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Improved Bounds on the Sum of Exponents of Runs in a String

Author Arkadiusz Czarkowski

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LIPIcs.CPM.2026.23.pdf
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Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
  • Mathematics of computing → Combinatorics on words
Keywords
  • strings
  • runs
  • sum of exponents of runs
  • Lyndon words
  • L-roots
  • maximal repetitions
  • combinatorics on words

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Abstract

A substring of a word is a run if it is at least twice as long as its minimum period and cannot be extended to either side with the same period. The exponent of a run is the quotient of its length and its minimum period. ρ(n) is the maximum number of runs in a string of length n, while σ(n) is the maximum sum of exponents of runs in a string of length n. While quite tight bounds on ρ(n) are known (0.944575712n ≤ ρ(n) ≤ n), the best upper bound on σ(n) is 3n whereas the best lower bound on σ(n) is 2.035n. In this paper, we improve the upper bound on σ(n) to 2.3n and the lower bound on σ(n) to 2.04448n. We also provide an improved upper bound on σ(n) of 2.2n in the case of a binary alphabet. Our results are achieved using a combination of theoretical and computer-based approaches.

Cite As Get BibTex

Arkadiusz Czarkowski. Improved Bounds on the Sum of Exponents of Runs in a String. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CPM.2026.23

Author Details

Arkadiusz Czarkowski
  • Institute of Informatics, University of Warsaw, Poland

Funding

Supported by the Polish National Science Center, grant no. 2022/46/E/ST6/00463.

Supplementary Materials

References

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