Dynamic Graphs:Conclusions

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Conclusions

In this chapter we have surveyed the algorithmic techniques underlying the fastest known dynamic graph algorithms for several problems, both on undirected and on directed graphs. Most of the algorithms that we have presented achieve bounds that are close to the best possible. In particular, we have presented fully dynamic algorithms with polylogarithmic amortized time bounds for connectivity and minimum spanning trees [22] on undirected graphs. It remains an interesting open problem to show whether polylogarithmic update bounds can be achieved also in the worst case: we recall that for both problems the current best worst-case bound is O(√n ) per update, and it is obtained with the sparsification technique [10] described in Section 36.2.

For directed graphs, we have shown how to achieve constant-time query bounds and nearly-quadratic update bounds for transitive closure and all pairs shortest paths. These bounds are close to optimal in the sense that one update can make as many as Ω(n2) changes to the transitive closure and to the all pairs shortest paths matrices. If one is willing to pay more for queries, Demetrescu and Italiano [6] have shown how to break the O(n2) barrier on the single-operation complexity of fully dynamic transitive closure for directed acyclic graphs. This also yields the first efficient update algorithm that maintains reachability in acyclic directed graphs between two fixed vertices s and t, or from s to all other vertices. However, in the case of general graphs or shortest paths, no solution better that the static is known for these problems. In general, it remains an interesting open problem to show whether effective query/update tradeoffs can be achieved for general graphs and for shortest paths problems.

Finally, dynamic algorithms for other fundamental problems such as matching and flow problems deserve further investigation.

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