Low-Congestion shortcuts without embedding

Abstract

Distributed optimization algorithms are frequently faced with solving sub-problems on disjoint connected parts of a network. Unfortunately, the diameter of these parts can be significantly larger than the diameter of the underlying network, leading to slow running times. This phenomenon can be seen as the broad underlying reason for the pervasive \(\tilde{\Omega }(\sqrt{n} + D)\) lower bounds that apply to most optimization problems in the CONGEST model. On the positive side, [Ghaffari and Hauepler; SODA’16] introduced low-congestion shortcuts as an elegant solution to circumvent this problem in certain topologies of interest. Particularly, they showed that there exist good shortcuts for any planar network and more generally any bounded genus network. This directly leads to fast \(O(D \log ^{O(1)} n)\) distributed algorithms for MST and Min-Cut approximation, given that one can efficiently construct these shortcuts in a distributed manner. Unfortunately, the shortcut construction of [Ghaffari and Hauepler; SODA’16] relies heavily on having access to a genus embedding of the network. Computing such an embedding distributedly, however, is a hard problem—even for planar networks. No distributed embedding algorithm for bounded genus graphs is in sight. In this work, we side-step this problem by defining tree-restricted shortcuts: a more structured and restricted form of shortcuts. We give a novel construction algorithm which efficiently finds such shortcuts that are, up to a logarithmic factor, as good as the best restricted shortcuts that exist for a given network. This new construction algorithm directly leads to an \(O(D \log ^{O(1)} n)\)-round algorithm for solving optimization problems like MST for any topology for which good restricted shortcuts exist—without the need to compute any embedding. This greatly simplifies the existing planar algorithms and includes the first efficient algorithm for bounded genus graphs.

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Fig. 1

Notes

  1. 1.

    The algorithm can be easily modified to run in \(O(\sqrt{n}\log ^* n + D)\) rounds of communication by growing components to size \(\sqrt{n / \log ^* n}\) in the first phase of the algorithm.

  2. 2.

    Throughout this paper, \(\widetilde{O}(\cdot )\), \(\widetilde{\Theta }(\cdot )\) and \(\widetilde{\Omega }(\cdot )\) hide polylogarithmic factors in n, the number of nodes in the network.

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Correspondence to Goran Zuzic.

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This work was supported in part by KAKENHI No. 15H00852 and 16H02878 as well as NSF grants CCF-1527110, CCF-1618280, CCF-1814603, CCF-1910588, NSF CAREER award CCF-1750808, a Sloan Research Fellowship and the 2018 DFINITY fellowship.

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Haeupler, B., Izumi, T. & Zuzic, G. Low-Congestion shortcuts without embedding. Distrib. Comput. 34, 79–90 (2021). https://doi.org/10.1007/s00446-020-00383-2

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