Tight Bound for the Number of Distinct Palindromes in a Tree

TL;DR

This paper establishes a tight bound of (n^{1.5}) for the maximum number of distinct palindromic substrings in a labeled tree and provides an algorithm to enumerate them efficiently.

Contribution

It proves the exact asymptotic bound for palindromic substrings in trees and introduces an (n^{1.5} \u2212} log n) algorithm for reporting all such palindromes.

Findings

Maximum palindromic substrings in trees are (n^{1.5})

Established tight bound matching previous lower bound

Provided an (n^{1.5} \u2212} log n) algorithm for enumeration

Abstract

For an undirected tree with $n$ edges labelled by single letters, we consider its substrings, which are labels of the simple paths between pairs of nodes. We prove that there are $O(n^{1.5})$ different palindromic substrings. This solves an open problem of Brlek, Lafreni\`ere, and Proven\c{c}al (DLT 2015), who gave a matching lower-bound construction. Hence, we settle the tight bound of $\Theta(n^{1.5})$ for the maximum palindromic complexity of trees. For standard strings, i.e., for paths, the palindromic complexity is $n+1$. We also propose $O(n^{1.5} \log{n})$-time algorithm for reporting all distinct palindromes in an undirected tree with $n$ edges.