Related papers: Integer Point Sets Minimizing Average Pairwise L1-…
We show that all perfect odd integer squares not divisible by 3, can be usefully written as sqrt(N) = a + 18p, where the constant a is determined by the basic properties of N. The equation can be solved deterministically by an efficient…
Approximate nearest-neighbor search is a fundamental algorithmic problem that continues to inspire study due its essential role in numerous contexts. In contrast to most prior work, which has focused on point sets, we consider…
A set of $N$ points is chosen randomly in a $D$-dimensional volume $V=a^D$, with periodic boundary conditions. For each point $i$, its distance $d_i$ is found to its nearest neighbour. Then, the maximal value is found, $d_{max}=max(d_i,…
We investigate how effectively finite metric spaces can be distinguished by distance-based invariants. As model spaces, we consider regular polygons, cycle graphs, and their generalization, circular metric spaces, and as invariants we…
We determine the maximal non-central hyperplane sections of the n-dimensional $l_1$-ball if the fixed distance of the hyperplane to the origin is between $1 / \sqrt 3$ and $1 / \sqrt 2$. This adds to a result of Liu and Tkocz who considered…
A longstanding puzzle in urban science is whether there's an intrinsic match between human populations and the mass of their built environments. Previous findings have revealed various urban properties scaling nonlinearly with population,…
We show that for $m$ points and $n$ lines in the real plane, the number of distinct distances between the points and the lines is $\Omega(m^{1/5}n^{3/5})$, as long as $m^{1/2}\le n\le m^2$. We also prove that for any $m$ points in the…
An annulus is, informally, a ring-shaped region, often described by two concentric circles. The maximum-width empty annulus problem asks to find an annulus of a certain shape with the maximum possible width that avoids a given set of $n$…
Let $(\{1,2,\ldots,n\},d)$ be a metric space. We analyze the expected value and the variance of $\sum_{i=1}^{\lfloor n/2\rfloor}\,d({\boldsymbol{\pi}}(2i-1),{\boldsymbol{\pi}}(2i))$ for a uniformly random permutation ${\boldsymbol{\pi}}$ of…
We present optimal measuring strategies for the estimation of the entanglement of unknown two-qubit pure states and of the degree of mixing of unknown single-qubit mixed states, of which N identical copies are available. The most general…
We identify optimal measurement strategies for phase estimation in different scenarios. For pure states of a single qubit, we show that optimal measurements form a broad set parametrized with a continuous variable. When the state is mixed…
Sampling-based motion-planning algorithms typically rely on nearest-neighbor (NN) queries when constructing a roadmap. Recent results suggest that in various settings NN queries may be the computational bottleneck of such algorithms.…
For an ordered point set in a Euclidean space or, more generally, in an abstract metric space, the ordered Nearest Neighbor Graph is obtained by connecting each of the points to its closest predecessor by a directed edge. We show that for…
The problem of clustering a set of points moving on the line consists of the following: given positive integers n and k, the initial position and the velocity of n points, find an optimal k-clustering of the points. We consider two…
Optimal transportation distances are a fundamental family of parameterized distances for histograms. Despite their appealing theoretical properties, excellent performance in retrieval tasks and intuitive formulation, their computation…
The goal of Point Distance Solving Problems is to find 2D or 3D placements of points knowing distances between some pairs of points. The common guideline is to solve them by a numerical iterative method (\emph{e.g.} Newton-Raphson method).…
Defining an objective boundary for a city is a difficult problem, which remains to be solved by an effective method. Recent years, new methods for identifying urban boundary have been developed by means of spatial search techniques (e.g.…
Recent theoretical and empirical studies have focused on the structural properties of complex relational networks in social, biological and technological systems. Here we study the basic properties of twenty 1-square-mile samples of street…
In some domestic professional sports leagues, the home stadiums are located in cities connected by a common train line running in one direction. For these instances, we can incorporate this geographical information to determine optimal or…
We present an iterative method to efficiently solve the optimal transportation problem for a class of strictly convex costs which includes quadratic and p-power costs. Given two probability measures supported on a discrete grid with n…