Kiichi Watanabe | Searcharxiv

arXiv2020

Palindromic Trees for a Sliding Window and Its Applications

The palindromic tree (a.k.a. eertree) for a string $S$ of length $n$ is a tree-like data structure that represents the set of all distinct palindromic substrings of $S$, using $O(n)$ space [Rubinchik and Shur, 2018]. It is known that, when $S$ is over an alphabet of size $σ$ and is given in an online manner, then the palindromic tree of $S$ can be constructed in $O(n\logσ)$ time with $O(n)$ space. In this paper, we consider the sliding window version of the problem: For a sliding window of length at most $d$, we present two versions of an algorithm which maintains the palindromic tree of size $O(d)$ for every sliding window $S[i..j]$ over $S$, where $1 \leq j-i+1 \leq d$. The first version works in $O(n\logσ')$ time with $O(d)$ space where $σ' \leq d$ is the maximum number of distinct characters in the windows, and the second one works in $O(n + dσ)$ time with $(d+2)σ+ O(d)$ space. We also show how our algorithms can be applied to efficient computation of minimal unique palindromic substrings (MUPS) and minimal absent palindromic words (MAPW) for a sliding window.

cs.DS↗

arXiv2020

Fast Algorithms for the Shortest Unique Palindromic Substring Problem on Run-Length Encoded Strings

For a string $S$, a palindromic substring $S[i..j]$ is said to be a \emph{shortest unique palindromic substring} ($\mathit{SUPS}$) for an interval $[s, t]$ in $S$, if $S[i..j]$ occurs exactly once in $S$, the interval $[i, j]$ contains $[s, t]$, and every palindromic substring containing $[s, t]$ which is shorter than $S[i..j]$ occurs at least twice in $S$. In this paper, we study the problem of answering $\mathit{SUPS}$ queries on run-length encoded strings. We show how to preprocess a given run-length encoded string $\mathit{RLE}_{S}$ of size $m$ in $O(m)$ space and $O(m \log σ_{\mathit{RLE}_{S}} + m \sqrt{\log m / \log\log m})$ time so that all $\mathit{SUPSs}$ for any subsequent query interval can be answered in $O(\sqrt{\log m / \log\log m} + α)$ time, where $α$ is the number of outputs, and $σ_{\mathit{RLE}_{S}}$ is the number of distinct runs of $\mathit{RLE}_{S}$. Additionaly, we consider a variant of the SUPS problem where a query interval is also given in a run-length encoded form. For this variant of the problem, we present two alternative algorithms with faster queries. The first one answers queries in $O(\sqrt{\log\log m /\log\log\log m} + α)$ time and can be built in $O(m \log σ_{\mathit{RLE}_{S}} + m \sqrt{\log m / \log\log m})$ time, and the second one answers queries in $O(\log \log m + α)$ time and can be built in $O(m \log σ_{\mathit{RLE}_{S}})$ time. Both of these data structures require $O(m)$ space.

cs.DS↗