Distributed exact shortest paths in sublinear time

The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in O(n) time, where n is the number of vertices in the input graph G. Peleg and Rubinovich, FOCS'99, showed a lower bound of Ω(D + √n) for this problem, where D is the hop-diameter of G. Whether or not this problem can be solved in o(n) time when D is relatively small is a major notorious open question. Despite intensive research that yielded near-optimal algorithms for the approximate variant of this problem, no progress was reported for the original problem. In this paper we answer this question in the affirmative. We devise an algorithm that requires O((n logn)5/6) time, for D = O(√n logn), and O(D1/3 #183; (n logn)2/3) time, for larger D. This running time is sublinear in n in almost the entire range of parameters, specifically, for D = o(n/log2 n). We also generalize our result in two directions. One is when edges have bandwidth b ≥ 1, and the other is the s-sources shortest paths problem. For the former problem, our algorithm provides an improved bound, compared to the unit-bandwidth case. In particular, we provide an all-pairs shortest paths algorithm that requires O(n5/3 #183; log2/3 n) time, even for b = 1, for all values of D. For the latter problem (of s sources), our algorithm also provides bounds that improve upon the previous state-of-the-art in the entire range of parameters. From the technical viewpoint, our algorithm computes a hopset G″ of a skeleton graph G′ of G without first computing G′ itself. We then conduct a Bellman-Ford exploration in G′ ∪ G″, while computing the required edges of G′ on the fly. As a result, our algorithm computes exactly those edges of G′ that it really needs, rather than computing approximately the entire G′.

[1]  David Peleg,et al.  Distributed verification and hardness of distributed approximation , 2010, STOC '11.

[2]  T. Lindvall ON A ROUTING PROBLEM , 2004, Probability in the Engineering and Informational Sciences.

[3]  Michael Elkin,et al.  Streaming and fully dynamic centralized algorithms for constructing and maintaining sparse spanners , 2007, TALG.

[4]  David Peleg,et al.  A near-tight lower bound on the time complexity of distributed MST construction , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[5]  Christoph Lenzen,et al.  Algebraic methods in the congested clique , 2015, Distributed Computing.

[6]  Dimitri P. Bertsekas,et al.  Data Networks , 1986 .

[7]  Sudipto Guha,et al.  Graph sketches: sparsification, spanners, and subgraphs , 2012, PODS.

[8]  A. Kemper,et al.  On Graph Problems in a Semi-streaming Model , 2015 .

[9]  Jeffrey M. Jaffe Distributed multi-destination routing: The constraints of local information , 1982, PODC '82.

[10]  Joan Feigenbaum,et al.  On graph problems in a semi-streaming model , 2005, Theor. Comput. Sci..

[11]  L. R. Ford,et al.  NETWORK FLOW THEORY , 1956 .

[12]  Michael Elkin,et al.  Unconditional lower bounds on the time-approximation tradeoffs for the distributed minimum spanning tree problem , 2004, STOC '04.

[13]  Hsin-Hao Su,et al.  Almost-Tight Distributed Minimum Cut Algorithms , 2014, DISC.

[14]  Joan Feigenbaum,et al.  Graph distances in the streaming model: the value of space , 2005, SODA '05.

[15]  Philip N. Klein,et al.  A Randomized Parallel Algorithm for Single-Source Shortest Paths , 1997, J. Algorithms.

[16]  Mihalis Yannakakis,et al.  High-probability parallel transitive closure algorithms , 1990, SPAA '90.

[17]  Kepa Korta Murua,et al.  Donostia - San Sebastián , 2009 .

[18]  Venkatesan Guruswami,et al.  Superlinear Lower Bounds for Multipass Graph Processing , 2012, Algorithmica.

[19]  Roger Wattenhofer,et al.  Networks cannot compute their diameter in sublinear time , 2012, SODA.

[20]  Baruch Awerbuch,et al.  Randomized distributed shortest paths algorithms , 1989, STOC '89.

[21]  Boaz Patt-Shamir,et al.  Distributed MST for constant diameter graphs , 2001, PODC '01.

[22]  Shay Kutten,et al.  Fast Distributed Construction of Small k-Dominating Sets and Applications , 1998, J. Algorithms.

[23]  Jian Zhang,et al.  Efficient algorithms for constructing (1+, varepsilon;, beta)-spanners in the distributed and streaming models. , 2004, PODC 2004.

[24]  David P. Dobkin,et al.  Generating Sparse Spanners for Weighted Graphs , 1990, SWAT.

[25]  Roger Wattenhofer,et al.  Optimal distributed all pairs shortest paths and applications , 2012, PODC '12.

[26]  David Peleg,et al.  Time-Optimal Leader Election in General Networks , 1990, J. Parallel Distributed Comput..

[27]  Michael Elkin,et al.  Hopsets with Constant Hopbound, and Applications to Approximate Shortest Paths , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[28]  Boaz Patt-Shamir,et al.  Fast Routing Table Construction Using Small Messages , 2012, ArXiv.

[29]  Jeffrey M. Jaffe Distributed Multi-Destination Routing: The Constraints of Local Information , 1985, SIAM J. Comput..

[30]  Michael Elkin,et al.  Distributed approximation: a survey , 2004, SIGA.

[31]  Boaz Patt-Shamir,et al.  Fast routing table construction using small messages: extended abstract , 2012, STOC '13.

[32]  Stephan Holzer,et al.  Approximation of Distances and Shortest Paths in the Broadcast Congest Clique , 2014, OPODIS.

[33]  Baruch Awerbuch,et al.  Distributed Shortest Paths Algorithms (Extended Abstract) , 1989, STOC 1989.

[34]  Surender Baswana,et al.  Dynamic Algorithms for Graph Spanners , 2006, ESA.

[35]  List of Open Problems in Sublinear Algorithms , .

[36]  Monika Henzinger,et al.  A deterministic almost-tight distributed algorithm for approximating single-source shortest paths , 2015, STOC.

[37]  Michael Elkin,et al.  Efficient algorithms for constructing very sparse spanners and emulators , 2017, SODA 2017.

[38]  Thomas H. Spencer,et al.  Time-Work Tradeoffs of the Single-Source Shortest Paths Problem , 1999, J. Algorithms.

[39]  Michael Elkin,et al.  Efficient algorithms for constructing (1+∊,β)-spanners in the distributed and streaming models , 2006, Distributed Computing.

[40]  Greg N. Frederickson,et al.  A Single Source Shortest Path Algorithm for a Planar Distributed Network , 1985, STACS.

[41]  Mihalis Yannakakis,et al.  High-Probability Parallel Transitive-Closure Algorithms , 1991, SIAM J. Comput..

[42]  Fabian Kuhn,et al.  Distributed Minimum Cut Approximation , 2013, DISC.

[43]  Polylog-time and near-linear work approximation scheme for undirected shortest paths , 2000, JACM.

[44]  David Peleg,et al.  Distributed Computing: A Locality-Sensitive Approach , 1987 .

[45]  Danupon Nanongkai,et al.  Distributed approximation algorithms for weighted shortest paths , 2014, STOC.

[46]  Christoph Lenzen,et al.  Approximate Undirected Transshipment and Shortest Paths via Gradient Descent , 2016, ArXiv.

[47]  Decision Systems.,et al.  Distributed minimum hop algorithms , 1982 .