A bounded space algorithm for online circle packing

doi:10.1016/j.ipl.2015.12.007

Information Processing Letters

Volume 116, Issue 5, May 2016, Pages 337-342

https://doi.org/10.1016/j.ipl.2015.12.007 Get rights and content

Highlights

•
We present results for the Online Circle Packing Problem.
•
A 2.4394-competitive bounded space algorithm.
•
A 2.292 lower bound on the competitive ratio of any online bounded space algorithm.
•
We combine Integer Linear Programming with Constraint Programming for computations.

Abstract

We study the Online Circle Packing Problem where we need to pack circles that arrive online in square bins with the objective to minimize the number of bins used. An online algorithm is said to have bounded space if at any given time, only a constant number of bins are open, circles are packed only in open bins and once a bin is closed it cannot be reopened. In particular, we present a 2.4394-competitive bounded space algorithm for this problem and a 2.2920 lower bound on the competitive ratio of any online bounded space algorithm for this problem.

Introduction

In the Online Circle Packing Problem, one has infinitely many square bins and receives a list of circles (given by its radii) in an online fashion. When a circle arrives, it must be packed in a bin, without intersecting other circles or the borders of the bin. Also, after packing the circle, it cannot be moved to another bin or another position in the bin. The objective is to minimize the number of bins used.

We say that an online algorithm A has an asymptotic competitive ratio of α if, for every instance I, $A (I) \leq α OPT (I) + C$ where $A (I)$ is the value of the solution produced by algorithm A, $OPT (I)$ is the value of an optimal offline solution and C is a constant. In this paper we present an online algorithm with asymptotic competitive ratio at most 2.4394. This algorithm has the nice property that it has bounded space, that is, at any time there is at most a constant number of open bins. After a bin is closed, it is not opened anymore and, hence, does not receive new circles. Also, we present a 2.2920 lower bound on the competitive ratio of any online bounded space algorithm for this problem.

Previous works The book of Szabo et al. [10] presents many results regarding finding the maximum common radius of k circles that can be packed in a unit square for several values of k along with other related problems. The website maintained by Specht [9] collects even more results, not only regarding the packing of circles in a unit square but also the packing of circles in a circle, in an isosceles right triangle, in a semicircle, in a circular quadrant and other problems. Some applications of circle packing includes obtaining a maximal coverage of radio towers in a geographical region [10] and construct photo collages [12]. A review on circle packing problems and methodologies can be found in [5].

For the offline circle packing problem, there is an asymptotic polynomial time approximation scheme by Miyazawa et al. [7] when we can augment the bin in one direction which can also be adapted to the circle strip packing problem. Note that, as shown by Demaine et al. [1], it is NP-hard to decide if a set of circles can be packed into a square bin.

This online problem is already studied in the literature when the objects to be packed are squares, rectangles and hyperboxes. Epstein and Van Stee [4] developed a bounded space online algorithm for the d-dimensional online hypercube packing, extended this algorithm for the d-dimensional online hyperbox packing, for the variable-sized d-dimensional bin packing problem and for the online bin packing with resource augmentation. Later on, Epstein and Van Stee [3] presented numerical lower and upper bounds for d-dimensional online bounded space hypercube for $d \in {2, \dots, 7}$ . Epstein [2] presented bounded and unbounded space algorithms for the two-dimensional online rectangle packing with orthogonal rotations.

Section snippets

An algorithm for Online Circle Packing

We start by presenting an algorithm for the Online Circle Packing Problem. For simplicity, we consider that a bin is a square of side length 1. The algorithm divides the circles into large circles and small circles. Given some positive integer constant M, a circle is said to be large if its radius is bigger than $1 / M$ and, otherwise, it is said to be small.

For every positive integer i, let $ρ_{i}^{⁎}$ be the largest value such that i circles of radius $ρ_{i}^{⁎}$ can be packed in a bin. For example, $ρ_{1}^{⁎} = 0.5$

Competitive ratio analysis

We compute an upper bound for the asymptotic competitive ratio of the algorithm as well a lower bound for every bounded space online algorithm using a weighting function, as previously done in [11], [6], [4] for other packing problems.

We start by defining a weighting function w. For a circle of radius r, if it is a large circle of type i then its weight is $1 / i$ and if it is a small circle of radius r, then its weight is $(π r^{2}) / α$ where $α = (1 - \frac{5.89}{M}) \frac{π}{\sqrt{12}} \frac{M^{2}}{{(M + 1)}^{2}}$ . Also, for a set $C$ of circles, let $w (C) =$

Numerical results

In order to obtain numerical results for the bounds presented in Sect. 3, we combine Integer Linear Programming with Constraint Programming. Constraint Satisfaction Problems (CSP) are defined by a set of variables $X = {x_{1}, \dots, x_{n}}$ , a finite set $Dom (x_{i})$ , for each variable $x_{i}$ , called domain of $x_{i}$ , with all possible values this variable can assume. Also, a set of constraints restricts the values variables can simultaneously assume. A solution for this problem is an assignment of values to all variables

Final remarks

In this paper we present a bounded space algorithm for the Online Circle Packing Problem, which has a competitive ratio of 2.4394. To our knowledge, this is the first algorithm for such problem in the literature. We also present a lower bound of 2.2920 for any bounded space algorithm for such problem. Notice that these bounds can possibly be improved using different techniques. Also, to our knowledge, this is the first paper to use Constrained Programming as heuristic to test the feasibility of

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Near-optimal bin packing algorithms
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There are more references available in the full text version of this article.

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Citation Excerpt :
We show a general idea of bounded space and unbounded space algorithms for packing circles/spheres which works for the three problems that we are considering and can work for other problems, if they present some characteristics which will be better described later. The bounded space algorithms are in part based on the one given by Hokama et al. [10], but they have a simpler analysis. Furthermore, we were able to increase the area/volume occupied by the items in one bin, we found a simpler way of subdividing a bin, and we used another method to find the numerical results, which are described next and summarized in Table 1.
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¹: This research was partially supported by grants #2013/21744-8 and #2011/13382-3, São Paulo Research Foundation (FAPESP), and grants 311499/2014-7 and 477692/2012-5, National Council for Scientific and Technological Development (CNPq).

View full text

A bounded space algorithm for online circle packing

Highlights

Abstract

Introduction

Section snippets

An algorithm for Online Circle Packing

Competitive ratio analysis

Numerical results

Final remarks

Theor. Comput. Sci.

Discrete Optim.

Circle packing for origami design is hard

Optimal online algorithms for multidimensional packing problems

SIAM J. Comput.

A literature review on circle and sphere packing problems: models and methodologies

Adv. Oper. Res.

Near-optimal bin packing algorithms