Seeing Far vs. Seeing Wide: Volume Complexity of Local Graph Problems
Pages 89 - 98
Abstract
Assume we have a graph problem that is locally checkable but not locally solvable---given a solution we can check that it is feasible by verifying all constant-radius neighborhoods, but to find a feasible solution each node needs to explore the input graph at least up to distance Ω (log n) in order to produce its own part of the solution.
Such problems have been studied extensively in the recent years in the area of distributed computing, where the key complexity measure has been distance: how far does a node need to see in order to produce its own part of the solution. However, if we are interested in e.g. sublinear-time centralized algorithms, a much more appropriate complexity measure would be volume: how large a subgraph does a node need to see in order to produce its own part of the solution.
In this work we study locally checkable graph problems on bounded-degree graphs and we give a number of constructions that exhibit different tradeoffs between deterministic distance, randomized distance, deterministic volume, and randomized volume:
• If the deterministic distance is linear, it is also known that randomized distance is near-linear. We show that volume complexity is fundamentally different: there are problems with a linear deterministic volume but only logarithmic randomized volume.
• We prove a volume hierarchy theorem for randomized complexity: Among problems with (near) linear deterministic volume complexity, there are infinitely many distinct randomized volume complexity classes between Ω(log n) and O(n). Moreover, this hierarchy persists even when restricting to problems whose randomized and deterministic distance complexities are Θ(log n).
• Similar hierarchies exist for polynomial distance complexities: we show that for any k, ℓ ∈ N with k ≤ ℓ, there are problems whose randomized and deterministic distance complexities are Θ(n1/ℓ), randomized volume complexities are [EQUATION], and whose deterministic volume complexities are [EQUATION].
We also consider connections between our volume model and massively parallel computation (MPC). We give a general simulation argument that any volume-efficient algorithm can be transformed into a space-efficient MPC algorithm.
References
[1]
Noga Alon, Ronitt Rubinfeld, Shai Vardi, and Ning Xie. 2012. Space-efficient Local Computation Algorithms. In Proc. 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2012). 1132--1139.
[2]
Alkida Balliu, Sebastian Brandt, Yi-Jun Chang, Dennis Olivetti, Mikaël Rabie, and Jukka Suomela. 2019. The Distributed Complexity of Locally Checkable Problems on Paths is Decidable. In Proc. 38th ACM Symposium on Principles of Distributed Computing (PODC 2019). ACM Press, 262--271. arXiv:1811.01672
[3]
Alkida Balliu, Sebastian Brandt, Dennis Olivetti, and Jukka Suomela. 2018. Almost global problems in the LOCAL model. In Proc. 32nd International Symposium on Distributed Computing (DISC 2018) (Leibniz International Proceedings in Informatics (LIPIcs)). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 9:1--9:16.
[4]
Alkida Balliu, Sebastian Brandt, Dennis Olivetti, and Jukka Suomela. 2019. How much does randomness help with locally checkable problems? arXiv:1902.06803 http://arxiv.org/abs/1902.06803
[5]
Alkida Balliu, Juho Hirvonen, Janne H Korhonen, Tuomo Lempiäinen, Dennis Olivetti, and Jukka Suomela. 2018. New classes of distributed time complexity. In Proc. 50th ACM Symposium on Theory of Computing (STOC 2018). ACM Press, 1307--1318.
[6]
Alkida Balliu, Juho Hirvonen, Christoph Lenzen, Dennis Olivetti, and Jukka Suomela. 2019. Locality of not-so-weak coloring. In Proc. 26th International Colloquium on Structural Information and Communication Complexity (SIROCCO 2019) (LNCS), Vol. 11639. Springer, 37--51. arXiv:1904.05627
[7]
Alkida Balliu, Juho Hirvonen, Dennis Olivetti, and Jukka Suomela. 2019. Hardness of Minimal Symmetry Breaking in Distributed Computing. In Proc. 38th ACM Symposium on Principles of Distributed Computing (PODC 2019). ACM Press, 369--378. arXiv:1811.01643
[8]
Sebastian Brandt. 2019. An Automatic Speedup Theorem for Distributed Problems. In Proc. 38th ACM Symposium on Principles of Distributed Computing (PODC 2019). ACM Press, 379--388. arXiv:1902.09958
[9]
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 Proc. 48th ACM Symposium on Theory of Computing (STOC 2016). ACM Press, 479--488.
[10]
Sebastian Brandt, Juho Hirvonen, Janne H Korhonen, Tuomo Lempiäinen, Patric R J Östergård, Christopher Purcell, Joel Rybicki, Jukka Suomela, and Przemysław Uznański. 2017. LCL problems on grids. In Proc. 36th ACM Symposium on Principles of Distributed Computing (PODC 2017). ACM Press, 101--110.
