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Asymptotically Optimal Representation of Palindromic Structure

Michael Itzhaki

TL;DR

This work addresses the problem of encoding the Manacher array in $O(n)$ bits while supporting constant-time access. It introduces a three-component compressed representation that separates periodic and non-periodic centers, leveraging smooth arrays and the Simple Dense Coding (SDC) framework to achieve space efficiency, and a boundary locator built from layered bitvectors to enable constant-time radius calculations. The main result is a lossless encoding of the Manacher array with $O(n)$ bits and $O(1)$-time access, realized via the sparse Manacher array $ ilde{ extsf A}$, the enriched center-period array $ ilde{ extsf L}$, and the boundary locator $ ilde{ extbf I}$. This advances compressed text indexing by exploiting palindromic structure, offering a practical pathway to efficient palindrome-aware pattern processing and indexing in strings.

Abstract

We introduce an asymptotically optimal representation of the Manacher array of a string that supports constant-time access. The approach relies on the combinatorial properties of palindromes, yielding a compact yet efficient structure. This work fits within the broader study of compressed text indexing and highlights structural aspects of palindromic substrings that may inspire further algorithmic applications.

Asymptotically Optimal Representation of Palindromic Structure

TL;DR

This work addresses the problem of encoding the Manacher array in bits while supporting constant-time access. It introduces a three-component compressed representation that separates periodic and non-periodic centers, leveraging smooth arrays and the Simple Dense Coding (SDC) framework to achieve space efficiency, and a boundary locator built from layered bitvectors to enable constant-time radius calculations. The main result is a lossless encoding of the Manacher array with bits and -time access, realized via the sparse Manacher array , the enriched center-period array , and the boundary locator . This advances compressed text indexing by exploiting palindromic structure, offering a practical pathway to efficient palindrome-aware pattern processing and indexing in strings.

Abstract

We introduce an asymptotically optimal representation of the Manacher array of a string that supports constant-time access. The approach relies on the combinatorial properties of palindromes, yielding a compact yet efficient structure. This work fits within the broader study of compressed text indexing and highlights structural aspects of palindromic substrings that may inspire further algorithmic applications.

Paper Structure

This paper contains 19 sections, 14 theorems, 34 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Preliminaries
  4. Computation model.
  5. Basic definitions.
  6. Periods and repetitions.
  7. Palindromes.
  8. Additional notation.
  9. Bitvector operations.
  10. SDC and Smooth arrays.
  11. Framework and Further Definitions
  12. Properties of Periodic Palindromes
  13. Compact Manacher Array
  14. High-level idea
  15. Description of the structures
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $S$ be a string of length $n$, and let $\mathsf{A}$ denote its Manacher array. The array $\mathsf{A}$ admits a lossless encoding of size $\mathcal{O}( n )$ bits that supports element access in constant time.

Figures (4)

  • Figure 1: Visual demonstration of repeat centers and internal centers. At the top, the palindromic run with period nine is described, below the three runs with period two, and lastly the run with period one.
  • Figure 2: A visualization of the center-period array $\mathsf L$ and the Manacher array $\mathsf A$. For each periodic center, the root $r_c$ is highlighted in red. The maximal palindrome is highlighted in blue. For clarity, each array is split into two subarrays -- the values in odd and in even indices.
  • Figure 3: Demonstration of the different cases in \ref{['lem:per-len']}.
  • Figure 4: The relation between internal centers of runs in $S$ to the layered bitvector $\mathbf{I}$. Above are the repeat centers of all palindromic runs with period $1$, i.e., repetitions $\textup{aaaaa}$. Below are the repeat centers of $\mathcal{R}_{15}$ and $\mathcal{R}_{32}$. For clarity, each center is classified with respect to a specific palindromic run.

Theorems & Definitions (37)

  • Theorem 1.1
  • Definition 2.1: Manacher array
  • Lemma 2.2: Folklore, Inenaga2014
  • Remark 2.3
  • Definition 2.4: Smooth array
  • Corollary 2.6
  • Lemma 3.1: Canonical decomposition of periodic palindromes Kosolobov2015
  • Definition 3.2: Palindromic Run
  • Definition 3.3: Palindromic run descriptor ($\mathrm{PRD}$)
  • Example 3.4
  • ...and 27 more