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The suffix array SAS[1 . . . n] of an n-length string S is a lexicographically sorted array of the suffixes of S. The suffix array is one of the most well known and widely used data structures in string algorithms. We present a data structure for maintaining a representation of the suffix array of a dynamic string which undergoes symbol substitutions, deletions, and insertions.For every string manipulation, our data structure can be updated in Õ(n

Lately, there is a growing interest in dynamic string matching problems. Specifically -the dynamic Longest Common Factor problem has been reserched and some interesting results has been reached.In this paper we examine another classic string problem in a dynamic setting -finding the longest palindrome substring of a given string. We show that the longest palindrome can be maintained in polylogarithmic time per symbol edit.

In vertex-cut sparsification, given a graph G = (V, E) with a terminal set T ⊆ V , we wish to construct a graph G ′ = (V ′ , E ′ ) with T ⊆ V ′ , such that for every two sets of terminals A, B ⊆ T , the size of a minimum (A, B)-vertex-cut in G ′ is the same as in G. In the most basic setting, G is unweighted and undirected, and we wish to bound the size of G ′ by a function of k = |T |. Kratsch and Wahlström [JACM 2020] proved that every graph G (possibly directed), admits a vertex-cut sparsifier G ′ with O(k 3 ) vertices, which can in fact be constructed in randomized polynomial time.We study (possibly directed) graphs G that are quasi-bipartite, i.e., every edge has at least one endpoint in T , and prove that they admit a vertex-cut sparsifier with O(k 2 ) edges and vertices, which can in fact be constructed in deterministic polynomial time. In fact, this bound naturally extends to all graphs with a small separator into bounded-size sets. Finally, we prove information-theoretically a nearly-matching lower bound, i.e., that Ω(k 2 ) edges are required to sparsify quasi-bipartite undirected graphs.