Faster Longest Common Extension Queries in Strings over General Alphabets

Longest common extension queries (often called longest common prefix queries) constitute a fundamental building block in multiple string algorithms, for example computing runs and approximate pattern matching. We show that a sequence of q LCE queries for a string of size n over a general ordered alphabet can be realized in O(q log log n + n log n) time making only O(q + n) symbol comparisons. Consequently, all runs in a string over a general ordered alphabet can be computed in O(n log log n) time making O(n) symbol comparisons. Our results improve upon a solution by Kosolobov (Information Processing Letters, 2016), who gave an algorithm with O(n log n) running time and conjectured that O(n) time is possible. We make a significant progress towards resolving this conjecture. Our techniques extend to the case of general unordered alphabets, when the time increases to O(q log n+ n log n). The main tools are difference covers and a variant of the disjoint-sets data structure by La Poutré (SODA 1990).