Δ-stepping: A parallelizable shortest path algorithm

Journal of Algorithms

Volumn 49, Issue 1, 2003, Pages 114-152

  Meyer, U ^a   Sanders, P ^a  

^a MAX PLANCK INSTITUTE FOR INFORMATICS   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

GRAPH THEORY; ITERATIVE METHODS; OPTIMIZATION; PROBABILITY; PROBLEM SOLVING; RANDOM ACCESS STORAGE;

DELTA-STEPPING; SINGLE-SOURCE SHORTEST-PATH PROBLEM;

PARALLEL ALGORITHMS;

EID: 0242583646     PISSN: 01966774     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0196-6774(03)00076-2     Document Type: Article

References (93)

1. P. Adamson, E. Tick, Greedy partitioned algorithms for the shortest path problem, Internat. J. Parallel Program. 20 (4) (1991) 271-298.
2. R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network flows: Theory, Algorithms, and Applications, Prentice Hall, 1993.
3. R.K. Ahuja, K. Mehlborn, J.B. Orlin, R.E. Tarjan, Faster algorithms for the shortest path problem, J. ACM 37 (2) (1990) 213-223.
4. P. Alefragis, P. Sanders, T. Takkula, D. Wedelin, Parallel integer optimization for crew scheduling, Ann. Oper. Res. 99 (1) (2000) 141-166.
5. N. Alon, J.H. Spencer, P. Erdõs, The Probabilistic Method, Wiley, 1992.
6. D. Angluin, L.G. Valiant, Fast probabilistic algorithms for Hamiltonian paths and matchings, J. Comput. System Sci. 18 (1979) 155-193.
7. M.J. Atallah, D.Z. Chen, O. Daescu, Efficient parallel algorithms for planar st-graphs, in: Proc. 8th Int. Symp. Algorithms and Computation, in: Lecture Notes in Comput. Sci., Vol. 1350, Springer-Verlag, 1997, pp. 223-232.
8. K.B. Athreya, Large deviation rates for branching processes-I, single type case, Ann. Appl. Probab. 4 (3) (1994) 779-790.
9. K. B. Athreya, P. Ney, Branching Processes, Springer-Verlag, 1972.
10. H. Bast, T. Hagerup, Fast and reliable parallel hashing, in: 3rd Symposium on Parallel Algorithms and Architectures, 1991, pp. 50-61.
11. R. Bellman, On a routing problem, Quart. Appl. Math. 16 (1958) 87-90.
12. B. Bollobás, Random Graphs, Academic Press, 1985.
13. G.S. Brodal, J.L. Tráff, C.D. Zarogliagis, A parallel priority queue with constant time operations, J. Parallel Distrib. Comput. 49 (1) (1998) 4-21.
14. K. M. Chandy, J. Misra, Distributed computation on graphs: Shortest path algorithms, Comm. ACM 25 (11) (1982) 833-837.
15. S. Chaudhuri, C.D. Zarogliagis, Shortest paths in digraphs of small treewidth. Part II: Optimal parallel algorithms, Theoret. Comput. Sci. 203 (2) (1998) 205-223.
16. S. Chaudhuri, C.D. Zarogliagis, Shortest paths in digraphs of small treewidth. Part I: Sequential algorithms, Algorithmica 27 (3-4) (2000) 212-226.
17. B.V. Cherkassky, A.V. Goldberg, T. Radzik, Shortest path algorithms: Theory and experimental evaluation, Math. Programming 73 (1996) 129-174.
18. A. Clementi, J. Rolim, E. Urland, Randomized parallel algorithms, in: Solving Combinatorial Problems in Parallel, in: Lecture Notes in Comput. Sci., Vol. 1054, 1996, pp. 25-50.
19. E. Cohen, Using selective path-doubling for parallel shortest-path computations, J. Algorithms 22 (1) (1997) 30-56.
20. E. Cohen, Polylog-time and near-linear work approximation scheme for undirected shortest paths, J. ACM 47 (2000) 132-166.
21. J. Cooper, A. Frieze, K. Mehlhorn, V. Priebe, Average-case complexity of shortest-paths problems in the vertex-potential model, Random Structures Algorithms 16 (2000) 33-46.
