 of that dish or drink consumed. The budget restriction and the nutrition recommendations can be converted into a system of linear inequalities like ∑︀𝑚 𝑖=1 cost𝑖 · amount𝑖 ≤ Budget, amount𝑖 ≥ 1000 and amount𝑖 − 2 · amount𝑗 ≥ 0, where amount𝑖 is the amount of 𝑖-th dish or drink consumed, cost𝑖 is the cost of one item of 𝑖-th dish or drink, and 𝐵𝑢𝑑𝑔𝑒𝑡 is your total budget for the diet. Of course, you can only eat a non-negative amount amount𝑖 of 𝑖-th item, so amount𝑖 ≥ 0. The goal to maximize total pleasure is reduced to the linear objective ∑︀𝑚 𝑖=1 amount𝑖 · pleasure𝑖 → max where pleasure𝑖 is the pleasure you get after consuming one unit of 𝑖-th dish or drink (some dishes like fish oil you don’t like at all, so pleasure𝑖 can be negative). Combined, all this is a linear programming problem which you need to solve now. Input Format. The first line of the input contains integers 𝑛 and 𝑚 — the number of restrictions on your diet and the number of all available dishes and drinks respectively. The next 𝑛 + 1 lines contain the coefficients of the linear inequalities in the standard form 𝐴𝑥 ≤ 𝑏, where 𝑥 = amount is the vector of length 𝑚 with amounts of each ingredient, 𝐴 is the 𝑛 × 𝑚 matrix with coefficients of inequalities and 𝑏 is the vector with the right-hand side of each inequality. Specifically, 𝑖-th of the next 𝑛 lines contains 𝑚 integers 𝐴𝑖1, 𝐴𝑖2, . . . , 𝐴𝑖𝑚, and the next line after those 𝑛 contains 𝑛 integers 𝑏1, 𝑏2, . . . , 𝑏𝑛. These lines describe 𝑛 inequalities of the form 𝐴𝑖1 · amount1 + 𝐴𝑖2 · amount2 + · · · + 𝐴𝑖𝑚 · amount𝑚 ≤ 𝑏𝑖 . The last line of the input contains 𝑚 integers — the pleasure for consuming one item of each dish and drink pleasure1 , pleasure2 , . . . , pleasure𝑚. Constraints. 1 ≤ 𝑛, 𝑚 ≤ 8; −100 ≤ 𝐴𝑖𝑗 ≤ 100; −1 000 000 ≤ 𝑏𝑖 ≤ 1 000 000; −100 ≤ cost𝑖 ≤ 100. Output Format. If there is no diet that satisfies all the restrictions, output “No solution” (without quotes). If you can get as much pleasure as you want despite all the restrictions, output “Infinity” (without quotes). If the maximum possible total pleasure is bounded, output two lines. On the first line, output “Bounded solution” (without quotes). On the second line, output 𝑚 real numbers — the optimal amounts for each dish and drink. Output all the numbers with at least 15 digits after the decimal point. The amounts you output will be inserted into the inequalities, and all the inequalities will be checked. An inequality 𝐿 ≤ 𝑅 will be considered satisfied if actually 𝐿 ≤ 𝑅 + 10−3 . The total pleasure of your 6 solution will be calculated and compared with the optimal value. Your output will be accepted if all the inequalities are satisfied and the total pleasure of your solution differs from the optimal value by at most 10−3 . We ask you to output at least 15 digits after the decimal point, although we will check the answer with precision of only 10−3 . This is because in the process of checking the inequalities we will multiply your answers with coefficients from the matrix 𝐴 and with the coefficients of the vector pleasure, and those coefficients can be pretty large, and computations with real numbers on a computer are not always precise. This way, the more digits after the decimal point you output for each amount — the less likely it is that your answer will be rejected because of precision issues. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time in seconds 1 1 2 30 1.5 2 30 30 4 Memory Limit. 