3, (30 points) Given a directed graph G - N. E), each edge eEhas weight We, 3, (30 points) Given a directed graph...
6. Prove that the following graphs are connected: (a) The 3 vertex cycle: (b) The following 4 vertex graph: (c) K 7. An edge e of a connected graph G is called a cut edge if the graph G obtained by deleting that edge (V(G) V(G) and E(G) E(G) \<ej) is not connected. Prove that if G1 and G2 are connected simple graphs which are isomorphic and if G1 has a cut edge, then G2 also has a cut edge....
Given a directed graph with positive edge lengths and a specified vertex v in the graph, the "all-pairs" v-constrained shortest path problem" is the problem of computing for each pair of vertices i and j the shortest path from i to j that goes through the vertex v. If no such path exists, the answer is . Describe an algorithm that takes a graph G= (V; E) and vertex v as input parameters and computes values L(i; j) that represent...
NP-completeness. We are given an undirected graph where each edge has a positive weight. Given (k, alpha), the problem asks whether there is a subgraph with k nodes such that the total weight of the edges in the subgraph is at least alpha. Prove this problem is NP-Complete.
Consider the following weighted, directed graph G. There are 7 vertices and 10 edges. The edge list E is as follows:The Bellman-Ford algorithm makes |V|-1 = 7-1 = 6 passes through the edge list E. Each pass relaxes the edges in the order they appear in the edge list. As with Dijkstra's algorithm, we record the current best known cost D[V] to reach each vertex V from the start vertex S. Initially D[A]=0 and D[V]=+oo for all the other vertices...
Viterbi algorithm We can use dynamic programming on a directed graph G = (V, E) for speech recognition. Each edge (u, v) in E is labeled with a sound s(u, v) from a finite set S of sounds. The labeled graph is a formal model of a person speaking a restricted language. Each path in the graph starting from a distinguished vertex v0 in V corresponds to a possible sequence of sounds produced by the model. The label of a...
Problem 3: Suppose you are given an undirected graph G and a specified starting node s and ending node t. The HaMILTONIAN PATH problem asks whether G contains a path beginning at s and ending at t that touches every node exactly once. The HAMILTONIAN CYCLE problem asks whether con- tains a cycle that touches every node exactly once (cycles don't have starting or ending points, so s and t are not used here) Assume that HaMIlTonian CYCLe is NP-Complete....
NAME . You are given a strongly connected directed graph G (V, E) with positive edge weights along with a particular node vo E V. Give an efficient algorithm for finding shortest paths between all pairs of nodes, with the one restriction that these paths must all pass through v (8 points)
Let G=(V, E) be a connected graph with a weight w(e) associated with each edge e. Suppose G has n vertices and m edges. Let E’ be a given subset of the edges of E such that the edges of E’ do not form a cycle. (E’ is given as part of input.) Design an O(mlogn) time algorithm for finding a minimum spanning tree of G induced by E’. Prove that your algorithm indeed runs in O(mlogn) time. A minimum...
Let G = (V, E, W) be a connected weighted graph where each edge e has an associated non-negative weight w(e). We call a subset of edges F subset of E unseparating if the graph G' = (V, E\F) is connected. This means that if you remove all of the edges F from the original edge set, this new graph is still connected. For a set of edges E' subset of E the weight of the set is just the...
6. Dijkstra's Algorithm assumes that all edge weights in a given weighted directed graph G = (VAE) are nonnegative. However, if we apply Dijkstra's Algorithm to the graph G where the edge weights may be negative, Dijkstra's Algorithm may produce incorrect answers. Show such an example where Dijkstra's Algorithm may produce incorrect answers. Then, explain why such incorrect answers happen. (15 pts]