An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

Yi Jun Chang, Tsvi Kopelowitz, Seth Pettie

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

85 Scopus citations

Abstract

Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. We prove that these exponential gaps are necessary and establish numerous connections between the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message: 1) Building on the recent randomized lower bounds of Brandt et al. [1], we prove that the randomized complexity of δ-coloring a tree with maximum degree δ is O(log δ log n + log∗n), for any δ > = 55, whereas its deterministic complexity is Ω(log δ n) for any δ > = 3. This also establishes a large separation between the deterministic complexity of δ-coloring and (δ+1)-coloring trees. 2) We prove that any deterministic algorithm for a natural class of problems that runs in O(1) + o(log δ n) rounds can be transformed to run in O(log∗n - log∗δ + 1) rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires Ω(log δ n) time deterministically. This gives an alternate proof that deterministically δ-coloring a tree with small δ takes Ω(log δ n) rounds. 3) We prove that the randomized complexity of any natural problem on instances of size n is at least its deterministic complexity on instances of size √log n. This shows that a deterministic Ω(log δ n) lower bound for any problem (δ-coloring a tree, for example) implies a randomized Ω(log δ log n) lower bound. It also illustrates that the graph shattering technique employed in recent randomized symmetry breaking algorithms is absolutely essential to the LOCAL model. For example, it is provably impossible to improve the 2O(√log log n) term in the complexities of the best MIS and (δ+1)-coloring algorithms without also improving the 2O(√log n)-round Panconesi-Srinivasan algorithm.

Original languageEnglish
Title of host publicationProceedings - 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016
PublisherIEEE Computer Society
Pages615-624
Number of pages10
ISBN (Electronic)9781509039333
DOIs
StatePublished - 14 Dec 2016
Externally publishedYes
Event57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016 - New Brunswick, United States
Duration: 9 Oct 201611 Oct 2016

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2016-December
ISSN (Print)0272-5428

Conference

Conference57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016
Country/TerritoryUnited States
CityNew Brunswick
Period9/10/1611/10/16

Bibliographical note

Publisher Copyright:
© 2016 IEEE.

Funding

This work is supported by NSF grants CCF-1217338, CNS-1318294, and CCF-1514383.

FundersFunder number
National Science Foundation1637546, CCF-1217338, 1514383, CNS-1318294, CCF-1514383

    Keywords

    • Coloring
    • Distributed algorithm
    • Local model
    • Symmetry breaking

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