Approximate distance oracles

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Approximate distance oracles

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Annual ACM Symposium on Theory of Computing archive
Proceedings of the thirty-third annual ACM symposium on Theory of computing table of contents

Hersonissos, Greece

Pages: 183 - 192

Year of Publication: 2001

ISBN:1-58113-349-9

Authors

Mikkel Thorup
Uri Zwick

Sponsor

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

ACM New York, NY, USA

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Downloads (6 Weeks): 14, Downloads (12 Months): 60, Citation Count: 42

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ABSTRACT

Let G=(V,E) be an undirected weighted graph with |V|=n and |E|=m. Let k\ge 1 be an integer. We show that G=(V,E) can be preprocessed in O(kmn^{1/k}) expected time, constructing a data structure of size O(kn^{1+1/k}), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k-1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k-1. We show that a 1963 girth conjecture of Erd{\H{o}}s, implies that &ohgr(n^{1+1/k}) space is needed in the worst case for any real stretch strictly smaller than 2k+1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name ``oracle''. Previously, data structures that used only O(n^{1+1/k}) space had a query time of &ohgr(n^{1/k}) and a slightly larger, non-optimal, stretch. Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.}

REFERENCES

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	1	D. Aingworth , C. Chekuri , P. Indyk , R. Motwani, Fast Estimation of Diameter and Shortest Paths (Without Matrix Multiplication), SIAM Journal on Computing, v.28 n.4, p.1167-1181, Aug. 1999 [doi>10.1137/S0097539796303421]
	2	N. Alon, S. Hoory, and N. Linial. The Moore bound for irregular graphs. Submitted for publication, 2000.
	3	N. Alon and M. Naor. Derandomization, witnesses for boolean matrix multiplication, and construction of perfect hash functions. Algorithmica, 16:434-449, 1996.
	4	N. Alon and J. Spencer. The probabilistic method. Wiley, 1992.
	5	Ingo Althöfer , Gautam Das , David Dobkin , Deborah Joseph , José Soares, On sparse spanners of weighted graphs, Discrete & Computational Geometry, v.9 n.1, p.81-100, 1993 [doi>10.1007/BF02189308]
	6	Srinivasa Rao Arikati , Danny Z. Chen , L. Paul Chew , Gautam Das , Michiel H. M. Smid , Christos D. Zaroliagis, Planar Spanners and Approximate Shortest Path Queries among Obstacles in the Plane, Proceedings of the Fourth Annual European Symposium on Algorithms, p.514-528, September 25-27, 1996
	7	Baruch Awerbuch , Bonnie Berger , Lenore Cowen , David Peleg, Near-Linear Time Construction of Sparse Neighborhood Covers, SIAM Journal on Computing, v.28 n.1, p.263-277, Feb. 1999 [doi>10.1137/S0097539794271898]
	8	Baruch Awerbuch , David Peleg, Routing with polynomial communication-space trade-off, SIAM Journal on Discrete Mathematics, v.5 n.2, p.151-162, May 1992 [doi>10.1137/0405013]
	9	Yair Bartal, On approximating arbitrary metrices by tree metrics, Proceedings of the thirtieth annual ACM symposium on Theory of computing, p.161-168, May 24-26, 1998, Dallas, Texas, United States [doi>10.1145/276698.276725]
	10	C. Benson. Minimal regular graphs of girth eight and twelve. Canadian Journal of Mathematics, 18:1091-1094, 1966.
	11	Bela Bollobas, Extremal Graph Theory, Dover Publications, Incorporated, 2004
	12	J. Bondy and M. Simonovits. Cycles of even length in graphs. Journal of Combinatorial Theory, Series B, 16:97-105, 1974.
	13	J. Bourgain. On libschitz embedding of finite metric spaces in hilbert space. Israel J. Math., 52:46-52, 1985.
