## Abstract

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 language | English |
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Pages (from-to) | 571-588 |

Number of pages | 18 |

Journal | Algorithmica |

Volume | 73 |

Issue number | 3 |

DOIs | |

State | Published - 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.

## Keywords

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

## ASJC Scopus subject areas

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