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# Brief Announcement: A Centralized Local Algorithm for the Sparse Spanning Graph Problem

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LIPIcs.DISC.2017.57.pdf
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## Cite As

Christoph Lenzen and Reut Levi. Brief Announcement: A Centralized Local Algorithm for the Sparse Spanning Graph Problem. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 57:1-57:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.DISC.2017.57

## Abstract

Constructing a sparse spanning subgraph is a fundamental primitive in graph theory. In this paper, we study this problem in the Centralized Local model, where the goal is to decide whether an edge is part of the spanning subgraph by examining only a small part of the input; yet, answers must be globally consistent and independent of prior queries. Unfortunately, maximally sparse spanning subgraphs, i.e., spanning trees, cannot be constructed efficiently in this model. Therefore, we settle for a spanning subgraph containing at most (1+epsilon)n edges (where n is the number of vertices and epsilon is a given approximation/sparsity parameter). We achieve a query complexity of O(poly(Delta/epsilon)n^(2/3)) (up to polylogarithmic factors in n) where Delta is the maximum degree of the input graph. Our algorithm is the first to do so on arbitrary bounded degree graphs. Moreover, we achieve the additional property that our algorithm outputs a spanner, i.e., distances are approximately preserved. With high probability, for each deleted edge there is a path of O(log n (Delta+log n)/epsilon) hops in the output that connects its endpoints.
##### Keywords
• local
• spanning graph
• sparse

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## References

1. Michael Elkin and Ofer Neiman. Efficient algorithms for constructing very sparse spanners and emulators. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 652-669, 2017.
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