	14	W. Brown. On graphs that do not contain a Thomsen graph. Canad. Math. Bull., 9:281-285, 1966.
	15	J. L. Carter and M. N. Wegman. Universal classes of hash functions. J. Comput. Syst. Sc., 18:143-154, 1979.
	16	S. Chaudhuri and C. Zaroliagis. Shortest paths in digraphs of small treewidth. Part I: Sequential algorithms. Algorithmica, 27:212-226, 2000.
	17	Yi-Jen Chiang , Joseph S. B. Mitchell, Two-point Euclidean shortest path queries in the plane, Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms, p.215-224, January 17-19, 1999, Baltimore, Maryland, United States
	18	Edith Cohen, Fast Algorithms for Constructing t-Spanners and Paths with Stretch t, SIAM Journal on Computing, v.28 n.1, p.210-236, Feb. 1999 [doi>10.1137/S0097539794261295]
	19	Edith Cohen , Uri Zwick, All-pairs small-stretch paths, Journal of Algorithms, v.38 n.2, p.335-353, Feb. 2001 [doi>10.1006/jagm.2000.1117]
	20	Thomas T. Cormen , Charles E. Leiserson , Ronald L. Rivest, Introduction to algorithms, MIT Press, Cambridge, MA, 1990
	21	Martin Dietzfelbinger , Torben Hagerup , Jyrki Katajainen , Martti Penttonen, A reliable randomized algorithm for the closest-pair problem, Journal of Algorithms, v.25 n.1, p.19-51, Oct. 1997 [doi>10.1006/jagm.1997.0873]
	22	M. Dietzfelbinger and M. Hune. A dictionary implementaion based on dynamic perfect hashing, 1996. DIMACS 1996 Implementation Challenge. To appear in AMS-DIMACS Series in Discrete Mathematics and Theoretical Computer Science.
	23	E. Dijkstra. A note on two problems in connexion with graphs. Numerische Mathematik, 1:269-271, 1959.
	24	Hristo Djidjev, On-Line Algorithms for Shortest Path Problems on Planar Digraphs, Proceedings of the 22nd International Workshop on Graph-Theoretic Concepts in Computer Science, p.151-165, June 12-14, 1996
	25	Dorit Dor , Shay Halperin , Uri Zwick, All-Pairs Almost Shortest Paths, SIAM Journal on Computing, v.29 n.5, p.1740-1759, March 2000 [doi>10.1137/S0097539797327908]
	26	M. Elkin, Computing Almost Shortest Paths,, Weizmann Science Press of Israel, Jerusalem, Israel, 2001
	27	Michael Elkin , David Peleg, (1 + &egr;&Bgr;)-spanner constructions for general graphs, Proceedings of the thirty-third annual ACM symposium on Theory of computing, p.173-182, July 2001, Hersonissos, Greece [doi>10.1145/380752.380797]
	28	P. Erd os. Extremal problems in graph theory. In Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), pages 29-36. Publ. House Czechoslovak Acad. Sci., Prague, 1964.
	29	P. Erdos, A. Renyi, and V. Sos. On a problem of graph theory. Studia Sci. Math. Hungar., 1:215-235, 1966.
	30	Michael L. Fredman , János Komlós , Endre Szemerédi, Storing a Sparse Table with 0(1) Worst Case Access Time, Journal of the ACM (JACM), v.31 n.3, p.538-544, July 1984 [doi>10.1145/828.1884]
	31	Michael L. Fredman , Robert Endre Tarjan, Fibonacci heaps and their uses in improved network optimization algorithms, Journal of the ACM (JACM), v.34 n.3, p.596-615, July 1987 [doi>10.1145/28869.28874]
	32	Cyril Gavoille , David Peleg , Stéphane Pérennes , Ran Raz, Distance labeling in graphs, Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, p.210-219, January 07-09, 2001, Washington, D.C., United States
	33	S. Halperin and U. Zwick. Unpublished result, 1996.
	34	Piotr Indyk, Sublinear time algorithms for metric space problems, Proceedings of the thirty-first annual ACM symposium on Theory of computing, p.428-434, May 01-04, 1999, Atlanta, Georgia, United States [doi>10.1145/301250.301366]
	35	David R. Karger , Daphne Koller , Steven J. Phillips, Finding the hidden path: time bounds for all-pairs shortest paths, SIAM Journal on Computing, v.22 n.6, p.1199-1217, Dec. 1993 [doi>10.1137/0222071]
