Binary Jumbled Pattern Matching on Trees and Tree-Like Structures

Travis Gagie, Danny Hermelin, Gad M. Landau, Oren Weimann

Research output: Contribution to journalArticlepeer-review


Binary jumbled pattern matching asks to preprocess a binary string $$S$$S in order to answer queries $$(i,j)$$(i,j) which ask for a substring of $$S$$S that is of length $$i$$i and has exactly $$j$$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 $$O(n^2 / \log ^2 n)$$O(n2/log2n)-time solution for trees, matching the currently best bound for (the simpler problem of) strings. We also give an $${O}({g^{2 / 3} n^{4 / 3}/(\log n)^{4/3}})$$O(g2/3n4/3/(logn)4/3)-time solution for strings that are compressed by a context-free grammar of size $$g$$g 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 $$w$$w of the graph, even for a constant number of different vertex-labels, thus improving the previous best $$n^{O(w)}$$nO(w) algorithm.

Original languageEnglish
Pages (from-to)571-588
Number of pages18
Issue number3
StatePublished - 1 Nov 2015

Bibliographical note

Funding Information:
Gad M. Landau: Supported in part by the National Science Foundation (NSF) Grant 0904246, the Israel Science Foundation (ISF) Grant 347/09, and the United States-Israel Binational Science Foundation (BSF) Grant 2008217. Oren Weimann: Supported in part by the Israel Science Foundation Grant 794/13.

Publisher Copyright:
© 2014, Springer Science+Business Media New York.


  • Grammar compression
  • Graph motifs
  • Pattern matching
  • Permutation pattern matching
  • Tree pattern matching

ASJC Scopus subject areas

  • Computer Science (all)
  • Computer Science Applications
  • Applied Mathematics


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