a. The total dispersion of a permutation f of a set {1, 2, … , n} is = | − |=1 . Write a backtracking algorithm that generates all permutations of {1, 2, … , n} where their total dispersion is ≤ k, for some given input positive integer k. b. Let A[1:n, 1:n] be matrix of real numbers. Write a backtracking algorithm that generates all permutations f of {1, 2, … , n} where , Problem 2: (15 points) a. Let 1 = (1,1) and 2 = (2, 2) be two undirected graphs, where 1 = {1, 2, … , } and 2 = {1, 2, … , } and ≥ . A monomorphism from 1 to 2 is any one-to-one function from 1 to 2 such that for all and in 1, if , is an edge in 1, then ( , ) is an edge in 2. Write a backtracking algorithm that generates all monomorphisms from 1 to 2. b. How would you use your algorithm in part (a) to generate all k-edge cycles in an input graph G, for a given input integer k? c. How would you use your algorithm in part (a) to generate all k-cliques in an input graph G, for a given input integer k? Problem 3 (25 points) a. A node weighted graph is a graph where every node x has a weight w(x). A k-node independent set of a graph G is a set of k nodes such that no two nodes in that set are adjacent in G. The weight of an independent set in a node-weighted graph is the sum of the weights of the nodes in that set. Give an approximate cost function (other than the “cost so far”) for a branch-and-bound algorithm that takes as input a node-weighted G and k, and returns a minimum-weight k-node independent set. Prove the validity of your . b. Apply your algorithm to derive a minimum-weight 3-node independent set in the following graph G=(V,E): V={1, 2, 3, 4, 5, 6}, E={(1,2), (2,3), (3,4), (4,5), (5,6), (6,1), (1,4), (3,6)}, and w(i)=10-i for all i=1, 2, … , 6 . Make sure you show the solution tree and the of each generated node. Problem 4: (20 points) Suppose you have an input of n points in the x-y plane: 1, 1 , 2, 2 , 3, 3 , … , , . Treat those points as a weighted graph G where (1) the nodes are the n points, (2) every pair of nodes is an edge (i.e., the graph is complete), and (3) the weight of every edge is the Euclidean 2 distance between its end points, that is, the weight of the edge between point , and , is ( − )2 + ( − )2. Such a graph is called a Euclidean graph. a. Give an approximate cost function (other than the “cost so far”) for a branch-and-bound algorithm that takes as input a Euclidean graph and returns a minimum-weight Hamiltonian cycle (where the weight of a Hamiltonian cycle is the sum of the weights of its edges). Prove the validity of your . b. Apply your B&B algorithm of (a) to derive a minimum-weight Hamiltonian cycle for the Euclidean graph made up of the following four points: (0,0), (1,0), (1, 5), and (0,1). Make sure you show the solution tree and the of each generated node. Problem 5: (25 points) a. Give a greedy algorithm that attempts to compute a minimum-weight Hamiltonian cycle in a Euclidean graph. b. Prove by a counterexample that the greedy solution is not necessarily optimal. c. Give a divide-and-conquer algorithm that attempts to compute a minimum-weight Hamiltonian cycle in a Euclidean graph. Analyze the time complexity of your algorithm. d. Prove by a counterexample that your divide-and-conquer solution is not necessarily optimal. Bonus Problem: (5 points) Derive the time complexity of the branch-and-bound algorithm for the job assignment problem if the approximate cost function is the cost function .本团队核心人员组成主要包括BAT一线工程师，精通德英语！我们主要业务范围是代做编程大作业、课程设计等等。我们的方向领域：window编程 数值算法 AI人工智能 金融统计 计量分析 大数据 网络编程 WEB编程 通讯编程 游戏编程多媒体linux 外挂编程 程序API图像处理 嵌入式/单片机 数据库编程 控制台 进程与线程 网络安全 汇编语言 硬件编程 软件设计 工程标准规等。其中代写编程、代写程序、代写留学生程序作业语言或工具包括但不限于以下范围:C/C++/C#代写Java代写IT代写Python代写辅导编程作业Matlab代写Haskell代写Processing代写Linux环境搭建Rust代写Data Structure Assginment 数据结构代写MIPS代写Machine Learning 作业 代写Oracle/SQL/PostgreSQL/Pig 数据库代写/代做/辅导Web开发、网站开发、网站作业ASP.NET网站开发Finance Insurace Statistics统计、回归、迭代Prolog代写Computer Computational method代做因为专业，所以值得信赖。如有需要，请加QQ：99515681 或邮箱：[email protected] 微信：codehelp QQ：99515681 或邮箱：[email protected] 微信：codehelp