22. A. Crauser, K. Mehlhorn, U. Meyer, P. Sanders, A parallelization of Dijkstra's shortest path algorithm, in: Proc. 23rd Symp. on Mathematical Foundations of Computer Science, in: Lecture Notes in Comput. Sci., Vol. 1450, Springer-Verlag, 1998, pp. 722-731.
23. D.E. Culler, R.M. Karp, D. Patternson, A. Sahay, E.E. Santos, K.E. Schauser, R. Subramanian, T. von Eicken, LogP: A practical model of parallel computation, Comm. ACM 39 (11) (1996) 78-85.
24. E. Dekel, D. Nassimi, S. Sahni, Parallel matrix and graph algorithms, SIAM J. Comput. 10 (4) (1981) 61-67.
25. E.V. Denardo, B.L. Fox, Shortest route methods: 1. Reaching pruning and buckets, Oper. Res. 27 (1979) 161-186.
26. R.B. Dial, Algorithm 360: Shortest-path forest with topological ordering, Comm. ACM 12 (11) (1969) 632-633.
27. R. B. Dial, F. Glover, D. Karney, D. Klingman, A computational analysis of alternative algorithms and labeling techniques for finding shortest path trees, Networks 9 (1979) 215-248.
28. 2, in: RANDOM: International Workshop on Randomization and Approximation Techniques in Computer Science, in: Lecture Notes in Comput. Sci., Vol. 1518, Springer-Verlag, 1998, pp. 294-306.
29. E.W. Dijkstra, A note on two problems in connexion with graphs, Num. Math. 1 (1959) 269-271.
30. E.A. Dinitz, Economical algorithms for finding shortest paths in a network, in: Transportation Modeling Systems, 1978, pp. 36-44.
31. J.R. Driscoll, H.N. Gabow, R. Shrairman, R.E. Tarjan, Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation, Comm. ACM 31 (11) (1988) 1343-1354.
32. D.P. Dubhashi, A. Panconesi, Concentration of measure for the analysis of randomized algorithms, Draft Manuscript, http://www.brics.dk/~ale/papers.html, October, 1998.
33. P. Erdõs, A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5(A) (1960) 17-61.
34. L.R. Ford, D. R. Fulkerson, Flows in Networks, Princeton Univ. Press, Princeton, NJ, 1963.
35. A. Formella, J. Keller, T. Walle, HPP: A high performance PRAM, in: Proc. Euro-Par 1996 Parallel Processing, in: Lecture Notes in Comput. Sci., Vol. 1124 II, Springer-Verlag, 1996, pp. 425-434.
36. S. Fortune, J. Wyllie, Parallelism in random access memories, in: Proc. 10th Symp. on the Theory of Computing, ACM, 1978, pp. 114-118.
37. M. L. Fredman, R.E. Tarjan, Fibonacci heaps their uses in improved network optimization algorithms, J. ACM 34 (1987) 596-615.
38. M.L. Fredman, D.E. Willard, Trans-dichotomous algorithms for minimum spanning trees and shortest paths, J. Comput. System Sci. 48 (1994) 533-551.
39. A.M. Frize, G.R. Grimmett, The shortest-path problem for graphs with random arc-lengths, Discrete Appl. Math. 10 (1985) 57-77.
40. A.M. Frieze, L. Rudolph, A parallel algorithm for all-pairs shortest paths in a random graph, in: Proc. 22nd Allerton Conference on Communication, Control and Computing, 1985, pp. 663-670.
41. G. Gallo, S. Pallotino, Shortest path methods: A unifying approach, Math. Programming Study 26 (1986) 38-64.
42. G. Gallo, S. Pallotino, Shortest path algorithms, Ann. Oper. Res. 13 (1988) 3-79.
43. A.V. Gerbessiotis, L.G. Valiant, Direct bulk-synchronous parallel algorithms, J. Parallel Distrib. Comput. 22 (2) (1994) 251-267.
44. F. Glover, R. Glover, D. Klingman, Computational study of an improved shortest path algorithm, Networks 14 (1984) 23-37.
45. A.V. Goldberg, A simple shortest path algorithm with linear average time, in: Proc. 9th Ann. European Symposium on Algorithms (ESA), in: Lecture Notes in Comput. Sci., Vol. 2161, Springer-Verlag, 2001, pp. 230-241.
