= Suppose we consider the special case of IS, WIS where the input graph is a tree T (V, E) with an identified root r € V (there are no other limitations on structure). This means that for every v E V(T), the subtree rooted at v is well-defined - we will refer to this subtree as Tv. In this section, you are asked to develop a polynomial-time algorithm for solving WIS (and including IS, when weights are all 1). For every v EV, we define the following: • Kv,o to be the total weight of the maximum-weight Independent set of subtree T, which does not include v itself. • Kv,1 to be the total weight of the maximum-weight Independent set of subtree T₂ with v belonging to the independent set. D(i) Develop a pair of recurrences which express the value of kv,o (and similarly of kå,1) in terms of the values of the child nodes of v. Include the "base case" whsn v is a leaf, and then describe how we can exploit the recurrences to design a polynomial-time algorithm to solve WIS for a tree. D(ii) In the earlier parts of this coursework, we developed algorithms for inputting general graphs into an Adjacency list Data Structure. We won't know whether some of these graphs were actually trees or not. Can you propose an algorithm/method to check self.graph to determine whether it is a tree? Discuss likely running-time, and how we would then convert the graph to a tree-like data structure in low polynomial-time. D(iii) In Part B we discussed the Greedy method for constructing Independent sets of a general (weighted or unweighted) graph, and saw that it does not always return an optimal result for general graphs. Consider the criterion (a) method that was implemented as GreedylS. Will this method result in a maximum independent set when the input graph is an unweighted tree? Justify your answer.
= Suppose we consider the special case of IS, WIS where the input graph is a tree T (V, E) with an identified root r € V (there are no other limitations on structure). This means that for every v E V(T), the subtree rooted at v is well-defined - we will refer to this subtree as Tv. In this section, you are asked to develop a polynomial-time algorithm for solving WIS (and including IS, when weights are all 1). For every v EV, we define the following: • Kv,o to be the total weight of the maximum-weight Independent set of subtree T, which does not include v itself. • Kv,1 to be the total weight of the maximum-weight Independent set of subtree T₂ with v belonging to the independent set. D(i) Develop a pair of recurrences which express the value of kv,o (and similarly of kå,1) in terms of the values of the child nodes of v. Include the "base case" whsn v is a leaf, and then describe how we can exploit the recurrences to design a polynomial-time algorithm to solve WIS for a tree. D(ii) In the earlier parts of this coursework, we developed algorithms for inputting general graphs into an Adjacency list Data Structure. We won't know whether some of these graphs were actually trees or not. Can you propose an algorithm/method to check self.graph to determine whether it is a tree? Discuss likely running-time, and how we would then convert the graph to a tree-like data structure in low polynomial-time. D(iii) In Part B we discussed the Greedy method for constructing Independent sets of a general (weighted or unweighted) graph, and saw that it does not always return an optimal result for general graphs. Consider the criterion (a) method that was implemented as GreedylS. Will this method result in a maximum independent set when the input graph is an unweighted tree? Justify your answer.
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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Transcribed Image Text:Suppose we consider the special case of IS, WIS where the input graph is a tree T (V, E) with an
identified root r ≤ V (there are no other limitations on structure). This means that for every v € V(T),
the subtree rooted at v is well-defined - we will refer to this subtree as Tv.
=
In this section, you are asked to develop a polynomial-time algorithm for solving WIS (and including
IS, when weights are all 1).
For every v € V, we define the following:
Kv,o to be the total weight of the maximum-weight Independent set of subtree T, which does not
include v itself.
• Kv,1 to be the total weight of the maximum-weight Independent set of subtree T with v belonging
to the independent set.
D(i) Develop a pair of recurrences which express the value of к,0 (and similarly of ×,₁) in terms of
the values of the child nodes of v. Include the "base case" whsn v is a leaf, and then describe
how we can exploit the recurrences to design a polynomial-time algorithm to solve WIS for a tree.
D(ii) In the earlier parts of this coursework, we developed algorithms for inputting general graphs into
an Adjacency list Data Structure. We won't know whether some of these graphs were actually
trees or not. Can you propose an algorithm/method to check self.graph to determine whether
it is a tree? Discuss likely running-time, and how we would then convert the graph to a tree-like
data structure in low polynomial-time.
D(iii) In Part B we discussed the Greedy method for constructing Independent sets of a general (weighted
or unweighted) graph, and saw that it does not always return an optimal result for general graphs.
Consider the criterion (a) method that was implemented as Greedy|S. Will this method result in a
maximum independent set when the input graph is an unweighted tree? Justify your answer.
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