512MB. Sample 1. Input: 3 2 -1 -1 1 0 0 1 -1 2 2 -1 2 Output: Bounded solution 0.000000000000000 2.000000000000000 Explanation: Here we have only two items, and we know that (−1) · amount1 + (−1) · amount2 ≤ −1 ⇒ amount1 + amount2 ≥ 1 from the first inequality, and also that amount1 ≤ 2 and amount2 ≤ 2 from the second and the third inequalities. We also know that all amounts are non-negative. We want to maximize (−1) · amount1 + 2 · amount2 under those restrictions — that is, we don’t like dish or drink number 1, and we twice as much like dish or drink number 2. It is optimal then to consume as few as possible of the first item and as much as possible of the second item. It turns out that we can avoid consuming the first item at all and take the maximum possible amount 2 of the second item, and all the restrictions will be satisfied! Clearly, this is a diet with the maximum possible total pleasure! Note that integers 0 and 2 in the output are printed with 15 digits after the decimal point. Don’t forget to print at least 15 digits after the decimal point, as the answers to some tests will be noninteger, and you don’t want to get your answer rejected only because of some rounding problems. 7 Sample 2. Input: 2 2 1 1 -1 -1 1 -2 1 1 Output: No solution Explanation: The first inequality gives amount1 + amount2 ≤ 1 and the second inequality gives (−1) · amount1 + (−1) · amount2 ≤ −2 ⇒ amount1 + amount2 ≥ 2. But amount1 + amount2 cannot be less than 1 and more than 2 simultaneously, so there is no solution in this case. Sample 3. Input: 1 3 0 0 1 3 1 1 1 Output: Infinity Explanation: The restrictions in this case are only that all amounts are non-negative (these restrictions are always there, because you cannot consume negative amount of a dish or a drink) and that amount3 ≤ 3. There is no restriction on how much to consume of items 1 and 2, and each of them has positive pleasure value, so you can take as much of items 1 and 2 as you want and receive as much total pleasure as you want. In this case, you should output “Infinite” (without quotes). Starter Files The starter solutions for this problem read the data from the input, pass it to a blank procedure and output the result. You need to implement this procedure if you are using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch. Filename: diet What To Do There are at most 8 inequalities (16 if you count in the inequalities amount𝑖 ≥ 0) with at most 8 variables. You can use this fact and the fact that the optimal solution is always in a vertex of the polyhedron corresponding to the linear programming problem. At least 𝑚 of the inequalities become equalities in each vertex of the polyhedron. If there are 𝑛 regular inequalities, 𝑚 variables and 𝑚 inequalities of the form amount𝑖 ≥ 0, you need to take each possible subset of size 𝑚 out of all the 𝑛 + 𝑚 inequalities, solve the system of linear equations where each equation is one of the selected inequalities changed to equality, check whether this solution satisfies all the other inequalities, and in the end select the solution with the largest value of the total pleasure out of those which satisfy all inequalities. The running time of this algorithm is 𝑂(2𝑛+𝑚(𝑚3 + 𝑚𝑛)), which is good enough to pass. 2 𝑛+𝑚 is to go through all the subsets of the inequalities (although you will need only subsets of size 𝑚), 𝑚3 is for Gaussian Elimination and 𝑚𝑛 is to check a solution of a system of linear equations against all the inequalities. Various ways to traverse all the subsets 8 of some set are described here and here. The only case that you would miss this way is the case when the correct answer is “Infinity”. There is a clever workaround for that. Add a new inequality to the initial set of inequalities. It is the inequality 1 · amount1 + 1 · amount2 + · · · + 1 · amount𝑚 ≤ VeryBigNumber, where VeryBigNumber is some number bigger than the sum of absolute values of all numbers in the input, you can take VeryBigNumber = 109 safely. The initial linear programming problem has bounded solution if and only if the newly added inequality holds strictly for the optimal solution of the augmented linear programming problem. So, if you solve the augmented problem, it has a solution, and it turns out that the vertex corresponding to the optimal solution has the last inequality among those 𝑚 that define the vertex, output “Infinity”, otherwise output “Bounded solution” and the solution of the augmented problem on the second line. Need Help? Ask a question or see the questions asked by other learners at this forum thread. 