Almost linear-time algorithm for the dense subset-sum problem

Zvi Galil, Oded Margalit

Research output: Contribution to journalArticlepeer-review

25 Scopus citations

Abstract

This paper describes a new approach for solving the subset-sum problem. It is useful for solving other NP-hard problems. The limits and potential of this approach are discussed. The approach yields an algorithm for solving the dense version of the subset-sum problem. It runs in time O(ℓ log ℓ), where ℓ is the bound on the size of the elements. But for dense enough inputs and target numbers near the middle sum, it runs in time O(m), where m is the number of elements. Consequently, it improves the previously best algorithms by at least one order of magnitude and sometimes by two. The algorithm yields a characterization of the set of subset sums as a collection of arithmetic progressions with the same difference. This characterization is derived by elementary number-theoretic and algorithmic techniques. Such a characterization was first obtained by using analytic number theory and yielded inferior algorithms.

Original languageEnglish
Pages (from-to)1157-1189
Number of pages33
JournalSIAM Journal on Computing
Volume20
Issue number6
DOIs
StatePublished - 1 Jan 1991
Externally publishedYes

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