Surender Baswana
Skip Abstract Section
Abstract
Spanner of an undirected graph G = (V,E) is a subgraph that is sparse and yet preserves all-pairs distances approximately. More formally, a spanner with stretch t ∈ ℕ is a subgraph (V,ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. Though G is trivially a t-spanner of itself, the research as well as applications of spanners invariably deal with a t-spanner that has as small number of edges as possible.
We present fully dynamic algorithms for maintaining spanners in centralized as well as synchronized distributed environments. These algorithms are designed for undirected unweighted graphs and use randomization in a crucial manner.
Our algorithms significantly improve the existing fully dynamic algorithms for graph spanners. The expected size (number of edges) of a t-spanner maintained at each stage by our algorithms matches, up to a polylogarithmic factor, the worst case optimal size of a t-spanner. The expected amortized time (or messages communicated in distributed environment) to process a single insertion/deletion of an edge by our algorithms is close to optimal.
- Abraham, I., Bartal, Y., Chan, H. T.-H., Dhamdhere, K., Gupta, A., Kleinberg, J. M., Neiman, O., and Slivkins, A. 2005. Metric embeddings with relaxed guarantees. In Proceedings of the 46th Annual (IEEE) Symposium on Foundations of Computer Science (FOCS). 83--100. Google Scholar
Digital Library
- Althöfer, I., Das, G., Dobkin, D. P., Joseph, D., and Soares, J. 1993. On sparse spanners of weighted graphs. Disc. Computat. Geom. 9, 81--100. Google ScholarDigital Library
- Ausiello, G., Franciosa, P. G., and Italiano, G. F. 2005. Small stretch spanners on dynamic graphs. In Proceedings of the 13th Annual European Symposium on Algorithms (ESA). 532--543. Google ScholarDigital Library
- Baswana, S. 2008. Streaming algorithm for graph spanners - single pass and constant processing time per edge. Inf. Process. Lett. 106, 3, 110--114. Google ScholarDigital Library
- Baswana, S. and Kavitha, T. 2010. Faster algorithms for all-pairs approximate shortest paths in undirected graphs. SIAM J. Comput. 39, 7, 2865--2896. Google ScholarDigital Library
- Baswana, S., Kavitha, T., Mehlhorn, K., and Pettie, S. 2010. Additive spanners and (alpha, beta)-spanners. ACM Trans. Algorithms 7, 1, 5. Google ScholarDigital Library
- Baswana, S. and Sen, S. 2007. A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. Rand. Struct. Algorithms 30, 532--563. Google ScholarDigital Library
- Bernstein, A. 2009. Fully dynamic (2 + epsilon) approximate all-pairs shortest paths with fast query and close to linear update time. In Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS). 693--702. Google ScholarDigital Library
- Bollobás, B., Coppersmith, D., and Elkin, M. 2003. Sparse distance preserves and additive spanners. In Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). 414--423. Google ScholarDigital Library
- Chernoff, H. 1952. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat. 23, 4, 493--507.Google ScholarCross Ref
- Cohen, E. 1998. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM J. Comput. 28, 210--236. Google ScholarDigital Library
- Coppersmith, D. and Elkin, M. 2005. Sparse source-wise and pairwise distance preservers. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). 660--669. Google ScholarDigital Library
- Cormen, T. H., Leiserson, C. E., and Rivest, R. L. 1990. In Introduction to Algorithms. The MIT Press. Google ScholarDigital Library
- Cowen, L. J. 2001. Compact routing with minimum stretch. J. Algorithms 38, 170--183. Google ScholarDigital Library
- Cowen, L. J. and Wagner, C. G. 2004. Compact roundtrip routing in directed networks. J. Algorithms 50, 79--95. Google ScholarDigital Library
- Demetrescu, C. and Italiano, G. F. 2004. A new approach to dynamic all pairs shortest paths. J. ACM 51, 6, 968--992. Google ScholarDigital Library
- Derbel, B., Gavoille, C., Peleg, D., and Viennot, L. 2008. On the locality of distributed sparse spanner construction. In Proceedings of the 27th Annual ACM Symposium on Principles of Distributed Computing (PODC). 273--282. Google ScholarDigital Library
- Dubhashi, D., Mei, A., Panconesi, A., Radhakrishnan, J., and Srinivasan, A. 2005. Fast distributed algorithms for (weakly) connected dominating sets and linear-size skeletons. J. Comput. Syst. Sci. 71, 467--479. Google ScholarDigital Library
- Elkin, M. 2005. Computing almost shortest paths. ACM Trans. Algorithms 1, 282--323. Google ScholarDigital Library
- Elkin, M. 2006. A near-optimal fully dynamic distributed algorithm for maintaining sparse spanners. CoRR abs/cs/0611001.Google Scholar
- Elkin, M. 2007. A near-optimal distributed fully dynamic algorithm for maintaining sparse spanners. In Proceedings of the 26th Annual ACM Symposium on Principles of Distributed Computing (PODC). 185--194. Google ScholarDigital Library
