Tight bounds on the maximum number of shortest unique substrings

Takuya Mieno, Shunsuke Inenaga, Hideo Bannai, Masayuki Takeda

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

A substring Q of a string S is called a shortest unique substring (SUS) for interval [s, t] in S, if Q occurs exactly once in S, this occurrence of Q contains interval [s, t], and every substring of S which contains interval [s, t] and is shorter than Q occurs at least twice in S. The SUS problem is, given a string S, to preprocess S so that for any subsequent query interval [s, t] all the SUSs for interval [s, t] can be answered quickly. When s = t, we call the SUSs for [s, t] as point SUSs, and when s ≤ t, we call the SUSs for [s, t] as interval SUSs. There exist optimal O(n)-time preprocessing scheme which answers queries in optimal O(k) time for both point and interval SUSs, where n is the length of S and k is the number of outputs for a given query. In this paper, we reveal structural, combinatorial properties underlying the SUS problem: Namely, we show that the number of intervals in S that correspond to point SUSs for all query positions in S is less than 1.5n, and show that this is a matching upper and lower bound. Also, we consider the maximum number of intervals in S that correspond to interval SUSs for all query intervals in S.

Original languageEnglish
Title of host publication28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Volume78
ISBN (Electronic)9783959770392
DOIs
Publication statusPublished - Jul 1 2017
Event28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017 - Warsaw, Poland
Duration: Jul 4 2017Jul 6 2017

Other

Other28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017
CountryPoland
CityWarsaw
Period7/4/177/6/17

All Science Journal Classification (ASJC) codes

  • Software

Cite this

Mieno, T., Inenaga, S., Bannai, H., & Takeda, M. (2017). Tight bounds on the maximum number of shortest unique substrings. In 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017 (Vol. 78). [24] Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.CPM.2017.24