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)