46. A.V. Goldberg, R.E. Tarjan, Expected performance of Dijkstra's shortest path algorithm, Technical Report TR-96-062, NEC Research, 1996.
47. L.M. Goldschlager, A unified approach to models of synchronous parallel machines, J. ACM 29 (4) (1982) 1073-1086.
48. Q.P. Gu, T. Takaoka, A sharper analysis of a parallel algorithm for the all pairs shortest path problem, Parallel Comput. 16 (1) (1990) 61-67.
49. T. Hagerup, Improved shortest paths on the word RAM, in: 27th Colloquium on Automata, Languages and Programming (ICALP), in: Lecture Notes in Comput. Sci., Vol. 1853, Springer-Verlag, 2000, pp. 61-72.
50. Y. Han, V. Pan, J. Reif, Efficient parallel algorithms for computing all pair shortest paths in directed graphs, Algorithmica 17 (4) (1997) 399-415.
51. T. Harris, The Theory of Branching Processes, Springer-Verlag, 1963.
52. R. Hassin, E. Zemel, On shortest paths in graphs with random weights, Math. Oper. Res. 10 (4) (1985) 557-564.
53. M.R. Henzinger, P. Klein, S. Rao, S. Subramanian, Faster shortest-path algorithms for planar graphs, J. Comput. System Sci. 55 (1) (1997) 3-23.
54. M.R. Hribar, V.E. Taylor, Performance study of parallel shortest path algorithms: Characteristics of good decompositions, in: Proc. 13th Ann. Conf. Intel Supercomputers Users Group (ISUG), 1997.
55. M.R. Hribar, V.E. Taylor, D.E. Boyce, Reducing the idle time of parallel shortest path algorithms, Technical Report CPDC-TR-9803-016, Northwestern University, Evanson, IL, 1998, J. Parallel Distrib. Comput. submitted for publication.
56. M.R. Hribar, V.E. Taylor, D.E. Boyce, Termination detection for parallel shortest path algorithms, J. Parallel Distrib. Comput. 55 (1998) 153-165.
57. M.S. Hung, J.J. Divoky, A computational study of efficient shortest path algorithms, Comput. Oper. Res. 15 (6) (1988) 567-576.
58. J. Jájá, An Introduction to Parallel Algorithms, Addison-Wesley, Reading, 1992.
59. R.M. Karp, The transitive closure of a random digraph, Random Structures Algorithms 1 (1) (1990) 73-93.
60. R.M. Karp, Y. Zhang, Bounded branching process and AND/OR tree evaluation, Random Structures Algorithms 7 (2) (1995) 97-116.
61. P.N. Klein, S. Subramanian, A linear-processor polylog-time algorithm for shortest paths in planar graphs, in: Proc. 34th Annual Symposium on Foundations of Computer Science, IEEE, 1993, pp. 259-270.
62. P.N. Klein, S. Subramanian, A randomized parallel algorithm for single-source shortest paths, J. Algorithms 25 (2) (1997) 205-220.
63. W.F. McColl, Universal computing, in: L. Bouge, P. Fraigniaud, A. Mignotte, Y. Robert (Eds.), Proc. Euro-Par '96 Parallel Processing, in: Lecture Notes in Comput. Sci., Vol. 1123, Springer-Verlag, 1996, pp. 25-36.
64. C. McDiarmid, Concentration, in: M. Habib, C. McDiarmid, J. Ramirez-Alfonsin, B. Reed (Eds.), in: Algorithms Combin., Vol. 16, Springer-Verlag, 1998, pp. 195-248.
65. K. Mehlhorn, V. Priebe, On the all-pairs shortest-path algorithm of Moffat and Takaoka, Random Structures Algorithms 10 (1997) 205-220.
66. U. Meyer, Single-source shortest-paths on arbitrary directed graphs in linear average-case time, in: Proc. 12th Ann. Symp. on Discrete Algorithms, ACM-SIAM, 2001, pp. 797-806.
67. Y. Meyer, P. Sanders, Δ-stepping: A parallel single source shortest path algorithm, in: Proc. 6th Ann. European Symposium on Algorithms (ESA), in: Lecture Notes in Comput. Sci., Vol. 1461, Springer-Verlag, 1998, pp. 393-404.
