62 research outputs found
The visible perimeter of an arrangement of disks
Given a collection of n opaque unit disks in the plane, we want to find a
stacking order for them that maximizes their visible perimeter---the total
length of all pieces of their boundaries visible from above. We prove that if
the centers of the disks form a dense point set, i.e., the ratio of their
maximum to their minimum distance is O(n^1/2), then there is a stacking order
for which the visible perimeter is Omega(n^2/3). We also show that this bound
cannot be improved in the case of a sufficiently small n^1/2 by n^1/2 uniform
grid. On the other hand, if the set of centers is dense and the maximum
distance between them is small, then the visible perimeter is O(n^3/4) with
respect to any stacking order. This latter bound cannot be improved either.
Finally, we address the case where no more than c disks can have a point in
common. These results partially answer some questions of Cabello, Haverkort,
van Kreveld, and Speckmann.Comment: 12 pages, 5 figure
Improved bounds and new techniques for Davenport-Schinzel sequences and their generalizations
Let lambda_s(n) denote the maximum length of a Davenport-Schinzel sequence of
order s on n symbols. For s=3 it is known that lambda_3(n) = Theta(n alpha(n))
(Hart and Sharir, 1986). For general s>=4 there are almost-tight upper and
lower bounds, both of the form n * 2^poly(alpha(n)) (Agarwal, Sharir, and Shor,
1989). Our first result is an improvement of the upper-bound technique of
Agarwal et al. We obtain improved upper bounds for s>=6, which are tight for
even s up to lower-order terms in the exponent. More importantly, we also
present a new technique for deriving upper bounds for lambda_s(n). With this
new technique we: (1) re-derive the upper bound of lambda_3(n) <= 2n alpha(n) +
O(n sqrt alpha(n)) (first shown by Klazar, 1999); (2) re-derive our own new
upper bounds for general s; and (3) obtain improved upper bounds for the
generalized Davenport-Schinzel sequences considered by Adamec, Klazar, and
Valtr (1992). Regarding lower bounds, we show that lambda_3(n) >= 2n alpha(n) -
O(n), and therefore, the coefficient 2 is tight. We also present a simpler
version of the construction of Agarwal, Sharir, and Shor that achieves the
known lower bounds for even s>=4.Comment: To appear in Journal of the ACM. 48 pages, 3 figure
The number of distinct distances from a vertex of a convex polygon
Erd\H{o}s conjectured in 1946 that every n-point set P in convex position in
the plane contains a point that determines at least floor(n/2) distinct
distances to the other points of P. The best known lower bound due to
Dumitrescu (2006) is 13n/36 - O(1). In the present note, we slightly improve on
this result to (13/36 + eps)n - O(1) for eps ~= 1/23000. Our main ingredient is
an improved bound on the maximum number of isosceles triangles determined by P.Comment: 11 pages, 4 figure
Upper bounds for centerlines
In 2008, Bukh, Matousek, and Nivasch conjectured that for every n-point set S
in R^d and every k, 0 <= k <= d-1, there exists a k-flat f in R^d (a
"centerflat") that lies at "depth" (k+1) n / (k+d+1) - O(1) in S, in the sense
that every halfspace that contains f contains at least that many points of S.
This claim is true and tight for k=0 (this is Rado's centerpoint theorem), as
well as for k = d-1 (trivial). Bukh et al. showed the existence of a (d-2)-flat
at depth (d-1) n / (2d-1) - O(1) (the case k = d-2).
In this paper we concentrate on the case k=1 (the case of "centerlines"), in
which the conjectured value for the leading constant is 2/(d+2). We prove that
2/(d+2) is an *upper bound* for the leading constant. Specifically, we show
that for every fixed d and every n there exists an n-point set in R^d for which
no line in R^d lies at depth larger than 2n/(d+2) + o(n). This point set is the
"stretched grid"---a set which has been previously used by Bukh et al. for
other related purposes.
Hence, in particular, the conjecture is now settled for R^3.Comment: This paper (without the appendix) has been published in Journal of
Computational Geometry 3:20--30, 2012. 17 pages; 10 figure
Eppstein's bound on intersecting triangles revisited
Let S be a set of n points in the plane, and let T be a set of m triangles
with vertices in S. Then there exists a point in the plane contained in
Omega(m^3 / (n^6 log^2 n)) triangles of T. Eppstein (1993) gave a proof of this
claim, but there is a problem with his proof. Here we provide a correct proof
by slightly modifying Eppstein's argument.Comment: Minor revision following referee's suggestions. To appear in Journal
of Combinatorial Theory, Series A. 5 pages, 1 figur
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