Programming Assignment 4: Coping with NP-completeness solution

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Introduction Welcome to your last programming assignment of the Advanced Algorithms and Complexity class! In this programming assignment, you will be practicing solving NP-complete problems. Of course, currently we don’t know efficient algorithms for solving these problems, but in some special cases there are efficient algorithms, and for some problems there are algorithms exponentially more efficient than the brute-force approach, although they still have exponential running time themselves.
Learning Outcomes Upon completing this programming assignment you will be able to: 1. design a part of integrated circuit; 2. plan a totally cool party; 3. find the optimal route for a school bus; 4. reschedule the exams.
Passing Criteria: 2 out of 4 Passing thisprogramming assignmentrequires passingat least2out of4code 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: Integrated Circuit Design 3
2 Problem: Plan a Fun Party 5
3 Problem: School Bus 8
4 Advanced Problem: Reschedule the Exams 11
5 General Instructions and Recommendations on Solving Algorithmic Problems 14 5.1 Reading the Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.2 Designing an Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.3 Implementing Your Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.4 Compiling Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.5 Testing Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.6 Submitting Your Program to the Grading System . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.7 Debugging and Stress Testing Your Program . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6 Frequently Asked Questions 17 6.1 I submit the program, but nothing happens. Why? . . . . . . . . . . . . . . . . . . . . . . . . 17 6.2 I submit the solution only for one problem, but all the problems in the assignment are graded. Why? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.3 What are the possible grading outcomes, and how to read them? . . . . . . . . . . . . . . . . 17 6.4 How to understand why my program fails and to fix it? . . . . . . . . . . . . . . . . . . . . . 18 6.5 Why do you hide the test on which my program fails? . . . . . . . . . . . . . . . . . . . . . . 18 6.6 My solution does not pass the tests? May I post it in the forum and ask for a help? . . . . . 19 6.7 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. . . 19
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1 Problem: Integrated Circuit Design Problem Introduction
In this problem, you will determine how to connect the modules of an integrated circuit with wires so that all the wires can be routed on the same layer of the circuit.
Problem Description Task. VLSI or Very Large-Scale Integration is a process of creating an integrated circuit by combining thousands of transistors on a single chip. You want to design a single layer of an integrated circuit. You know exactly what modules will be used in this layer, and which of them should be connected by wires. The wires will be all on the same layer, but they cannot intersect with each other. Also, each wire can only be bent once, in one of two directions — to the left or to the right. If you connect two modules with a wire, selecting the direction of bending uniquely defines the position of the wire. Of course, some positions of some pairs of wires lead to intersection of the wires, which is forbidden. You need to determine a position for each wire in such a way that no wires intersect. This problem can be reduced to 2-SAT problem — a special case of the SAT problem in which each clause contains exactly 2 variables. For each wire i, denote by xi a binary variable which takes value 1 if the wire is bent to the right and 0 if the wire is bent to the left. For each i, xi must be either 0 or 1. Also, some pairs of wires intersect in some positions. For example, it could be so that if wire 1 is bent to the left and wire 2 is bent to the right, then they intersect. We want to write down a formula which is satisfied only if no wires intersect. In this case, we will add the clause (x1 OR x2) to the formula which ensures that either x1 (the first wire is bent to the right) is true or x2 (the second wire is bent to the left) is true, and so the particular crossing when wire 1 is bent to the left AND wire 2 is bent to the right doesn’t happen whenever the formula is satisfied. We will add such a clause for each pair of wires and each pair of their positions if they intersect when put in those positions. Of course, if some pair of wires intersects in any pair of possible positions, we won’t be able to design a circuit. Your task is to determine whether it is possible, and if yes, determine the direction of bending for each of the wires. Input Format. The input represents a 2-CNF formula. The first line contains two integers V and C — the number of variables and the number of clauses respectively. Each of the next C lines contains two non-zero integers i and j representing a clause in the CNF form. If i 0, it represents xi, otherwise if i < 0, it represents x−i, and the same goes for j. For example, a line “2 3” represents a clause (x2 OR x3), line “1 -4” represents (x1 OR x4), line “-1 -3” represents (x1 OR x3), and line “0 2” cannot happen, because i and j must be non-zero. Constraints. 