68. U. Meyer, P. Sanders, Parallel shortest path for arbitrary graphs, in: Proc. Euro-Par 2000 Parallel Processing, in: Lecture Notes in Comput. Sci., Vol. 1900, Springer-Verlag, 2000, pp. 461-470.
69. G.L. Miller, J.H. Reif, Parallel tree contraction, Part 1: Fundamentals, in: S. Micali (Ed.), Advances in Computing Research 5: Randomness and Computation, JAI Press, 1989.
70. 2log n), SIAM J. Comput. 16 (1987) 1023-1031.
71. J.-F. Mondou, T.G. Crainic, S. Nugyen, Shortest path algorithms: A computational study with the C programming language, Comput. Oper. Res. 18 (1991) 767-786.
72. R. Motwani, P. Raghavan, Randomized Algorithms, Cambridge Univ. Press, 1995.
73. K. Mulmuley, P. Shah, A lower bound for the shortest path problem, in: Proc. 15th Annual Conference on Computational Complexity, IEEE, 2000, pp. 14-21.
74. K. Noshita, A theorem on the expected complexity of Dijkstra's shortest path algorithm, J. Algorithms 6 (1985) 400-408.
75. R.C. Paige, C.P. Kruskal, Parallel algorithms for shortest path problems, in: International Conference on Parallel Processing, IEEE, 1985, pp. 14-20.
76. S. Rajasekaran, J.H. Reif, Optimal and sublogarithmic time randomized parallel sorting algorithms, SIAM J. Comput. 18 (3) (1989) 594-607.
77. R. Raman, Priority queues: Small, monotone and trans-dichotomous, in: 4th Annual European Symposium on Algorithms (ESA), in: Lecture Notes in Comput. Sci., Vol. 1136, Springer-Verlag 1996, pp. 121-137.
78. R. Raman, Recent results on the single-source shortest paths problem, ACM SIGACT News 28 (2) (1997) 81-87.
79. J. Reif, P. Spirakis, Expected parallel time and sequential space complexity of graph and digraph problems. Algorithmica 7 (1992) 597-630.
80. R. Sedgewick, J.S. Vitter, Shortest paths in euclidean graphs, Algorithmica 1 (1986) 31-48.
81. H. Shi, T.H. Spencer, Time-work tradeoffs of the single-source shortest paths problem, J. Algorithms 30 (1) (1999) 19-32.
82. M. Snir, S.W. Otto, S. Huss-Lederman, D.W. Walker, J. Dongarra, MPI-The Complete Reference, MIT Press, 1996.
83. 2n), SIAM J. Comput. 2 (1973) 28-32.
84. S. Subramanian, R. Tamassia, J.S. Vitter, An efficient parallel algorithm for shortest paths in planar layered digraphs, Algorithmica 14 (4) (1995) 322-339.
85. M. Thorup, Undirected single-source shortest paths with positive integer weights in linear time, J. ACM 46 (1999) 362-394.
86. M. Thorup, Floats, integers, and single source shortest paths, J. Algorithms 35 (2000) 189-201.
87. M. Thorup, On RAM priority queues, SIAM J. Comput. 30 (2000) 86-109.
88. J.L. Träff, An experimental comparison of two distributed single-source shortest path algorithms, Parallel Comput. 21 (1995) 1505-1532.
89. J.L. Träff, C.D. Zaroliagis, A simple parallel algorithm for the single-source shortest path problem on planar digraphs, J. Parallel Distrib. Comput. 60 (9) (2000) 1103-1124.
90. J.D. Ullman, M. Yannakakis, High-probability parallel transitive closure algorithms, SIAM J. Comput. 20(1) (1991) 100-125.
91. L.G. Valiant, A bridging model for parallel computation, Comm. ACM 33 (8) (1990) 103-111.
92. L.G. Valiant, General purpose parallel architectures, in: J. van Leeuwen (Ed.), Handbook of Theoretical Computer Science, Vol. A: Algorithms and Complexity, Elsevier, 1990, pp. 943-971.
93. F.B. Zhan, C.E. Noon, Shortest path algorithms: An evaluation using real road networks, Transp. Sci. 32 (1998) 65-73.