Fully Persistent Dynamic LCE via AVL Trees and AVL Grammars

2026-07-01

AI Summary

This work proposes FeAVL, a fully persistent dynamic string data structure that efficiently supports split, concatenate, single-character update, equality testing, and longest common extension (LCE) queries. By integrating AVL trees with path-copying persistence for the first time, FeAVL achieves worst-case O(log n) time per update operation and O(log n + log²ℓ) time for LCE queries, creating only O(log n) new nodes per update. The design is further enhanced with an AVL-grammar-based compression scheme, yielding a total space complexity of O(g₀ + I + U log nₘₐₓ) after U updates, where g₀ denotes the initial grammar size, I the number of insertions, and nₘₐₓ the maximum string length. This approach avoids pitfalls of amortized analysis while maintaining strong theoretical guarantees and practical compression performance.

Abstract

We study fully persistent dynamic strings with equality and longest common extension (LCE) queries. Straightforward full persistence is problematic for the splay-based FeST structure, since the same unbalanced past version can be reused indefinitely and the usual amortized analysis no longer applies. We give a fully persistent dynamic LCE structure, called FeAVL, based on path copying over AVL trees. For an operation involving string(s) of total length $n$, it supports split, concatenate, and single-character updates in worst-case $O(\log n)$ time, equality in worst-case $O(\log n)$ time w.h.p., and LCE in worst-case $O(\log n+\log^2\ell)$ time w.h.p., where $\ell$ is the answer; each update creates only $O(\log n)$ new permanent nodes. We also give a grammar-compressed instantiation via AVL grammars: starting from an initial grammar of size $g_0$, after $U$ updates, the total number of permanent grammar nodes is $O(g_0+I+U\log n_{\max})$, where $I$ is the number of inserted fresh characters and $n_{\max}$ is the maximum string length appearing during the update sequence.

Problem

• fully persistent
• dynamic strings
• longest common extension
• LCE queries
• equality queries

Innovation

• fully persistent
• AVL trees
• longest common extension
• dynamic strings
• grammar compression