Abstract
Let S be a set of n points in the general position, that is, no three points in S are collinear. A simple k-gon with all corners in S such that its interior avoids any point of S is called a k-hole. In this paper, we present the first algorithm that counts the number of non-convex 5-holes in S. To our best knowledge, prior to this work there was no known algorithm in the literature except a trivial brute force algorithm. Our algorithm runs in time O(T+Q), where T denotes the number of 3-holes, or empty triangles, in S and Q that denotes the number of non-convex 4-holes in S. Note that T+Q ranges from Ω(n2) to O(n3), while its expected number is Θ(n2logn) when the points in S are chosen uniformly and independently at random from a convex and bounded body in the plane.
Highlights
Let S be a set of n points in the plane that are in general position; that is, no three points in S are collinear
We focus on the number of k-holes in S for a fixed k ≥ 3, and address the problem of counting non-convex k-holes in a given set S, in particular for k = 5
We presented the first algorithm that counts non-convex 5-holes in a given set S of n points in general position
Summary
Let S be a set of n points in the plane that are in general position; that is, no three points in S are collinear. Dobkin et al [1] in 1990 presented an algorithm that enumerates all convex k-holes in time O(h3(S) + k · hk(S)), where hk(S) denotes the number of k-holes in S. This algorithm, in particular, for k = 3, 4 takes time O(hk(S)), proportional to the number of reported k-holes. Rote et al [2] studied the problem of counting convex k-gons in S and presented an O(nk−2)-time algorithm. This was improved to O(kn3) time for k ≥ 6 by Mitchell et al [3] after Rote and Woeginger [4]. The number h3(S) of empty triangles in S ranges between Ω(n2) and O(n3), while the expected value of h3(S) is proven to be Θ(n2) if the points in S are uniformly chosen from a convex and bounded body in the plane [6,7]
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