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Decentralized Low-Stretch Trees via Low Diameter Graph Decompositions

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Title: Decentralized Low-Stretch Trees via Low Diameter Graph Decompositions
Authors: Becker, Ruben, Emek, Yuval, Ghaffari, Mohsen, Lenzen, Christoph
Contributors: Becker, Ruben, Emek, Yuval, Ghaffari, Mohsen, Lenzen, Christoph
Publication Year: 2024
Collection: Università Ca’ Foscari Venezia: ARCA (Archivio Istituzionale della Ricerca)
Subject Terms: distributed graph algorithm, parallel graph algorithm, (semi-)streaming graph algorithm, metric tree embedding, low-stretch tree, graph decompositions, Settore INF/01 - Informatica
Description: We study the problem of approximating the distances in an undirected weighted graph G by the distances in trees based on the notion of stretch. Focusing on decentralized models of computation such as the CONGEST, PRAM, and semi -streaming models, our main results are as follows: (1) We develop a simple randomized algorithm that constructs a spanning tree such that the expected stretch of every edge is O(log(3) n), where n is the number of nodes in G. If G is unweighted, then this algorithm can be implemented to run in O(hop(G)) rounds in the CONGEST model, where hop(G) is the hop -diameter of G; thus our algorithm is asymptotically optimal in this case. In the weighted case, the run-time of the algorithm matches the currently best known bound for exact single source shortest path (SSSP) computations, which despite recent progress is still separated from the lower bound of Omega(root n + hop(G)) by polynomial factors. A naive attempt to replace exact SSSP computations with approximate ones in order to improve the complexity in the weighted case encounters a fundamental challenge, as the underlying decomposition technique fails to work under distance approximation. (2) We overcome this obstacle by developing a technique termed blurry ball growing. This technique, in combination with a clever algorithmic idea of Miller, Peng, and Xu (SPAA 2013), allows us to obtain low diameter graph decompositions with small edge cutting probabilities based solely on approximate SSSP computations. (3) Using these decompositions, we in turn obtain metric tree embedding algorithms in the vein of the celebrated work of Bartal (FOCS 1996), whose computational complexity is optimal up to polylogarithmic factors not only in the CONGEST model but also in the PRAM and semi -streaming models. Our embeddings have the additional useful property that the tree can be mapped back to the original graph such that each edge is "used" only logarithmically many times. This property is of interest for capacitated problems and for simulating CONGEST ...
Document Type: article in journal/newspaper
Language: English
Relation: info:eu-repo/semantics/altIdentifier/wos/WOS:001185608700001; volume:53; issue:2; firstpage:247; lastpage:286; numberofpages:40; journal:SIAM JOURNAL ON COMPUTING; https://hdl.handle.net/10278/5068732
DOI: 10.1137/22m1489034
Availability: https://hdl.handle.net/10278/5068732
https://doi.org/10.1137/22m1489034
Accession Number: edsbas.2CCB29EA
Database: BASE
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  – Url: https://hdl.handle.net/10278/5068732#
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  Data: Decentralized Low-Stretch Trees via Low Diameter Graph Decompositions
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  Data: <searchLink fieldCode="AR" term="%22Becker%2C+Ruben%22">Becker, Ruben</searchLink><br /><searchLink fieldCode="AR" term="%22Emek%2C+Yuval%22">Emek, Yuval</searchLink><br /><searchLink fieldCode="AR" term="%22Ghaffari%2C+Mohsen%22">Ghaffari, Mohsen</searchLink><br /><searchLink fieldCode="AR" term="%22Lenzen%2C+Christoph%22">Lenzen, Christoph</searchLink>
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  Data: Becker, Ruben<br />Emek, Yuval<br />Ghaffari, Mohsen<br />Lenzen, Christoph
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  Data: 2024
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  Data: Università Ca’ Foscari Venezia: ARCA (Archivio Istituzionale della Ricerca)
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  Data: <searchLink fieldCode="DE" term="%22distributed+graph+algorithm%22">distributed graph algorithm</searchLink><br /><searchLink fieldCode="DE" term="%22parallel+graph+algorithm%22">parallel graph algorithm</searchLink><br /><searchLink fieldCode="DE" term="%22%28semi-%29streaming+graph+algorithm%22">(semi-)streaming graph algorithm</searchLink><br /><searchLink fieldCode="DE" term="%22metric+tree+embedding%22">metric tree embedding</searchLink><br /><searchLink fieldCode="DE" term="%22low-stretch+tree%22">low-stretch tree</searchLink><br /><searchLink fieldCode="DE" term="%22graph+decompositions%22">graph decompositions</searchLink><br /><searchLink fieldCode="DE" term="%22Settore+INF%2F01+-+Informatica%22">Settore INF/01 - Informatica</searchLink>
– Name: Abstract
  Label: Description
  Group: Ab
  Data: We study the problem of approximating the distances in an undirected weighted graph G by the distances in trees based on the notion of stretch. Focusing on decentralized models of computation such as the CONGEST, PRAM, and semi -streaming models, our main results are as follows: (1) We develop a simple randomized algorithm that constructs a spanning tree such that the expected stretch of every edge is O(log(3) n), where n is the number of nodes in G. If G is unweighted, then this algorithm can be implemented to run in O(hop(G)) rounds in the CONGEST model, where hop(G) is the hop -diameter of G; thus our algorithm is asymptotically optimal in this case. In the weighted case, the run-time of the algorithm matches the currently best known bound for exact single source shortest path (SSSP) computations, which despite recent progress is still separated from the lower bound of Omega(root n + hop(G)) by polynomial factors. A naive attempt to replace exact SSSP computations with approximate ones in order to improve the complexity in the weighted case encounters a fundamental challenge, as the underlying decomposition technique fails to work under distance approximation. (2) We overcome this obstacle by developing a technique termed blurry ball growing. This technique, in combination with a clever algorithmic idea of Miller, Peng, and Xu (SPAA 2013), allows us to obtain low diameter graph decompositions with small edge cutting probabilities based solely on approximate SSSP computations. (3) Using these decompositions, we in turn obtain metric tree embedding algorithms in the vein of the celebrated work of Bartal (FOCS 1996), whose computational complexity is optimal up to polylogarithmic factors not only in the CONGEST model but also in the PRAM and semi -streaming models. Our embeddings have the additional useful property that the tree can be mapped back to the original graph such that each edge is "used" only logarithmically many times. This property is of interest for capacitated problems and for simulating CONGEST ...
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  Data: info:eu-repo/semantics/altIdentifier/wos/WOS:001185608700001; volume:53; issue:2; firstpage:247; lastpage:286; numberofpages:40; journal:SIAM JOURNAL ON COMPUTING; https://hdl.handle.net/10278/5068732
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  Data: 10.1137/22m1489034
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        Value: 10.1137/22m1489034
    Languages:
      – Text: English
    Subjects:
      – SubjectFull: distributed graph algorithm
        Type: general
      – SubjectFull: parallel graph algorithm
        Type: general
      – SubjectFull: (semi-)streaming graph algorithm
        Type: general
      – SubjectFull: metric tree embedding
        Type: general
      – SubjectFull: low-stretch tree
        Type: general
      – SubjectFull: graph decompositions
        Type: general
      – SubjectFull: Settore INF/01 - Informatica
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      – TitleFull: Decentralized Low-Stretch Trees via Low Diameter Graph Decompositions
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