In this paper we study proximity structures for geometric graphs. The study of these structures was recently motivated by topology control for wireless networks [6, 7]. We obtain the following results: (i) We prove that if G is a D1-graph on n vertices, then it has O(n 3/2) edges. (ii) We show that for any n there exist D1-graphs with n vertices and Ω(n 4/3) edges. (iii) We prove that if G is a D2-graph on n vertices, then it has O(n) edges. This bound is worst-case asymptotically tight. As an application of the first result, we show that: (iv) The maximum size of a family of pairwise non-overlapping lenses in an arrange-ment of n unit circles in the plane is O(n3/2). The first two results improve the best previously known upper and lower bounds of O(n5/3) and Ω(n) respectively (see ). The third result improves the best previously known upper bound of O(n log n) (). Finally, our last result improves the best previously known upper bound (for the more general case of not necessarily unit circles) of O(n3/2κ(n)) (see ), where κ(n) = (log n)O(α 2(n)) and where α(n) is the extremely slowly growing inverse Ackermann’s function.
|State||Published - 2004|
|Event||Thirty-Sixth Annual ACM Symposium on Theory of Computing (STOC 2004) - Chicago, United States|
Duration: 13 Jun 2004 → 15 Jun 2004
|Conference||Thirty-Sixth Annual ACM Symposium on Theory of Computing (STOC 2004)|
|Period||13/06/04 → 15/06/04|