Brief Announcement: Local Advice and Local Decompression
Authors: 
Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Krzysztof Nowicki, Dennis Olivetti, Eva Rotenberg, Jukka SuomelaAuthors Info & Claims
Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Krzysztof Nowicki, Dennis Olivetti, Eva Rotenberg, Jukka SuomelaAuthors Info & ClaimsPages 117 - 120
Abstract
In this work we study local computation with advice: the goal is to solve a graph problem Π with a distributed algorithm in f (Δ) communication rounds, for some function f that only depends on the maximum degree Δ of the graph, and the key question is how many bits of advice per node are needed. Our main results are:
(1) Any locally checkable labeling problem (LCL) can be solved in graphs with sub-exponential growth with only 1 bit of advice per node. Moreover, we can make the set of nodes that carry advice bits arbitrarily sparse.
(2) The assumption of sub-exponential growth is necessary: assuming the Exponential-Time Hypothesis (ETH), there are LCLs that cannot be solved in general with any constant number of bits per node.
(3) In any graph we can find an almost-balanced orientation (indegrees and outdegrees differ by at most one) with 1 bit of advice per node, and again we can make the advice arbitrarily sparse.
(4) As a corollary, we can also compress an arbitrary subset of edges so that a node of degree d stores only d/2 + 2 bits, and we can decompress it locally, in f(Δ) rounds.
(5) In any graph of maximum degree Δ, we can find a Δ-coloring (if it exists) with 1 bit of advice per node, and again, we can make the advice arbitrarily sparse.
(6) In any 3-colorable graph, we can find a 3-coloring with 1 bit of advice per node. Here, it remains open whether we can make the advice arbitrarily sparse.
References
[1]
Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Krzysztof Nowicki, Dennis Olivetti, Eva Rotenberg, and Jukka Suomela. 2024. Local Advice and Local Decompression. arXiv:2405.04519 [cs.DC]
[2]
Alkida Balliu, Sebastian Brandt, Dennis Olivetti, and Jukka Suomela. 2020. How much does randomness help with locally checkable problems?. In PODC '20: ACM Symposium on Principles of Distributed Computing, Virtual Event, Italy, August 3-7, 2020, Yuval Emek and Christian Cachin (Eds.). ACM, 299--308.
[3]
Alkida Balliu, Sebastian Brandt, Dennis Olivetti, and Jukka Suomela. 2021. Almost global problems in the LOCAL model. Distributed Computing 34, 4 (2021), 259--281.
[4]
Alkida Balliu, Keren Censor-Hillel, Yannic Maus, Dennis Olivetti, and Jukka Suomela. 2021. Locally Checkable Labelings with Small Messages. In 35th International Symposium on Distributed Computing, DISC 2021, October 4-8, 2021, Freiburg, Germany (Virtual Conference) (LIPIcs, Vol. 209), Seth Gilbert (Ed.). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 8:1--8:18.
[5]
Alkida Balliu, Juho Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, Dennis Olivetti, and Jukka Suomela. 2018. New classes of distributed time complexity. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, Ilias Diakonikolas, David Kempe, and Monika Henzinger (Eds.). ACM, 1307--1318.
[6]
Leonid Barenboim, Michael Elkin, and Uri Goldenberg. 2022. Locally-iterative Distributed (Δ + 1)-coloring and Applications. J. ACM 69, 1 (2022), 5:1--5:26.
[7]
Sebastian Brandt, Orr Fischer, Juho Hirvonen, Barbara Keller, Tuomo Lempiäinen, Joel Rybicki, Jukka Suomela, and Jara Uitto. 2016. A lower bound for the distributed Lovász local lemma. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, Daniel Wichs and Yishay Mansour (Eds.). ACM, 479--488.
[8]
Yi-Jun Chang, Tsvi Kopelowitz, and Seth Pettie. 2019. An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model. SIAM J. Comput. 48, 1 (2019), 122--143.
[9]
Yi-Jun Chang and Seth Pettie. 2019. A Time Hierarchy Theorem for the LOCAL Model. SIAM J. Comput. 48, 1 (2019), 33--69.
