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Exact Methods for Recursive Circle Packing [article]

Ambros Gleixner, Stephen Maher, Benjamin Müller, João Pedro Pedroso
2017 arXiv   pre-print
This problem is known as the Recursive Circle Packing Problem (RCPP).  ...  We present the first dedicated method for solving RCPP that provides strong dual bounds based on an exact Dantzig--Wolfe reformulation of a nonconvex mixed-integer nonlinear programming formulation.  ...  All responsibilty for the content of this publication is assumed by the authors.  ... 
arXiv:1702.07799v1 fatcat:dmuwwpruanbyhnm6zk4fcgynce

Circle Packing for Origami Design Is Hard [article]

Erik D. Demaine, Sandor P. Fekete, Robert J. Lang
2010 arXiv   pre-print
On the positive side, we show that any set of circles of total area 1 can be packed into a square of size 4/√(pi)=2.2567...  ...  We show that deciding whether a given set of circles can be packed into a rectangle, an equilateral triangle, or a unit square are NP-hard problems, settling the complexity of these natural packing problems  ...  Acknowledgments We thank Ron Graham for several helpful hints concerning the state of the art on packing circles. We also thank Vinayak Pathak for pointing out a numerical typo related to Figure 13 .  ... 
arXiv:1008.1224v2 fatcat:pkp57wi5vrhi5nvnyxdmb7dqoy

Circle Packing for Origami Design Is Hard [chapter]

Erik Demaine, Sandor Fekete, Robert Lang
2011 Origami 5  
Acknowledgments We thank Ron Graham for several helpful hints concerning the state of the art on packing circles. We also thank Vinayak Pathak for pointing out a numerical typo related to Figure 13 .  ...  Now a recursive subdivision of S into sub-squares of progressively smaller size can be used to pack all squares S i , showing that all circles C i can be packed.  ...  One of the first robust algorithms for origami design was the circle/river method (also called the tree method) developed independently by Lang [7] [8] [9] and Meguro [12, 13] .  ... 
doi:10.1201/b10971-52 fatcat:jlrpzg6z25acbmczzcxm2celfq

Tree Search and Simulation [chapter]

João Pedro Pedroso, Rui Rei
2015 Applied Simulation and Optimization  
Target problems are either difficult to express as mixed integer optimization models, or have models which provide rather loose bounds; in both cases, traditional, exact methods typically fail.  ...  This method has been used with considerable success in game playing, but has received very limited attention as a tool for optimization.  ...  Acknowledgments This work was partially funded by PhD grant SFRH/BD/66075/2009 from the Portuguese Foundation for Science and Technology (FCT).  ... 
doi:10.1007/978-3-319-15033-8_4 fatcat:egdp6m7mozeudnkv4odjxh2d4i

A Reallocation Algorithm for Online Split Packing of Circles [article]

Shunhao Oh, Seth Gilbert
2018 arXiv   pre-print
If only insertions are considered, our algorithm is also able to pack to critical density, with an amortised reallocation cost of O(c1/c) for squares, and O(c(1+s^2)_1+s^21/c) for right angled triangles  ...  The algorithm can be used to pack circles into squares and right angled triangles.  ...  Many heuristic methods have also been developed for offline circle packing.  ... 
arXiv:1802.05873v3 fatcat:5mbv3yac3ndhtoxcmqr3ixzrhe

Split Packing: Algorithms for Packing Circles with Optimal Worst-Case Density [article]

Sándor P. Fekete, Sebastian Morr, Christian Scheffer
2018 arXiv   pre-print
It employs an elegant subdivision scheme that recursively splits the circles into two groups and then packs these into subcontainers. We call this algorithm "Split Packing".  ...  In this paper, we present new sufficient conditions for packing circles into square and triangular containers, using only the sum of the circles' areas: For square containers, it is possible to pack any  ...  We also thank several anonymous reviewers for their helpful comments.  ... 
arXiv:1705.00924v2 fatcat:hnj63jnq7jg3tbxfl4nb2gipxm

Apollonian packings as physical fractals

Francesco Varrato, Giuseppe Foffi
2011 Molecular Physics  
The Apollonian packings (APs) are fractals that result from a space-filling procedure with spheres.  ...  We discuss the finite size effects for finite intervals s∈[s_min,s_max] between the largest and the smallest sizes of the filling spheres.  ...  For example, it has been calculated for the plane tiled with circles obtained by the circular inversion method [12] . In d = 3, a generalized inversion algorithm has been adopted [13] .  ... 
doi:10.1080/00268976.2011.630598 fatcat:t6f3gdrhjjgm3jal36btbn32sm

ENUMERATION OF DIMER CONFIGURATIONS ON A FRACTAL LATTICE

Dušanka Marčetić, Sunčica Elezović Hadžić, Ivan Živić
2018 Contemporary Materials  
Self-similarity of the fractal lattice enables exact recursive enumeration of all close-packed dimer configurations at every stage of fractal construction.  ...  Originally, it was introduced as a model for adsorption of diatomic molecules on surfaces.  ...  After that, we introduce a close-packed dimer model and develop a recursive method for the enumeration of all close-packed dimer configurations on MR lattice.  ... 
doi:10.7251/comen1802115m fatcat:vwsbaxh2sre4dlw24sq4yueh34

