Related papers: Square packing with $O(x^{0.6})$ wasted area
We present a novel iterative algorithm for approximating the linear least squares solution with low complexity. After a motivation of the algorithm we discuss the algorithm's properties including its complexity, and we present theoretical…
We study the d-dimensional hypercube knapsack problem where we are given a set of d-dimensional hypercubes with associated profits, and a knapsack which is a unit d-dimensional hypercube. The goal is to find an axis-aligned non-overlapping…
We construct a hyperbolic sextic surface in P^3(C).
We prove polynomial-time solvability of a large class of clustering problems where a weighted set of items has to be partitioned into clusters with respect to some balancing constraints. The data points are weighted with respect to…
We propose a new method for constructing small-bias spaces through a combination of Hermitian codes. For a class of parameters our multisets are much faster to construct than what can be achieved by use of the traditional algebraic…
We study several variations of line segment covering problem with axis-parallel unit squares in $I\!\!R^2$. 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…
In this paper, new schemes for a squarer, multiplier and divider of complex numbers are proposed. Traditional structural solutions for each of these operations require the presence some number of general-purpose binary multipliers. The…
Suppose that $I$ is a unit square. Let $T$ (resp. $\Delta$) be an isosceles right triangle (resp. an equilateral triangle). We prove that any collection of triangles homothetic to $T$ (resp. $\Delta$), whose total area does not exceed…
We filled a void with a regular or asymmetric pattern without overlap using a time-dependent packing method termed random sequential adsorption (RSA). In the infinite-time limit, the density of coverage frequently hits a limit. This study…
We study several consequences of the packing problem, a conjecture from combinatorial optimization, using algebraic invariants of square-free monomial ideals. While the packing problem is currently unresolved, we successfully settle the…
In this paper we study the hard sphere packing problem in the Hamming space by the cavity method. We show that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically…
Let \Sigma_g be a closed orientable surface of genus g \geq 2 and \tau a graph on \Sigma_g with one vertex which lifts to a triangulation of the universal cover. We have shown that the cross ratio parameter space \mathcal{C}_\tau associated…
The set of 2-dimensional packing problems builds an important class of optimization problems and Strip Packing together with 2-dimensional Bin Packing and 2-dimensional Knapsack is one of the most famous of these problems. Given a set of…
We adapt a construction of Klee (1981) to find a packing of unit balls in $\ell_p$ ($1\leq p<\infty$) which is efficient in the sense that enlarging the radius of each ball to any $R>2^{1-1/p}$ covers the whole space. We show that the value…
We consider a decluttering problem where multiple rigid convex polygonal objects rest in randomly placed positions and orientations on a planar surface and must be efficiently transported to a packing box using both single and multi-object…
The properties of the number of iterations in random sequential adsorption protocol needed to generate finite saturated random packing of spherically symmetric shapes were studied. Numerical results obtained for one, two, and three…
We propose a space-efficient algorithm for hidden surface removal that combines one of the fastest previous algorithms for that problem with techniques based on bit manipulation. Such techniques had been successfully used in other settings,…
We recall a proof of the mixing for almost all Ornstein's stochastic rank one constructions, replace stochastic spacers by special algebraic ones and prove the mixing in this new situation.
In this paper, we study the 3D strip packing problem in which we are given a list of 3-dimensional boxes and required to pack all of them into a 3-dimensional strip with length 1 and width 1 and unlimited height to minimize the height used.…
Functions with discontinuities appear in many applications such as image reconstruction, signal processing, optimal control problems, interface problems, engineering applications and so on. Accurate approximation and interpolation of these…