37 research outputs found
Tight Hardness Results for Maximum Weight Rectangles
Given weighted points (positive or negative) in dimensions, what is
the axis-aligned box which maximizes the total weight of the points it
contains?
The best known algorithm for this problem is based on a reduction to a
related problem, the Weighted Depth problem [T. M. Chan, FOCS'13], and runs in
time . It was conjectured [Barbay et al., CCCG'13] that this runtime is
tight up to subpolynomial factors. We answer this conjecture affirmatively by
providing a matching conditional lower bound. We also provide conditional lower
bounds for the special case when points are arranged in a grid (a well studied
problem known as Maximum Subarray problem) as well as for other related
problems.
All our lower bounds are based on assumptions that the best known algorithms
for the All-Pairs Shortest Paths problem (APSP) and for the Max-Weight k-Clique
problem in edge-weighted graphs are essentially optimal
Minimum Convex Partitions and Maximum Empty Polytopes
Let be a set of points in . A Steiner convex partition
is a tiling of with empty convex bodies. For every integer ,
we show that admits a Steiner convex partition with at most tiles. This bound is the best possible for points in general
position in the plane, and it is best possible apart from constant factors in
every fixed dimension . We also give the first constant-factor
approximation algorithm for computing a minimum Steiner convex partition of a
planar point set in general position. Establishing a tight lower bound for the
maximum volume of a tile in a Steiner convex partition of any points in the
unit cube is equivalent to a famous problem of Danzer and Rogers. It is
conjectured that the volume of the largest tile is .
Here we give a -approximation algorithm for computing the
maximum volume of an empty convex body amidst given points in the
-dimensional unit box .Comment: 16 pages, 4 figures; revised write-up with some running times
improve
On the Number of Maximum Empty Boxes Amidst n Points
We revisit the following problem (along with its higher dimensional variant): Given a set S of n points inside an axis-parallel rectangle U in the plane, find a maximum-area axis-parallel sub-rectangle that is contained in U but contains no points of S.
1. We prove that the number of maximum-area empty rectangles amidst n points in the plane is O(n log n 2^alpha(n)), where alpha(n) is the extremely slowly growing inverse of Ackermann\u27s function. The previous best bound, O(n^2), is due to Naamad, Lee, and Hsu (1984).
2. For any d at least 3, we prove that the number of maximum-volume empty boxes amidst n points in R^d is always O(n^d) and sometimes Omega(n^floor(d/2)).
This is the first superlinear lower bound derived for this problem.
3. We discuss some algorithmic aspects regarding the search for a maximum empty box in R^3. In particular, we present an algorithm that finds a (1-epsilon)-approximation of the maximum empty box amidst n points in O(epsilon^{-2} n^{5/3} log^2{n}) time
Applications of Computational Geometry and Computer Vision
Recent advances in machine learning research promise to bring us closer to the original goals of artificial intelligence. Spurred by recent innovations in low-cost, specialized hardware and incremental refinements in machine learning algorithms, machine learning is revolutionizing entire industries. Perhaps the biggest beneficiary of this progress has been the field of computer vision. Within the domains of computational geometry and computer vision are two problems: Finding large, interesting holes in high dimensional data, and locating and automatically classifying facial features from images. State of the art methods for facial feature classification are compared and new methods for finding empty hyper-rectangles are introduced. The problem of finding holes is then linked to the problem of extracting features from images and deep learning methods such as convolutional neural networks. The performance of the hole-finding algorithm is measured using multiple standard machine learning benchmarks as well as a 39 dimensional dataset, thus demonstrating the utility of the method for a wide range of data
Faster Algorithms for Largest Empty Rectangles and Boxes
We revisit a classical problem in computational geometry: finding the
largest-volume axis-aligned empty box (inside a given bounding box) amidst
given points in dimensions. Previously, the best algorithms known have
running time for (by Aggarwal and Suri [SoCG'87]) and near
for . We describe faster algorithms with running time (i)
for , (ii) time for ,
and (iii) time for any constant .
To obtain the higher-dimensional result, we adapt and extend previous
techniques for Klee's measure problem to optimize certain objective functions
over the complement of a union of orthants.Comment: full version of a SoCG 2021 pape