Abstract
We consider the problem of maintaining a minimum spanning tree within a graph with dynamically changing edge weights. An online algorithm is confronted with an input sequence of edge weight changes and has to choose a minimum spanning tree after each such change in the graph. The task of the algorithm is to perform as few changes in its minimum spanning tree as possible.
We compare the number of changes in the minimum spanning tree produced by an online algorithm and that produced by an optimal offline algorithm. The number of changes is counted in the number of edges changed between spanning trees in consecutive rounds.
For any graph with n vertices we provide a deterministic algorithm achieving a competitive ratio of \(\mathcal{O}(n^2)\). We show that this result is optimal up to a constant. Furthermore we give a lower bound for randomized algorithms of Ω(logn). We show a randomized algorithm achieving a competitive ratio of \(\mathcal{O}(n\log n)\) for general graphs and \(\mathcal{O}(\log n)\) for planar graphs.
Partially supported by the EU within the 6th Framework Programme under contract 001907 (DELIS) and by the DFG-Sonderforschungsbereich SPP 1183: “Organic Computing. Smart Teams: Local, Distributed Strategies for Self-Organizing Robotic Exploration Teams”.
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References
Agarwal, P.K., Eppstein, D., Guibas, L.J., Henzinger, M.R.: Parametric and Kinetic Minimum Spanning Trees. In: Proceedings of the 39th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Washington (1998)
Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)
Chazelle, B.: A Minimum Spanning Tree Algorithm with Inverse-Ackermann Type Complexity. J. ACM 47(6), 1028–1047 (2000)
Cheriton, D., Tarjan, R.E.: Finding Minimum Spanning Trees. SIAM Journal of Computing 5 (1976)
Chin, F., Houck, D.: Algorithms for Updating Minimal Spanning Trees. Journal of Computer and System Sciences 16, 333–344 (1978)
Chrobak, M., Larmore, L.L., Lund, C., Reingold, N.: A Better Lower Bound on the Competitive Ratio of the Randomized 2-Server Problem. Information Processing Letters 63(2), 79–83 (1997)
Eppstein, D.: Spanning Trees and Spanners. Technical Report ICS-TR-96-16 (1996)
Eppstein, D., Galil, Z., Italiano, G.F., Nissenzweig, A.: Sparsification a Technique for Speeding up Dynamic Graph Algorithms. J. ACM 44(5), 669–696 (1997)
Frederickson, G.N.: Data Structures for On-Line Updating of Minimum Spanning Trees. In: STOC’83: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pp. 252–257. ACM Press, New York (1983)
Henzinger, M.R., King, V.: Maintaining Minimum Spanning Trees in Dynamic Graphs. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 594–604. Springer, Heidelberg (1997)
Kruskal, J.B.: On the Shortest Spanning Subtree of a Graph and the Traveling Salesman Problem. Proceedings of the American Mathematics Society 7, 48–50 (1956)
Pendarakis, D.E., Shi, S., Verma, D.C., Waldvogel, M.: Almi: An Application Level Multicast Infrastructure. In: 3rd USENIX Symposium on Internet Technologies and Systems, pp. 49–60 (2001)
Pettie, S., Ramachandran, V.: An Optimal Minimum Spanning Tree Algorithm. J. ACM 49(1), 16–34 (2002)
Prim, R.C.: Shortest Connection Networks and Some Generalizations. Bell System Technical Journal 36, 1389–1401 (1957)
Wan, P.-J., Calinescu, G., Li, X.-Y., Frieder, O.: Minimum-Energy Broadcasting in Static Ad Hoc Wireless Networks. Wireless Networks 8(6), 607–617 (2002)
Young, A., Chen, J., Ma, Z., Krishnamurthy, A., Peterson, L., Wang, R.Y.: Overlay Mesh Construction Using Interleaved Spanning Trees. In: IEEE INFOCOM, pp. 396–407 (2004)
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Dynia, M., Korzeniowski, M., Kutyłowski, J. (2007). Competitive Maintenance of Minimum Spanning Trees in Dynamic Graphs. In: van Leeuwen, J., Italiano, G.F., van der Hoek, W., Meinel, C., Sack, H., Plášil, F. (eds) SOFSEM 2007: Theory and Practice of Computer Science. SOFSEM 2007. Lecture Notes in Computer Science, vol 4362. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69507-3_21
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DOI: https://doi.org/10.1007/978-3-540-69507-3_21
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