Binary jumbled pattern matching on trees and tree-like structures

Binary jumbled pattern matching asks to preprocess a binary string S in order to answer queries (i,j) which ask for a substring of S that is of length i and has exactly j 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 /log² n)-time solution for trees, matching the currently best bound for (the simpler problem of) strings. We also give an O(g ^2/3 n ^4/3/(logn) ^4/3)-time solution for strings that are compressed by a grammar of size g. This solution improves the known bounds when the string is compressible under many popular compression schemes. Finally, we prove that the problem is fixed-parameter tractable with respect to the treewidth w of the graph, even for a constant number of different vertex-labels, thus improving the previous best n^O(w) algorithm.