21.1-2 Professor Sabatier conjectures the following converse of Theorem 21.1. Let G = (V,E) be a connected, undirected graph with a real-valued weight function w defined on E. Let A be a subset of E that is included in some minimum spanning tree for G, let (S, V-S) be any cut of G that respects A, and let (u, v) be a safe edge for A crossing (S, V-S). Then, (u, v) is a light edge for the cut. Show that the professor's conjecture is incorrect by giving a counterexample. - Theorem 21.1 provides insight into how the GENERIC-MST method works on a connected graph G = (V,E). As the method proceeds, the set 4 is always acyclic, since it is a subset of a minimum spanning tree and a tree may not contain a cycle. At any point in the execution, the graph G (V, 4) is a forest, and each of the connected components of G is a tree. (Some of the trees may contain just one vertex, as is the case, for example, when the method begins: A is empty and the forest contains trees, one for each vertex.) Moreover, any safe edge (u, v) for 4 connects distinct components of G, since 4 U {(u, v)) must be acyclic. Figure 21.3 The proof of Theorem 21.1. Orange vertices belong to S, and tan vertices belong to V-S. Only edges in the minimum spanning tree Tare shown, along with edge (u, v), which does not lie in 7. The edges in 4 are blue, and (v) is a light edge crossing the cut (S, -5). The edge (x,y) is an edge on the unique simple path p from u to v in T. To form a minimum spanning tree that contains (u, v), remove the edge (x,y) from T and add the edge (1) The while loop in lines 2-4 of GENERIC-MST executes -1 times because it finds one of the 1-1 edges of a minimum spanning tree in each iteration. Initially, when A-0, there are trees in G, and each iteration reduces that number by 1. When the forest contains only a single tree, the method terminates. The two algorithms in Section 21.2 use the following corollary to Theorem 21.1. |
21.1-2 Professor Sabatier conjectures the following converse of Theorem 21.1. Let G = (V,E) be a connected, undirected graph with a real-valued weight function w defined on E. Let A be a subset of E that is included in some minimum spanning tree for G, let (S, V-S) be any cut of G that respects A, and let (u, v) be a safe edge for A crossing (S, V-S). Then, (u, v) is a light edge for the cut. Show that the professor's conjecture is incorrect by giving a counterexample. - Theorem 21.1 provides insight into how the GENERIC-MST method works on a connected graph G = (V,E). As the method proceeds, the set 4 is always acyclic, since it is a subset of a minimum spanning tree and a tree may not contain a cycle. At any point in the execution, the graph G (V, 4) is a forest, and each of the connected components of G is a tree. (Some of the trees may contain just one vertex, as is the case, for example, when the method begins: A is empty and the forest contains trees, one for each vertex.) Moreover, any safe edge (u, v) for 4 connects distinct components of G, since 4 U {(u, v)) must be acyclic. Figure 21.3 The proof of Theorem 21.1. Orange vertices belong to S, and tan vertices belong to V-S. Only edges in the minimum spanning tree Tare shown, along with edge (u, v), which does not lie in 7. The edges in 4 are blue, and (v) is a light edge crossing the cut (S, -5). The edge (x,y) is an edge on the unique simple path p from u to v in T. To form a minimum spanning tree that contains (u, v), remove the edge (x,y) from T and add the edge (1) The while loop in lines 2-4 of GENERIC-MST executes -1 times because it finds one of the 1-1 edges of a minimum spanning tree in each iteration. Initially, when A-0, there are trees in G, and each iteration reduces that number by 1. When the forest contains only a single tree, the method terminates. The two algorithms in Section 21.2 use the following corollary to Theorem 21.1. |
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
Related questions
Question
Please solve and show all work.
21.1-2
Professor Sabatier conjectures the following converse of Theorem 21.1. Let G = (V, E) be a connected, undirected graph with a real-valued weight function w defined on E. Let A be a subset of E that is included in some minimum spanning tree for G, let (S, V – S) be any cut of G that respects A, and let (u, v) be a safe edge for A crossing (S, V – S). Then, (u, v) is a light edge for the cut. Show that the professor’s conjecture is incorrect by giving a counterexample.
Transcribed Image Text:21.1-2
Professor Sabatier conjectures the following converse of Theorem 21.1. Let G = (V,E) be a
connected, undirected graph with a real-valued weight function w defined on E. Let A be a
subset of E that is included in some minimum spanning tree for G, let (S, V-S) be any cut of G
that respects A, and let (u, v) be a safe edge for A crossing (S, V-S). Then, (u, v) is a light edge
for the cut. Show that the professor's conjecture is incorrect by giving a counterexample.
-
Theorem 21.1 provides insight into how the GENERIC-MST method works on a connected graph G = (V,E). As the method proceeds, the set 4 is always acyclic, since it
is a subset of a minimum spanning tree and a tree may not contain a cycle. At any point in the execution, the graph G (V, 4) is a forest, and each of the connected
components of G is a tree. (Some of the trees may contain just one vertex, as is the case, for example, when the method begins: A is empty and the forest contains trees,
one for each vertex.) Moreover, any safe edge (u, v) for 4 connects distinct components of G, since 4 U {(u, v)) must be acyclic.
Figure 21.3 The proof of Theorem 21.1. Orange vertices belong to S, and tan vertices belong to V-S. Only edges in the minimum spanning tree Tare shown, along with edge (u, v), which does not lie in 7. The edges in 4 are
blue, and (v) is a light edge crossing the cut (S, -5). The edge (x,y) is an edge on the unique simple path p from u to v in T. To form a minimum spanning tree that contains (u, v), remove the edge (x,y) from T and add the
edge (1)
The while loop in lines 2-4 of GENERIC-MST executes -1 times because it finds one of the 1-1 edges of a minimum spanning tree in each iteration. Initially, when
A-0, there are trees in G, and each iteration reduces that number by 1. When the forest contains only a single tree, the method terminates.
The two algorithms in Section 21.2 use the following corollary to Theorem 21.1.
|
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
C How to Program (8th Edition)
Computer Science
ISBN:
9780133976892
Author:
Paul J. Deitel, Harvey Deitel
Publisher:
PEARSON
Database Systems: Design, Implementation, & Manag…
Computer Science
ISBN:
9781337627900
Author:
Carlos Coronel, Steven Morris
Publisher:
Cengage Learning
Programmable Logic Controllers
Computer Science
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education