Apply the shortest-augmenting path algorithm to find a maximum flow in the following networks.

9. In a weighted graph, assume that the shortest path from a source 'S' to a destination 'T' is correctly calculated using Dijkstra's shortest path algorithm. If we increase weight of every edge by 1, does the shortest path always remain the same? If so, prove it. If not, give a counterexample.

Find the cost of a shortest path between a and d in the given weighted graph.

Compute the maximum flow in the following flow network using Ford-Fulkerson's algorithm. 12 8 14 15 11 2 5 1 3 d 7 f 6 What is the maximum flow value? A minimum cut has exactly which vertices on one side? 4.

Show the final flow that the Ford-Fulkerson Algorithm finds for this network, given that it proceeds to completion from the flow rates you have given in your answer to part, and augments flow along the edges (?,?1,?3,?) and (?,?2,?5,?). Identify a cut of the network that has a cut capacity equal to the maximum flow of the network.

Consider the following network: (a) Which of the following most accurately describes the connectedness of this network? Strongly Weakly Disconnected None of the above   (b) When discussing path lengths on a weighted graph, one must first define how the weights are related to the length of a path between two nodes is then the sum of the distances of the links in that path. Consider the previous network and assume that the link weights represent distances. Using this distance metric, what is the shortest path between nodes 1 and 6? (c) A common way to define the distance between two nodes is the inverse (or reciprocal) of the link weight. Consider the previous network and assume that the distance between two adjacent nodes is defined as the reciprocal of the link weight. Using this distance metric, what is the shortest path between node 1 and 6?

Prove that The number of augmenting paths needed in the shortest-augmenting-path implementation of the Ford-Fulkerson maxflow algorithm for a flow network with V vertices and E edges is at most EV /2.

Using given matrix P produced by Floyd2 Algorithm restore the path from v9 to V2 that has the minimum cost (shortest path). Hint: use path(r, q) function. P v1 "2 V3 v4 V5 V6 V7 8 90 19 v1 0 6 5 0 0 9 22 0 0 5 0 1 0 6 6 9 5 0 6 V3 6 9 0 6 9 9 9 9 0 v4 0 0 5 0 0 15 0 9 0 9 0 9 0 6 6 9 7 5 4. 3 0. 9 8 9 8 V6 17 18 19 8 8 0 0 2 7 8 5 0 1 0 0 9 0 8 0 8 8 0 9 8 0 0 8 9 0 3 0 6 7 0

Need help with bottleneck flow problem

Given flow graph G below, find the max-flow. Show the residual graphs G† and updated flow for each of the steps. (a) use the shortest path method to get the augmenting paths (b) use the fattest path method to get the augmenting paths (c) show the min-cut that corresponds to the max-flow for each of the above -15 15 10 2 14- -10- -15- 10 2 -15- 2. 2.