1 ≤ V,C ≤ 1 000 000; −V ≤ i,j ≤ V; i,j ̸= 0. Output Format. If the 2-CNF formula in the input is unsatisfiable, output just the word “UNSATISFIABLE” (without quotes, using capital letters). If the 2-CNF formula in the input is satisfiable, output the word “SATISFIABLE” (without quotes, using capital letters) on the first line and the corresponding assignment of variables on the second line. For each xi in order from x1 to xV , output i if xi = 1 or −i if xi = 0. For example, if a formula is satisfied by assignment x1 = 0,x2 = 1,x3 = 0, 3 output “-1 2 -3” on the second line (without quotes). If there are several possible assignments satisfying the input formula, output any one of them. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time in seconds 1 1 18 16 1.5 2 16 16 36 Memory Limit. 512MB. Sample 1. Input: 3 3 1 -3 -1 2 -2 -3 Output: SATISFIABLE 1 2 -3 Explanation: The input formula is (x1 OR x3) AND (x1 OR x2) AND (x2 OR x3), and the assignment x1 = 1,x2 = 1,x3 = 0 satisfies it. Sample 2. Input: 1 2 1 1 -1 -1 Output: UNSATISFIABLE Explanation: The input formula is (x1 OR x1) AND (x1 OR x1), and it is unsatisfiable. Starter Files The starter solutions for this problem read the data from the input, pass it to a brute-force procedure that checks all possible variable assignments, and output the result. You need to implement a more efficient algorithm in this procedure if you’re using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch. Filename: circuit_design What To Do You need to implement an algorithm described in the lectures. Need Help? Ask a question or see the questions asked by other learners at this forum thread. 4 2 Problem: Plan a Fun Party Problem Introduction In this problem, you will design and implement an efficient algorithm to plan invite the coolest people from your company to a party in such a way that everybody is relaxed, because the direct boss of any invited person is not invited. Problem Description Task. You’re planning a company party. You’d like to invite the coolest people, and you’ve assigned each one of them a fun factor — the more the fun factor, the cooler is the person. You want to maximize the total fun factor (sum of the fun factors of all the invited people). However, you can’t invite everyone, because if the direct boss of some invited person is also invited, it will be awkward. Find out what is the maximum possible total fun factor. Input Format. The first line contains an integer n — the number of people in the company. The next line contains n numbers fi — the fun factors of each of the n people in the company. Each of the next n−1 lines describes the subordination structure. Everyone but for the CEO of the company has exactly one direct boss. There are no cycles: nobody can be a boss of a boss of a ... of a boss of himself. So, the subordination structure is a regular tree. Each of the n−1 lines contains two integers u and v, and you know that either u is the boss of v or vice versa (you don’t really need to know which one is the boss, but you can invite only one of them or none of them). Constraints. 1 ≤ n ≤ 100 000; 1 ≤ fi ≤ 1 000; 1 ≤ u,v ≤ n; u ̸= v. Output Format. Output the maximum possible total fun factor of the party (the sum of fun factors of all the invited people). 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. Sample 1. Input: 1 1000 Output: 1000 Explanation: There is only one person in the company, the CEO, and the fun factor is 1000. We can just invite the CEO and get the total fun factor of 1000. 5 Sample 2. Input: 2 1 2 1 2 Output: 2 Explanation: There are two people, and one of them is the boss of another one. We can invite only one of them. If we invite the second one, the total fun factor is 2, and it is bigger than total fun factor of 1 that we get in case we invite the first one. Sample 3. Input: 5 1 5 3 7 5 5 4 2 3 4 2 1 2 Output: 11 Explanation: A possible subordination structure: 1 2 3 4 5 We can invite 1, 3 and 4 for a total fun factor of 11. If we invite 2, we cannot invite 1, 3 or 4, so the total fun factor will be at most 10, thus we don’t invite 2 in the optimal solution. If we don’t invite 4 also, we will get a fun factor of at most 1+3+5 = 9, so we must invite 4. But then we can’t invite 5, so we invite also 1 and 3 and get the total fun factor of 11. Starter Files The starter solutions for this problem read the data from the input, pass it to a template procedure that implements depth-first search but doesn’t compute anything, and output the result. You need to augment the template procedure if you are using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch. Filename: plan_party What To Do You need to implement an algorithm described in the lectures. 6 Need Help? Ask a question or see the questions asked by other learners at this forum thread. 7 3 Problem: School Bus Problem Introduction In this problem, you will determine the fastest route for a school bus to start from the depot, visit all the children’s homes, get them to school and return back to depot. Problem Description Task. A school bus needs to start from the depot early in the morning, pick up all the children from their homes in some order, get them all to school and return to the depot. You know the time it takes to get from depot to each home, from each home to each other home, from each home to the school and from the school to the depot. You want to define the order in which to visit children’s homes so as to minimize the total time spent on the route. This is an instance of a classical NP-complete problem called Traveling Salesman Problem. Given a graph with weighted edges, you need to find the shortest cycle visiting each vertex exactly once. Vertices correspond to homes, the school and the depot. Edges weights correspond to the time to get from one vertex to another one. Some vertices may not be connected by an edge in the general case. Input Format. The first line contains two integers n and m — the number of vertices and the number of edges in the graph. The vertices are numbered from 1 through n. Each of the next m lines contains three integers u, v and t representing an edge of the graph. This edge connects vertices u and v, and it takes time t to get from u to v. The edges are bidirectional: you can go both from u to v and from v to u in time t using this edge. No edge connects a vertex to itself. No two vertices are connected by more than one edge. Constraints. 