9 3 Advanced Problem: Online Advertisement Allocation We strongly recommend you start solving advanced problems only when you are done with the basic problems (for some advanced problems, algorithms are not covered in the video lectures and require additional ideas to be solved; for some other advanced problems, algorithms are covered in the lectures, but implementing them is a more challenging task than for other problems). Problem Introduction Online and mobile advertising is one of the most profitable businesses in the world. Google and Facebook are generating many billions of dollars of revenue each year, and around 90% of their revenues come from advertisement. In this problem you will help an online advertising system like Google AdSense or Yandex Direct to allocate the ad impressions in its Advertising Network so as to maximize revenue while satisfying all the advertisers’ requirements. Problem Description Task. You have 𝑛 clients, they are advertisers, and each of them wants to show their ads to some number of internet users specified in the contract (or more) next month. Your online advertising network has 𝑚 placements overall on all the sites connected to the network. You know how many users each advertiser wants to reach, how many users will see each of the 𝑚 ad placements next month, and how much each advertiser is willing to pay for one user who sees their ad through each particular ad placement (different placements can be on different sites attracting different types of users, and each advertiser is more interested in the visitors of some sites than the others). You can show different ads of different advertisers in the same ad placement throughout the next month or show always the same ad of the same advertiser, but the total number of users that will see some ad in that placement is estimated and fixed. You want to maximize your total revenue which is the sum of amounts each advertiser will pay you for all the users who have seen their ads. If we denote by 𝑥𝑖𝑗 the number of users who have seen an ad of advertiser 𝑖 in the ad placement 𝑗, then all the restrictions can be written as linear equalities and inequalities in 𝑥𝑖𝑗 . For example, if the total number of users that will see ad placement 𝑗 is 𝑆𝑗 , then we add an equality ∑︀𝑚 𝑖=1 𝑥𝑖𝑗 = 𝑆𝑗 . If the 𝑖-th advertiser wants to show the ad to at least 𝑈𝑖 users, we add an inequality ∑︀𝑛 𝑗=1 𝑥𝑖𝑗 ≥ 𝑈𝑖 ⇔ ∑︀𝑛 𝑗=1 (−1)·𝑥𝑖𝑗 ≤ −𝑈𝑖 . Of course, each 𝑥𝑖𝑗 is non-negative: 𝑥𝑖𝑗 ≥ 0. If advertiser 𝑖 wishes to pay 𝑐𝑖𝑗 cents for each user who sees her advertisement through ad placement 𝑗, then the goal to maximize the total revenue is given by linear objective ∑︀𝑛 𝑖=1 ∑︀𝑚 𝑗=1 𝑐𝑖𝑗𝑥𝑖𝑗 → max. This leads to a linear programming problem which you need to solve. This time it will contain more variables and inequalities, because the number of advertisers and the number of different ad placements can be large. Input Format. You are given the ad allocation problem reduced to a linear programming problem of the form 𝐴𝑥 ≤ 𝑏, 𝑥 ≥ 0, ∑︀𝑞 𝑖=1 𝑐𝑖𝑥𝑖 → max, where 𝐴 is a matrix 𝑝 × 𝑞, 𝑏 is a vector of length 𝑝, 𝑐 is a vector of length 𝑞 and 𝑥 is the unknown vector of length 𝑞. The first line of the input contains integers 𝑝 and 𝑞 — the number of inequalities in the system and the number of variables respectively. The next 𝑝 + 1 lines contain the coefficients of the linear inequalities in the standard form 𝐴𝑥 ≤ 𝑏. Specifically, 𝑖-th of the next 𝑝 lines contains 𝑞 integers 𝐴𝑖1, 𝐴𝑖2, . . . , 𝐴𝑖𝑞, 10 and the next line after those 𝑝 contains 𝑝 integers 𝑏1, 𝑏2, . . . , 𝑏𝑝. These lines describe 𝑝 inequalities of the form 𝐴𝑖1 · 𝑥1 + 𝐴𝑖2 · 𝑥2 + · · · + 𝐴𝑖𝑞 · 𝑥𝑞 ≤ 𝑏𝑖 . The last line of the input contains 𝑞 integers — the coefficients 𝑐𝑖 of the objective ∑︀𝑞 𝑖=1 𝑐𝑖𝑥𝑖 → max. Constraints. 