Question

1. Figure 1 shows the contents of Δ5 and 115, the final distance and path matrices after the execution of the Floyd-Warshall all-pairs shortest path algorithm on a weighted directed graph D 0 4 16 18 1 10 0 22 24 7 Δ5-115502 115-123ф32 50 40 35 0 20 30 20 15 17 0 Figure 1: Distance(Δ) and Path(11) Matrices (a) Give π(2, 1), the full shortest path from vertex 2 to vertex 1 , as a sequence of vertices, and δ (2, 1), the shortest distance from vertex 2 to vertex 1 . [5 points] (b) Give the sequence of vertices for the full shortest path of the round trip from vertex 1to vertex 4 and back to vertex 1. Also, list all the edges traversed with their corre- sponding weights in the order in which they are traversed along the shortest path for the round-trip. What is the shortest distance of the round trip from vertex 1 to vertex 4 and back to vertex 1? [10 points] (c) Add weighted directed edges to the graph in Figure 2 so that it represents a digraph that is minimal with respect to edges that generates the distance and path matrices in Figure 1. [10 points 2 4 Figure 2: A Weighted Digraph with Missing Edges

Answer 1

-fino distonte matrix 8 I 2.2 15 SJ 15 S 5o 40 3s 召口 2- 2- 3 3 3 S 0 20 15 distance = 1- Vertex ( IS l o 20 2-0 2-O