Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, October 27, 2022, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Saeed Ilchi

Deterministic Distributed Sparse and Ultra-Sparse Spanners and Connectivity Certificates

We present efficient distributed algorithms for a number of fundamental problems in the area of graph sparsification: We provide the first deterministic distributed algorithm that computes an ultra-sparse spanner in polylog(n) rounds in weighted graphs. Concretely, our algorithm outputs a spanning subgraph with only n+o(n) edges in which the pairwise distances are stretched by a factor of at most O(log n⋅2^O(log∗n)).

We provide a polylog(n)-round deterministic distributed algorithm that computes a spanner with stretch (2k−1) and O(nk+n^1+1/klog k) edges in unweighted graphs and with O(n^1+1/kk) edges in weighted graphs.

We present the first polylog(n)-round randomized distributed algorithm that computes a sparse connectivity certificate. For an n-node graph G, a certificate for connectivity k is a spanning subgraph H that is k-edge-connected if and only if G is k-edge-connected, and this subgraph H is called sparse if it has O(nk) edges. Our algorithm achieves a sparsity of (1+o(1))nk edges, which is within a 2(1+o(1)) factor of the best possible.

This talk is based on a joint work with Marcel Bezdrighin, Michael Elkin, Mohsen Ghaffari, Christoph Grunau, Bernhard Haeupler, and Václav Rozhoň:
https://arxiv.org/abs/2204.14086

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