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 15.2, Problem 4E
Program Plan Intro
To find the vertices and edges for matrix-chain multiplication with an input chain of length n .
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Write a Java program to find the Adjacency Matrix Representation using Directed Graph.
а.
Insert new nodes and directed edge between two nodes
b. Display the representation
Draw a simple, connected, weighted graph with 8 vertices and 16 edges, each with unique edge weights. Identify one vertex as a “start” vertex and illustrate a running of Dijkstra’s algorithm on this graph.
Problem R-14.23 in the photo
Adjacency list representations for weighted graphs have drawbacks.
Chapter 15 Solutions
Introduction to Algorithms
Ch. 15.1 - Prob. 1ECh. 15.1 - Prob. 2ECh. 15.1 - Prob. 3ECh. 15.1 - Prob. 4ECh. 15.1 - Prob. 5ECh. 15.2 - Prob. 1ECh. 15.2 - Prob. 2ECh. 15.2 - Prob. 3ECh. 15.2 - Prob. 4ECh. 15.2 - Prob. 5E
Ch. 15.2 - Prob. 6ECh. 15.3 - Prob. 1ECh. 15.3 - Prob. 2ECh. 15.3 - Prob. 3ECh. 15.3 - Prob. 4ECh. 15.3 - Prob. 5ECh. 15.3 - Prob. 6ECh. 15.4 - Prob. 1ECh. 15.4 - Prob. 2ECh. 15.4 - Prob. 3ECh. 15.4 - Prob. 4ECh. 15.4 - Prob. 5ECh. 15.4 - Prob. 6ECh. 15.5 - Prob. 1ECh. 15.5 - Prob. 2ECh. 15.5 - Prob. 3ECh. 15.5 - Prob. 4ECh. 15 - Prob. 1PCh. 15 - Prob. 2PCh. 15 - Prob. 3PCh. 15 - Prob. 4PCh. 15 - Prob. 5PCh. 15 - Prob. 6PCh. 15 - Prob. 7PCh. 15 - Prob. 8PCh. 15 - Prob. 9PCh. 15 - Prob. 10PCh. 15 - Prob. 11PCh. 15 - Prob. 12P
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- Discuss the difference between the adjacency list representation and the adjacency matrix representation of graphs. Describe the advantages and disadvantages of each method.arrow_forwardB Write a topological order of vertices of above graph and illustrate your method.arrow_forwardThink about the problems with representing weighted graphs using adjacency lists.arrow_forward
- How many entries would the adjacency matrix representation for the following graph contain? D B E A F C Garrow_forwardDoes Dijkastra's Algorithm work on every kind of graph ?arrow_forwardLet n be an even positive integer. (a) Show that if the degree of every vertex in a simple graph G on n vertices is at least (i.e., greater than or equal to) n/2, then G must be connected. (b) Give an example of a disconnected simple graph on n vertices in which every vertex has degree (n/2) – 1.arrow_forward
- Describe a method that computes the strong connected component that contains a given vertex (v) in linear time. Describe a straightforward quadratic algorithm for calculating a digraph's strong components based on that method.arrow_forwarddraw the graph that represents said matrix and find out, using Python, the number of 3-paths that connect the vertices v1 and v3 of the same.arrow_forwardIn Computer Science a Graph is represented using an adjacency matrix. Ismatrix is a square matrix whose dimension is the total number of vertices.The following example shows the graphical representation of a graph with 5 vertices, its matrixof adjacency, degree of entry and exit of each vertex, that is, the total number ofarrows that enter or leave each vertex (verify in the image) and the loops of the graph, that issay the vertices that connect with themselvesTo program it, use Object Oriented Programming concepts (Classes, objects, attributes, methods), it can be in Java or in Python.-Declare a constant V with value 5-Declare a variable called Graph that is a VxV matrix of integers-Define a MENU procedure with the following textGRAPHS1. Create Graph2.Show Graph3. Adjacency between pairs4.Input degree5.Output degree6.Loops0.exit-Validate MENU so that it receives only valid options (from 0 to 6), otherwise send an error message and repeat the reading-Make the MENU call in the main…arrow_forward
- Sketch an undirected graph of the above designed adjacency matrix.arrow_forwardYou are given a bipartite graph G=(U,V,E), and an integer K. U and V are the two bipartitions of the graph such that |U| = |V| = N , and E is the edge set. The vertices of U are {1,2,...,N } and that of V are {N+1,N+2,...,2N }. You need to find out whether the total number of different perfect matchings in G is strictly greater than K or not. Recall that a perfect matching is a subset of E such that every vertex of the graph belongs to exactly one edge in the subset. Two perfect matchings are considered to be different even if one edge is different. Write a program in C++ programming language that prints a single line containing “Perfect” if the number of perfect matchings is greater than K, and “Not perfect” in other cases.Sample Input:3 5 21 42 62 53 53 5Output:Not Perfectarrow_forwardLet G (V, E) be a digraph in which every vertex is a source, or a sink, or both a sink and a source. (a) Prove that G has neither self-loops nor anti-parallel edges.arrow_forward
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