Global unit squares and local unit squares

Translative Packing of Unit Squares into Squares

Packing Unit Squares in Squares: A Survey and New Results

Let s(n) be the side of the smallest square into which we can pack n unit squares. We improve the best known upper bounds for s(n) when n = 26, 37, 39, 50, 54, 69, 70, 85, 86, and 88. We present relatively simple proofs for the values of s(n) when n = 2, 3, 5, 8, 15, 24, and 35, and more complicated proofs for n=7 and 14. We also prove many other lower bounds for various s(n). We also give the ...

Covering Segments with Unit Squares

We study several variations of line segment covering problem with axis-parallel unit squares in IR. A set S of n line segments is given. The objective is to find the minimum number of axis-parallel unit squares which cover at least one end-point of each segment. The variations depend on the orientation and length of the input segments. We prove some of these problems to be NP-complete, and give...

Geometric Unique Set Cover on Unit Disks and Unit Squares

We study the Unique Set Cover problem on unit disks and unit squares. For a given set P of n points and a set D of m geometric objects both in the plane, the objective of the Unique Set Cover problem is to select a subset D′ ⊆ D of objects such that every point in P is covered by at least one object in D′ and the number of points covered uniquely is maximized, where a point is covered uniquely ...

Ranking Unit Squares with Few Visibilities

Given a set of n unit squares in the plane, the goal is to rank them in space in such a way that only few squares see each other vertically. We prove that ranking the squares according to the lexicographic order of their centers results in at most 3n-7 pairwise visibilities for n at least 4. We also show that this bound is best possible, by exhibiting a set of n squares with at least 3n-7 pairw...