Longest Common Extension of a Dynamic String in Parallel Constant Time
A longest common extension (LCE) query on a string computes the length of the longest common suffix or prefix at two given positions. A dynamic LCE algorithm maintains a data structure that allows efficient LCE queries on a string that can change via character insertions and deletions. A dynamic parallel constant-time algorithm is presented that can maintain LCE queries on a common CRCW PRAM with $\mathcal{O}(n^{\epsilon})$ work, for any $\epsilon > 0$. The algorithm maintains a string synchronizing sets hierarchy, which it uses to answer substring equality queries, which it in turn uses to answer LCE queries. To achieve constant runtime, the algorithm allows parts of its information to become outdated by up to $\log n \log^* n$ updates. It answers queries by combining this slightly outdated information with a list of the recent changes. Two applications of this dynamic LCE algorithm are shown. Firstly, a dynamic parallel constant-time algorithm can maintain membership in a Dyck language $D_k, k > 0$ with $\mathcal{O}(n^{\epsilon})$ work for any $\epsilon > 0$. Secondly, a dynamic parallel constant-time algorithm can maintain squares with $\mathcal{O}(n^{\epsilon})$ work for any $\epsilon > 0$.