2 ≤ n ≤ 17; 1 ≤ m ≤ n(n−1) 2 ; 1 ≤ u,v ≤ n; u ̸= v; 1 ≤ t ≤ 1 000 000. Output Format. If it is possible to start in some vertex, visit each other vertex exactly once in some order going by edges of the graph and return to the starting vertex, output two lines. On the first line, output the minimum possible time to go through such circular route visiting all vertices exactly once (apart from the first vertex which is visited twice — in the beginning and in the end). On the second line, output the order in which you should visit the vertices to get the minimum possible time on the route. That is, output the numbers from 1 through n in the order corresponding to visiting the vertices. Don’t output the starting vertex second time. However, account for the time to get from the last vertex back to the starting vertex. If there are several solutions, output any one of them. If there is no such circular route, output just −1 on a single line. Note that for n = 2 it is considered a correct circular route to go from one vertex to another by an edge and then return back by the same edge. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time in seconds 1 1 1.5 45 1.5 2 45 45 3 Memory Limit. 512MB. 8 Sample 1. Input: 4 6 1 2 20 1 3 42 1 4 35 2 3 30 2 4 34 3 4 12 Output: 97 1 4 3 2 Explanation: 1 2 34 20 35 30 12 42 34 The suggested route starts in the vertex 1, goes to 4 in 35 minutes, from there to 3 in 12 minutes, from there to 2 in 30 minutes, from there back to 1 in 20 minutes, totalling in 35 + 12 + 30 + 20 = 97 minutes. Check yourself that any other circular route visiting each vertex exactly once is longer. Sample 2. Input: 4 4 1 2 1 2 3 4 3 4 5 4 2 1 Output: -1 Explanation: 1 2 34 1 4 5 1 There is no way to visit all the vertices exactly once, as there is only one edge from the vertex 1 (going to 2), so after leaving it you cannot return without visiting 2 twice. 9 Starter Files The starter solutions for this problem read the data from the input, pass it to a brute-force procedure that tries each possible order of visit, and output the result. You need to change the brute-force procedure to implement some more efficient algorithm if you are using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch. Filename: school_bus What To Do You need to implement an algorithm described in the lectures. Need Help? Ask a question or see the questions asked by other learners at this forum thread. 10 4 Advanced Problem: Reschedule the Exams 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 In this problem, you will design and implement an efficient algorithm to reschedule the exams in such a way that every student can come to the exam she is assigned to, and no two friends will be passing the exam the same day. Problem Description Task. The new secretary at your Computer Science Department has prepared a schedule of exams for CS101: each student was assigned to his own exam date. However, it’s a disaster: not only some pairs of students known to be close friends may have been assigned the same date, but also NONE of the students can actually come to the exam at the day they were assigned (there was a misunderstanding between the secretary who asked to specify available dates and the students who understood they needed to select the date at which they cannot come). There are three different dates the professors are available for these exams, and these dates cannot be changed. The only thing that can be changed is the assignment of students to the dates of exams. You know for sure that each student can’t come at the currently scheduled date, but also each student definitely can come at any of the two other possible dates. Also, you must make sure that no two known close friends are assigned to the same exam date. You need to determine whether it is possible or not, and if yes, suggest a specific assignment of the students to the dates. This problem can be reduced to a graph problem called 3-recoloring. You are given a graph, and each vertex is colored in one of the 3 possible colors. You need to assign another color to each vertex in such a way that no two vertices connected by and edge are assigned the same color. Here, possible colors correspond to the possible exam dates, vertices correspond to students, colors of the vertices correspond to the assignment of students to the exam dates, and edges correspond to the pairs of close friends. Input Format. The first line contains two integers n and m — the number of vertices and the number of edges of the graph. The vertices are numbered from 1 through n. The next line contains a string of length n consisting only of letters R, G and B representing the current color assignments. For each position i (1-based) in the string, if it is R, then the vertex i is colored red; if it’s G, the vertex i is colored green; if it’s B, the vertex i is colored blue. These are the current color assignments, and each of them must be changed. Each of the next m lines contains two integers u and v — vertices u and v are connected by an edge (it is possible that u = v). Constraints. 