On the modality of convex polygons

Discrete & Computational Geometry, Aug 1990

Under two definitions of random convex polygons, the expected modality of a random convex polygon grows without bound as the number of vertices grows. This refutes a conjecture of Aggarwal and Melville.

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On the modality of convex polygons

Discrete Comput Geom On the Modality of Convex Polygons 0 Computer Science Department, Washington State University , Pullman, WA 99164-1210 , USA Under two definitions of random convex polygons, the expected modality of a random convex polygon grows without bound as the number of vertices grows. This refutes a conjecture of Aggarwal and Melville. 1. Introduction Algorithms in computational geometry tend to be quite difficult, and to involve a large n u m b e r o f special cases. Some o f the cases might be very rare. Algorithm design can be simplified by either excluding rare and difficult cases, o r b y designing an efficient algorithm for the c o m m o n cases, and combining it with a less efficient algorithm for the rare cases. The problem is to recognize the rare cases. For algorithms which operate on convex polygons, one useful condition on the input is unimodality. The modality of a vertex o f a polygon is the n u m b e r of local m a x i m a in the sequence o f distances from that vertex to the other vertices, in their natural order around the polygon. The modality of a polygon is the m a x i m u m of the modalities of its vertices. A polygon is unimodai if its modality is 1; otherwise, it is multimodai. Figure 1 shows polygons o f modality 1 and 2. When drawing a " r a n d o m " convex polygon, it is quite c o m m o n to find that the selected polygon is unimodal. In fact, Dobkin and Snyder [ 4 ] and Snyder and Tang [ 7 ] each assume that all convex polygons are unimodal in designing algorithms to find the diameter o f a convex polygon. The fact that there exist multimodal convex polygons, pointed out by Avis et al. [ 3 ], thus at least initially seems counterintuitive. Aggarwal and Melville [ 2 ] conjecture that, under any reasonable definition o f a r a n d o m convex polygon, the probability that a r a n d o m convex n-gon is unimodal tends to 1 as n ~ oo. (They do not say just what a reasonable definition would be.) I f true, the conjecture would imply that multimodality is one o f those rare features which it might pay to ignore. Aggarwal and Melville [ 2 ] also give a linear-time algorithm to determine the modality o f a convex polygon. Using that algorithm, an algorithm which runs in linear time, but assumes a unimodal input, can be used to construct an algorithm which is correct for any convex polygon, but runs in linear time on unimodal inputs. An example is the linear-time all-furthest-neighbors algorithm o f Toussaint [ 8 ] for unimodal convex polygons. The conjecture o f Aggarwal and Melville, if true, would suggest that all-furthest-neighbors for convex polygons can be solved in linear time for the expected case input. (In fact, it can be solved in linear time for any convex polygon, by the algorithm o f Aggarwal et al. [ 1 ].) This paper refutes the conjecture for two definitions of random convex polygons, and in fact shows that the probability that a random convex polygon is unimodal approaches zero quite rapidly. Thus, algorithms which perform well on unimodal convex polygons can, unfortunately, be expected to be weak. On the other hand, the expected modality is not large. It may be that algorithms can exploit the fact that an input has a small modality. The two chosen definitions o f a random convex polygon are (1) the convex hull o f n points drawn uniformly from a disk in the plane, and (2) the convex hull of n points drawn from a two-dimensional normal distribution. The expected modality is O(log n / l o g log n) in case (1) and O(log log n/log log log n) in case (2). Interestingly, the expected number of vertices on the convex hull is O(n I/3) in case (1) and O(log ~/2 n) in case (2) [ 5 ], [ 6 ]. So in the cases studied here, the expected modality is 19(1og h / l o g log h), where h is the expected number of hull vertices. The proofs of our results apply only to very large n. What of moderate size n? Table 1 was computed using a nonlinear additive feedback pseudorandom number generator, with 1000 samples generated for each n. Note that n is the number o f randomly selected points. The convex hulls generally had considerably fewer than n vertices. For n = 1024, the convex hulls averaged 34 vertices in the uniform case, and 11 vertices in the normal case. The table indicates that convex hulls of modality 2 (which tend to be somewhat oblong) are prevalent for moderate size point sets. No convex hulls of modality exceeding 3 were generated. 2. Lower Bounds on Expected Modality This section introduces the basic techniques, and establishes lower bounds on the expected modality. The proofs are fairly crude, but relatively simple. All logarithms are base e, and all angles in radians. Theorem 1. When n points are drawn at random uniformly from a disk in the plane, the expected modality of their convex hull is f](log n/log log n) and the probability that their convex hull is unimodal approaches 0 as n --)oo. Proof Assume that n is very large. Write x ~ y wh (...truncated)


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Karl Abrahamson. On the modality of convex polygons, Discrete & Computational Geometry, 1990, pp. 409-419, Volume 5, Issue 4, DOI: 10.1007/BF02187802