SWAT 2012: Algorithm Theory – SWAT 2012 pp 316-327

Connectivity Oracles for Planar Graphs

• Seth Pettie
• Christian Wulff-Nilsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)

Abstract

We consider dynamic subgraph connectivity problems for planar undirected graphs. In this model there is a fixed underlying planar graph, where each edge and vertex is either “off” (failed) or “on” (recovered). We wish to answer connectivity queries with respect to the “on” subgraph. The model has two natural variants, one in which there are d edge/vertex failures that precede all connectivity queries, and one in which failures/recoveries and queries are intermixed.

We present a d-failure connectivity oracle for planar graphs that processes any d edge/vertex failures in sort(d,n) time so that connectivity queries can be answered in pred(d,n) time. (Here sort and pred are the time for integer sorting and integer predecessor search over a subset of [n] of size d.) Our algorithm has two discrete parts. The first is an algorithm tailored to triconnected planar graphs. It makes use of Barnette’s theorem, which states that every triconnected planar graph contains a degree-3 spanning tree. The second part is a generic reduction from general (planar) graphs to triconnected (planar) graphs. Our algorithm is, moreover, provably optimal. An implication of Pǎtraşcu and Thorup’s lower bound on predecessor search is that no d-failure connectivity oracle (even on trees) can beat pred(d,n) query time.

We extend our algorithms to the subgraph connectivity model where edge/vertex failures (but no recoveries) are intermixed with connectivity queries. In triconnected planar graphs each failure and query is handled in O(logn) time (amortized), whereas in general planar graphs both bounds become O(log2 n).

Keywords

Span Tree Planar Graph Subgraph Connectivity Primal Graph Connectivity Query
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. 1.
Afshani, P., Chan, T.M.: Dynamic connectivity for axis-parallel rectangles. Algorithmica 53(4), 474–487 (2009)
2. 2.
Alstrup, S., Gavoille, C., Kaplan, H., Rauhe, T.: Nearest common ancestors: a survey and a new distributed algorithm. In: Proceedings 14th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pp. 258–264 (2002)Google Scholar
3. 3.
Alstrup, S., Husfeldt, T., Rauhe, T.: Marked ancestor problems. In: Proceedings 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 534–544 (1998)Google Scholar
4. 4.
Andersson, A., Hagerup, T., Nilsson, S., Raman, R.: Sorting in linear time? J. Comput. Syst. Sci. 57(1), 74–93 (1998)
5. 5.
Andersson, A., Thorup, M.: Dynamic ordered sets with exponential search trees. J. ACM 54(3), 13 (2007)
6. 6.
Barnette, D.: Trees in polyhedral graphs. Canadian Journal of Mathematics 18, 731–736 (1966)
7. 7.
Bender, M.A., Farach-Colton, M.: The LCA Problem Revisited. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 88–94. Springer, Heidelberg (2000)
8. 8.
Bernstein, A., Karger, D.: A nearly optimal oracle for avoiding failed vertices and edges. In: Proceedings 41st Annual ACM Symposium on Theory of Computing (STOC), pp. 101–110 (2009)Google Scholar
9. 9.
Borradaile, G., Pettie, S., Wulff-Nilsen, C.: Connectivity Oracles for Planar Graphs. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 316–327. Springer, Heidelberg (2012)Google Scholar
10. 10.
Chan, T.: Dynamic subgraph connectivity with geometric applications. SIAM J. Comput. 36(3), 681–694 (2006)
11. 11.
Chan, T.M., Patrascu, M., Roditty, L.: Dynamic connectivity: Connecting to networks and geometry. SIAM J. Comput. 40(2), 333–349 (2011)
12. 12.
Demetrescu, C., Thorup, M., Chowdhury, R.A., Ramachandran, V.: Oracles for distances avoiding a failed node or link. SIAM J. Comput. 37(5), 1299–1318 (2008)
13. 13.
Duan, R.: New Data Structures for Subgraph Connectivity. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010, Part I. LNCS, vol. 6198, pp. 201–212. Springer, Heidelberg (2010)
14. 14.
Duan, R., Pettie, S.: Dual-failure distance and connectivity oracles. In: Proceedings 20th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 506–515 (2009)Google Scholar
15. 15.
Duan, R., Pettie, S.: Connectivity oracles for failure prone graphs. In: Proceedings 42nd ACM Symposium on Theory of Computing, pp. 465–474 (2010)Google Scholar
16. 16.
Edmonds, J.: A combinatorial representation for polyhedral surfaces. Notices of the American Mathematical Society 7, 646 (1960)Google Scholar
17. 17.
Eppstein, D., Galil, Z., Italiano, G., Nissenzweig, A.: Sparsification — a technique for speeding up dynamic graph algorithms. J. ACM 44(5), 669–696 (1997)
18. 18.
Eppstein, D., Italiano, G.F., Tamassia, R., Tarjan, R.E., Westbrook, J., Yung, M.: Maintenance of a minimum spanning forest in a dynamic plane graph. J. Algor. 13(1), 33–54 (1992)
19. 19.
Frederickson, G.: Data structures for on-line updating of minimum spanning trees, with applications. SIAM J. Comput. 14(4), 781–798 (1985)
20. 20.
Fredman, M.L., Willard, D.E.: Trans-dichotomous algorithms for minimum spanning trees and shortest paths. J. Comput. Syst. Sci. 48(3), 533–551 (1994)
21. 21.
Frigioni, D., Italiano, G.F.: Dynamically switching vertices in planar graphs. Algorithmica 28(1), 76–103 (2000)
22. 22.
Han, Y.: Deterministic sorting in O(n loglogn) time and linear space. J. Algorithms 50, 96–105 (2004)
23. 23.
Han, Y., Thorup, M.: Integer sorting in $${O}(n\sqrt{\log\log n})$$ expected time and linear space. In: Proceedings of the 43rd Symposium on Foundations of Computer Science, FOCS 2002, pp. 135–144. IEEE Computer Society, Washington, DC (2002)Google Scholar
24. 24.
Harel, D., Tarjan, R.: Fast algorithms for finding nearest common ancestors. SIAM J. Comput. 13(2), 338–355 (1984)
25. 25.
Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM 48(4), 723–760 (2001)
26. 26.
Pǎtraşcu, M., Thorup, M.: Time-space trade-offs for predecessor search. In: Proceedings 38th ACM Symposium on Theory of Computing (STOC), pp. 232–240 (2006)Google Scholar
27. 27.
Pǎtraşcu, M., Thorup, M.: Planning for fast connectivity updates. In: Proceedings 48th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 263–271 (2007)Google Scholar
28. 28.
Strothmann, W.: Bounded degree spanning trees. PhD thesis, Universität-Gesamthochschule Paderborn (1997)Google Scholar
29. 29.
van Emde Boas, P., Kaas, R., Zijlstra, E.: Design and implementation of an efficient priority queue. Math. Syst. Theory 10, 99–127 (1977)
30. 30.
Whitney, H.: Planar graphs. Fundamenta mathematicae 21, 73–84 (1933)Google Scholar
31. 31.
Willard, D.E.: Log-logarithmic worst-case range queries are possible in space θ(n). Info. Proc. Lett. 17(2), 81–84 (1983)