Binary Jumbled Pattern Matching via All-Pairs Shortest Paths

In binary jumbled pattern matching we wish to preprocess a binary string S in order to answer queries (i, j) which ask for a substring of S that is of size i and has exactly j 1-bits. The problem naturally generalizes to node-labeled trees and graphs by replacing “substring” with “connected subgraph”. In this paper, we give an n/2 n/ log log n) 1/2 time solution for both strings and trees. This odd-looking time complexity improves the state of the art O(n/ log n) solutions by more than any polylogarithmic factor. It originates from the recent seminal algorithm of Williams for min-plus matrix multiplication. We obtain the result by giving a black box reduction from trees to strings. This is then combined with a reduction from strings to min-plus matrix multiplications.