Suppose you have n items with non-negative weights w1, ... , wn. For each subset S of the items, define S's weight as w(S) = [_{jES) wj, the sum of the weights of the items in S. In the problem SPLIT-EVEN, the goal is to split the items into sets S and S = {1,..., n} \ S such that max{w(S), w(S)} is minimized. Greedy Split(w1, ..., wn) A=Ø, B=Ø for i=1 to n if w(A) ≤ w(B) A = AU {i} else B = BU {i} return (A, B) Let OPT denote the minimum, over all subsets S of the items, of max{w(S), W(S)} Prove that OPT 2 maxj wj and OPT ≥ 1/2 [_{j=1}^{n} wj
Suppose you have n items with non-negative weights w1, ... , wn. For each subset S of the items, define S's weight as w(S) = [_{jES) wj, the sum of the weights of the items in S. In the problem SPLIT-EVEN, the goal is to split the items into sets S and S = {1,..., n} \ S such that max{w(S), w(S)} is minimized. Greedy Split(w1, ..., wn) A=Ø, B=Ø for i=1 to n if w(A) ≤ w(B) A = AU {i} else B = BU {i} return (A, B) Let OPT denote the minimum, over all subsets S of the items, of max{w(S), W(S)} Prove that OPT 2 maxj wj and OPT ≥ 1/2 [_{j=1}^{n} wj
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 you have n items with non-negative weights w1, ... , wn. For each subset S of the items, define S's weight as
W(S) = _{jES} wj, the sum of the weights of the items in S. In the problem SPLIT-EVEN, the goal is to split the items into
sets S and S = {1,..., n} \ S such that max{w(S), w(S)} is minimized.
Greedy Split(w1, ..., wn)
A = Ø, B = Ø
for i=1 to n
if w(A) ≤ w(B)
A = AU {i}
else B = BU {i}
return (A, B)
Let OPT denote the minimum, over all subsets S of the items, of max{w(S), w(S)}
Prove that OPT ≥ maxj wj and OPT≥ 1/2 Σ_{j=1}^{n} wj
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