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 thisprogramming assignmentrequires passingat least2out of3code problemsfrom thisassignment. 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.
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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.
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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.
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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
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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.
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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. 2n+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
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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. Theinitiallinearprogrammingproblemhasboundedsolutionifandonlyifthenewlyaddedinequality 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.
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3 Advanced Problem: Online Advertisement Allocation Westronglyrecommendyoustartsolvingadvancedproblemsonlywhenyouaredonewiththebasicproblems (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,
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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.
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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.
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4 General Instructions and Recommendations on Solving Algorithmic Problems Your main goal in an algorithmic problem is to implement a program that solves a given computational problem in just few seconds even on massive datasets. Your program should read a dataset from the standard input and write an answer to the standard output. Below we provide general instructions and recommendations on solving such problems. Before reading them, go through readings and screencasts in the first module that show a step by step process of solving two algorithmic problems: link.
4.1 Reading the Problem Statement You start by reading the problem statement that contains the description of a particular computational task as well as time and memory limits your solution should fit in, and one or two sample tests. In some problems your goal is just to implement carefully an algorithm covered in the lectures, while in some other problems you first need to come up with an algorithm yourself.
4.2 Designing an Algorithm If your goal is to design an algorithm yourself, one of the things it is important to realize is the expected running time of your algorithm. Usually, you can guess it from the problem statement (specifically, from the subsection called constraints) as follows. Modern computers perform roughly 108–109 operations per second. So, if the maximum size of a dataset in the problem description is n = 105, then most probably an algorithm with quadratic running time is not going to fit into time limit (since for n = 105, 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.
4.3 Implementing Your Algorithm When you have an algorithm in mind, you start implementing it. Currently, you can use the following programming languages to implement a solution to a problem: C, C++, C#, Haskell, Java, JavaScript, Python2, Python3, Ruby, Scala. For all problems, we will be providing starter solutions for C++, Java, and Python3. If you are going to use one of these programming languages, use these starter files. For other programming languages, you need to implement a solution from scratch.
4.4 Compiling Your Program For solving programming assignments, you can use any of the following programming languages: C, C++, C#, Haskell, Java, JavaScript, Python2, Python3, Ruby, and Scala. However, we will only be providing starter solution files for C++, Java, and Python3. The programming language of your submission is detected automatically, based on the extension of your submission. We have reference solutions in C++, Java and Python3 which solve the problem correctly under the given restrictions, and in most cases spend at most 1/3 of the time limit and at most 1/2 of the memory limit. You can also use other languages, and we’ve estimated the time limit multipliers for them, however, we have no guarantee that a correct solution for a particular problem running under the given time and memory constraints exists in any of those other languages. Your solution will be compiled as follows. We recommend that when testing your solution locally, you use the same compiler flags for compiling. This will increase the chances that your program behaves in the
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same way on your machine and on the testing machine (note that a buggy program may behave differently when compiled by different compilers, or even by the same compiler with different flags). ∙ C (gcc 5.2.1). File extensions: .c. Flags: gcc -pipe -O2 -std=c11
