Competitive Maintenance of Minimum Spanning Trees in Dynamic Graphs

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 nvertices 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 i¾?(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.

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