Extremal Configurations and Levels in Pseudoline Arrangements

Abstract

This paper studies a variety of problems involving certain types of extreme configurations in arrangements of (x-monotone) pseudo-lines. For example, we obtain a very simple proof of the bound O(nk ^1/3) on the maximum complexity of the k-th level in an arrangement of n pseudo-lines, which becomes even simpler in the case of lines. We thus simplify considerably the previous proof by Tamaki and Tokuyama (and also simplify Dey’s proof for lines).

We also consider diamonds and anti-diamonds in (simple) pseudo-line arrangements, where a diamond (resp., an anti-diamond) is a pair u,v of vertices, so that u lies in the double wedge of v and vice versa (resp., neither u nor v lies in the other double wedge). We show that the maximum size of a diamond-free set of vertices in an arrangement of n pseudo-lines is 3n—6, by showing that the induced graph (where each vertex of the arrangement is regarded as an edge connecting the two incident curves) is planar, simplifying considerably a previous proof of the same fact by Tamaki and Tokuyama. Similarly, we show that the maximum size of an anti-diamond-free set of vertices in an arrangement of n pseudo-lines is 2n—2. We also obtain several additional results, which are listed in the introduction.

In some of our results, we use a recent duality transform between points and pseudo-lines due to Agarwal and Sharir, which extends an earlier transform by Goodman (that applied only in the projective plane). We show that this transform maps a set of vertices in a pseudo-line arrangement to a topological graph whose edges are drawn as x-monotone arcs that connect pairs of the dual points, and form a set of extendible pseudo-segments (they are pieces of curves that form a pseudo-line arrangement in the dual plane). This allows us (a) to ‘import’ known results on this kind of topological graphs to the context of pseudo-lines; (b) to extend techniques that have been originally applied only for geometric graphs (whose edges are drawn as straight segments), thereby obtaining new results for pseudo-line arrangements, or for the above-class of x-monotone topological graphs; and (c) to derive new techniques, facilitated by the passage to the dual setting, that apply in the more general pseudo-line context, and extend and simplify considerably the earlier proofs. This paper contains examples of all three kinds of results.

Work on this paper has been supported by a grant from the Israel Science Fund (for a Center of Excellence in Geometric Computing). Work by Micha Sharir has also been supported by NSF Grants CCR-97-32101 and CCR-00-98246, by a grant from the U.S.-Israeli Binational Science Foundation, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.