Maintaining dynamic sequences under equality tests in polylogarithmic time

Abstract

We present a randomized and a deterministic data structure for maintaining a dynamic family of sequences under equality tests of pairs of sequences and creations of new sequences by joining or splitting existing sequences. Both data structures support equality tests in O (1) time. The randomized version supports new sequence creations in O(log 2 n) expected time where n is the length of the sequence created. The deterministic solution supports sequence creations in O (log n (log m log* m +log n)) time for the mth operation.

Dynamic Algorithms for the Dyck Languages

“…We use a technique for maintaining dynamic sequences under equality tests by Mehlhorn, Sundar, and Uhrig [10], which also gives (Las Vegasstyle) randomised algorithms that run in slightly better expected time: O(log 3 n).…”

“…Mehlhorn, Sundar, and Uhrig [10] present a data structure for dynamically maintaining a family of strings under equality tests. We use a slightly modified set of updates that is better suited to our problem.…”

“…We use a slightly modified set of updates that is better suited to our problem. More precisely, we want to maintain an initially empty family S of strings from a finite alphabet Σ under the following operations: The techniques from [10] can easily be modified to cope with the above updates. The time bounds are summarised in the following lemma:…”

Locally Consistent Parsing and Applications to Approximate String Comparisons

“…Two surprising applications of the DCT appeared (almost simultaneously) in 1994, which generalized DCT to strings with character repetitions [13,15,16]. LCP is this generalization of DCT to string partitioning.…”

“…LCP is this generalization of DCT to string partitioning. In [13], an efficient data structure for maintaining a dynamic collection of strings that allow equality tests, concatenation and split operations is described. In [15,16], a novel algorithm for building the suffix tree of a given string in polylogarithmic parallel time while performing a total of O(n) operations is given.…”