[10]
Dariusz Dereniowski and Andrzej Pelc. 2012. Drawing maps with advice. J. Parallel and Distrib. Comput. 72, 2 (2012), 132--143.
[11]
Stefan Dobrev, Rastislav Královic, and Euripides Markou. 2012. Online Graph Exploration with Advice. In Structural Information and Communication Complexity - 19th International Colloquium, SIROCCO 2012, Reykjavik, Iceland, June 30-July 2, 2012, Revised Selected Papers (Lecture Notes in Computer Science, Vol. 7355), Guy Even and Magnús M. Halldórsson (Eds.). Springer, 267--278.
[12]
Laurent Feuilloley and Pierre Fraigniaud. 2017. Survey of Distributed Decision. arXiv:1606.04434 [cs.DC]
[13]
Manuela Fischer and Mohsen Ghaffari. 2017. Sublogarithmic Distributed Algorithms for Lovász Local Lemma, and the Complexity Hierarchy. In 31st International Symposium on Distributed Computing, DISC 2017, October 16-20, 2017, Vienna, Austria (LIPIcs, Vol. 91), Andréa W. Richa (Ed.). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 18:1--18:16.
[14]
Pierre Fraigniaud, Cyril Gavoille, David Ilcinkas, and Andrzej Pelc. 2009. Distributed computing with advice: information sensitivity of graph coloring. Distributed Computing 21, 6 (2009), 395--403.
[15]
Pierre Fraigniaud, Marc Heinrich, and Adrian Kosowski. 2016. Local Conflict Coloring. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, Irit Dinur (Ed.). IEEE Computer Society, 625--634.
[16]
Pierre Fraigniaud, David Ilcinkas, and Andrzej Pelc. 2008. Tree exploration with advice. Information and Computation 206, 11 (2008), 1276--1287.
[17]
Pierre Fraigniaud, David Ilcinkas, and Andrzej Pelc. 2010. Communication algorithms with advice. J. Comput. System Sci. 76, 3-4 (2010), 222--232.
[18]
Pierre Fraigniaud, Amos Korman, and Emmanuelle Lebhar. 2007. Local MST computation with short advice. In SPAA 2007: Proceedings of the 19th Annual ACM Symposium on Parallelism in Algorithms and Architectures, San Diego, California, USA, June 9-11, 2007, Phillip B. Gibbons and Christian Scheideler (Eds.). ACM, 154--160.
[19]
Emanuele G. Fusco and Andrzej Pelc. 2011. Trade-offs Between the Size of Advice and Broadcasting Time in Trees. Algorithmica 60, 4 (2011), 719--734.
[20]
Emanuele G. Fusco, Andrzej Pelc, and Rossella Petreschi. 2016. Topology recognition with advice. Information and Computation 247 (2016), 254--265.
[21]
Mohsen Ghaffari, David G. Harris, and Fabian Kuhn. 2018. On Derandomizing Local Distributed Algorithms. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, Mikkel Thorup (Ed.). IEEE Computer Society, 662--673.
[22]
Mohsen Ghaffari, Juho Hirvonen, Fabian Kuhn, Yannic Maus, Jukka Suomela, and Jara Uitto. 2020. Improved distributed degree splitting and edge coloring. Distributed Computing 33, 3-4 (2020), 293--310.
[23]
Mohsen Ghaffari and Hsin-Hao Su. 2017. Distributed Degree Splitting, Edge Coloring, and Orientations. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, Philip N. Klein (Ed.). SIAM, 2505--2523.
[24]
Christian Glacet, Avery Miller, and Andrzej Pelc. 2017. Time vs. Information Tradeoffs for Leader Election in Anonymous Trees. ACM Transactions on Algorithms 13, 3 (2017), 31:1--31:41.
[25]
Mika Göös and Jukka Suomela. 2016. Locally Checkable Proofs in Distributed Computing. Theory of Computing 12, 1 (2016), 1--33.
