Binary Jumbled Pattern Matching on Trees and Tree-Like Structures

Binary jumbled pattern matching asks to preprocess a binary string SS in order to answer queries (i,j)(i,j) which ask for a substring of SS that is of length ii and has exactly jj 1-bits. This problem naturally generalizes to vertex-labeled trees and graphs by replacing “substring” with “connected subgraph”. In this paper, we give an O(n2/log2n)O(n2/log2⁡n)-time solution for trees, matching the currently best bound for (the simpler problem of) strings. We also give an O(g2/3n4/3/(logn)4/3)O(g2/3n4/3/(log⁡n)4/3)-time solution for strings that are compressed by a context-free grammar of size gg in Chomsky normal form. This solution improves the known bounds when the string is compressible under many popular compression schemes. Finally, we prove that on graphs the problem is fixed-parameter tractable with respect to the treewidth ww of the graph, even for a constant number of different vertex-labels, thus improving the previous best nO(w)nO(w) algorithm.