Fully Persistent Dynamic LCE via AVL Trees and AVL Grammars

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.