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 4E
Program Plan Intro
To show that using a single bit to store each vertex color suffices by arguing that the DFS procedure would produce the same result even the line 8 of DFS-VISIT is removed.
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Enhance the Python-coded Luby MIS algorithm by randomly choosing a vertex, determining its neighbours, and then identifying the vertex with the lowest assigned random value to be used in the MIS.
Consider the following graph and Dijkstra's algorithm to find the shortest paths from the vertex A.
(See image attached)
The distances / weights in the incident edges at the vertex F are given in the table:
edge: (B, F) (C, F) (E, F)distance: 8 11 4.5
At the end of the algorithm, what is the value of the distance for the vertex F?
In order to test connectivity, Python code uses the function Connectivity Undirected, which then calls the BFS algorithm, which returns a list of visited vertices and sorts them as shown below. The connectivity is then calculated by comparing this list to every node in the network in the main function. Be aware that in order to match the visited vertices with the sorted array of vertices, we must first sort the visited vertices. However, ordering can be avoided if sets are used in place of lists.
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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- Starting at vertex 1, list the order in which the vertices of the graph below are visited using DFS.If there are multiple adjacent vertices, assume that DFS visits them in ascending order of their id’s. For example, at vertex 3, DFSwould visit vertex 4 before visiting vertex 5.arrow_forwardBy randomly selecting a vertex, identifying its neighbours, and then determining the vertex with the lowest assigned random value to be included in the MIS, Luby's Python-coded MIS algorithm can be improved.arrow_forwardDuring the execution of DFS, give the conditions under which there is an edge from a vertex u with color c1 has an edge to a vertex v with color c2. Consider the following color combinations, (c1, c2)= (w,w), (w,g), (g, b) and (b,g).arrow_forward
- Provide a class to assess DAGs for mathematical expressions. Use a vertex-indexed array to store values for every vertex. Suppose that values for leaves have already been determined. Explain a family of mathematical expressions where the expression tree's size is exponentially bigger than the matching DAG's size (so the running time of your programme for the DAG is proportional to the logarithm of the running time for the tree).arrow_forwardFor each graph representation, select the appropriate worst-case time complexity for printing the vertex label of all the neighbors of a given vertex. Assume that vertex label retrieval from a typical integer vertex representation is O(1). Adjacency Matrix: ________ Edge List: ________ Adjacency List: _________ Choices: O(V+E), O(E^2), O(V^2), O(E)arrow_forwardProve Proposition (continued). DFS allows us to provide clients with a path from a given source to anymarked vertex in time proportional its length.arrow_forward
- Assume for an undirected graph with 6 vertices and 6 edges that a vertex index requires 3 bytes, a pointer requires 5 bytes, and that edge weights require 3 bytes. Since the graph is undirected, each undirected edge is represented by two directed edges. Calculate the byte requirements for an adjacency matrix.arrow_forwardLet’s consider the directed graph with unweighted links presented in Figure 4.12. Thisgraph is similar to the previous graph except by one missing link between A and E.Write networking code to describes how to create the new links dataset and then search for the existing cycles within the directed graph. ans in 20 min.or asaparrow_forwardProve that every connected graph contains a vertex (together with all nearby edges) whose removal will not cause the graph to become disconnected. Then, provide a DFS technique to locate this vertex.arrow_forward
- create a pseudo-code that uses the proc network to calculate the closeness centrality based on an undirected graph with unweighted links.arrow_forwardConsider the graph shown in Figure 2 consisting of vertices 1, 2, .., 9. Construct a matrix with 9 rows (one for each vertex) and 10 columns labeled k= 0, 1, 2, ..., 9. Fill the table with the distance results of the Djikstra algorithm when running it on iterations k = 0, 1, 2,...,9. The source vertex is 1.arrow_forwardpseudo code invokes proc network to compute the closeness centrality based on a directed graph using unweighted links.arrow_forward
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