8 research outputs found
An upper bound on the k-modem illumination problem
A variation on the classical polygon illumination problem was introduced in
[Aichholzer et. al. EuroCG'09]. In this variant light sources are replaced by
wireless devices called k-modems, which can penetrate a fixed number k, of
"walls". A point in the interior of a polygon is "illuminated" by a k-modem if
the line segment joining them intersects at most k edges of the polygon. It is
easy to construct polygons of n vertices where the number of k-modems required
to illuminate all interior points is Omega(n/k). However, no non-trivial upper
bound is known. In this paper we prove that the number of k-modems required to
illuminate any polygon of n vertices is at most O(n/k). For the cases of
illuminating an orthogonal polygon or a set of disjoint orthogonal segments, we
give a tighter bound of 6n/k + 1. Moreover, we present an O(n log n) time
algorithm to achieve this bound.Comment: 9 pages, 4 figure
New Frameworks for Offline and Streaming Coreset Constructions
A coreset for a set of points is a small subset of weighted points that
approximately preserves important properties of the original set. Specifically,
if is a set of points, is a set of queries, and is a cost function, then a set with weights
is an -coreset for some parameter if
is a multiplicative approximation to
for all . Coresets are used to solve fundamental
problems in machine learning under various big data models of computation. Many
of the suggested coresets in the recent decade used, or could have used a
general framework for constructing coresets whose size depends quadratically on
what is known as total sensitivity .
In this paper we improve this bound from to . Thus our
results imply more space efficient solutions to a number of problems, including
projective clustering, -line clustering, and subspace approximation.
Moreover, we generalize the notion of sensitivity sampling for sup-sampling
that supports non-multiplicative approximations, negative cost functions and
more. The main technical result is a generic reduction to the sample complexity
of learning a class of functions with bounded VC dimension. We show that
obtaining an -sample for this class of functions with appropriate
parameters and suffices to achieve space efficient
-coresets.
Our result implies more efficient coreset constructions for a number of
interesting problems in machine learning; we show applications to
-median/-means, -line clustering, -subspace approximation, and the
integer -projective clustering problem
Extending the Centerpoint Theorem to Multiple Points
The centerpoint theorem is a well-known and widely used result in discrete geometry. It states that for any point set P of n points in R^d, there is a point c, not necessarily from P, such that each halfspace containing c contains at least n/(d+1) points of P. Such a point c is called a centerpoint, and it can be viewed as a generalization of a median to higher dimensions. In other words, a centerpoint can be interpreted as a good representative for the point set P. But what if we allow more than one representative? For example in one-dimensional data sets, often certain quantiles are chosen as representatives instead of the median.
We present a possible extension of the concept of quantiles to higher dimensions. The idea is to find a set Q of (few) points such that every halfspace that contains one point of Q contains a large fraction of the points of P and every halfspace that contains more of Q contains an even larger fraction of P. This setting is comparable to the well-studied concepts of weak epsilon-nets and weak epsilon-approximations, where it is stronger than the former but weaker than the latter. We show that for any point set of size n in R^d and for any positive alpha_1,...,alpha_k where alpha_1 <= alpha_2 <= ... <= alpha_k and for every i,j with i+j <= k+1 we have that (d-1)alpha_k+alpha_i+alpha_j <= 1, we can find Q of size k such that each halfspace containing j points of Q contains least alpha_j n points of P. For two-dimensional point sets we further show that for every alpha and beta with alpha <= beta and alpha+beta <= 2/3 we can find Q with |Q|=3 such that each halfplane containing one point of Q contains at least alpha n of the points of P and each halfplane containing all of Q contains at least beta n points of P. All these results generalize to the setting where P is any mass distribution. For the case where P is a point set in R^2 and |Q|=2, we provide algorithms to find such points in time O(n log^3 n)
Algorithms for Geometric Facility Location: Centers in a Polygon and Dispersion on a Line
We study three geometric facility location problems in this thesis.
First, we consider the dispersion problem in one dimension. We are given an ordered list
of (possibly overlapping) intervals on a line. We wish to choose exactly one point from
each interval such that their left to right ordering on the line matches the input order.
The aim is to choose the points so that the distance between the closest pair of points is
maximized, i.e., they must be socially distanced while respecting the order. We give a new
linear-time algorithm for this problem that produces a lexicographically optimal solution.
We also consider some generalizations of this problem.
For the next two problems, the domain of interest is a simple polygon with n vertices.
The second problem concerns the visibility center. The convention is to think of a polygon
as the top view of a building (or art gallery) where the polygon boundary represents opaque
walls. Two points in the domain are visible to each other if the line segment joining them
does not intersect the polygon exterior. The distance to visibility from a source point to a
target point is the minimum geodesic distance from the source to a point in the polygon
visible to the target. The question is: Where should a single guard be located within the
polygon to minimize the maximum distance to visibility? For m point sites in the polygon,
we give an O((m + n) log (m + n)) time algorithm to determine their visibility center.
Finally, we address the problem of locating the geodesic edge center of a simple polygon—a
point in the polygon that minimizes the maximum geodesic distance to any edge. For a
triangle, this point coincides with its incenter. The geodesic edge center is a generalization
of the well-studied geodesic center (a point that minimizes the maximum distance to any
vertex). Center problems are closely related to farthest Voronoi diagrams, which are well-
studied for point sites in the plane, and less well-studied for line segment sites in the plane.
When the domain is a polygon rather than the whole plane, only the case of point sites has
been addressed—surprisingly, more general sites (with line segments being the simplest
example) have been largely ignored. En route to our solution, we revisit, correct, and
generalize (sometimes in a non-trivial manner) existing algorithms and structures tailored
to work specifically for point sites. We give an optimal linear-time algorithm for finding
the geodesic edge center of a simple polygon