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Maximal Closed Substrings

Golnaz Badkobeh, Alessandro De Luca, Gabriele Fici, Simon Puglisi

TL;DR

The paper studies maximal closed substrings (MCS), a broad generalization of runs and related to maximal gapped repeats, and aims to enumerate all MCSs in a string. It introduces a bottom-up algorithm on the binary suffix tree $T_B(S)$, merging child leaf lists with AVL trees and tracking right-border candidates to output MCSs for each border $\\hat{v}$ in $O(n\\log n)$ time. The main results establish an $O(n\\log n)$ upper bound on the number of MCSs and prove a matching order of growth lower bound $\\Omega(n)$, with additional observations about starting-position collisions and open problems for tighter bounds and a constructive proof. This work connects combinatorics on words with efficient data-structure-based enumeration, yielding a near-optimal method to list a large class of repetitive substrings and linking MCSs to runs and gapped repeats.

Abstract

A string is closed if it has length 1 or has a nonempty border without internal occurrences. In this paper we introduce the definition of a \emph{maximal closed substring} (MCS), which is an occurrence of a closed substring that cannot be extended to the left nor to the right into a longer closed substring. MCSs with exponent at least $2$ are commonly called \emph{runs}; those with exponent smaller than $2$, instead, are particular cases of \emph{maximal gapped repeats}. We provide an algorithm that, given a string of length $n$ locates all MCSs the string contains in $\mathcal O(n\log n)$ time.

Maximal Closed Substrings

TL;DR

The paper studies maximal closed substrings (MCS), a broad generalization of runs and related to maximal gapped repeats, and aims to enumerate all MCSs in a string. It introduces a bottom-up algorithm on the binary suffix tree , merging child leaf lists with AVL trees and tracking right-border candidates to output MCSs for each border in time. The main results establish an upper bound on the number of MCSs and prove a matching order of growth lower bound , with additional observations about starting-position collisions and open problems for tighter bounds and a constructive proof. This work connects combinatorics on words with efficient data-structure-based enumeration, yielding a near-optimal method to list a large class of repetitive substrings and linking MCSs to runs and gapped repeats.

Abstract

A string is closed if it has length 1 or has a nonempty border without internal occurrences. In this paper we introduce the definition of a \emph{maximal closed substring} (MCS), which is an occurrence of a closed substring that cannot be extended to the left nor to the right into a longer closed substring. MCSs with exponent at least are commonly called \emph{runs}; those with exponent smaller than , instead, are particular cases of \emph{maximal gapped repeats}. We provide an algorithm that, given a string of length locates all MCSs the string contains in time.
Paper Structure (5 sections, 3 theorems, 2 equations, 1 figure)

This paper contains 5 sections, 3 theorems, 2 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An Algorithm for Locating All MCSs
  4. Bounds on the Number of MCSs
  5. Concluding Remarks

Key Result

Lemma 2

Two AVL trees of size at most $n$ and $m$ can be merged in time $\mathcal{O}(\log \binom{n+m}{n})$.

Figures (1)

  • Figure 1: Example of a step for turning $T(S)$ into $T_B(S)$.

Theorems & Definitions (5)

  • Definition 1: Suffix tree
  • Lemma 2: Brown and Tarjan BT79
  • Lemma 3: Brodal et al. BLPS99, Lemma 3.3
  • Theorem 4
  • Example 1