Constant-time edge label and leaf pointer maintenance on sliding suffix trees

TL;DR

This paper introduces a novel method for maintaining leaf pointers and edge labels in sliding suffix trees, achieving constant-time updates and simplifying correctness proofs, which improves efficiency over previous approaches.

Contribution

It presents a new approach that maintains leaf pointers without credit-based arguments, enabling constant-time updates and simplifying the analysis of sliding suffix trees.

Findings

Edge index-pairs can be derived in constant time from leaf pointers.

Leaf pointer and edge label maintenance per update is reduced to O(1) time.

The new method simplifies correctness proofs compared to previous credit-based approaches.

Abstract

Sliding suffix trees (Fiala & Greene, 1989) for an input text $T$ over an alphabet of size $\sigma$ and a sliding window $W$ of $T$ can be maintained in $O(|T| \log \sigma)$ time and $O(|W|)$ space. The two previous approaches that achieve this can be categorized into the credit-based approach of Fiala and Greene (1989) and Larsson (1996, 1999), or the batch-based approach proposed by Senft (2005). Brodnik and Jekovec (2018) showed that the sliding suffix tree can be supplemented with leaf pointers in order to find all occurrences of an online query pattern in the current window, and that leaf pointers can be maintained by credit-based arguments as well. The main difficulty in the credit-based approach is in the maintenance of index-pairs that represent each edge. In this paper, we show that valid edge index-pairs can be derived in constant time from leaf pointers, thus reducing the maintenance of edge index-pairs to the maintenance of leaf pointers. We further propose a new simple method that maintains leaf pointers without using credit-based arguments. The lack of credit-based arguments allow a simpler proof of correctness compared to the credit-based approach, whose analyses were initially flawed (Senft 2005). In addition, our method reduces the worst-case time of leaf pointer and edge label maintenance per leaf insertion and deletion from $\Theta(|W|)$ time to $O(1)$ time.