Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 22.3, Problem 11E
Program Plan Intro
To explain a vertex of a directed graph can end up in a depth-first tree containing only that vertex, even though vertex has both incoming and outgoing edge in graph.
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Explain how a vertex u of a directed graph G can end up in a depth-first tree containing only u, even though u has both incoming and outgoing edges in G.
Explain how a directed graph's vertex u, which includes both incoming and outgoing edges in G, may end up in a depth-first tree that only contains u.
Given a graph that is a tree (connected and acyclic).
(1) Pick any vertex v.
(II) Compute the shortest path from v to every other vertex. Let w be the vertex with
the largest shortest path distance.
(III) Compute the shortest path from w to every other vertex. Let x be the vertex
with the largest shortest path distance.
Consider the path p from w to x. Which of the following are true
a. p is the longest path in the graph
b. p is the shortest path in the graph
c. p can be calculated in time linear in the number of edges/vertices
a,c
a,b
a,b,c
b.c
Chapter 22 Solutions
Introduction to Algorithms
Ch. 22.1 - Prob. 1ECh. 22.1 - Prob. 2ECh. 22.1 - Prob. 3ECh. 22.1 - Prob. 4ECh. 22.1 - Prob. 5ECh. 22.1 - Prob. 6ECh. 22.1 - Prob. 7ECh. 22.1 - Prob. 8ECh. 22.2 - Prob. 1ECh. 22.2 - Prob. 2E
Ch. 22.2 - Prob. 3ECh. 22.2 - Prob. 4ECh. 22.2 - Prob. 5ECh. 22.2 - Prob. 6ECh. 22.2 - Prob. 7ECh. 22.2 - Prob. 8ECh. 22.2 - Prob. 9ECh. 22.3 - Prob. 1ECh. 22.3 - Prob. 2ECh. 22.3 - Prob. 3ECh. 22.3 - Prob. 4ECh. 22.3 - Prob. 5ECh. 22.3 - Prob. 6ECh. 22.3 - Prob. 7ECh. 22.3 - Prob. 8ECh. 22.3 - Prob. 9ECh. 22.3 - Prob. 10ECh. 22.3 - Prob. 11ECh. 22.3 - Prob. 12ECh. 22.3 - Prob. 13ECh. 22.4 - Prob. 1ECh. 22.4 - Prob. 2ECh. 22.4 - Prob. 3ECh. 22.4 - Prob. 4ECh. 22.4 - Prob. 5ECh. 22.5 - Prob. 1ECh. 22.5 - Prob. 2ECh. 22.5 - Prob. 3ECh. 22.5 - Prob. 4ECh. 22.5 - Prob. 5ECh. 22.5 - Prob. 6ECh. 22.5 - Prob. 7ECh. 22 - Prob. 1PCh. 22 - Prob. 2PCh. 22 - Prob. 3PCh. 22 - Prob. 4P
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- What is the Edges of the minimum spanning tree of the graph shown below?arrow_forwardGiven a graph that is a tree (connected and acyclic). (I) Pick any vertex v.(II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance.(III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are truea. p is the longest path in the graphb. p is the shortest path in the graphc. p can be calculated in time linear in the number of edges/verticesarrow_forwardDraw a tree with 14 vertices Draw a directed acyclic graph with 6 vertices and 14 edges Suppose that your computer only has enough memory to store 40000 entries. Which best graph data structure(s) – you can choose more than 1 -- should you use to store a simple undirected graph with 200 vertices, 19900 edges, and the existence of edge(u,v) is frequently asked? - Adjacency Matrix - Adjacency List - Edge Listarrow_forward
- Design an algorithm for finding a maximum spanning tree—a spanning tree with the largest possible edge weight—of a weighted connected graph. (Look at Kruskal’s Algorithm for supplemental information)arrow_forwardKruskal's minimum spanning tree algorithm is executed on the following graph. Select all edges from edgeList that belong to the minimum spanning tree. edgeList result List AD BC BE CF EF DG EG EH GH D 7 9 G 1 B 5 3 A E H 4 2 6 8 Farrow_forwardShow that the smallest edge must be present in a graph component's minimal spanning tree.arrow_forward
- 1 A spanning tree for an undirected graph is a sub-graph which includes all vertices but has no cycles. There can be several spanning trees for a graph. A weighted undirected graph can have several spanning trees One of the spanning trees has smallest sum of all the weights associated with the edges. This tree is called minimum spanning tree (MST). Find the MST in the following graph using Kruskal?s Algorithm. V={a, b, c, d, e, f, g, h, i, j} and E={(a, b, 12), (a, b, 3), (a, j, 13), (b, c, 12), (b, d, 2), (b, h, 4), (b, i, 25), (c, d, 7), (c, j, 5), (e, f, 11), (e, j, 9), (f, g, 15), (f, h, 14), (g, h, 6), (h, d, 20), (h, i, 1) }. Note: Numeric value is the weight of the corresponding edge.arrow_forwardStatement: Let G be a graph having distinct edge weights. Then, there exists a unique shortest-path tree in G. Question: Is the statement NEVER TRUE?arrow_forwardGiven the graph below, what is the edges of its minimum spanning tree?arrow_forward
- All the vertices in a directed, connected graph are linked by the edges of the minimal weight subgraph. Several spanning trees are possible in a graph. The total weight of a spanning tree is the sum of the weights of its individual edges. The number of edges in an MST is (the number of vertices minus 1).arrow_forwardQ: Dijkstra's algorithm is an algorithm for finding the shortest paths between nodes in a weighted graph. Given a graph and a source vertex in the graph, find shortest paths from source vertex (E) to all vertices in the graph below.arrow_forwardCreate a weighted connected graph with the following characteristics: 11 vertices are alphabetically labeled starting with A. Vertex A is not adjacent to vertex G. At least two vertices have a degree greater than 2. All weights are greater than 0. No weights are the same.arrow_forward
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