[11]
Andrea Campagna, Alan Guo, and Ronitt Rubinfeld. 2013. Local Reconstructors and Tolerant Testers for Connectivity and Diameter. In Proc. APPROX/RANDOM 2013 (LNCS), Vol. 8096. Springer, 411--424.
[12]
Yi-Jun Chang, Qizheng He, Wenzheng Li, Seth Pettie, and Jara Uitto. 2018. The Complexity of Distributed Edge Coloring with Small Palettes. In Proc. 29th ACM-SIAM Symposium on Discrete Algorithms (SODA 2018). Society for Industrial and Applied Mathematics, 2633--2652.
[13]
Yi-Jun Chang, Tsvi Kopelowitz, and Seth Pettie. 2016. An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model. In Proc. 57th IEEE Symposium on Foundations of Computer Science (FOCS 2016). IEEE, 615--624.
[14]
Yi-Jun Chang and Seth Pettie. 2019. A Time Hierarchy Theorem for the LOCAL Model. SIAM J. Comput. 48, 1 (2019), 33--69.
[15]
Richard Cole and Uzi Vishkin. 1986. Deterministic coin tossing with applications to optimal parallel list ranking. Information and Control 70, 1 (1986), 32--53.
[16]
Talya Eden and Will Rosenbaum. 2018. Lower Bounds for Approximating Graph Parameters via Communication Complexity. In Proc. APPROX/RANDOM 2018 (Leibniz International Proceedings in Informatics (LIPIcs)), Vol. 116. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 11:1--11:18.
[17]
Guy Even, Moti Medina, and Dana Ron. 2014. Deterministic Stateless Centralized Local Algorithms for Bounded Degree Graphs. In Proc. 22th Annual European Symposium on Algorithms (ESA 2014). Springer, 394--405.
[18]
Guy Even, Moti Medina, and Dana Ron. 2018. Best of two local models: Centralized local and distributed local algorithms. Information and Computation 262 (2018), 69--89.
[19]
Uriel Feige, Yishay Mansour, and Robert Schapire. 2015. Learning and inference in the presence of corrupted inputs. In Proc. 28th Conference on Learning Theory (COLT 2015) (Proceedings of Machine Learning Research). 637--657.
[20]
Manuela Fischer and Mohsen Ghaffari. 2017. Sublogarithmic Distributed Algorithms for Lovász Local Lemma, and the Complexity Hierarchy. In Proc. 31st International Symposium on Distributed Computing (DISC 2017). 18:1--18:16.
[21]
Mohsen Ghaffari, David G Harris, and Fabian Kuhn. 2018. On Derandomizing Local Distributed Algorithms. In Proc. 59th IEEE Symposium on Foundations of Computer Science (FOCS 2018). 662--673. arXiv:1711.02194
[22]
Mohsen Ghaffari, Juho Hirvonen, Fabian Kuhn, and Yannic Maus. 2018. Improved Distributed Δ-Coloring. In Proc. 37th ACM Symposium on Principles of Distributed Computing (PODC 2018). ACM, 427--436.
[23]
Mohsen Ghaffari and Hsin-Hao Su. 2017. Distributed Degree Splitting, Edge Coloring, and Orientations. In Proc. 28th ACM-SIAM Symposium on Discrete Algorithms (SODA 2017). Society for Industrial and Applied Mathematics, 2505--2523.
[24]
Mohsen Ghaffari and Jara Uitto. 2019. Sparsifying Distributed Algorithms with Ramifications in Massively Parallel Computation and Centralized Local Computation. In Proc. 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019). Society for Industrial and Applied Mathematics, Philadelphia, PA, 1636--1653.
[25]
Michael T. Goodrich, Nodari Sitchinava, and Qin Zhang. 2011. Sorting, Searching, and Simulation in the MapReduce Framework. In Proc. 22nd International Symposium on Algorithms and Computation (ISAAC 2011) (LNCS), Vol. 7074. Springer, 374--383.
[26]
Mika Göös, Juho Hirvonen, Reut Levi, Moti Medina, and Jukka Suomela. 2016. Non-local Probes Do Not Help with Many Graph Problems. In Proc. 30th International Symposium on Distributed Computing (DISC 2016) (Lecture Notes in Computer Science), Vol. 9888. Springer, 201--214.
[27]
Bala Kalyanasundaram and Georg Schintger. 1992. The Probabilistic Communication Complexity of Set Intersection. SIAM Journal on Discrete Mathematics 5, 4 (1992), 545--557.
[28]
Howard Karloff, Siddharth Suri, and Sergei Vassilvitskii. 2010. A Model of Computation for MapReduce. In Proc. 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010). SIAM, Philadelphia, PA, 938--948.
[29]
Christoph Lenzen and Roger Wattenhofer. 2010. Brief announcement: Exponential speed-up of local algorithms using non-local communication. In Proc. 29th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC 2010). ACM Press, 295--296.
