On the complexity of the sub-permutation problem

Shlomo Ahal, Yuri Rabinovich

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We study various computational aspects of the problem of determining whether a given order contains a given sub-order. Formally, given a permutation π on k elements, and a permutation σ on n > k elements, the goal is to determine whether there exists a strictly increasing function f from [1.k] to [1.n] which is order preserving, i.e., f satisfies σ(f (i)) > σ(f(j)) whenever π(i) > π(j). We call this decision problem the Sub-Permutation Problem. The study falls into two parts. In the first part we develop and analyze an algorithm (or, rather, an algorithmic paradigm) for this problem. We show that the complexity of this algorithm is at most O(n1+C(π)), where C(π) is a naturally defined function of the permutation π. In the second part we study C(π). In particular, we show that C(π) ≤0.35k + o(k), implying that the complexity of the Sub-Permutation problem is O(ck + n0.35k+o(k)). On the other hand, we prove that for most π’s, C(π) = Ω(k), establishing a lower bound for our algorithm. In addition, we develop a fast polylogarithmic approximation algorithm for computing C(π), and bound the value of this parameter for some interesting families of permutations.

Original languageEnglish
Title of host publicationAlgorithm Theory - SWAT 2000 - 7th Scandinavian Workshop on Algorithm Theory, 2000, Proceedings
EditorsMagnús M. Halldórsson
PublisherSpringer Verlag
Number of pages14
ISBN (Print)3540676902, 9783540676904
StatePublished - 2000
Event7th Scandinavian Workshop on Algorithm Theory, SWAT 2000 - Bergen, Norway
Duration: 5 Jul 20007 Jul 2000

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference7th Scandinavian Workshop on Algorithm Theory, SWAT 2000

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2000.

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science (all)


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