37 research outputs found
Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes
Given a set of spheres in , with and
odd, having a fixed number of distinct radii , we
show that the worst-case combinatorial complexity of the convex hull
of is
, where
is the number of spheres in with radius .
To prove the lower bound, we construct a set of spheres in
, with odd, where spheres have radius ,
, and , such that their convex hull has combinatorial
complexity
.
Our construction is then generalized to the case where the spheres have
distinct radii.
For the upper bound, we reduce the sphere convex hull problem to the problem
of computing the worst-case combinatorial complexity of the convex hull of a
set of -dimensional convex polytopes lying on parallel hyperplanes
in , where odd, a problem which is of independent
interest. More precisely, we show that the worst-case combinatorial complexity
of the convex hull of a set
of -dimensional convex polytopes lying on parallel hyperplanes of
is
, where
is the number of vertices of .
We end with algorithmic considerations, and we show how our tight bounds for
the parallel polytope convex hull problem, yield tight bounds on the
combinatorial complexity of the Minkowski sum of two convex polytopes in
.Comment: 22 pages, 5 figures, new proof of upper bound for the complexity of
the convex hull of parallel polytopes (the new proof gives upper bounds for
all face numbers of the convex hull of the parallel polytopes
The maximum number of faces of the Minkowski sum of two convex polytopes
We derive tight expressions for the maximum number of -faces,
, of the Minkowski sum, , of two
-dimensional convex polytopes and , as a function of the number
of vertices of the polytopes.
For even dimensions , the maximum values are attained when and
are cyclic -polytopes with disjoint vertex sets. For odd dimensions
, the maximum values are attained when and are
-neighborly -polytopes, whose vertex sets are
chosen appropriately from two distinct -dimensional moment-like curves.Comment: 37 pages, 8 figures, conference version to appear at SODA 2012; v2:
fixed typos, made stylistic changes, added figure
A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes
We derive tight expressions for the maximum number of -faces,
, of the Minkowski sum, , of convex
-polytopes in , where and , as a
(recursively defined) function on the number of vertices of the polytopes.
Our results coincide with those recently proved by Adiprasito and Sanyal [2].
In contrast to Adiprasito and Sanyal's approach, which uses tools from
Combinatorial Commutative Algebra, our approach is purely geometric and uses
basic notions such as - and -vector calculus and shellings, and
generalizes the methodology used in [15] and [14] for proving upper bounds on
the -vector of the Minkowski sum of two and three convex polytopes,
respectively.
The key idea behind our approach is to express the Minkowski sum
as a section of the Cayley polytope of the
summands; bounding the -faces of reduces to bounding the
subset of the -faces of that contain vertices from each
of the polytopes.
We end our paper with a sketch of an explicit construction that establishes
the tightness of the upper bounds.Comment: 43 pages; minor changes (mostly typos
The maximum number of faces of the Minkowski sum of three convex polytopes
We derive tight expressions for the maximum
number of -faces, , of the
Minkowski sum, , of three -dimensional convex polytopes , and in ,
as a function of the number of vertices of the polytopes, for any .
Expressing the Minkowski sum as a section of the Cayley polytope of its summands, counting the -faces of reduces to counting the -faces of which meet the vertex sets of the three polytopes.
In two dimensions our expressions reduce to known results,
while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of -polytopes in , where .
For , the maximum values are attained when
, and are -polytopes, whose vertex sets are chosen appropriately from three distinct -dimensional moment-like curves
Analysis of the Incircle predicate for the Euclidean Voronoi diagram of axes-aligned line segments
In this paper we study the most-demanding predicate for computing the
Euclidean Voronoi diagram of axes-aligned line segments, namely the Incircle
predicate. Our contribution is two-fold: firstly, we describe, in algorithmic
terms, how to compute the Incircle predicate for axes-aligned line segments,
and secondly we compute its algebraic degree. Our primary aim is to minimize
the algebraic degree, while, at the same time, taking into account the amount
of operations needed to compute our predicate of interest.
In our predicate analysis we show that the Incircle predicate can be answered
by evaluating the signs of algebraic expressions of degree at most 6; this is
half the algebraic degree we get when we evaluate the Incircle predicate using
the current state-of-the-art approach. In the most demanding cases of our
predicate evaluation, we reduce the problem of answering the Incircle predicate
to the problem of computing the sign of the value of a linear polynomial (in
one variable), when evaluated at a known specific root of a quadratic
polynomial (again in one variable). Another important aspect of our approach is
that, from a geometric point of view, we answer the most difficult case of the
predicate via implicitly performing point locations on an appropriately defined
subdivision of the place induced by the Voronoi circle implicated in the
Incircle predicate.Comment: 17 pages, 4 figures, work presented in the paper is part of M.
Kamarianakis' M.S. thesi
Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphs
AbstractIn this paper we consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs.We transform the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a properly defined triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in its 2-dominating set. We show that: (1) ⌊n+13⌋ diagonal guards are always sufficient and sometimes necessary, and (2) ⌊2n+15⌋ edge guards are always sufficient and sometimes necessary, in order to 2-dominate a triangulation graph. Furthermore, we show how to compute: (1) a diagonal 2-dominating set of size ⌊n+13⌋ in linear time and space, (2) an edge 2-dominating set of size ⌊2n+15⌋ in O(n2) time and O(n) space, and (3) an edge 2-dominating set of size ⌊3n7⌋ in O(n) time and space.Based on the above-mentioned results, we prove that, for piecewise-convex polygons, we can compute: (1) a mobile guard set of size ⌊n+13⌋ in O(nlogn) time, (2) an edge guard set of size ⌊2n+15⌋ in O(n2) time, and (3) an edge guard set of size ⌊3n7⌋ in O(nlogn) time. All space requirements are linear. Finally, we show that ⌊n3⌋ mobile or ⌈n3⌉ edge guards are sometimes necessary.When restricting our attention to monotone piecewise-convex polygons, the bounds mentioned above drop: ⌈n+14⌉ edge or mobile guards are always sufficient and sometimes necessary; such an edge or mobile guard set, of size at most ⌈n+14⌉, can be computed in O(n) time and space