a. Use dynamic programming to solve the shortest route problem for the network below where node 1 is the origin and 7 is the destination. Branches are distances in km. 20 18 22 3 2 39 28 36 18 6 5 13 10 7
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- 1. The following diagram shows the number of minutes needed to travel between various intersections in the city of Ottawa along one-way streets. What is the fastest route from the intersection at node A to the intersection at node K? Formulate and solve as a network flow program. You may use a computer program such as LINDO for the solution. 1 23 B 23 21 17 14 K 10 10 18 21 18 14 27 14 10 11 H. 20 19 33 16 10 L. 16 14 2.Use the provided data to create the Dijkstra Shortest Weighted Path method.Precondition: S is a node in the weighted (directed or undirected) network G.Post-cond: specifies the shortest weighted route between each node of G and s, and d specifies the lengths of those paths.Major Luna is sending military troops from his base (A) to the town of G. However, there is no direct route between A and G. Instead, the troops must pass through other places, namely: B, C, D, E, and F. The following routes are possible: A to B, A to C, B to F, C to D, C to E, D to G, E to G, and F to G. For some reason, such places can only accommodate a number of soldiers going through these places. B, C, D, E, and F has a capacity of 3, 3, 9, 6, and 4 soldiers. b. Solve the problem using the Ford-Fulkerson algorithm.
- Dijkstra’s algorithm computes the least-cost path from one node (the source, which we will refer to as ?u) to all other nodes in the network. Dijkstra’s algorithm is iterative and has the property that after the k-th iteration of the algorithm, the least-cost paths are known to k destination nodes, and among the least-cost paths to all destination nodes, these k paths will have the k smallest costs. Let - D(v): cost of the least-cost path from the source node to destination ?v as of this iteration of the algorithm. - p(v): previous node (neighbor of v) along the current least-cost path from the source to v. - N′: a subset of nodes; v is in N′ if the least-cost path from the source to ?v is definitely known. The centralized routing algorithm consists of an initialization step followed by a loop. The number of times the loop is executed is equal to the number of nodes in the network. Upon termination, the algorithm will have calculated the shortest paths from the source…Create an algorithm for the Dijkstra Shortest Weighted Path based on the provided data.G is a weighted (directed or undirected) network, and s is a node in it.post-cond: specifies the shortest weighted route from s to each node of G, and d the lengths of those paths.The input to this problem is a sequence of n points p1; ; pn in the Euclidean plane. You are to nd the shortest routes for two taxis to service these requests in order. The two taxis start at the origin p1. If a taxi visits a point pi before pj then it must be the case that i < j. Each point must be visited by one of the two taxis. The cost of a routing is the total distance traveled by the rst taxi plus the total distance traveled by the second taxi. Design an ecient dynamic programming algorithm to nd the minimum cost routing such that each point is covered by one of the two taxis.
- Let a network be given by nodes V = {S, A, B,C, D,T} and arcs, capacities e and a flow f according to the table: SA SC AB Bт АC DC BD DT AT CT 3 ? 4 1 1 ? 3 4 3 5 ? 3 9. 3 5 1 1 (a) Complete f and e such that f is an admissible flow from S to T. Justify your answer. What is the value of the flow?Solve the following all-pairs shortest path problem using the dynamic programming method. 9 3 8 1C. A 2 B 5 с 10 3 D 4 E 11 F i. What is the shortest cost or distance to travel from node A to node F? ii. Give the weight of each edges that is traversed to get from A to F. [ iii. What would be the longest path from node A to node B?
- The Voyager spacecraft has been transported by an alien being to the Delta Quadrant of space, millions of light-years from Earth. Captain Janeway and her crew are attempting to visit all the planets on the Quadrant to collect soil samples by connecting them as a minimum spanning tree. Following is a network of the Delta Quadrant, where the nodes are different worlds, planetary systems, and space anomalies, and the values on the branches are the actual times, in years, between nodes. Determine the minimum possible way to connect all planets so the spacecraft can travel. Start from the node 1, where the Voyager is currently located at. The Earth does not have to be the last stop as long as all planets are connected.Let a network be given by nodes V = {S, A, B,C, D,T} and ares, capacities e and a flow f according to the table: SA SC AB вТ АС DC BD DT AT CT ? ? 1 3 ? 3 4 1 0 5 1 1 9 3 2 3 4 3 5 (a) Complete f and e such that f is an admissible flow from S to T. Justify your answer. What is the value of the flow? (b) Determine a maximal flow from S to T and a minimal cut with its capacity by the augmenting path method starting by the flow f from (a).Let a network be given by nodes V = {S, A, B,C, D,T} and ares, capacities e and a flow f according to the table: SA SC AB вТ АС DC BD DT AT CT 3 4 1 0 3 5 1 1 ? ? 1 3 ? 2 3 4 3 5 9 (a) Complete f and e such that f is an admissible flow from S to T. Justify your answer. What is the value of the flow? (b) Determine a maximal flow from S to T and a minimal cut with its capacity by the augmenting path method starting by the flow f from (a). (c) What is the relation between each admissible flow and each cut? (d) What is the relation between the maximum flow and the minimum cut?