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Maintaining the Size of LZ77 on Semi-Dynamic Strings

Authors Hideo Bannai , Panagiotis Charalampopoulos , Jakub Radoszewski

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Author Details

Hideo Bannai
  • M&D Data Science Center, Tokyo Medical and Dental University (TMDU), Japan
Panagiotis Charalampopoulos
  • Birkbeck, University of London, UK
Jakub Radoszewski
  • Institute of Informatics, University of Warsaw, Poland

Funding

  • Bannai, Hideo: JSPS KAKENHI Grant Number JP20H04141.
  • Radoszewski, Jakub: Supported by the Polish National Science Center, grant no. 2022/46/E/ST6/00463.

Cite AsGet BibTex

Hideo Bannai, Panagiotis Charalampopoulos, and Jakub Radoszewski. Maintaining the Size of LZ77 on Semi-Dynamic Strings. In 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 296, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CPM.2024.3

Abstract

We consider the problem of maintaining the size of the LZ77 factorization of a string S of length at most n under the following operations: (a) appending a given letter to S and (b) deleting the first letter of S. Our main result is an algorithm for this problem with amortized update time Õ(√n). As a corollary, we obtain an Õ(n√n)-time algorithm for computing the most LZ77-compressible rotation of a length-n string - a naive approach for this problem would compute the LZ77 factorization of each possible rotation and would thus take quadratic time in the worst case. We also show an Ω(√n) lower bound for the additive sensitivity of LZ77 with respect to the rotation operation. Our algorithm employs dynamic trees to maintain the longest-previous-factor array information and depends on periodicity-based arguments that bound the number of the required updates and enable their efficient computation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
  • Theory of computation → Data compression
Keywords
  • Lempel-Ziv
  • compression
  • LZ77
  • semi-dynamic algorithm
  • cyclic rotation

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