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. -- 2
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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:
must demonstrate the design, implementation and experiments of generateRandomGraph repeatedDikstra and floydAlgorithm.
SUBMISSION
o The soft copy of the report in both WORD and PDF format;
o The source codes and executables for the program;
o The raw data from the experiments;
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