1 ≤ 𝑛, 𝑚 ≤ 100; −100 ≤ 𝐴𝑖𝑗 ≤ 100; −1 000 000 ≤ 𝑏𝑖 ≤ 1 000 000; −100 ≤ 𝑐𝑖 ≤ 100. Output Format. If there is no allocation that satisfies all the requirements, output “No solution” (without quotes). If you can get as much revenue as you want despite all the requirements, output “Infinity” (without quotes). If the maximum possible revenue is bounded, output two lines. On the first line, output “Bounded solution” (without quotes). On the second line, output 𝑞 real numbers — the optimal values of the vector 𝑥 (recall that 𝑥 = 𝑥𝑖𝑗 is how many users will see the ad of advertiser 𝑖 through the placement 𝑗, but we changed the numbering of variables to 𝑥1, 𝑥2, . . . , 𝑥𝑞). Output all the numbers with at least 15 digits after the decimal point. Your solution will be accepted if all the inequalities are satisfied and the answer has absolute error of at most 10−3 . See the previous problem output format description for the explanation of what this means and why do we ask to output at least 15 digits after the decimal point. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time in seconds 1 1 1.5 6 1.5 2 6 6 3 Memory Limit. 512MB. Sample 1. Input: 3 2 -1 -1 1 0 0 1 -1 2 2 -1 2 Output: Bounded solution 0.000000000000000 2.000000000000000 Explanation: Here we have only two variables, and we know that (−1)· 𝑥1 + (−1)· 𝑥2 ≤ −1 ⇒ 𝑥1 + 𝑥2 ≥ 1 from the first inequality, and also that 𝑥1 ≤ 2 and 𝑥2 ≤ 2 from the second and the third inequalities. We also know that all amounts are non-negative. We want to maximize (−1)·𝑥1+2·𝑥2 under those restrictions. It is optimal to minimize 𝑥1 and maximize 𝑥2. It turns out that we can set 𝑥1 = 0 (the minimum possible) and 𝑥2 = 2 (the maximum possible), and all the requirements will be satisfied. Note that integers 0 and 2 in the output are printed with 15 digits after the decimal point. Don’t forget to print at least 15 digits after the decimal point, as the answers to some tests will be non-integer, and you don’t want to get your answer rejected only because of some rounding problems. 11 Sample 2. Input: 2 2 1 1 -1 -1 1 -2 1 1 Output: No solution Explanation: The first inequality gives 𝑥1 + 𝑥2 ≤ 1 and the second inequality gives (−1) · 𝑥1 + (−1) · 𝑥2 ≤ −2 ⇒ 𝑥1 + 𝑥2 ≥ 2. But 𝑥1 + 𝑥2 cannot be less than 1 and more than 2 simultaneously, so there is no solution in this case. Sample 3. Input: 1 3 0 0 1 3 1 1 1 Output: Infinity Explanation: The restrictions in this case are only that all amounts are non-negative (these restrictions are always there, because you cannot show an ad to negative number of users) and that 𝑥3 ≤ 3. There is no upper bound on 𝑥1 and 𝑥2, and both 𝑐1 and 𝑐2 are positive, so you can set 𝑥1 and 𝑥2 big enough and generate as much revenue as you want. In this case, you should output “Infinite” (without quotes). Starter Files The starter solutions for this problem read the data from the input, pass it to a blank procedure and output the result. You need to implement this procedure if you are using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch. Filename: ad_allocation What To Do You will need to implement the Simplex Method from the lectures to solve this problem. Need Help? Ask a question or see the questions asked by other learners at this forum thread. 12 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 13 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 ++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 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
java – Xmx1024m
∙ 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.
References
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