	36	J. Kruskal. On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical Society, 7:48-50, 1956.
	37	Felix Lazebnik , Vasiliy A. Ustimenko, New examples of graphs without small cycles and of large size, European Journal of Combinatorics, v.14 n.5, p.445-460, Sept. 1993 [doi>10.1006/eujc.1993.1048]
	38	F. Lazebnik, V. Ustimenko, and A. Woldar. A new series of dense graphs of high girth. Bulletin of the American Mathematical Society (New Series), 32(1):73-79, 1995.
	39	F. Lazebnik , V. A. Ustimenko , A. J. Woldar, A characterization of the components of the graphs D(k,q), Discrete Mathematics, v.157 n.1-3, p.271-283, Oct. 1, 1996
	40	N. Linial, E. London, and Y. Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15:215-245, 1995.
	41	A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8:261-277, 1988.
	42	G. Margulis. Explicit group-theoretical construction of combinatorial schemes and their application to the design of expanders and concentrators. Problems of Information Transmission, 24:39-46, 1988.
	43	C. McGeoch. All-pairs shortest paths and the essential subgraph. Algorithmica, 13:426-461, 1995.
	44	David Peleg, Distributed computing: a locality-sensitive approach, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2000
	45	D. Peleg. Proximity-preserving labeling schemes. J. Graph Theory, 33:167-176, 2000.
	46	D. Peleg and A. Schaffer. Graph spanners. J. Graph Theory, 13:99-116, 1989.
	47	I. Reiman. Uber ein Problem von K. Zarankiewicz. Acta. Math. Acad. Sci. Hungar., 9:269-273, 1958.
	48	Avi Shoshan , Uri Zwick, All Pairs Shortest Paths in Undirected Graphs with Integer Weights, Proceedings of the 40th Annual Symposium on Foundations of Computer Science, p.605, October 17-18, 1999
	49	Mikkel Thorup, Undirected single-source shortest paths with positive integer weights in linear time, Journal of the ACM (JACM), v.46 n.3, p.362-394, May 1999 [doi>10.1145/316542.316548]
	50	Mikkel Thorup, Even strongly universal hashing is pretty fast, Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms, p.496-497, January 09-11, 2000, San Francisco, California, United States
	51	Mikkel Thorup, Floats, integers, and single source shortest paths, Journal of Algorithms, v.35 n.2, p.189-201, May 2000 [doi>10.1006/jagm.2000.1080]
	52	M. Thorup. Compact oracles for reachability and approximate distances in planar digraphs, 2001. Submitted.
	53	Mikkel Thorup, Quick k-Median, k-Center, and Facility Location for Sparse Graphs, Proceedings of the 28th International Colloquium on Automata, Languages and Programming,, p.249-260, July 08-12, 2001
	54	Mikkel Thorup , Uri Zwick, Compact routing schemes, Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures, p.1-10, July 2001, Crete Island, Greece [doi>10.1145/378580.378581]
	55	J. Tits. Sur la trialite et certains groupes qui s'en d eduisent. Publ. Math. I.H.E.S., 2:14-20, 1959.
	56	R. Wenger, Extremal graphs with no C4,s, or C10,s, Journal of Combinatorial Theory Series B, v.52 n.1, p.113-116, May 1991 [doi>10.1016/0095-8956(91)90097-4]
	57	J. Williams. Heapsort. Comm. ACM, 7(5):347-348, 1964.
	58	A. Woldar and V. Ustimenko. An application of group theory to extremal graph theory. In Group theory, Proceedings of the Ohio State-Denison Conference, pages 293-298. World Sci. Publishing, River Edge, NJ, 1993.
	59	Uri Zwick, All Pairs Shortest Paths in weighted directed graphs ? exact and almost exact algorithms, Proceedings of the 39th Annual Symposium on Foundations of Computer Science, p.310, November 08-11, 1998

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