Assignment Report for Algorithm and Advanced Data Structure

All Pairs Shortest Paths Problem (APSPP) Consider a weighted complete graph G with vertex set G.V = {v0, v1, v2, …, vn-1}. The weight of the edge from vi and vj is denoted as G.w(i, j). It is assumed that the weights of the edges are non-negative. In other words, the weights satisfy the following constraints: G.w(i, j) > 0 if i ≠ j G.w(i, j) = 0 if i = j The All Pairs Shortest Paths Problem (APSPP) is, given G, to find the distance network D which is a weighted complete graph such that (i) D has the same vertex set as G.V. In other words, D.V=G.V= {v0, v1, v2, …, vn-1}; (ii) The weights of the edges in D represents the lengths of the shortest paths in G, In other words, D.w(i, j)=length of the shortest path from vi and vj APSPP problem can be solved by the following approaches: Approach A (Dijkstra’s algorithm): Repeatedly solving the Single Source Shortest Paths Problem (SSSPP) using Dijkstra’s algorithm which is a well known greedy algorithm. Approach B (Floyd Algorithm): This approach solves APSPP using Dynamic Programming. It finds all the constrained shortest paths in the graph that only go via intermediate nodes {v0, v1, v2, …, vk}, for k=0, 1,2,.. n-1. When k=n-1, there is no more constraint. Thus all-pairs shortest paths problem is solved when k=n-1. TASKS 1. Implement the following function, Graph generateRandomGraph (int n) that will generate a non-negative weighted complete graph with n vertices. 2. Implement the following functions that solve APSPP using Approach A and Approach B respectively. The headings of the function are as follows: Graph repeatedDikstra (Graph G) Graph floydAlgorithm (Graph G) Input to the functions is a weighted complete graph G. The output of the functions is the distance network D 3. Write a main program to test Approach A and Approach B. o The program will generate a non-negative weighted complete graph G with the number of vertices specified interactively by the end user. o The program will generate the distance network using repeatedDikstra(G), and measures the time taken to run the function. o The program will generate the distance network using floydAlgorithm(G), and measures the time taken to run the function. 4. Write a report on your work. The report should cover the following issues: (i) Data structure design, especially the representation of complete graph; (ii) Pseudo Codes and activity diagrams for repeatedDikstra and floydAlgorithm; (iii) Test plan and test results for the correctness of repeatedDikstra and floydAlgorithm; (iv) Comparison of the execution time for Approach A and Approach B. Programming language: recommend Java. DEMONSTRATION: The report must demonstrate the design, implementation and experiments of generateRandomGraph repeatedDikstra and floydAlgorithm.