On k-convex polygons

Abstract

We introduce a notion of k-convexity and explore polygons in the plane that have this property. Polygons which are k-convex can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of 2-convex polygons, a particularly interesting class, and show how to recognize them in O(n log n) time. A description of their shape is given as well, which leads to Erdős-Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric permutations, and show that their number can be exponential in the number of 2-convex objects considered.

Threesomes, Degenerates, and Love Triangles

“…Throughout the paper we shall refer to the ingenious observation that a`b ă c`d iff a´c ă d´b as Fredman's trick. 4 In order to shave off polyplog nq factors in runtime we apply the geometric domination technique invented by Chan [15] and developed further by Bremner, Chan, Demaine, Erickson, Hurtado, Iacono, Langerman, Pǎtraşcu, and Taslakian [12].…”

“…It is often conjectured that 3SUM requires Ωpn 2 q time and that Integer3SUM requires Ωpn 2´op1q q time [32,3]. These conjectures have been shown to imply strong lower bounds on numerous problems in computational geometry [25,4,9,35] dynamic graph algorithms [32,3], and pattern matching [1,6,14,17]. For example, the 3SUM conjecture implies that the following problems require at least Ωpn 2 q time.…”

Straight Skeletons and Mitered Offsets of Nonconvex Polytopes

“…However, when only the information present in Q is used, one will in general not end up with the initial polytope Q when expanding Q again, even when all events so far have been unique: Some of the offset planes may have become redundant in the shrinking process, and then cannot be restored any more. Such a 'loss' of planes can only happen for edge events, and there only for E 3 , E 5 , and E 6 , where cells of the skeleton get completed. The inverse of a triangle collapse (where a new triangle is born), or of a tetrahedron collapse (where a new tetrahedron is born), thus do not occur in the shrinking process.…”

“…In particular, the afore-mentioned ambiguous case for parallel edges is excluded (where we have 1 = 2 , hence α 1 = α 2 ). 5 The vertex u is not generated in the left and the right case, respectively. In the case shown in the middle, the skeleton faces sharing the arc a continue in a monotone way In summary, there is a unique way to proceed in the construction of SK(v) after each event.…”

“…However, if weighting is allowed then no efficient method is known even in the planar case. A discussion of several generalized notions of convexity for polygons (being different from near-convexity) can be found in [5].…”