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Is tme use of Essentially, by processing the vertices in a random order, you can potentially avoid (with a high probability) any particularly bad orderings. So, consider the following randomized algorithm for constructing independent sets:

Is tme use of Essentially, by processing the vertices in a random order, you can potentially avoid (with a high probability) any particularly bad orderings. So, consider the following randomized algorithm for constructing independent sets:

Database System Concepts
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
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L. I need help with this discrete math problem (question 7). Thanks in advance!

One way of trying to avoid dependence on ordering is the use of randomized algorithms. Essentially, by processing the vertices in a random order, you can potentially avoid (with a high probability) any particularly bad orderings. So, consider the following randomized algorithm for constructing independent sets: First, starting with an empty set I, add each vertex of G to I independently with probability p Next, for any edges with both vertices in I, delete one of the two vertices from I (at random) Note – in the second step, deleting one vertex from I may remove multiple edges from I! Return the final set I Question 7: Argue that the expected size of I after Step 2 is greater than or equal to the following: p|V] – p?|E|
Transcribed Image Text:One way of trying to avoid dependence on ordering is the use of randomized algorithms. Essentially, by processing the vertices in a random order, you can potentially avoid (with a high probability) any particularly bad orderings. So, consider the following randomized algorithm for constructing independent sets: First, starting with an empty set I, add each vertex of G to I independently with probability p Next, for any edges with both vertices in I, delete one of the two vertices from I (at random) Note – in the second step, deleting one vertex from I may remove multiple edges from I! Return the final set I Question 7: Argue that the expected size of I after Step 2 is greater than or equal to the following: p|V] – p?|E|
Let's consider an undirected graph G that for any vertices, i, j E I and there is no edge between i and į in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent set is not necessarily the largest independent set in G. Let a(G) denote the size of the largest maximal independent set in G. = (V,E). An independent subset is a subset I cV such
Transcribed Image Text:Let's consider an undirected graph G that for any vertices, i, j E I and there is no edge between i and į in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent set is not necessarily the largest independent set in G. Let a(G) denote the size of the largest maximal independent set in G. = (V,E). An independent subset is a subset I cV such
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