### Abstract

Given a vertex-colored graph H and a multiset M of colors as input, the graph motif problem asks us to decide whether H has a connected induced subgraph whose multiset of colors agrees with M. The graph motif problem is NP-complete but known to admit randomized algorithms based on constrained multilinear sieving over GF(2^b) that run in time O(2^kk^2m {M({2^b})}) and with a false-negative probability of at most k/2^{b-1} for a connected m-edge input and a motif of size k. On modern CPU microarchitectures such algorithms have practical edge-linear scalability to inputs with billions of edges for small motif sizes, as demonstrated by Björklund, Kaski, Kowalik, and Lauri [ALENEX'15]. This scalability to large graphs prompts the dual question whether it is possible to scale to large motif sizes. We present a vertex-localized variant of the constrained multilinear sieve that enables us to obtain, in time O(2^kk^2m{M({2^b})}) and for every vertex simultaneously, whether the vertex participates in at least one match with the motif, with a per-vertex probability of at most k/2^{b-1} for a false negative. Furthermore, the algorithm is easily vector-parallelizable for up to 2^k threads, and parallelizable for up to 2^kn threads, where n is the number of vertices in H. Here {M({2^b})} is the time complexity to multiply in GF(2^b). We demonstrate with an open-source implementation that our variant of constrained multilinear sieving can be engineered for vector-parallel microarchitectures to yield hardware utilization that is bound by the available memory bandwidth. Our main engineering contributions are (a) a version of the recurrence for tightly labeled arborescences that can be executed as a sequence of memory-and-arithmetic coalescent parallel workloads on multiple GPUs, and (b) a bit-sliced low-level implementation for arithmetic in characteristic 2 to support (a).

Original language | English |
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Title of host publication | 17th Symposium on Experimental Algorithms, SEA 2018 |

Editors | Gianlorenzo D'Angelo |

Pages | 1-19 |

ISBN (Electronic) | 978-3-95977-070-5 |

DOIs | |

Publication status | Published - 2018 |

MoE publication type | A4 Article in a conference publication |

Event | International Symposium on Experimental Algorithms - Gran Sasso Science Institute, L'Aquila, Italy Duration: 27 Jun 2018 → 29 Jun 2018 Conference number: 17 |

### Publication series

Name | Leibniz International Proceedings in Informatics (LIPIcs) |
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Publisher | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |

Volume | 103 |

ISSN (Electronic) | 1868-8969 |

### Conference

Conference | International Symposium on Experimental Algorithms |
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Abbreviated title | SEA |

Country | Italy |

City | L'Aquila |

Period | 27/06/2018 → 29/06/2018 |

### Keywords

- algorithm engineering
- constrained multilinear sieving
- graph motif problem
- multi-GPU
- vector-parallel
- vertex-localization

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## Equipment

## Cite this

Kaski, P., Lauri, J., & Muniyappa, S. (2018). Engineering Motif Search for Large Motifs. In G. D'Angelo (Ed.),

*17th Symposium on Experimental Algorithms, SEA 2018*(pp. 1-19). [28] (Leibniz International Proceedings in Informatics (LIPIcs); Vol. 103). https://doi.org/10.4230/LIPIcs.SEA.2018.28