Related papers: Adaptive Computation of the Klee's Measure in High…
Union volume estimation is a classical algorithmic problem. Given a family of objects $O_1,\ldots,O_n \subseteq \mathbb{R}^d$, we want to approximate the volume of their union. In the special case where all objects are boxes (also known as…
Klee's measure problem (computing the volume of the union of $n$ axis-parallel boxes in $\mathbb{R}^d$) is well known to have $n^{\frac{d}{2}\pm o(1)}$-time algorithms (Overmars, Yap, SICOMP'91; Chan FOCS'13). Only recently, a conditional…
Motivated by the analysis of range queries in databases, we introduce the computation of the Depth Distribution of a set $\mathcal{B}$ of axis aligned boxes, whose computation generalizes that of the Klee's Measure and of the Maximum Depth.…
Klee's Measure Problem (KMP) asks for the volume of the union of n axis-aligned boxes in d-space. Omitting logarithmic factors, the best algorithm has runtime O*(n^{d/2}) [Overmars,Yap'91]. There are faster algorithms known for several…
We revisit a classical problem in computational geometry: finding the largest-volume axis-aligned empty box (inside a given bounding box) amidst $n$ given points in $d$ dimensions. Previously, the best algorithms known have running time…
We consider the computation of the volume of the union of high-dimensional geometric objects. While showing that this problem is #P-hard already for very simple bodies (i.e., axis-parallel boxes), we give a fast FPRAS for all objects where…
We introduce space-efficient plane-sweep algorithms for basic planar geometric problems. It is assumed that the input is in a read-only array of $n$ items and that the available workspace is $\Theta(s)$ bits, where $\lg n \leq s \leq n…
In this paper, we focus on lower bounds and algorithms for some basic geometric problems in the one-pass (insertion only) streaming model. The problems considered are grouped into three categories: (i) Klee's measure (ii) Convex body…
We present a (combinatorial) algorithm with running time close to $O(n^d)$ for computing the minimum directed $L_\infty$ Hausdorff distance between two sets of $n$ points under translations in any constant dimension $d$. This substantially…
Let $B$ be a set of $n$ axis-parallel boxes in $\mathbb{R}^d$ such that each box has a corner at the origin and the other corner in the positive quadrant of $\mathbb{R}^d$, and let $k$ be a positive integer. We study the problem of…
We give the first nontrivial upper and lower bounds on the maximum volume of an empty axis-parallel box inside an axis-parallel unit hypercube in $\RR^d$ containing $n$ points. For a fixed $d$, we show that the maximum volume is of the…
Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of an $n$-dimensional convex body within…
The union volume estimation problem asks to $(1\pm\varepsilon)$-approximate the volume of the union of $n$ given objects $X_1,\ldots,X_n \subset \mathbb{R}^d$. In their seminal work in 1989, Karp, Luby, and Madras solved this problem in…
The efficient estimation of frequency moments of a data stream in one-pass using limited space and time per item is one of the most fundamental problem in data stream processing. An especially important estimation is to find the number of…
In this paper, we study the problem of computing the diameter of a set of $n$ points in $d$-dimensional Euclidean space for a fixed dimension $d$, and propose a new $(1+\varepsilon)$-approximation algorithm with $O(n+ 1/\varepsilon^{d-1})$…
We show that the volume of a convex body in $\mathbb{R}^{n}$ in the general membership oracle model can be computed to within relative error $\varepsilon$ using $\widetilde{O}(n^{3.5}\psi^{2} + n^3/\varepsilon^{2})$ oracle queries, where…
We consider the problem of computing Shapley values for points in the plane, where each point is interpreted as a player, and the value of a coalition is defined by the area of usual geometric objects, such as the convex hull or the minimum…
We give new polynomial lower bounds for a number of dynamic measure problems in computational geometry. These lower bounds hold in the Word-RAM model, conditioned on the hardness of either 3SUM, APSP, or the Online Matrix-Vector…
The curse of dimensionality in the realm of association rules is twofold. Firstly, we have the well known exponential increase in computational complexity with increasing item set size. Secondly, there is a \emph{related curse} concerned…
We consider the problem of augmenting an $n$-vertex tree with one shortcut in order to minimize the diameter of the resulting graph. The tree is embedded in an unknown space and we have access to an oracle that, when queried on a pair of…