ABSTRACT
We use exponential start time clustering to design faster parallel graph algorithms involving distances. Previous algorithms usually rely on graph decomposition routines with strict restrictions on the diameters of the decomposed pieces. We weaken these bounds in favor of stronger local probabilistic guarantees. This allows more direct analyses of the overall process, giving:
Linear work parallel algorithms that construct spanners with O(k) stretch and size O(n1+1/k) in unweighted graphs, and size O(n1+1/k log k) in weighted graphs.
Hopsets that lead to the first parallel algorithm for approximating shortest paths in undirected graphs with O(m poly log n) work.
- Ingo Althöfer, Gautam Das, David Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discrete Comput. Geom., 9(1):81--100, January 1993. Google ScholarDigital Library
- N. Alon, R. Karp, D. Peleg, and D. West. A graph-theoretic game and its application to the k-server problem. SIAM J. Comput., 24(1):78--100, 1995. Google ScholarDigital Library
- Baruch Awerbuch. Complexity of network synchronization. J. Assoc. Comput. Mach., 32(4):804--823, 1985. Google ScholarDigital Library
- Yair Bartal. Probabilistic approximation of metric spaces and its algorithmic applications. In Foundations of Computer Science, 1996. Proceedings., 37th Annual Symposium on, pages 184--193. IEEE, 1996. Google ScholarDigital Library
- Guy E. Blelloch, Anupam Gupta, Ioannis Koutis, Gary L. Miller, Richard Peng, and Kanat Tangwongsan. Nearly-linear work parallel SDD solvers, low-diameter decomposition, and low-stretch subgraphs. Theory Comput. Syst., 55(3):521--554, 2014. Google ScholarDigital Library
- Surender Baswana, Telikepalli Kavitha, Kurt Mehlhorn, and Seth Pettie. New constructions of (α,β)-spanners and purely additive spanners. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '05, pages 672--681, Philadelphia, PA, USA, 2005. Society for Industrial and Applied Mathematics. Google ScholarDigital Library
- Surender Baswana, Telikepalli Kavitha, Kurt Mehlhorn, and Seth Pettie. Additive spanners and (α, β)-spanners. ACM Transactions on Algorithms (TALG), 7(1):5, 2010. Google ScholarDigital Library
- Surender Baswana and Sandeep Sen. A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. Random Struct. Algorithms, 30(4):532--563, July 2007. Google ScholarDigital Library
- Michael B. Cohen, Gary L. Miller, Jakub W. Pachocki, Richard Peng, and Shen Chen Xu. Stretching stretch. CoRR, abs/1401.2454, 2014.Google Scholar
- Edith Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. Comput., 28(1):210--236, 1998. Google ScholarDigital Library
- Edith Cohen. Polylog-time and near-linear work approximation scheme for undirected shortest paths. J. ACM, 47(1):132--166, 2000. Google ScholarDigital Library
- Dorit Dor, Shay Halperin, and Uri Zwick. All-pairs almost shortest paths. SIAM J. Comput., 29(5):1740--1759, March 2000. Google ScholarDigital Library
- Joseph Gil, Yossi Matias, and Uzi Vishkin. Towards a theory of nearly constant time parallel algorithms. In FOCS, pages 698--710. IEEE Computer Society, 1991. Google ScholarDigital Library
- Ioannis Koutis. Simple parallel and distributed algorithms for spectral graph sparsification. In Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '14, pages 61--66, New York, NY, USA, 2014. ACM. Google ScholarDigital Library
- Philip N Klein and Sairam Subramanian. A randomized parallel algorithm for single-source shortest paths. Journal of Algorithms, 25(2):205--220, 1997. Google ScholarDigital Library
- Howard Karloff, Siddharth Suri, and Sergei Vassilvitskii. A model of computation for mapreduce. In Proceedings of the Twenty-first Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '10, pages 938--948, Philadelphia, PA, USA, 2010. Society for Industrial and Applied Mathematics. Google ScholarDigital Library
- Gary L. Miller, Richard Peng, Adrian Vladu, and Shen Chen Xu. Improved parallel algorithms for spanners and hopsets. CoRR, abs/1309.3545, 2014.Google Scholar
- Gary L. Miller, Richard Peng, and Shen Chen Xu. Parallel graph decompositions using random shifts. In Proceedings of the 25th ACM symposium on Parallelism in algorithms and architectures, SPAA '13, pages 196--203, New York, NY, USA, 2013. ACM. Google ScholarDigital Library
- David Peleg. Distributed Computing: A Locality-sensitive Approach. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2000. Google ScholarCross Ref
- Seth Pettie. Distributed algorithms for ultrasparse spanners and linear size skeletons. In Proceedings of the Twenty-seventh ACM Symposium on Principles of Distributed Computing, PODC '08, pages 253--262, New York, NY, USA, 2008. ACM. Google ScholarDigital Library
- David Peleg and Alejandro A. Schäffer. Graph spanners. Journal of Graph Theory, 13(1):99--116, 1989.Google ScholarCross Ref
- Liam Roditty, Mikkel Thorup, and Uri Zwick. Deterministic constructions of approximate distance oracles and spanners. In Proceedings of the 32nd International Conference on Automata, Languages and Programming, ICALP'05, pages 261--272, Berlin, Heidelberg, 2005. Springer-Verlag. Google ScholarDigital Library
- Julian Shun, Laxman Dhulipala, and Guy Blelloch. A simple and practical linear-work parallel algorithm for connectivity. In Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '14, pages 143--153, New York, NY, USA, 2014. ACM. Google ScholarDigital Library
- Hanmao Shi and Thomas H. Spencer. Time-work tradeoffs of the single-source shortest paths problem. J. Algorithms, 30(1):19--32, January 1999. Google ScholarDigital Library
- Mikkel Thorup and Uri Zwick. Approximate distance oracles. J. ACM, 52(1):1--24, Jan 2005. Google ScholarDigital Library
- J. Ullman and M. Yannakakis. High-probability parallel transitive-closure algorithms. SIAM Journal on Computing, 20(1):100--125, 1991. Google ScholarDigital Library
Index Terms
- Improved Parallel Algorithms for Spanners and Hopsets
Recommendations
Parallel Planar Subgraph Isomorphism and Vertex Connectivity
SPAA '20: Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and ArchitecturesWe present the first parallel fixed-parameter algorithm for subgraph isomorphism in planar graphs, bounded-genus graphs, and, more generally, all minor-closed graphs of locally bounded treewidth. Our randomized low depth algorithm has a near-linear work ...
L(2,1)-labeling of dually chordal graphs and strongly orderable graphs
An L(2,1)-labeling of a graph G=(V,E) is a function f:V(G)->{0,1,2,...} such that |f(u)-f(v)|>=2 whenever uv@__ __E(G) and |f(u)-f(v)|>=1 whenever u and v are at distance two apart. The span of an L(2,1)-labeling f of G, denoted as SP"2(f,G), is the ...
Fast Connected Components Algorithms for the EREW PRAM
We present fast and efficient parallel algorithms for finding the connected components of an undirected graph. These algorithms run on the exclusive-read, exclusive-write (EREW) PRAM. On a graph with n vertices and m edges, our randomized algorithm ...
Comments