NDLI: A New Lower Bound for the Set-Partitioning Problem

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	John, John Welliaveetil
Copyright Year	1988
Abstract	Let S be a set of n elements that allows all possible total ordering and on which a total order exists that is initially not known. The problem of determining the t largest elements of S using binary comparisons between elements of S is the set-partitioning problem. In this paper we establish a new lower bound on the number of comparisons required for the set-partitioning problem. For sufficiently large n, our lower bound improves on all previous lower bounds for this problem for almost all values of t. Our technique is based on estimating the number of leaves of a decision tree, say T, for the problem. Let P denote the set of total orderings allowed on S. Let $P'_i ,P'_2 , \cdots ,P'_m $ be a partition of P such that the set of leaves of T corresponding to $P'_i $ is disjoint from those corresponding to $P'_j $ whenever $i \ne j$. We determine subsets $P_1 ,P_2 , \cdots ,P_r $ of P and an integer k, such that for each i, $1 \leqq i \leqq m$, $P'_i \cap P_j \ne \emptyset $ holds for at most k distinct $P_j $s. For each subset $P_j $, $1 \leqq j \leqq r$, we establish a lower bound, say $t_j $, on the number of leaves of a decision tree for the set-partitioning problem when S is restricted to a total order from $P_j $. Then T has at least $\sum\nolimits_{j = 1}^r {{{t_j } / k}} $ leaves.
Starting Page	640
Ending Page	647
Page Count	8
File Format	PDF
ISSN	00975397
DOI	10.1137/0217040
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4
Volume Number	17
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-13
Access Restriction	Subscribed
Subject Keyword	decision tree analysis of algorithm partitioning problem lower bounds
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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