An approximate method for circle packing and disc covering

Shuangmin Chen, Yuen-Shan Leung, Shiqing Xin, Ying He, Yuanfeng Zhou, Changhe Tu
2018 Communications in Information and Systems  
In this paper, we develop an approximate method for packing a large number of circles (of similar sizes).  ...  Circle packing is to optimize the arrangement of circles (of equal or varying sizes) on a given domain with the maximal packing density such that no overlapping occurs.  ...  Acknowledgement We are very grateful to the editors and anonymous reviewers for their insightful comments and suggestions.  ... 
doi:10.4310/cis.2018.v18.n2.a1 fatcat:ratgru6f4zdhljyhithr7ffxey

Preface to the Special Issue on Cutting and Packing

A. Miguel Gomes, José Fernando Gonçalves, Ramón Alvarez-Valdés, J. Valério de Carvalho
2015 International Transactions in Operational Research  
The seventh paper co-authored by Silva, Oliveira and Wäscher presents a review of solution methods and computational experiments for the pallet loading problem.  ...  The fourth paper co-authored by Wei, Qin and Cheang proposes an efficient algorithm for the two-dimensional strip packing problem.  ...  for the recursive circle packing problem We would like to thank the members of the ITOR editorial office, especially, Professor Celso C.  ... 
doi:10.1111/itor.12208 fatcat:t6p3h7mgbrhmhn5waxhifrozve

Barycentric fixed-mass method for multifractal analysis

Y. Kamer, G. Ouillon, D. Sornette
2013 Physical Review E  
Our tests support the conclusion of a genuine but weak multifractality of the central core of DLA clusters, with Dq decreasing from 1.75+/-0.01 for q=-10 to 1.65+/-0.01 for q=+10.  ...  Implementation of the method on synthetic benchmarks demonstrates the superior performance of the proposed method compared with existing alternatives routinely used in the literature.  ...  synthetic datasets are constructed by recursive replicati (See Figure 3 for an example).  ... 
doi:10.1103/physreve.88.022922 pmid:24032916 fatcat:eumr6sdoefggtcqs63emkpsk4e

The number of spanning trees in Apollonian networks

Zhongzhi Zhang, Bin Wu, Francesc Comellas
2014 Discrete Applied Mathematics  
In this paper we find an exact analytical expression for the number of spanning trees in Apollonian networks.  ...  An Apollonian packing can be used to design a graph, when each circle is associated to a vertex of the graph and vertices are connected if their corresponding circles are tangent.  ...  Apollonian networks In the process known as Apollonian packing [9] , which dates back to Apollonius of Perga (c262-c190 BC), we start with three mutually tangent circles, and draw their inner Soddy circle  ... 
doi:10.1016/j.dam.2014.01.015 fatcat:uifskwk75ncxpc3w33xtscddce

The Number of Spanning Trees in Apollonian Networks [article]

Zhongzhi Zhang, Bin Wu, Francesc Comellas
2014 arXiv   pre-print
In this paper we find an exact analytical expression for the number of spanning trees in Apollonian networks.  ...  An Apollonian packing can be used to design a graph, when each circle is associated to a vertex of the graph and vertices are connected if their corresponding circles are tangent.  ...  Apollonian networks In the process known as Apollonian packing [9] , which dates back to Apollonius of Perga (c262-c190 BC), we start with three mutually tangent circles, and draw their inner Soddy circle  ... 
arXiv:1210.0090v2 fatcat:5jcvdyy4zjgibfnvafwu6ryjre

Construction of Circular Quadrature Amplitude Modulations (CQAM)

Johannes Van Wonterghem, Joseph J. Boutros, Marc Moeneclaey
2018 2018 IEEE International Conference on the Science of Electrical Engineering in Israel (ICSEE)  
We propose an exact construction method via recursive equations for bidimensional CQAM constellations. We show that CQAMs are subsets of the hexagonal lattice A2 for a particular alphabet size.  ...  Circular quadrature amplitude modulations (CQAM) are introduced as an alternative to mono-dimensional ASK constellations (and their QAM Cartesian product) for probabilistic shaping with non-binary error-correcting  ...  ACKNOWLEDGMENT Johannes Van Wonterghem would like to thank the Research Foundation in Flanders (FWO) for funding his PhD fellowship.  ... 
doi:10.1109/icsee.2018.8646035 fatcat:y7dkanbtivhojktbkpbstbqfci

High-dimensional Apollonian networks

Zhongzhi Zhang, Francesc Comellas, Guillaume Fertin, Lili Rong
2006 Journal of Physics A: Mathematical and General  
We derive analytical expressions for the degree distribution, the clustering coefficient and the diameter of the networks, which are determined by their dimension.  ...  Support for F.C. was provided by the Secretaria de Estado de Universidades e Investigación (Ministerio de Educación y Ciencia), Spain, and the European Regional Development Fund (ERDF) under project TIC2002  ...  However, new deterministic models with fixed degree distributions constructed by recursive methods have been recently introduced [6, 7, 8, 9, 12, 10, 11] .  ... 
doi:10.1088/0305-4470/39/8/003 fatcat:khupyiavlbdypltzzawxczgb5q
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