- Elkin, M. 2011. Streaming and fully dynamic centralized algorithms for constructing and maintaining sparse spanners. ACM Trans. Algorithms 7, 2, 20. Google ScholarDigital Library
- Elkin, M., Emek, Y., Spielman, D. A., and Teng, S.-H. 2008. Lower-stretch spanning trees. SIAM J. Comput. 38, 608--628. Google ScholarDigital Library
- Elkin, M. and Peleg, D. 2004. (1 + ε,β)-spanner construction for general graphs. SIAM J. Comput. 33, 608--631. Google ScholarDigital Library
- Erdős, P. 1964. Extremal problems in graph theory. In Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Publ. House Czechoslovak Acad. Sci., Prague, 29--36.Google Scholar
- Even, S. and Shiloach, Y. 1981. An on-line edge-deletion problem. J. ACM 28, 1--4. Google ScholarDigital Library
- Feigenbaum, J., Kannan, S., McGregor, A., Suri, S., and Zhang, J. 2008. Graph distances in the data-stream model. SIAM J. Comput. 38, 5, 1709--1727. Google ScholarDigital Library
- Frederickson, G. N. 1985. Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput. 14, 781--798.Google ScholarDigital Library
- Halperin, S. and Zwick, U. 1996. Linear time deterministic algorithm for computing spanners for unweighted graphs. unpublished manuscript.Google Scholar
- Henzinger, M. R. and King, V. 1999. Randomized fully dynamic graph algorithms with polylogarithmic time per operation. J. ACM 46, 502--516. Google ScholarDigital Library
- Holm, J., de Lichtenberg, K., and Thorup, M. 2001. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM 48, 723--760. Google ScholarDigital Library
- Kruskal, J. 1956. On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. AMS 7. 48--50.Google ScholarCross Ref
- Motwani, R. and Raghavan, P. 1995. Randomized Algorithms. Cambridge University Press, New York. Google ScholarDigital Library
- Pagh, R. and Rodler, F. F. 2004. Cuckoo hashing. J. Algorithms 51, 122--144. Google ScholarDigital Library
- Peleg, D. and Schaffer, A. A. 1989. Graph spanners. J. Graph Theory 13, 99--116.Google ScholarCross Ref
- Peleg, D. and Upfal, E. 1989. A trade-off between space and efficiency for routing tables. J. ACM 36(3), 510--530. Google ScholarDigital Library
- Pettie, S. 2010. Distributed algorithms for ultrasparse spanners and linear size skeletons. Distrib. Comput. 22, 3, 147--166.Google ScholarDigital Library
- Roditty, L., Thorup, M., and Zwick, U. 2005. Deterministic construction of approximate distance oracles and spanners. In Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (ICALP). 261--272. Google ScholarDigital Library
- Roditty, L., Thorup, M., and Zwick, U. 2008. Roundtrip spanners and roundtrip routing in directed graphs. ACM Trans. Algorithms 4, 29:1--29:17. Google ScholarDigital Library
- Roditty, L. and Zwick, U. 2004a. Dynamic approximate all-pairs shortest paths in undirected graphs. In Proceedings of the 45th Annual Symposium on Foundations of Computer Science (FOCS). 499--508. Google ScholarDigital Library
- Roditty, L. and Zwick, U. 2004b. On dynamic shortest paths problems. In Proceedings of the 12th Annual European Symposium on Algorithms (ESA). 580--591.Google Scholar
- Roditty, L. and Zwick, U. 2008. Improved dynamic reachability algorithms for directed graphs. SIAM J. Comput. 37, 5, 1455--1471. Google ScholarDigital Library
- Thorup, M. 2004. Fully-dynamic all-pairs shortest paths: Faster and allowing negative cycles. In Proceedings of the 9th Scandinavian Workshop on Algorithm Theory (SWAT). 384--396.Google ScholarCross Ref
- Thorup, M. and Zwick, U. 2005. Approximate distance oracles. J. ACM 52, 1--24. Google ScholarDigital Library
- Thorup, M. and Zwick, U. 2006. Spanners and emulators with sublinear distance errors. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). 802--809. Google ScholarDigital Library
- Woodruff, D. P. 2010. Additive spanners in nearly quadratic time. In Proceedings of the 37th International Colloquium Conference on Automata, Languages and Programming (ICALP). 463--474. Google ScholarDigital Library
Index Terms
- Fully dynamic randomized algorithms for graph spanners
Recommendations
Edge-disjoint spanners in tori
A spanning subgraph S=(V,E^') of a connected graph G=(V,E) is an (x+c)-spanner if for any pair of vertices u and v, d"S(u,v)@?d"G(u,v)+c where d"G and d"S are the usual distance functions in G and S, respectively. The parameter c is called the delay of ...
Additive Tree Spanners
A spanning tree of a graph is a k-additive tree spanner whenever the distance of every two vertices in the graph differs from that in the tree by at most k. In this paper we show that certain classes of graphs, such as distance-hereditary graphs, ...
Dynamic algorithms for graph spanners
ESA'06: Proceedings of the 14th conference on Annual European Symposium - Volume 14Let G =( V,E ) be an undirected weighted graph on | V |= n vertices and | E |= m edges. For the graph G, A spanner with stretch t is a subgraph ( V,E S ), E s ⊆ E , such that the distance between any pair of vertices in this subgraph ...
Comments