research-article

Parallel Metric Tree Embedding based on an Algebraic View on Moore-Bellman-Ford

Authors:

Stephan Friedrichs,

Christoph LenzenAuthors Info & Claims

SPAA '16: Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures

Pages 455 - 466

https://doi.org/10.1145/2935764.2935777

Published: 11 July 2016 Publication History

Abstract

A metric tree embedding of expected stretch α maps a weighted n-node graph G = (V, E, w) to a weighted tree T = (V_T, E_T, w_T) with V ⊆ V_T, and dist(v, w, G) ≤ dist(v, w, T) and E[dist(v, w, T)] ≤ α dist(v, w, G) for all v, w ∈ V. Such embeddings are highly useful for designing fast approximation algorithms, as many hard problems are easy to solve on tree instances. However, to date the best parallel polylog n depth algorithm that achieves an asymptotically optimal expected stretch of α ∈ Ω(log n) uses Ω(n²) work and requires a metric as input.

In this paper, we show how to achieve the same guarantees using Ω(m^1+ε) work, where $m$ is the number of edges of G and ε >0 is an arbitrarily small constant. Moreover, one may reduce the work further to Ω(m + n^1+ε), at the expense of increasing the expected stretch α to Ω(ε^-1 log n) using the spanner construction of Baswana and Sen as preprocessing step.

Our main tool in deriving these parallel algorithms is an algebraic characterization of a generalization of the classic Moore-Bellman-Ford algorithm. We consider this framework, which subsumes a large variety of previous "Moore-Bellman-Ford-flavored" algorithms, to be of independent interest.

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Cited By

Elkin MNeiman O(2022)Linear-Size hopsets with small hopbound, and constant-hopbound hopsets in RNCDistributed Computing10.1007/s00446-022-00431-z35:5(419-437)Online publication date: 29-Jun-2022
https://doi.org/10.1007/s00446-022-00431-z
Chechik SMukhtar D(2021)Single-source shortest paths in the CONGEST model with improved boundsDistributed Computing10.1007/s00446-021-00412-835:4(357-374)Online publication date: 27-Nov-2021
https://doi.org/10.1007/s00446-021-00412-8
Chechik SMukhtar DEmek YCachin C(2020)Single-Source Shortest Paths in the CONGEST Model with Improved BoundProceedings of the 39th Symposium on Principles of Distributed Computing10.1145/3382734.3405729(464-473)Online publication date: 31-Jul-2020
https://dl.acm.org/doi/10.1145/3382734.3405729
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Index Terms

Parallel Metric Tree Embedding based on an Algebraic View on Moore-Bellman-Ford
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Approximation algorithms
      2. Graph algorithms
2. Theory of computation
  1. Computational complexity and cryptography
    1. Circuit complexity
  2. Design and analysis of algorithms

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cover image ACM Conferences

SPAA '16: Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures

July 2016

492 pages

ISBN:9781450342100

DOI:10.1145/2935764

General Chair:
Christian Scheideler
University of Paderborn
,
Program Chair:
Seth Gilbert
National University of Singapore

Copyright © 2016 ACM.

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Published: 11 July 2016

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SPAA '16: 28th ACM Symposium on Parallelism in Algorithms and Architectures

July 11 - 13, 2016

California, Pacific Grove, USA

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Overall Acceptance Rate 447 of 1,461 submissions, 31%

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Cited By

Elkin MNeiman O(2022)Linear-Size hopsets with small hopbound, and constant-hopbound hopsets in RNCDistributed Computing10.1007/s00446-022-00431-z35:5(419-437)Online publication date: 29-Jun-2022
https://doi.org/10.1007/s00446-022-00431-z
Chechik SMukhtar D(2021)Single-source shortest paths in the CONGEST model with improved boundsDistributed Computing10.1007/s00446-021-00412-835:4(357-374)Online publication date: 27-Nov-2021
https://doi.org/10.1007/s00446-021-00412-8
Chechik SMukhtar DEmek YCachin C(2020)Single-Source Shortest Paths in the CONGEST Model with Improved BoundProceedings of the 39th Symposium on Principles of Distributed Computing10.1145/3382734.3405729(464-473)Online publication date: 31-Jul-2020
https://dl.acm.org/doi/10.1145/3382734.3405729
Elkin MFiltser ANeiman OEmek YCachin C(2020)Distributed Construction of Light NetworksProceedings of the 39th Symposium on Principles of Distributed Computing10.1145/3382734.3405701(483-492)Online publication date: 31-Jul-2020
https://dl.acm.org/doi/10.1145/3382734.3405701
Elkin MNeiman OScheideler CBerenbrink P(2019)Linear-Size Hopsets with Small Hopbound, and Constant-Hopbound Hopsets in RNCThe 31st ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3323165.3323177(333-341)Online publication date: 17-Jun-2019
https://dl.acm.org/doi/10.1145/3323165.3323177
Friedrichs SLenzen C(2018)Parallel Metric Tree Embedding Based on an Algebraic View on Moore-Bellman-FordJournal of the ACM10.1145/323159165:6(1-55)Online publication date: 22-Nov-2018
https://dl.acm.org/doi/10.1145/3231591

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