Let G = (V, E) be an undirected and connected graph, where each edge (u, v) E E has a weight wt(u, v) > 0. Moreover, assume that all weights are equal. (a) G. The running time of your algorithm must be faster than the running time of Kruskal's algorithm. Design an algorithm to compute a minimum spanning tree (MST) of You must describe your algorithm in plain English (no pseudocode). You must write the running time of your algorithm and explain why you get this running time. (b) In at most 50 words, explain why your algorithm is correct.

Let G = (V, E) be an undirected and connected graph, where each edge (u, v) E E has a weight wt(u, v) > 0. Moreover, assume that all weights are equal. (a) G. The running time of your algorithm must be faster than the running time of Kruskal's algorithm. Design an algorithm to compute a minimum spanning tree (MST) of You must describe your algorithm in plain English (no pseudocode). You must write the running time of your algorithm and explain why you get this running time. (b) In at most 50 words, explain why your algorithm is correct.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Let G = (V, E) be a connected, undirected graph, and let s be a fixed vertex in G. Let TB be the spanning tree of G found by a bread first search starting from s, and similarly TD the spanning tree found by depth first search, also starting at s. (As in problem 1, these trees are just sets of edges; the order in which they were traversed by the respective algorithms is irrelevant.) Using facts, prove that TB = TD if and only if TB = TD = E, i.e., G is itself a tree.
Given a graph G = (V, E), let us call G an almost-tree if G is connected and G contains at most n + 12 edges, where n = |V |. Each edge of G has an associated cost, and we may assume that all edge costs are distinct. Describe an algorithm that takes as input an almost-tree G and returns a minimum spanning tree of G. Your algorithm should run in O(n) time.
Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Prove by contradiction or counterexample. Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.
Consider an undirected graph on 8 vertices, with 12 edges given as shown below: Q2.1 Give the result of running Kruskal's algorithm on this edge sequence (specify the order in which the edges are selected). Q2.2 For the same graph, exhibit a cut that certifies that the edge ry is in the minimum spanning tree. Your answer should be in the form E(S, V/S) for some vertex set S. Specifically, you should find S.
We recollect that Kruskal's Algorithm is used to find the minimum spanning tree in a weighted graph. Given a weighted undirected graph G = (V , E, W), with n vertices/nodes, the algorithm will first sort the edges in E according to their weights. It will then select (n-1) edges with smallest weights that do not form a cycle. (A cycle in a graph is a path along the edges of a graph that starts at a node and ends at the same node after visiting at least one other node and not traversing any of the edges more than once.) Use Kruskal's Algorithm to nd the weight of the minimum spanning tree for the following graph.
Kruskal's algorithm can return different spanning trees for the same input G. Use task part 1 to show that there is only one minimum spanning tree for a graph in which all the edge weights are different. Does the inverse also hold? Part 1 was: Show that for each minimal spanning tree T of G, there is an execution of the algorithm that returns T as a result.
We have a connected graph G=(V,E), and a specific vertex u∈V. Suppose we compute a depth-first search tree rooted at u, and obtain a tree T that includes all nodes of G. Suppose we then compute a breadth-first search tree rooted at u, and obtain the same tree T. Prove that G=T. (In other words, if T is both a depth-first search tree and a breadth-first search tree rooted at u, then G cannot contain any edges that do not belong to T.)
Hey, Kruskal's algorithm can return different spanning trees for the input Graph G.Show that for every minimal spanning tree T of G, there is an execution of the algorithm that gives T as a result. How can i do that? Thank you in advance!
a) Draw the connected subgraph of the given graph above which contains only four nodes ACGB and is also a minimum spanning tree with these four nodes. What is its weighted sum? Draw the adjacency matrix representation of this subgraph (use boolean matrix with only 0 or 1, to show its adjacency in this case).b) Find the shortest path ONLY from source node D to destination node G of the given graph above, using Dijkstra’s algorithm. Show your steps with a table as in our course material, clearly indicating the node being selected for processing in each step.c) Draw ONLY the shortest path obtained above, and indicate the weight in each edge in the diagram. Also determine the weighted sum
Given the graph below, what should be the souce node such that in finding the shortest path tree, the result would be the same whe the minimum spanning tree is searched?
. Let G be a weighted, connected, undirected graph, and let V1 and V2 be a partition of the vertices of G into two disjoint nonempty sets. Furthermore, let e be an edge in the minimum spanning tree for G such that e has one endpoint in V1 and the other in V2. Give an example that shows that e is not necessarily the smallest- weight edge that has one endpoint in V1 and the other in V2.
We have undirected graph K, with two distinct vertices z, k. And let O be a minimum spanning tree of K. Prove that the path from z to k along O is a minmax path. Assume that K has distinct edge weights. (Assume for contradiction that the minmax path is not completely on the minimum spanning tree.) Note: A MinMax path in an undirected Graph K is a path between two vertices z, k that minimizes the maximum weight of the edges on the pat h.h.