Ultra-Sparse Near-Additive Emulators
Pages 235 - 246
Abstract
Near-additive (aka (1+ε,β)β-) emulators and spanners are a fundamental graph-algorithmic construct, with numerous applications for computing approximate shortest paths and related problems in distributed, streaming and dynamic settings.
Known constructions of near-additive emulators enable one to trade between their sparsity (i.e., number of edges) and the additive stretch β. Specifically, for any pair of parameters ε >0, κ=1,2,..., one can have a (1+ε,β)-emulator with O(n^1+1/κ ) edges, with β = łeft(\fracłog κ ε \right)^łog κ . At their sparsest, these emulators employ c.n edges, for some constant c≥ 2. We tighten this bound, and show that in fact precisely n^1+1/κ edges suffice.
Supplementary Material
We prove that ultra-sparse near-additive emulators exist and describe efficient centralized as well as distributed CONGEST algorithms that construct such emulators. Given an unweighted, undirected graph G=(V,E), a graph H=(V,E?) is a near-additive emulator for G if for every pair of vertices u,v in V it holds that: d_G(u,v) \leq d_H(u,v) \leq (1+\epsilon)d_G(u,v) +\beta. We show that for every n-vertex graph G, there exists near-additive emulator that is ultra-sparse, i.e., with n+o(n) edges. We also show how one can construct such emulators efficiently in the centralized and in the distributed CONGEST model.
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References
[1]
Amir Abboud and Greg Bodwin. 2016. The 4/3 additive spanner exponent is tight. 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, 351--361. https://doi.org/10.1145/2897518.2897555
[2]
Amir Abboud, Greg Bodwin, and Seth Pettie. 2018. A Hierarchy of Lower Bounds for Sublinear Additive Spanners. SIAM J. Comput. 47, 6 (2018), 2203--2236. https: //doi.org/10.1137/16M1105815
[3]
Ingo Althöfer, Gautam Das, David P. Dobkin, Deborah Joseph, and José Soares. 1993. On Sparse Spanners of Weighted Graphs. Discrete & Computational Geometry 9 (1993), 81--100. https://doi.org/10.1007/BF02189308
[4]
Alexandr Andoni, Clifford Stein, and Peilin Zhong. 2020. Parallel approximate undirected shortest paths via low hop emulators. In Proccedings 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, 322--335. https: //doi.org/10.1145/3357713.3384321
[5]
Aaron Bernstein and Liam Roditty. 2011. Improved Dynamic Algorithms for Maintaining Approximate Shortest Paths Under Deletions. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23--25, 2011. 1355--1365. https: //doi.org/10.1137/1.9781611973082.104
[6]
Keren Censor-Hillel, Michal Dory, Janne H. Korhonen, and Dean Leitersdorf. 2019. Fast Approximate Shortest Paths in the Congested Clique. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, PODC 2019, Toronto, ON, Canada, July 29 - August 2, 2019. 74--83. https://doi.org/10.1145/ 3293611.3331633
[7]
Edith Cohen. 1994. Polylog-time and near-linear work approximation scheme for undirected shortest paths. In Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, 23--25 May 1994, Montréal, Québec, Canada. 16--26. https://doi.org/10.1145/195058.195089
[8]
Michael B. Cohen, Rasmus Kyng, Gary L. Miller, Jakub W. Pachocki, Richard Peng, Anup B. Rao, and Shen Chen Xu. 2014. Solving SDD linear systems in nearly mlog1/2n time. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, David B. Shmoys (Ed.). ACM, 343--352. https://doi.org/10.1145/2591796.2591833
[9]
Bilel Derbel, Mohamed Mosbah, and Akka Zemmari. 2006. Fast distributed graph partition and application. In 20th International Parallel and Distributed Processing Symposium (IPDPS 2006), Proceedings, 25--29 April 2006, Rhodes Island, Greece. IEEE. https://doi.org/10.1109/IPDPS.2006.1639362
[10]
Michal Dory and Merav Parter. 2020. Exponentially Faster Shortest Paths in the Congested Clique. In PODC '20: ACM Symposium on Principles of Distributed Computing, Virtual Event, Italy, August 3--7, 2020, Yuval Emek and Christian Cachin (Eds.). ACM, 59--68. https://doi.org/10.1145/3382734.3405711
[11]
Devdatt P. Dubhashi, Alessandro Mei, Alessandro Panconesi, Jaikumar Radhakrishnan, and Aravind Srinivasan. 2005. Fast distributed algorithms for (weakly) connected dominating sets and linear-size skeletons. J. Comput. Syst. Sci. 71, 4 (2005), 467--479. https://doi.org/10.1016/j.jcss.2005.04.002
[12]
