## Abstract

Spanners are fundamental graph structures that sparsify graphs at the cost of small stretch. In particular, in recent years, many sequential algorithms constructing additive all-pairs spanners were designed, providing very sparse small-stretch subgraphs. Remarkably, it was then shown that the known (+6)-spanner constructions are essentially the sparsest possible, that is, larger additive stretch cannot guarantee a sparser spanner, which brought the stretch-sparsity trade-off to its limit. Distributed constructions of spanners are also abundant. However, for additive spanners, while there were algorithms constructing (+2) and (+4)-all-pairs spanners, the sparsest case of (+6)-spanners remained elusive. We remedy this by designing a new sequential algorithm for constructing a (+6)-spanner with the essentially-optimal sparsity of O˜(n^{4/3}) edges. We then show a distributed implementation of our algorithm, answering an open problem in [12]. A main ingredient in our distributed algorithm is an efficient construction of multiple weighted BFS trees. A weighted BFS tree is a BFS tree in a weighted graph, that consists of the lightest among all shortest paths from the root to each node. We present a distributed algorithm in the CONGEST model, that constructs multiple weighted BFS trees in |S|+D−1 rounds, where S is the set of sources and D is the diameter of the network graph.

Original language | English |
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Pages (from-to) | 33-44 |

Number of pages | 12 |

Journal | Theoretical Computer Science |

Volume | 840 |

DOIs | |

State | Published - 6 Nov 2020 |

## Keywords

- Additive spanners
- Congest model
- Distributed graph algorithms
- Weighted BFS trees

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)