[26]
Barun Gorain and Andrzej Pelc. 2019. Deterministic Graph Exploration with Advice. ACM Transactions on Algorithms 15, 1 (2019), 8:1--8:17.
[27]
David Ilcinkas, Dariusz R. Kowalski, and Andrzej Pelc. 2010. Fast radio broadcasting with advice. Theoretical Computer Science 411, 14-15 (2010), 1544--1557.
[28]
Dennis Komm, Rastislav Královic, Richard Královic, and Jasmin Smula. 2015. Treasure Hunt with Advice. In Structural Information and Communication Complexity - 22nd International Colloquium, SIROCCO 2015, Montserrat, Spain, July 14-16, 2015, Post-Proceedings (Lecture Notes in Computer Science, Vol. 9439), Christian Scheideler (Ed.). Springer, 328--341.
[29]
Amos Korman and Shay Kutten. 2006. On Distributed Verification. In Distributed Computing and Networking, 8th International Conference, ICDCN 2006, Guwahati, India, December 27-30, 2006 (Lecture Notes in Computer Science, Vol. 4308), Soma Chaudhuri, Samir R. Das, Himadri S. Paul, and Srikanta Tirthapura (Eds.). Springer, 100--114.
[30]
Amos Korman and Shay Kutten. 2007. Distributed verification of minimum spanning trees. Distributed Computing 20, 4 (2007), 253--266.
[31]
Amos Korman, Shay Kutten, and David Peleg. 2010. Proof labeling schemes. Distributed Computing 22, 4 (2010), 215--233.
[32]
Amos Korman, David Peleg, and Yoav Rodeh. 2010. Constructing Labeling Schemes through Universal Matrices. Algorithmica 57, 4 (2010), 641--652.
[33]
Nathan Linial. 1992. Locality in Distributed Graph Algorithms. SIAM J. Comput. 21, 1 (1992), 193--201.
[34]
Yannic Maus and Tigran Tonoyan. 2022. Linial for lists. Distributed Computing 35, 6 (2022), 533--546.
[35]
Avery Miller and Andrzej Pelc. 2015. Fast rendezvous with advice. Theoretical Computer Science 608 (2015), 190--198.
[36]
Avery Miller and Andrzej Pelc. 2015. Tradeoffs between cost and information for rendezvous and treasure hunt. J. Parallel and Distrib. Comput. 83 (2015), 159--167.
[37]
Avery Miller and Andrzej Pelc. 2016. Election vs. Selection: How Much Advice is Needed to Find the Largest Node in a Graph?. In Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2016, Asilomar State Beach/Pacific Grove, CA, USA, July 11-13, 2016, Christian Scheideler and Seth Gilbert (Eds.). ACM, 377--386.
[38]
Moni Naor and Larry J. Stockmeyer. 1995. What Can be Computed Locally? SIAM J. Comput. 24, 6 (1995), 1259--1277.
[39]
Nicolas Nisse and David Soguet. 2009. Graph searching with advice. Theoretical Computer Science 410, 14 (2009), 1307--1318.
[40]
Alessandro Panconesi and Aravind Srinivasan. 1992. Improved Distributed Algorithms for Coloring and Network Decomposition Problems. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing, May 4-6, 1992, Victoria, British Columbia, Canada, S. Rao Kosaraju, Mike Fellows, Avi Wigderson, and John A. Ellis (Eds.). ACM, 581--592.
[41]
Václav Rozhon and Mohsen Ghaffari. 2020. Polylogarithmic-time deterministic network decomposition and distributed derandomization. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy (Eds.). ACM, 350--363.
Index Terms
- Brief Announcement: Local Advice and Local Decompression
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Published In
June 2024
570 pages
ISBN:9798400706684
DOI:10.1145/3662158
- Chair:
- Ran Gelles,
- Proceedings Chair:
- Dennis Olivetti,
- Program Chair:
- Petr Kuznetsov
This work is licensed under a Creative Commons Attribution International 4.0 License.
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Association for Computing Machinery
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Published: 17 June 2024
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