# Towards a complete perspective on labeled tree indexing: new size bounds, efficient constructions, and beyond

## TL;DR

This paper provides a comprehensive analysis of labeled tree indexing structures, including size bounds, efficient construction algorithms, and pattern matching capabilities for forward and backward tries, advancing understanding of their space and time complexities.

## Contribution

It offers new size bounds, a compact representation of DAWGs, and extends indexing techniques to bidirectional pattern searches within linear space.

## Key findings

- DAWG size for forward trie is Ω(σn)
- Compact O(n)-space DAWG representation exists
- Supports bidirectional pattern searches in linear space

## Abstract

A labeled tree (or a trie) is a natural generalization of a string, which can also be seen as a compact representation of a set of strings. This paper considers the labeled tree indexing problem, and provides a number of new results on space bound analysis, and on algorithms for efficient construction and pattern matching queries. Kosaraju [FOCS 1989] was the first to consider the labeled tree indexing problem, and he proposed the suffix tree for a backward trie, where the strings in the trie are read in the leaf-to-root direction. In contrast to a backward trie, we call a usual trie as a forward trie. Despite a few follow-up works after Kosaraju's paper, indexing forward/backward tries is not well understood yet. In this paper, we show a full perspective on the sizes of indexing structures such as suffix trees, DAWGs, CDAWGs, suffix arrays, affix trees, affix arrays for forward and backward tries. Some of them take $O(n)$ space in the size $n$ of the input trie, while the others can occupy $O(n^2)$ space in the worst case. In particular, we show that the size of the DAWG for a forward trie with $n$ nodes is $\Omega(\sigma n)$, where $\sigma$ is the number of distinct characters in the trie. This becomes $\Omega(n^2)$ for an alphabet of size $\sigma = \Theta(n)$. Still, we show that there is a compact $O(n)$-space implicit representation of the DAWG for a forward trie, whose space requirement is independent of the alphabet size. This compact representation allows for simulating each DAWG edge traversal in $O(\log \sigma)$ time, and can be constructed in $O(n)$ time and space over any integer alphabet of size $O(n)$. In addition, this readily extends to the first indexing structure that permits bidirectional pattern searches over a trie within linear space in the input trie size.