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Space-Efficient Online Computation of String Net Occurrences

Authors Takuya Mieno , Shunsuke Inenaga

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • string net occurrences
  • suffix trees
  • CDAWGs
  • maximal repeats
  • minimal unique substrings (MUSs)

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Abstract

A substring u of a string T is said to be a repeat if u occurs at least twice in T. An occurrence [i..j] of a repeat u in T is said to be a net occurrence if each of the substrings aub = T[i-1..j+1], au = T[i-1..j], and ub = T[i..j+1] occurs exactly once in T. The occurrence [i-1..j+1] of aub is said to be an extended net occurrence of u. Let T be an input string of length n over an alphabet of size σ, and let ENO(T) denote the set of extended net occurrences of repeats in T. Guo et al. [SPIRE 2024] presented an online algorithm which can report ENO(T[1..i]) in T[1..i] in O(nσ²) time, for each prefix T[1..i] of T. Very recently, Inenaga [arXiv 2024] gave a faster online algorithm that can report ENO(T[1..i]) in optimal O(#ENO(T[1..i])) time for each prefix T[1..i] of T, where #S denotes the cardinality of a set S. Both of the aforementioned data structures can be maintained in O(n log σ) time and occupy O(n) space, where the O(n)-space requirement comes from the suffix tree data structure. In particular, Inenaga’s recent algorithm is based on Weiner’s right-to-left online suffix tree construction. In this paper, we show that one can modify Ukkonen’s left-to-right online suffix tree construction algorithm in O(n) space, so that ENO(T[1..i]) can be reported in optimal O(#ENO(T[1..i])) time for each prefix T[1..i] of T. This is an improvement over Guo et al.’s method that is also based on Ukkonen’s algorithm. Further, this leads us to the two following space-efficient alternatives:  
- A sliding-window algorithm of O(d) working space that can report ENO(T[i-d+1..i]) in optimal O(#ENO(T[i-d+1..i])) time for each sliding window T[i-d+1..i] of size d in T. 
- A CDAWG-based online algorithm of O(𝖾) working space that can report ENO(T[1..i]) in optimal O(#ENO(T[1..i])) time for each prefix T[1..i] of T, where 𝖾 < 2n is the number of edges in the CDAWG for T.  All of our proposed data structures can be maintained in O(n log σ) time for the input online string T. We also discuss that the extended net occurrences of repeats in T can be fully characterized in terms of the minimal unique substrings (MUSs) in T.

Cite As Get BibTex

Takuya Mieno and Shunsuke Inenaga. Space-Efficient Online Computation of String Net Occurrences. In 36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 331, pp. 23:1-23:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CPM.2025.23

Author Details

Takuya Mieno
  • Department of Computer and Network Engineering, University of Electro-Communications, Chofu, Japan
Shunsuke Inenaga
  • Department of Informatics, Kyushu University, Fukuoka, Japan

Funding

  • Mieno, Takuya: JSPS KAKENHI Grant Number JP24K20734.
  • Inenaga, Shunsuke: JSPS KAKENHI Grant Numbers JP23K24808, JP23K18466, JP20H05964.

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