Does an edge of the smallest possible value in a weighted connected graph G always belong to a minimum spanning tree of G? Discuss this for a number of cases.
Does an edge of the smallest possible value in a weighted connected graph G always belong...
Let G = (V, E) be a weighted undirected connected graph that contains a cycle. Let k ∈ E be the edge with maximum weight among all edges in the cycle. Prove that G has a minimum spanning tree NOT including k.
Let (u, v) be a minimum-weight edge in a connected graph G. Show that (u, v) belongs to some minimum spanning tree of G.
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...
Suppose you are given a connected graph G, with edge costs that you may assume are all distinct. G has n vertices and m edges. A particular edge e of G is specified. Give an algorithm with running time O(m + n) to decide whether e is contained in a minimum spanning tree of G.
You are given an undirected graph G with weighted edges and a minimum spanning tree T of G. Design an algorithm to update the minimum spanning tree when the weight of a single edge is increased. The input to your algorithm should be the edge e and its new weight: your algorithm should modify T so that it is still a MST. Analyze the running time of your algorithm and prove its correctness.
You are given an undirected graph G with weighted edges and a minimum spanning tree T of G. Design an algorithm to update the minimum spanning tree when the weight of a single edge is decreased. The input to your algorithm should be the edge e and its new weight; your algorithm should modify T so that it is still a MST. Analyze the running time of your algorithm and prove its correctness.
Problem 3's picture are given below. 5. (a) Let G = (V, E) be a weighted connected undirected simple graph. For n 1, let cycles in G. Modify {e1, e2,.. . ,en} be a subset of edges (from E) that includes no Kruskal's algorithm in order to obtain a spanning tree of G that is minimal among all the spanning trees of G that include the edges e1, e2, . . . , Cn. (b) Apply your algorithm in (a)...
Let G be the weighted graph (a) Find a minimum spanning tree for G using Prim's algorithm, showing all interme- diate steps. What is the cost of this tree? (b) Find a minimum spanning tree for G using Kruskal's algorithm, showing all inter- mediate steps. What is the cost of this tree?
1) Professor Sabatier conjectures the following converse of Theorem 23.1. Let G=(V,E) be a connected, undirected graph with a real-valued weight function w defined on E. Let A be a subset of E that is included in some minimum spanning tree for G, let (S,V−S) be any cut of G that respects A, and let (u,v) be a safe edge for A crossing (S,V−S). Then, (u,v) is a light edge for the cut. Show that the professor's conjecture is incorrect...
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.