Sublinear Graph Augmentation for Fast Query Implementation

We introduce the problem of augmenting graphs with sublinear memory in order to speed up replies to queries. As a concrete example, we focus on the following problem: the input is an (unpartitioned) bipartite graph \(G=(V,E)\). Given a query \(q \in V\), the algorithm’s goal is to output q’s color in some legal 2-coloring of G, using few probes to the graph. All replies have to be consistent with the same 2-coloring. We show that if a linear amount of preprocessing is allowed, there is a randomized algorithm that, for any \(\alpha \), uses \(O\left( \frac{m}{\alpha }\right) \) probes and \(\tilde{O}(\alpha )\) memory, where m is the number of edges in the graph. On the negative side, we show that for a natural family of algorithms that we call probe-first local computation algorithms, this trade-off is optimal even with unbounded preprocessing.

[1]  Eli Upfal,et al.  A trade-off between space and efficiency for routing tables , 1989, JACM.

[2]  Mark Jerrum,et al.  Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains , 1987, International Workshop on Graph-Theoretic Concepts in Computer Science.

[3]  David P. Dobkin,et al.  On sparse spanners of weighted graphs , 1993, Discret. Comput. Geom..

[4]  Moni Naor,et al.  What can be computed locally? , 1993, STOC.

[5]  David R. Karger,et al.  Approximating s-t minimum cuts in Õ(n2) time , 1996, STOC '96.

[6]  Dana Ron,et al.  Property Testing in Bounded Degree Graphs , 2002, STOC '97.

[7]  Dana Ron,et al.  Property Testing in Bounded Degree Graphs , 1997, STOC.

[8]  Dana Ron,et al.  A Sublinear Bipartiteness Tester for Bounded Degree Graphs , 1998, STOC '98.

[9]  Jonathan Katz,et al.  On the efficiency of local decoding procedures for error-correcting codes , 2000, STOC '00.

[10]  Mikkel Thorup,et al.  Compact routing schemes , 2001, SPAA '01.

[11]  Dana Ron,et al.  Tight Bounds for Testing Bipartiteness in General Graphs , 2004, SIAM J. Comput..

[12]  Mikkel Thorup,et al.  Approximate distance oracles , 2005, J. ACM.

[13]  Bernard Chazelle,et al.  Property-Preserving Data Reconstruction , 2004, Algorithmica.

[14]  Michael E. Saks,et al.  Local Monotonicity Reconstruction , 2010, SIAM J. Comput..

[15]  Ronitt Rubinfeld,et al.  Fast Local Computation Algorithms , 2011, ICS.

[16]  Debmalya Panigrahi,et al.  A general framework for graph sparsification , 2010, STOC '11.

[17]  Sergey Yekhanin,et al.  Locally Decodable Codes , 2012, Found. Trends Theor. Comput. Sci..

[18]  Krzysztof Onak,et al.  Planar Graphs: Random Walks and Bipartiteness Testing , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[19]  Nikhil Srivastava,et al.  Graph Sparsification by Effective Resistances , 2011, SIAM J. Comput..

[20]  Noga Alon,et al.  Space-efficient local computation algorithms , 2011, SODA.

[21]  Yishay Mansour,et al.  Converting Online Algorithms to Local Computation Algorithms , 2012, ICALP.

[22]  Christian Wulff-Nilsen,et al.  Approximate distance oracles with improved preprocessing time , 2011, SODA.

[23]  Yishay Mansour,et al.  A Local Computation Approximation Scheme to Maximum Matching , 2013, APPROX-RANDOM.

[24]  Dana Ron,et al.  Best of Two Local Models: Local Centralized and Local Distributed Algorithms , 2014, ArXiv.

[25]  Shiri Chechik,et al.  Approximate Distance Oracle with Constant Query Time , 2013, ArXiv.

[26]  Ronitt Rubinfeld,et al.  Local Algorithms for Sparse Spanning Graphs , 2014, APPROX-RANDOM.

[27]  Avinatan Hassidim,et al.  Local computation mechanism design , 2013, EC.

[28]  Boaz Patt-Shamir,et al.  Constant-Time Local Computation Algorithms , 2015, WAOA.

[29]  Ronitt Rubinfeld,et al.  Local Computation Algorithms for Graphs of Non-Constant Degrees , 2015, SPAA.

[30]  Omer Reingold,et al.  New techniques and tighter bounds for local computation algorithms , 2014, J. Comput. Syst. Sci..

[31]  Pierre Fraigniaud,et al.  Local Conflict Coloring , 2015, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[32]  Shay Solomon,et al.  The Greedy Spanner is Existentially Optimal , 2016, PODC.

[33]  Ronitt Rubinfeld,et al.  A Local Algorithm for Constructing Spanners in Minor-Free Graphs , 2016, APPROX-RANDOM.

[34]  Moti Medina,et al.  Non-local Probes Do Not Help with Many Graph Problems , 2016, DISC.

[35]  Mohsen Ghaffari,et al.  An Improved Distributed Algorithm for Maximal Independent Set , 2015, SODA.

[36]  Adam Wierman,et al.  Distributed Optimization via Local Computation Algorithms , 2017, SIGMETRICS Perform. Evaluation Rev..

[37]  Christoph Lenzen,et al.  A Local Algorithm for the Sparse Spanning Graph Problem , 2017, ArXiv.

[38]  Boaz Patt-Shamir,et al.  Constant-Time Local Computation Algorithms , 2017, Theory of Computing Systems.

[39]  Boaz Patt-Shamir,et al.  On the Probe Complexity of Local Computation Algorithms , 2018, ICALP.