21 research outputs found
Output-Sensitive Tools for Range Searching in Higher Dimensions
Let be a set of points in . A point is
\emph{-shallow} if it lies in a halfspace which contains at most points
of (including ). We show that if all points of are -shallow, then
can be partitioned into subsets, so that any hyperplane
crosses at most subsets. Given such
a partition, we can apply the standard construction of a spanning tree with
small crossing number within each subset, to obtain a spanning tree for the
point set , with crossing number . This allows us to extend the construction of Har-Peled
and Sharir \cite{hs11} to three and higher dimensions, to obtain, for any set
of points in (without the shallowness assumption), a
spanning tree with {\em small relative crossing number}. That is, any
hyperplane which contains points of on one side, crosses
edges of . Using a
similar mechanism, we also obtain a data structure for halfspace range
counting, which uses space (and somewhat higher
preprocessing cost), and answers a query in time , where is the output size
Finding Small Hitting Sets in Infinite Range Spaces of Bounded VC-Dimension
We consider the problem of finding a small hitting set in an infinite range space F=(Q,R) of bounded VC-dimension. We show that, under reasonably general assumptions, the infinite-dimensional convex relaxation can be solved (approximately) efficiently by multiplicative weight updates. As a consequence, we get an algorithm that finds, for any delta>0, a set of size O(s_F(z^*_F)) that hits (1-delta)-fraction of R (with respect to a given measure) in time proportional to log(1/delta), where s_F(1/epsilon) is the size of the smallest epsilon-net the range space admits, and z^*_F is the value of the fractional optimal solution. This exponentially improves upon previous results which achieve the same approximation guarantees with running time proportional to poly(1/delta). Our assumptions hold, for instance, in the case when the range space represents the visibility regions of a polygon in the plane, giving thus a deterministic polynomial-time O(log z^*_F)-approximation algorithm for guarding (1-delta)-fraction of the area of any given simple polygon, with running time proportional to polylog(1/delta)
Lower Bounds for Semialgebraic Range Searching and Stabbing Problems
In the semialgebraic range searching problem, we are to preprocess points
in s.t. for any query range from a family of constant complexity
semialgebraic sets, all the points intersecting the range can be reported or
counted efficiently. When the ranges are composed of simplices, the problem can
be solved using space and with query time with and this trade-off is almost tight. Consequently, there exists
low space structures that use space with query
time and fast query structures that use space with
query time. However, for the general semialgebraic ranges, only low space
solutions are known, but the best solutions match the same trade-off curve as
the simplex queries. It has been conjectured that the same could be done for
the fast query case but this open problem has stayed unresolved.
Here, we disprove this conjecture. We give the first nontrivial lower bounds
for semilagebraic range searching and related problems. We show that any data
structure for reporting the points between two concentric circles with
query time must use space, meaning, for
, space must be used. We also study
the problem of reporting the points between two polynomials of form
where are given at the
query time. We show . So
for , we must use space. For
the dual semialgebraic stabbing problems, we show that in linear space, any
data structure that solves 2D ring stabbing must use query
time. This almost matches the linearization upper bound. For general
semialgebraic slab stabbing problems, again, we show an almost tight lower
bounds.Comment: Submitted to SoCG'21; this version: readjust the table and other
minor change
A Nearly Quadratic Bound for the Decision Tree Complexity of k-SUM
We show that the k-SUM problem can be solved by a linear decision tree of depth O(n^2 log^2 n),improving the recent bound O(n^3 log^3 n) of Cardinal et al. Our bound depends linearly on k, and allows us to conclude that the number of linear queries required to decide the n-dimensional Knapsack or SubsetSum problems is only O(n^3 log n), improving the currently best known bounds by a factor of n. Our algorithm extends to the RAM model, showing that the k-SUM problem can be solved in expected polynomial time, for any fixed k, with the above bound on the number of linear queries. Our approach relies on a new point-location mechanism, exploiting "Epsilon-cuttings" that are based on vertical decompositions in hyperplane arrangements in high dimensions.
A major side result of the analysis in this paper is a sharper bound on the complexity of the vertical decomposition of such an arrangement (in terms of its dependence on the dimension). We hope that this study will reveal further structural properties of vertical decompositions in hyperplane arrangements
Efficient NC algorithms for set cover with applications to learning and geometry
In this paper we give the first NC approximation algorithms for the unweighted and weighted set cover problems. Our algorithms use a linear number of processors and give a cover that has at most log n times the optimal size/weight, thus matching the performance of the best sequential algorithms. We apply our set cover algorithm to learning theory, giving an NC algorithm to learn the concept class obtained by taking the closure under finite union or finite intersection of any concept class of finite VC-dimension that has an NC hypothesis finder. In addition, we give a linear-processor NC algorithm for a variant of the set cover problem first proposed by Chazelle and Friedman and use it to obtain NC algorithms for several problems in computational geometry
Effizient algorithms for generalized intersection searching on non-iso-oriented objects
In a generalized intersection searching problem, a set of colored geometric objects is to be preprocessed so that, given a query object , the distinct colors of the objects of that are intersected by can be reported or counted efficiently. These problems generalize the well-studied standard intersection searching problems and are rich in applications. Unfortunately, the solutions known for the standard problems do not yield efficient solutions to the generalized problems. Recently, efficient solutions have been given for generalized problems where the input and query objects are iso-oriented, i.e., axes-parallel, or where the color classes satisfy additional properties, e.g., connectedness. In this paper, efficient algorithms are given for several generalized problems involving non-iso-oriented objects. These problems include: generalized halfspace range searching in , for any fixed , segment intersection searching, triangle stabbing, and triangle range searching in . The techniques used include: computing suitable sparse representations of the input, persistent data structures, and filtering search
Fast algorithms for collision and proximity problems involving moving geometric objects
Consider a set of geometric objects, such as points, line segments, or axes-parallel hyperrectangles in \IR^d, that move with constant but possibly different velocities along linear trajectories. Efficient algorithms are presented for several problems defined on such objects, such as determining whether any two objects ever collide and computing the minimum inter-point separation or minimum diameter that ever occurs. The strategy used involves reducing the given problem on moving objects to a different problem on a set of static objects, and then solving the latter problem using techniques based on sweeping, orthogonal range searching, simplex composition, and parametric search