Algorithms for Jumbled Indexing, Jumbled Border and Jumbled Square on run-length encoded strings

In this paper we investigate jumbled (Abelian) versions of three classical strings problems. In all these problems we assume the input string S[1.n] is given in its run-length format S^′[1.r]. The Jumbled Indexing problem is the problem of indexing a string S^′[1.r] over |Σ| for histogram queries, i.e. given a pattern P, we want to find all substrings of S that are permutations of P. We provide an algorithm that constructs an index of size O(r²|Σ|) in time O(r²(log⁡r+|Σ|log⁡|Σ|)), which allows answering histogram queries in O(|Σ|³log⁡r)-time. The Jumbled Border problem is the problem of finding for every location j in S, the longest proper prefix of S[1.j] that is also a permutation of a proper suffix of S[1.j], if such exists. We provide an algorithm that solves this problem in O(|Σ|(r²+n)) time, and O(|Σ|n) space. A Jumbled Square is a string of the form xx¯, where x¯ is a permutation of x. The Jumbled Square problem is the problem of finding for every location j in S, the longest jumbled square that ends in j, if such exists. We provide an algorithm that solves this problem in O(|Σ|(r²+n)) time, and O(|Σ|n) space.