TY - GEN

T1 - Tight bounds on the maximum number of shortest unique substrings

AU - Mieno, Takuya

AU - Inenaga, Shunsuke

AU - Bannai, Hideo

AU - Takeda, Masayuki

N1 - Publisher Copyright:
© Takuya Mieno, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2017/7/1

Y1 - 2017/7/1

N2 - 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.

AB - 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.

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U2 - 10.4230/LIPIcs.CPM.2017.24

DO - 10.4230/LIPIcs.CPM.2017.24

M3 - Conference contribution

AN - SCOPUS:85027271339

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017

A2 - Radoszewski, Jakub

A2 - Karkkainen, Juha

A2 - Radoszewski, Jakub

A2 - Rytter, Wojciech

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017

Y2 - 4 July 2017 through 6 July 2017

ER -