# Algorithms for Programmers: Ideas and Source Code by Jörg Arndt PDF

By Jörg Arndt

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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.

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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.