Michael Elkin. 2001. Computing almost shortest paths. In Proceedings of the Twentieth Annual ACM Symposium on Principles of Distributed Computing, PODC 2001, Newport, Rhode Island, USA, August 26--29, 2001. 53--62. https://doi.org/10. 1145/383962.383983
[13]
Michael Elkin and Shaked Matar. 2019. Near-Additive Spanners In Low Polynomial Deterministic CONGEST Time. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, PODC 2019, Toronto, ON, Canada, July 29 - August 2, 2019. 531--540. https://doi.org/10.1145/3293611.3331635
[14]
Michael Elkin and Shaked Matar. 2021. Ultra-Sparse Near-Additive Emulators. CoRR (2021). arXiv:cs.DS/2106.01036 https://arxiv.org/abs/2106.01036
[15]
Michael Elkin and Ofer Neiman. 2016. Hopsets with Constant Hopbound, and Applications to Approximate Shortest Paths. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9--11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA. 128--137. https://doi.org/10.1109/FOCS.2016.22
[16]
Michael Elkin and Ofer Neiman. 2016. On Efficient Distributed Construction of Near Optimal Routing Schemes: Extended Abstract. In Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, Chicago, IL, USA, July 25--28, 2016, George Giakkoupis (Ed.). ACM, 235--244. https://doi.org/ 10.1145/2933057.2933098
[17]
Michael Elkin and Ofer Neiman. 2017. Efficient Algorithms for Constructing Very Sparse Spanners and Emulators. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16--19. 652--669. https://doi.org/10.1137/1.9781611974782.41
[18]
Michael Elkin and Ofer Neiman. 2017. Linear-Size Hopsets with Small Hopbound, and Distributed Routing with Low Memory. CoRR abs/1704.08468 (2017). arXiv:1704.08468 http://arxiv.org/abs/1704.08468
[19]
Michael Elkin and Ofer Neiman. 2020. Near-Additive Spanners and Near-Exact Hopsets, A Unified View. Bull. EATCS 130 (2020). http://bulletin.eatcs.org/index. php/beatcs/article/view/608/624
[20]
Michael Elkin and David Peleg. 2001. (1+epsilon, beta)-spanner constructions for general graphs. In Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6--8, 2001, Heraklion, Crete, Greece. 173--182. https://doi.org/10. 1145/380752.380797
[21]
Michael Elkin and Seth Pettie. 2015. A Linear-Size Logarithmic Stretch PathReporting Distance Oracle for General Graphs. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4--6, 2015. 805--821. https://doi.org/10.1137/1.9781611973730.55
[22]
Michael Elkin and Jian Zhang. 2004. Efficient algorithms for constructing (1+, varepsilon;, beta)-spanners in the distributed and streaming models. In Proceedings of the Twenty-Third Annual ACM Symposium on Principles of Distributed Computing, PODC 2004, St. John's, Newfoundland, Canada, July 25- 28, 2004, Soma Chaudhuri and Shay Kutten (Eds.). ACM, 160--168. https: //doi.org/10.1145/1011767.1011791
[23]
Shay Halperin and Uri Zwick. 1996. Optimal randomized EREW PRAM Algorithms for Finding Spanning Forests and for other Basic Graph Connectivity Problems. In Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, 28--30 January 1996, Atlanta, Georgia, USA, Éva Tardos (Ed.). ACM/SIAM, 438--447. http://dl.acm.org/citation.cfm?id=313852.314099
[24]
Monika Henzinger, Sebastian Krinninger, and Danupon Nanongkai. 2016. A deterministic almost-tight distributed algorithm for approximating single-source shortest paths. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18--21, 2016. 489--498. https://doi.org/10.1145/2897518.2897638
[25]
Monika Henzinger, Sebastian Krinninger, and Danupon Nanongkai. 2018. Decremental Single-Source Shortest Paths on Undirected Graphs in Near-Linear Total Update Time. J. ACM 65, 6 (2018), 36:1--36:40. https://doi.org/10.1145/3218657
[26]
Shang-En Huang and Seth Pettie. 2017. Thorup-Zwick Emulators are Universally Optimal Hopsets. CoRR abs/1705.00327 (2017). arXiv:1705.00327 http://arxiv. org/abs/1705.00327
[27]
Shang-En Huang and Seth Pettie. 2018. Lower Bounds on Sparse Spanners, Emulators, and Diameter-reducing shortcuts. In 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018, June 18--20, 2018, Malmö, Sweden (LIPIcs), David Eppstein (Ed.), Vol. 101. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 26:1--26:12. https://doi.org/10.4230/LIPIcs.SWAT.2018.26
[28]
Arun Jambulapati and Aaron Sidford. 2020. Ultrasparse Ultrasparsifiers and Faster Laplacian System Solvers. CoRR abs/2011.08806 (2020). arXiv:2011.08806 https://arxiv.org/abs/2011.08806
