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By Arora S., Karakostas G.

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If they are in different components but inside the same class vertex , we return “yes” iff is not an isolated vertex in H. Dynamic Connectivity for Axis-Parallel Rectangles 25 Otherwise, we return “yes” iff the vertex containing u and the vertex containing v are connected in the graph H. Since we know all the connected components of H, the query time is O(1). Thus we have proved the following theorem. Theorem 1. 910 ) amordynamic data structure which performs updates in O(n tized time and answers connectivity queries in constant time.

Workshop on Virtual Reality Interactions and Physical Simulations, pages 81-90, 2005. 10. J. Erickson, L. Guibas, J. Stolfi, and L. Zhang. Separation-sensitive collision detection for convex objects. In Proc. 10th ACM-SIAM Symposium on Discrete Algorithms, pages 327–336, 1999. 11. L. Guibas. Kinetic data structures: A state of the art report. In Proc. 3rd Workshop on Algorithmic Foundations of Robotics, pages 191–209, 1998. 12. L. Guibas. Motion. In J. Goodman and J. O’Rourke, editors, Handbook of Discrete and Computational Geometry, pages 1117–1134.

We mention that a slightly weaker bound can be obtained by using VCdimension techniques [12]. It is possible to show (by a K5 -avoidance argument) that the set system defined by the curves (where the ground set is R and each curve defines the set of regions it intersects) has VC-dimension four. This implies that the number of curves is O(n4 ). We omit the details, as the approach we now give produces a better bound: Lemma 1. Assume C is a set of pairwise non-crossing curves with common endpoints p and q, p = q.

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A 2 + approximation algorithm for the k-MST problem by Arora S., Karakostas G.

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