## Abstract

Given a constant number d of n×n matrices with total m non-infinity entries, we show how to construct in (essentially optimal) O˜(mn+n^{2}) time a data structure that can compute in (essentially optimal) O˜(n^{2}) time the distance product of these matrices after incrementing the value (possibly to infinity) of a constant number k of entries. Our result is obtained by designing an oracle for single source replacement paths that is suited for short distances: Given a graph G=(V,E), a source vertex s, and a shortest paths tree T of depth d rooted at s, the oracle can be constructed in O˜(md^{k}) time for any constant k. Then, given an arbitrary set S of at most k=O(1) edges and an arbitrary vertex t, the oracle in O˜(1) time either reports the length of the shortest s-to-t path in (V,E∖S) or otherwise reports that any such shortest path must use more than d edges.

Original language | English |
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Article number | 105977 |

Journal | Information Processing Letters |

Volume | 161 |

DOIs | |

State | Published - Sep 2020 |

### Bibliographical note

Funding Information:Supported partially by ISF grant 592/17.Supported partially by ISF grant 1082/16.

Publisher Copyright:

© 2020 Elsevier B.V.

## Keywords

- Data structures
- Distance product
- Fault tolerance
- Graph algorithms
- Shortest paths

## ASJC Scopus subject areas

- Theoretical Computer Science
- Signal Processing
- Information Systems
- Computer Science Applications