1 ≤ n ≤ 1 000; 0 ≤ m ≤ 20 000; 1 ≤ u,v ≤ n. Output Format. If it is impossible to reassign the students to the dates of exams in such a way that no two friends are going to pass the exam the same day, and each student’s assigned date has changed, output just one word “Impossible” (without quotes). Otherwise, output one string consisting of n characters R, G and B representing the new coloring of the vertices. Note that the color of each vertex must be 11 different from the initial color of this vertex. The vertices connected by an edge must have different colors. Time Limits. language C C++ Java Python C# Haskell JavaScript Ruby Scala time in seconds 1 1 3 5 1.5 2 5 5 3 Memory Limit. 512MB. Sample 1. Input: 4 5 RRRG 1 3 1 4 3 4 2 4 2 3 Output: GGBR Explanation: The initial coloring and the new coloring: 1 2 34 1 2 34 Note that the vertices 1 and 2 are ok to be of the same color, as they are not connected by an edge. Sample 2. Input: 4 5 RGRR 1 3 1 4 3 4 2 4 2 3 Output: Impossible 12 Explanation: The initial coloring: 1 2 34 It is impossible to recolor the vertices properly, because it means that none of the vertices 1, 3 and 4 can be red, but that leaves only two choices of colors to them — blue and green — but each pair of them is connected by an edge, so we need at least three different colors to color them properly. Starter Files The starter solutions for this problem read the data from the input, pass it to a procedure that just tries to set the colors of the vertices in a fixed way without looking at the edges or the current colors. You need to change the main procedure to solve the problem correctly if you are using C++, Java, or Python3. For other programming languages, you need to implement a solution from scratch. Filename: reschedule_exams What To Do You need to reduce this problem to another problem you already know how to solve. Need Help? Ask a question or see the questions asked by other learners at this forum thread. 13 5 General Instructions and Recommendations on Solving Algorithmic Problems Your main goal in an algorithmic problem is to implement a program that solves a given computational problem in just few seconds even on massive datasets. Your program should read a dataset from the standard input and write an answer to the standard output. Below we provide general instructions and recommendations on solving such problems. Before reading them, go through readings and screencasts in the first module that show a step by step process of solving two algorithmic problems: link. 5.1 Reading the Problem Statement You start by reading the problem statement that contains the description of a particular computational task as well as time and memory limits your solution should fit in, and one or two sample tests. In some problems your goal is just to implement carefully an algorithm covered in the lectures, while in some other problems you first need to come up with an algorithm yourself. 5.2 Designing an Algorithm If your goal is to design an algorithm yourself, one of the things it is important to realize is the expected running time of your algorithm. Usually, you can guess it from the problem statement (specifically, from the subsection called constraints) as follows. Modern computers perform roughly 108–109 operations per second. So, if the maximum size of a dataset in the problem description is n = 105, then most probably an algorithm with quadratic running time is not going to fit into time limit (since for n = 105, n2 = 1010) while a solution with running time O(nlogn) will fit. However, an O(n2) solution will fit if n is up to 103 = 1000, and if n is at most 100, even O(n3) solutions will fit. In some cases, the problem is so hard that we do not know a polynomial solution. But for n up to 18, a solution with O(2nn2) running time will probably fit into the time limit. To design an algorithm with the expected running time, you will of course need to use the ideas covered in the lectures. Also, make sure to carefully go through sample tests in the problem description. 5.3 Implementing Your Algorithm When you have an algorithm in mind, you start implementing it. Currently, you can use the following programming languages to implement a solution to a problem: C, C++, C#, Haskell, Java, JavaScript, Python2, Python3, Ruby, Scala. For all problems, we will be providing starter solutions for C++, Java, and Python3. If you are going to use one of these programming languages, use these starter files. For other programming languages, you need to implement a solution from scratch. 5.4 Compiling Your Program For solving programming assignments, you can use any of the following programming languages: C, C++, C#, Haskell, Java, JavaScript, Python2, Python3, Ruby, and Scala. However, we will only be providing starter solution files for C++, Java, and Python3. The programming language of your submission is detected automatically, based on the extension of your submission. We have reference solutions in C++, Java and Python3 which solve the problem correctly under the given restrictions, and in most cases spend at most 1/3 of the time limit and at most 1/2 of the memory limit. You can also use other languages, and we’ve estimated the time limit multipliers for them, however, we have no guarantee that a correct solution for a particular problem running under the given time and memory constraints exists in any of those other languages. Your solution will be compiled as follows. We recommend that when testing your solution locally, you use the same compiler flags for compiling. This will increase the chances that your program behaves in the 14 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