| Abstract (2007) | |||||||||||||||
Abstract | |||||||||||||||
| Dynamic connectivity is an extremely well-studied problem, but so far the most compelling progress has been confined to the edge-update model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two equally natural and fundamental problems which are even more challenging. Subgraph connectivity asks to maintain an understanding of connectivity under node updates: updates can turn nodes on and off, and queries refer to the subgraph induced by on nodes. (For instance, this is closer to applications in networks of routers, where node faults may occur.) We describe a data structure supporting vertex updates in � O(m 2/3) amortized time, where m denotes the number of edges in the graph. This is a major improvement over the previous result [Chan, STOC’02], which required fast matrix multiplication and had an update time of O(m 0.94). The new data structure is also surprisingly simple. Geometric connectivity asks to maintain a dynamic set of n geometric objects, and query connectivity in their intersection graph. (For instance, the intersection graph of balls describes connectivity in a network of sensors with bounded transmission radius.) Previously, nontrivial fully dynamic results were known only for special cases like axis-parallel line segments and rectangles. We not only provide similarly improved update times, � O(n2/3), for these special cases, but show how to obtain sublinear update bounds for virtually all families of geometric objects which allow sublinear-time range queries. In particular, we obtain the first sublinear update 1 d(2d+1)); and for time for arbitrary 2D line segments: � O(n 9/10); for d-dimensional simplices: � O(n 1− d-dimensional balls: � O(n 1 − (d+1)(2d+3)). 1 | |||||||||||||||
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