[30]
Reut Levi, Guy Moshkovitz, Dana Ron, Ronitt Rubinfeld, and Asaf Shapira. 2017. Constructing near spanning trees with few local inspections. Random Structures & Algorithms 50, 2 (2017), 183--200.
[31]
Reut Levi, Dana Ron, and Ronitt Rubinfeld. 2014. Local Algorithms for Sparse Spanning Graphs. In Proc. APPROX/RANDOM 2014 (Leibniz International Proceedings in Informatics (LIPIcs)), Vol. 28. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 826--842.
[32]
Reut Levi, Ronitt Rubinfeld, and Anak Yodpinyanee. 2017. Local Computation Algorithms for Graphs of Non-constant Degrees. Algorithmica 77, 4 (2017), 971--994.
[33]
Nathan Linial. 1992. Locality in Distributed Graph Algorithms. SIAM J. Comput. 21, 1 (1992), 193--201.
[34]
Yishay Mansour, Aviad Rubinstein, Shai Vardi, and Ning Xie. 2012. Converting Online Algorithms to Local Computation Algorithms. In Proc. 39th International Colloquium on Automata, Languages, and Programming (ICALP 2012) (LNCS), Vol. 7391. Springer, 653--664.
[35]
Yishay Mansour and Shai Vardi. 2013. A Local Computation Approximation Scheme to Maximum Matching. In Proc. APPROX/RANDOM 2013 (LNCS), Vol. 8096. Springer, 260--273.
[36]
Michael Mitzenmacher and Eli Upfal. 2005. Probability and Computing. Cambridge University Press, Cambridge.
[37]
Moni Naor. 1991. A lower bound on probabilistic algorithms for distributive ring coloring. SIAM Journal on Discrete Mathematics 4, 3 (1991), 409--412.
[38]
Moni Naor and Larry Stockmeyer. 1995. What Can be Computed Locally? SIAM J. Comput. 24, 6 (1995), 1259--1277.
[39]
Alessandro Panconesi and Aravind Srinivasan. 1995. The local nature of Δ-coloring and its algorithmic applications. Combinatorica 15, 2 (1995), 255--280.
[40]
Michal Parnas and Dana Ron. 2007. Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms. Theoretical Computer Science 381, 1-3 (2007), 183--196.
[41]
David Peleg. 2000. Distributed Computing: A Locality-Sensitive Approach. Society for Industrial and Applied Mathematics.
[42]
A.A. Razborov. 1992. On the distributional complexity of disjointness. Theoretical Computer Science 106, 2 (1992), 385--390.
[43]
Omer Reingold and Shai Vardi. 2016. New techniques and tighter bounds for local computation algorithms. J. Comput. System Sci. 82, 7 (2016), 1180--1200.
[44]
Will Rosenbaum and Jukka Suomela. 2019. Seeing Far vs. Seeing Wide: Volume Complexity of Local Graph Problems. arXiv:cs.DC/1907.08160
[45]
Václav Rozhoň and Mohsen Ghaffari. 2020. Polylogarithmic-Time Deterministic Network Decomposition and Distributed Derandomization. In Proc. 52nd Annual ACM Symposium on Theory of Computing (STOC 2020). arXiv:1907.10937 http://arxiv.org/abs/1907.10937
[46]
Ronitt Rubinfeld, Gil Tamir, Shai Vardi, and Ning Xie. 2011. Fast Local Computation Algorithms. In Proc. 2nd Symposium on Innovations in Computer Science (ICS 2011). 223--238.
Index Terms
- Seeing Far vs. Seeing Wide: Volume Complexity of Local Graph Problems
Recommendations
Node and edge averaged complexities of local graph problems
AbstractWe continue the recently started line of work on the distributed node-averaged complexity of distributed graph algorithms. The node-averaged complexity of a distributed algorithm running on a graph is the average over the times at which ...
Node and Edge Averaged Complexities of Local Graph Problems
PODC'22: Proceedings of the 2022 ACM Symposium on Principles of Distributed ComputingWe continue the recently started line of work on the distributed node-averaged complexity of distributed graph algorithms. The node-averaged complexity of a distributed algorithm running on a graph G=(V,E) is the average over the times at which the ...
An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model
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 ...
Comments
Information & Contributors
Information
Published In
July 2020
539 pages
Copyright © 2020 ACM.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].
Sponsors
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 31 July 2020
Check for updates
Author Tags
Qualifiers
- Research-article
Conference
PODC '20: ACM Symposium on Principles of Distributed Computing
August 3 - 7, 2020
Virtual Event, Italy
Acceptance Rates
Overall Acceptance Rate 740 of 2,477 submissions, 30%
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 128Total Downloads
- Downloads (Last 12 months)18
- Downloads (Last 6 weeks)2
Reflects downloads up to 10 Aug 2024
Other Metrics
Citations
Cited By
View allView Options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in