Abstract
Distance oracles and graph spanners are excerpts of a graph that allow to compute approximate shortest paths. Here, we consider the situation where it is possible to access the original graph in addition to the graph excerpt while computing paths. This allows for asymptotically much smaller excerpts than distance oracles or spanners. The quality of an algorithm in this setting is measured by the size of the excerpt (in bits), by how much of the original graph is accessed (in number of edges), and the stretch of the computed path (as the ratio between the length of the path and the distance between its end points). Because these three objectives are conflicting goals, we are interested in a good trade-off. We measure the number of accesses to the graph relative to the number of edges in the computed path.
We present a parametrized construction that, for constant stretches, achieves excerpt sizes and number of accessed edges that are both sublinear in the number of graph vertices. We also show that within these limits, a stretch smaller than 5 cannot be guaranteed.
This work has been partially supported under the EU programme COST 295 (DYNAMO) and by the Swiss SBF under contract number C05.0047.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aleksandrov, L., Maheshwari, A., Sack, J.-R.: Determining approximate shortest paths on weighted polyhedral surfaces. Journal of the ACM 52(1), 25–53 (2005)
Alon, N., Spencer, J.: The Probabilistic Method. John Wiley, New York (1992)
Althöfer, I., Das, G., Dobkin, D.P., Joseph, D.: Generating sparse spanners for weighted graphs. In: Gilbert, J.R., Karlsson, R. (eds.) SWAT 1990. LNCS, vol. 447, pp. 26–37. Springer, Heidelberg (1990)
Baswana, S., Sen, S.: Approximate distance oracles for unweighted graphs in Õ (n 2) time. In: SODA 2004, pp. 271–280 (2004)
Bollobás, B.: Extremal Graph Theory. Academic Press, San Diego (1978)
Chvátal, V.: A greedy heuristic for the set-covering problem. Mathematics of Operations Research 4, 233–235 (1979)
Cohen, E., Halperin, E., Kaplan, H., Zwick, U.: Reachability and distance queries via 2-hop labels. SIAM Journal on Computing 32(5), 1338–1355 (2003)
Demetrescu, C., Finocchi, I., Ribichini, A.: Trading off space for passes in graph streaming problems. In: SODA 2006, pp. 714–723 (2006)
Demetrescu, C., Goldberg, A., Johnson, D. (eds.): 9th DIMACS Challenge on Shortest Paths, available at http://www.dis.uniroma1.it/~challenge9 (to appear)
Dijkstra, E.W.: A note on two problems in connection with graphs. Numerische Mathematik 1, 269–271 (1959)
Feigenbaum, J., Kannan, S., McGregor, A., Suri, S., Zhang, J.: Graph distances in the streaming model: the value of space. In: SODA 2005, pp. 745–754 (2005)
Garey, M.R., Johnson, D.S.: Computer and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)
Goldberg, A.V., Harrelson, C.: Computing the shortest path: A* search meets graph theory. In: SODA 2005, vol. 16, pp. 156–165 (2005)
Katriel, I., Meyer, U.: Elementary graph algorithms in external memory. In: Algorithms for Memory Hierarchies, pp. 62–84 (2002)
Meyer, U., Zeh, N.: I/O-efficient undirected shortest paths. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 434–445. Springer, Heidelberg (2003)
Peleg, D., Schäffer, A.A.: Graph spanners. Journal of Graph Theory 13(1), 99–116 (1989)
Prüfer, H.: Neuer Beweis eines Satzes über Permutationen. Arch. Math. Phys. 27, 742–744 (1918)
Regev, H.: The weight of the greedy graph spanner. Technical Report CS95-22, Weizmann Institute Of Science (July 1995)
Sanders, P., Schultes, D.: Highway hierarchies hasten exact shortest path queries. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 568–579. Springer, Heidelberg (2005)
Thorup, M., Zwick, U.: Approximate distance oracles. Journal of the ACM 52(1), 1–24 (2005)
Zwick, U.: Exact and approximate distances in graphs - a survey. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 33–48. Springer, Heidelberg (2001)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Derungs, J., Jacob, R., Widmayer, P. (2007). Approximate Shortest Paths Guided by a Small Index. In: Dehne, F., Sack, JR., Zeh, N. (eds) Algorithms and Data Structures. WADS 2007. Lecture Notes in Computer Science, vol 4619. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73951-7_48
Download citation
DOI: https://doi.org/10.1007/978-3-540-73951-7_48
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73948-7
Online ISBN: 978-3-540-73951-7
eBook Packages: Computer ScienceComputer Science (R0)