Formulate and solve the multi-target shortest path problem on this network (from the source node 1 to all other nodes). Next formulate and solve the dual of the shortest path problem and compare your results with that of the primal problem.

Question 7. We have computed the shortest path between all the nodes in a weighted graph. Then all the edges increase their costs by adding the same value c > 0. Will the shortest paths change? If your answer is NO, prove so, if the answer is YES, give an example of a graph where at least one shortest path changes.

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.

After how many time ticks, will L's distance vector (i.e., its DV that it exchanges with its neighbor) reflect this change in topology?

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?

Problem 2: Compute the shortest path from s to t in the following network using the Bellman-Ford algorithm. 2 (² 5 3 2 1 -2 2 -1 (3 t -4

Let f be an s,t-flow in an s, t-network D = (V, A) with capacities c : A → R>o, and assume that there is no augmenting path. Let X be the set of vertices that can be reached from s by unsaturated paths. Let (a, b) E A(X,V\ X). Explain why f(a,b) = c(a, b).

Consider the Network N in Figure Q3. The two numbers on an arc indicate the minimum and maximum capacities of the arc. a) Find a feasible flow from the source vertex S to the terminal vertex T ? Show your working and write down a feasible flow in Network N.

Use proof by induction for the Shortest Path Distance algorithm of graph G where u and in S(u,v) cannot = infinity on graph G