Homework Answers
If there is an efficient algorithm ALG to find a simple
path more than or equal to k then we can use this alorithm
to find simple paths starting width k = 2 and continue till
we get a value of k for which we don't get a longest path
>= k. Then the path corresponding to k will be the longest
path
in the undirected graoh.
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2. Recall the language: LONG-PATH-(〈G, k〉 | k is an integer; G is an undirected graph...
Note: For the following problems, you can assume that INDEPENDENT SET, VERTEX COVER, 3-SAT, HAMILTONIAN PATH, and GRAPH COLORING are NP-complete. You, of course, may look up the defini- tions of the...
Note: For the following problems, you can assume that INDEPENDENT SET, VERTEX COVER, 3-SAT, HAMILTONIAN PATH, and GRAPH COLORING are NP-complete. You, of course, may look up the defini- tions of the above problems online. 5. The LONGEST PATH problem asks, given an undirected graph G (V, E), and a positive integer k , does G contain a simple path (a path visiting no vertex more than once) with k or more edges? Prove that LONGEST PATH is NP-complete.
Note:...
2. Let G be an undirected graph. For every u,vE V(G), let dc(u,v) be the length of the shoertest ...
2. Let G be an undirected graph. For every u,vE V(G), let dc(u,v) be the length of the shoertest path from u to v. The diameter of G is he maximum distance bet In other words: max (de(u, v) u,vEV(G) the running time of your algorithm
2. Let G be an undirected graph. For every u,vE V(G), let dc(u,v) be the length of the shoertest path from u to v. The diameter of G is he maximum distance bet In...
Reachability. You are given a connected undirected graph G = (V, E ) as an adjacency...
Reachability. You are given a connected undirected graph G = (V, E ) as an adjacency list. The graph G might not be connected. You want to fill-in a two-dimensional array R[,] so that R[u,v] is 1 if there is a path from vertex u to vertex v. If no such path exists, then R[u,v] is 0. From this two-dimensional array, you can determine whether vertex u is reachable from vertex v in O(1) time for any pair of vertices...
Write a program that specifies a simple undirected graph by its “adjacency matrix”. Recall that that...
Write a program that specifies a simple undirected graph by its “adjacency matrix”. Recall that that the adjacency matrix A is such that A(i, j) = 1 if nodes i and j are adjacent and A(i, j) = 0 otherwise. Let αk(i ,j) be the number of paths of length k between nodes i and j. For instance, the number of paths of length-1 between nodes i and j in a simple undirected graph is 1 if they are adjacent...
Prove the claim. Consider an undirected graph G with minimum degree δ(G) ≥ 2. Then G...
Prove the claim. Consider an undirected graph G with minimum degree δ(G) ≥ 2. Then G has a path of length δ(G) and a cycle with at least δ(G) + 1 vertices.
Problem 5. (15 marks) Given a connected, undirected, weighted graph G- (V, E), define the cost...
Problem 5. (15 marks) Given a connected, undirected, weighted graph G- (V, E), define the cost of a spanning tree to be the maximum weight among the weights associated with the edges of the spanning tree. Design an efficient algorithm to find the spanning tree of G which maximize above defined cost What is the complexity of your algorithm.
Q1: Here we consider finding the length of the shortest path between all pairs of nodes...
Q1: Here we consider finding the length of the shortest path between all pairs of nodes in an undirected, weighted graph G. For simplicity, assume that the n nodes are labeled 1; 2; : : : ; n, that the weight wij of any edge e = (i; j) is positive and that there is an edge between every pair of nodes. In this question, the goal is to solve this via dynamic programming. Note that the algorithm you will...
2. Let G = (V, E) be an undirected connected graph with n vertices and with...
2. Let G = (V, E) be an undirected connected graph with n vertices and with an edge-weight function w : E → Z. An edge (u, v) ∈ E is sparkling if it is contained in some minimum spanning tree (MST) of G. The computational problem is to return the set of all sparkling edges in E. Describe an efficient algorithm for this computational problem. You do not need to use pseudocode. What is the asymptotic time complexity of...
Say that we have an undirected graph G(V, E) and a pair of vertices s, t and a vertex v that we call a a desired middle vertex . We wish to find out if there exists a simple path (every vertex appears...
Say that we have an undirected graph G(V, E) and a pair of vertices s, t and a vertex v that we call a a desired middle vertex . We wish to find out if there exists a simple path (every vertex appears at most once) from s to t that goes via v. Create a flow network by making v a source. Add a new vertex Z as a sink. Join s, t with two directed edges of capacity...
Consider a directed acyclic graph G = (V, E) without edge lengths and a start vertex...
Consider a directed acyclic graph G = (V, E) without edge lengths and a start vertex s E V. (Recall, the length of a path in an graph without edge lengths is given by the number of edges on that path). Someone claims that the following greedy algorithm will always find longest path in the graph G starting from s. path = [8] Ucurrent = s topologically sort the vertices V of G. forall v EV in topological order do...
Note: For the following problems, you can assume that INDEPENDENT SET, VERTEX COVER, 3-SAT, HAMILTONIAN PATH, and GRAPH COLORING are NP-complete. You, of course, may look up the defini- tions of the above problems online. 5. The LONGEST PATH problem asks, given an undirected graph G (V, E), and a positive integer k , does G contain a simple path (a path visiting no vertex more than once) with k or more edges? Prove that LONGEST PATH is NP-complete.
Note:...
2. Let G be an undirected graph. For every u,vE V(G), let dc(u,v) be the length of the shoertest path from u to v. The diameter of G is he maximum distance bet In other words: max (de(u, v) u,vEV(G) the running time of your algorithm
2. Let G be an undirected graph. For every u,vE V(G), let dc(u,v) be the length of the shoertest path from u to v. The diameter of G is he maximum distance bet In...
Problem 5. (15 marks) Given a connected, undirected, weighted graph G- (V, E), define the cost of a spanning tree to be the maximum weight among the weights associated with the edges of the spanning tree. Design an efficient algorithm to find the spanning tree of G which maximize above defined cost What is the complexity of your algorithm.
Consider a directed acyclic graph G = (V, E) without edge lengths and a start vertex s E V. (Recall, the length of a path in an graph without edge lengths is given by the number of edges on that path). Someone claims that the following greedy algorithm will always find longest path in the graph G starting from s. path = [8] Ucurrent = s topologically sort the vertices V of G. forall v EV in topological order do...
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