[29]
Michael Kapralov and Rina Panigrahy. 2012. Spectral sparsification via random spanners. In Innovations in Theoretical Computer Science 2012, Cambridge, MA, USA, January 8--10, 2012, Shafi Goldwasser (Ed.). ACM, 393--398. https://doi.org/ 10.1145/2090236.2090267
[30]
Jonathan A. Kelner, Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. 2014. An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations. In Proceedings of the TwentyFifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5--7, 2014, Chandra Chekuri (Ed.). SIAM, 217--226. https: //doi.org/10.1137/1.9781611973402.16
[31]
Alexandra Kolla, Yury Makarychev, Amin Saberi, and Shang-Hua Teng. 2010. Subgraph sparsification and nearly optimal ultrasparsifiers. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5--8 June 2010. 57--66. https://doi.org/10.1145/1806689.1806699
[32]
Fabian Kuhn, Yannic Maus, and Simon Weidner. 2018. Deterministic Distributed Ruling Sets of Line Graphs. In Structural Information and Communication Complexity - 25th International Colloquium, SIROCCO 2018, Ma'ale HaHamisha, Israel, June 18--21, 2018, Revised Selected Papers. 193--208. https: //doi.org/10.1007/978--3-030-01325--7_19
[33]
Jakub Lacki and Yasamin Nazari. 2020. Near-Optimal Decremental Approximate Multi-Source Shortest Paths. CoRR abs/2009.08416 (2020). arXiv:2009.08416 https://arxiv.org/abs/2009.08416
[34]
Christoph Lenzen and Boaz Patt-Shamir. 2015. Fast Partial Distance Estimation and Applications. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, PODC 2015, Donostia-San Sebastián, Spain, July 21 - 23, 2015, Chryssis Georgiou and Paul G. Spirakis (Eds.). ACM, 153--162. https: //doi.org/10.1145/2767386.2767398
[35]
Christoph Lenzen and David Peleg. 2013. Efficient distributed source detection with limited bandwidth. In ACM Symposium on Principles of Distributed Computing, PODC '13, Montreal, QC, Canada, July 22--24, 2013, Panagiota Fatourou and Gadi Taubenfeld (Eds.). ACM, 375--382. https://doi.org/10.1145/2484239.2484262
[36]
David Peleg. 2000. Distributed Computing: A Locality-Sensitive Approach. (01 2000). https://doi.org/10.1137/1.9780898719772
[37]
Seth Pettie. 2007. Low Distortion Spanners. In Automata, Languages and Programming, 34th International Colloquium, ICALP 2007, Wroclaw, Poland, July 9--13, 2007, Proceedings (Lecture Notes in Computer Science), Lars Arge, Christian Cachin, Tomasz Jurdzinski, and Andrzej Tarlecki (Eds.), Vol. 4596. Springer, 78--89. https://doi.org/10.1007/978--3--540--73420--8_9
[38]
Seth Pettie. 2008. Distributed algorithms for ultrasparse spanners and linear size skeletons. In Proceedings of the Twenty-Seventh Annual ACM Symposium on Principles of Distributed Computing, PODC 2008, Toronto, Canada, August 18--21, 2008. 253--262. https://doi.org/10.1145/1400751.1400786
[39]
Seth Pettie. 2009. Low distortion spanners. ACM Trans. Algorithms 6, 1 (2009), 7:1--7:22. https://doi.org/10.1145/1644015.1644022
[40]
Seth Pettie. 2010. Distributed algorithms for ultrasparse spanners and linear size skeletons. Distributed Computing 22, 3 (2010), 147--166. https://doi.org/10.1007/ s00446-009-0091--7
[41]
Liam Roditty and Uri Zwick. 2004. On Dynamic Shortest Paths Problems. In Algorithms - ESA 2004, 12th Annual European Symposium, Bergen, Norway, September 14--17, 2004, Proceedings. 580--591. https://doi.org/10.1007/978--3--540--30140-0_52
[42]
Johannes Schneider, Michael Elkin, and Roger Wattenhofer. 2013. Symmetry breaking depending on the chromatic number or the neighborhood growth. Theor. Comput. Sci. 509 (2013), 40--50. https://doi.org/10.1016/j.tcs.2012.09.004
[43]
Jonah Sherman. 2013. Nearly Maximum Flows in Nearly Linear Time. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26- 29 October, 2013, Berkeley, CA, USA. IEEE Computer Society, 263--269. https: //doi.org/10.1109/FOCS.2013.36
[44]
Daniel A. Spielman and Shang-Hua Teng. 2004. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13--16, 2004, László Babai (Ed.). ACM, 81--90. https://doi.org/10.1145/1007352. 1007372
[45]
Mikkel Thorup and Uri Zwick. 2006. Spanners and emulators with sublinear distance errors. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, Miami, Florida, USA, January 22--26, 2006. 802--809. http://dl.acm.org/citation.cfm?id=1109557.1109645
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