-4.1 2 -2 -1 -1 .7 a (a) Use Prim's algorithm to compute the minimum spanning tree for the weighted graph. Start the algorithm at vertex a. Show the order in which the edges are added to the tree. (b) Now suppose that the weight of edge {e,f} is decreased from 7 to -3. Is there a way to find the minimum spanning tree of the altered graph without running Prim's algorithm from scratch?

-4.1 2 -2 -1 -1 .7 a (a) Use Prim's algorithm to compute the minimum spanning tree for the weighted graph. Start the algorithm at vertex a. Show the order in which the edges are added to the tree. (b) Now suppose that the weight of edge {e,f} is decreased from 7 to -3. Is there a way to find the minimum spanning tree of the altered graph without running Prim's algorithm from scratch?

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

See similar textbooks

Similar questions

Use Prim's Algorithm starting at vertex SS to construct a minimum spanning tree of the graph below. If we tie-break alphabetically, what is the order of edges we construct our spanning tree? a. {S,T}, {S,V}, {V,Z}, {W,Z}, {Y,Z}, {W,X}, {U,Y} b. {S,T}, {S,W}, {W,Z}, {V,Z}, {Y,Z}, {W,X}, {U,Y} c. {S,T}, {V,Z}, {W,Z}, {Y,Z}, {S,V}, {W,X}, {U,Y} d. {S,T}, {S,V}, {S,W}, {V,Z}, {W,X}, {Y,Z}, {U,Y}
Fill in the blank Dijkstra's algorithm works because, on every shortest path p from a source vertex u to a target vertex v, there is a (predecessor) vertex w in p immediately before v such that removing v from p yields the shortest path from u to w. In other words, the path through the previous vertex is also the shortest path. Thus, choosing an edge from the previous vertex that brings us to v with the __ cost always yields the shortest path to v.
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? 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.
A mathematician applies Prim’s Algorithm to find a minimum spanning tree for the weighted graph starting at vertex D. The order of the edges picked so far is AD, AB, BE, and EF. The next edge selected when applying Prim’s Algorithm should be BF FG CG CE What is the total weight of the tree that uses the edges DE, CE, CG, FG and BF? 93 92 91 90 The tree described in problem #8 is a spanning True False
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.)
R-13.7 - Let G be a graph whose vertices are the integers 1 through 8, and let the adjacent vertices of each vertex be given by the table below: Vertex: 1 2 3 4 5 6 7 8 Adjacent Vertices: (2, 3, 4) (1, 3, 4) (1, 2, 4) (1, 2, 3, 6) (6, 7, 8) (4, 5, 7) (5, 6, 8) (5, 7) Assume that, in a traversal of G, the adjacent vertices of a given vertex are returned in the same order as they are listed in the table above. a. Draw G. b. Give the sequence of vertices of G visited using a DFS traversal starting at vertex 1. c. Give the sequence of vertices visited using a BFS traversal starting at vertex 1.
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 the graph attached Find a minimum-cost spanning tree by Prim's algorithm. Find a minimum-cost spanning tree by Kruskal's algorithm. Find a depth-first spanning tree starting at a and at d. Find a breadth-first spanning tree starting at a and at d. Describe an algorithm to insert and delete edges in the adjacency list representation for an undirected graph. Remember that an edge (i, j) appears on the adjacency list for both vertex i and vertex j. Implement both Prim's algorithm and Kruskal's algorithm. Compare the running times of your programs on a set of "random" graphs. Write a program to find all the connected components of a graph. Consider the following list of integers: [1,2,3,4,5,6,7,8,9,10]. Show how this list is sorted by the following algorithms: bubble sort selection sort insertion sort shell sort (you decide on the increments) merge sort quick sort (you decide on the pivot value
Use depth-first search starting at vertex SS to construct a spanning tree of the graph below. If we tie-break alphabetically, what is the order of edges we construct our spanning tree? a. {S,T}, {T,X}, {W,X}, {W,Z}, {V,Z}, {X,Y}, {U,Y} b. {S,T}, {S,V}, {V,Z}, {W,Z}, {W,X}, {T,X}, {X,Y}, {U,Y} c. {S,T}, {T,X}, {W,X}, {S,V}, {V,Z}, {Y,Z}, {U,Y} d. {S,T}, {T,X}, {W,X}, {W,Z}, {V,Z}, {Z,Y}, {U,Y}
Let DIST (u, v) denote the distance between vertex u and v. It is well known that distances in graphs satisfy the triangle inequality. That is, for any three vertices u, v, w, DIST (u, v) ≤ DIST (u, w) + DIST (w, v). Let D∗ denote the distance between the two farthest nodes in G. Show that for any vertex s D∗ ≤ 2 max DIST (s, v).
We have the following directed graph G, where the number on each edge is the cost of the edge. 1. Step through Dijkstra’s Algorithm on the graph starting from vertex s, and complete the table below to show what the arrays d and p are at each step of the algorithm. For any vertex x, d[x] stores the current shortest distance from s to x, and p[x] stores the current parent vertex of x. 2. After you complete the table, provide the returned shortest path from s to t and the cost of the path.
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? Let P be the shortest path between two nodes s, t. Now, suppose we replace each edge weight ℓ(e) withℓ(e)^2, then P is still a shortest path between s and t.