# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

__Mittagsseminar Talk Information__ | |

**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/k}log k) edges in unweighted graphs and with O(n^{1+1/k}k) 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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