Related papers: On Ray Shooting for Triangles in 3-Space and Relat…
Recently, Ezra and Sharir [ES22a] showed an $O(n^{3/2+\sigma})$ space and $O(n^{1/2+\sigma})$ query time data structure for ray shooting among triangles in $\mathbb{R}^3$. This improves the upper bound given by the classical…
A common representation of a three dimensional object in computer applications, such as graphics and design, is in the form of a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy…
The {\it matrix-chain multiplication} problem is a classic problem that is widely taught to illustrate dynamic programming. The textbook solution runs in $\theta(n^3)$ time. However, there is a complex $O(n \log n)$-time method \cite{HU82},…
We investigate the problem of partitioning a rectilinear polygon $P$ with $n$ vertices and no holes % with no holes into rectangles using disjoint line segments drawn inside $P$ under two optimality criteria. In the minimum ink partition,…
We study quantum algorithms for problems in computational geometry, such as POINT-ON-3-LINES problem. In this problem, we are given a set of lines and we are asked to find a point that lies on at least $3$ of these lines. POINT-ON-3-LINES…
Given $n$ non-vertical lines in 3-space, their vertical depth (above/below) relation can contain cycles. We show that the lines can be cut into $O(n^{3/2}\mathop{\mathrm{polylog}} n)$ pieces, such that the depth relation among these pieces…
We study the problem of computing the minimum area triangle that circumscribes a given $n$-sided convex polygon touching edge-to-edge. In other words, we compute the minimum area triangle that is the intersection of 3 half-planes out of $n$…
While well-known methods to list the intersections of either a list of segments or a complex polygon aim at achieving optimal time-complexity they often do so at the cost of memory comsumption and complex code. Real-life software…
Assume we are given a set of parallel line segments in the plane, and we wish to place a point on each line segment such that the resulting point set maximizes or minimizes the area of the largest or smallest triangle in the set. We analyze…
We show that the three-dimensional layers-of-maxima problem can be solved in $o(n\log n)$ time in the word RAM model. Our algorithm runs in $O(n(\log \log n)^3)$ deterministic time or $O(n(\log\log n)^2)$ expected time and uses O(n) space.…
We present new iterative algorithms for solving a square linear system $Ax=b$ in dimension $n$ by employing the {\it Triangle Algorithm} \cite{kal12}, a fully polynomial-time approximation scheme for testing if the convex hull of a finite…
Given a set of $m$ points and a set of $n$ lines in the plane, we consider the problem of computing the faces of the arrangement of the lines that contain at least one point. In this paper, we present an $O(m^{2/3}n^{2/3}+(n+m)\log n)$ time…
The study of extremal problems on triangle areas was initiated in a series of papers by Erd\H{o}s and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that…
Finding, counting and/or listing triangles (three vertices with three edges) in large graphs are natural fundamental problems, which received recently much attention because of their importance in complex network analysis. We provide here a…
In this paper, we propose new techniques for solving geometric optimization problems involving interpoint distances of a point set in the plane. Given a set $P$ of $n$ points in the plane and an integer $1 \leq k \leq \binom{n}{2}$, the…
We consider the Max-$3$-Section problem, where we are given an undirected graph $ G=(V,E)$ equipped with non-negative edge weights $w :E\rightarrow \mathbb{R}_+$ and the goal is to find a partition of $V$ into three equisized parts while…
The article proposes a new method for finding the triangle-triangle intersection in 3D space, based on the use of computer graphics algorithms -- cutting off segments on the plane when moving and rotating the beginning of the coordinate…
We revisit the algorithmic problem of finding a triangle in a graph: We give a randomized combinatorial algorithm for triangle detection in a given $n$-vertex graph with $m$ edges running in $O(n^{7/3})$ time, or alternatively in…
Modeling of microlensing events poses computational challenges for the resolution of the lens equation and the high dimensionality of the parameter space. In particular, numerical noise represents a severe limitation to fast and efficient…
We improve upon the running time for finding a point in a convex set given a separation oracle. In particular, given a separation oracle for a convex set $K\subset \mathbb{R}^n$ contained in a box of radius $R$, we show how to either find a…