Related papers: Ranking Unit Squares with Few Visibilities

Let $S$ be a set of $n$ weighted points in the plane and let $R$ be a query range in the plane. In the range closest pair problem, we want to report the closest pair in the set $R \cap S$. In the range minimum weight problem, we want to…

We report results from an experiment on ranking visual markers and node positioning techniques for network visualizations. Inspired by prior ranking studies, we rethink the ranking when the dataset size increases and when the markers are…

Consider an arrangement of $k$ lines intersecting the unit square. There is some minimum scaling factor so that any placement of a rectangle with aspect ratio $1 \times p$ with $p\geq 1$ must non-transversely intersect some portion of the…

We analyze the problem of packing squares in an online fashion: Given a semi-infinite strip of width 1 and an unknown sequence of squares of side length in [0,1] that arrive from above, one at a time. The objective is to pack these items as…

We study unit disk visibility graphs, where the visibility relation between a pair of geometric entities is defined by not only obstacles, but also the distance between them. That is, two entities are not mutually visible if they are too…

To determine whether an $n\times n$-matrix has rank at most $r$ it suffices to check that the $(r+1)\times (r+1)$-minors have rank at most $r$. In other words, to describe the set of $n\times n$-matrices with the property of having rank at…

We construct partitions of $\mathbb{R}^n$ into three sets $\{\mathscr{X}(1),\mathscr{X}(2),\mathscr{X}(3)\}$ that locally minimize interfacial area among compactly supported volume preserving variations and that blow down at infinity to…

At the time of writing, the general problem of finding the maximal Waring rank for homogeneous polynomials of fixed degree and number of variables (or, equivalently, the maximal symmetric rank for symmetric tensors of fixed order and in…

We consider the problem of identifying n points in the plane using disks, i.e., minimizing the number of disks so that each point is contained in a disk and no two points are in exactly the same set of disks. This problem can be seen as an…

Tile the unit square with $n$ small squares. We determine the minimum of the sum of the side lengths of the $n$ small squares, where the minimum is taken over all tilings of the unit square with $n$ squares.

It is classically known that a real cubic surface in the real projective 3-space cannot have more than one solitary point (locally given by x^2+y^2+z^2=0) whereas it can have up to four nodes (x^2+y^2-z^2=0). We show that on any surface of…

Many clustering problems in computer vision and other contexts are also classification problems, where each cluster shares a meaningful label. Subspace clustering algorithms in particular are often applied to problems that fit this…

Given an undirected graph representing similarities between a set of items and an additive measure evaluating the items, we treat the position of a special subset of items in an ordinal ranking through a collection of combinatorial…

We consider the problem of packing a large square with nonoverlapping unit squares. Let $W(x)$ be the minimum wasted area when a large square of side length $x$ is packed with unit squares. In Roth and Vaughan's paper that proves the lower…

Unit square (grid) visibility graphs (USV and USGV, resp.) are described by axis-parallel visibility between unit squares placed (on integer grid coordinates) in the plane. We investigate combinatorial properties of these graph classes and…

A methodology is presented to rank universities on the basis of the lists of programmes the students applied for. We exploit a crucial feature of the centralised assignment system to higher education in Hungary: a student is admitted to the…

Given a finite family of squares in the plane, the packing problem asks for the maximum number $\nu$ of pairwise disjoint squares among them, while the hitting problem for the minimum number $\tau$ of points hitting all of them. Clearly,…

Given a set $P$ of points and a set $U$ of axis-parallel unit squares in the Euclidean plane, a minimum ply cover of $P$ with $U$ is a subset of $U$ that covers $P$ and minimizes the number of squares that share a common intersection,…

It is unknown at present whether a magic square of squared integers exists. Such an object is defined to be a 3 by 3 grid of 9 distinct integer squares, such that the entries of each row, column, and two main diagonals sum to the same…

A graph is called a $k$-planar unit distance graph if it can be drawn in the plane such that every edge is a unit line segment and is involved in at most $k$ crossings. We investigate $u_k(n)$, the maximum number of edges of such graphs on…