Online Packing to Minimize Area or Perimeter

We consider online packing problems where we get a stream of axis-parallel rectangles. The rectangles have to be placed in the plane without overlapping, and each rectangle must be placed without knowing the subsequent rectangles. The goal is to minimize the perimeter or the area of the axis-parallel bounding box of the rectangles. We either allow rotations by 90^? or translations only. For the perimeter version we give algorithms with an absolute competitive ratio slightly less than 4 when only translations are allowed and when rotations are also allowed. We then turn our attention to minimizing the area and show that the competitive ratio of any algorithm is at least ?(?n), where n is the number of rectangles in the stream, and this holds with and without rotations. We then present algorithms that match this bound in both cases and the competitive ratio is thus optimal to within a constant factor. We also show that the competitive ratio cannot be bounded as a function of Opt. We then consider two special cases. The first is when all the given rectangles have aspect ratios bounded by some constant. The particular variant where all the rectangles are squares and we want to minimize the area of the bounding square has been studied before and an algorithm with a competitive ratio of 8 has been given [Fekete and Hoffmann, Algorithmica, 2017]. We improve the analysis of the algorithm and show that the ratio is at most 6, which is tight. The second special case is when all edges have length at least 1. Here, the ?(?n) lower bound still holds, and we turn our attention to lower bounds depending on Opt. We show that any algorithm for the translational case has a competitive ratio of at least ?(?{Opt}). If rotations are allowed, we show a lower bound of ?(?{Opt}). For both versions, we give algorithms that match the respective lower bounds: With translations only, this is just the algorithm from the general case with competitive ratio O(?n) = O(?{Opt}). If rotations are allowed, we give an algorithm with competitive ratio O(min{?n,?{Opt}}), thus matching both lower bounds simultaneously

Phi-Functions for 2D Objects Formed by Line Segments and Circular Arcs

We study the cutting and packing (C&P) problems in two dimensions by using phi-functions. Our phi-functions describe the layout of given objects; they allow us to construct a mathematical model in which C&P problems become constrained optimization problems. Here we define (for the first time) a complete class of basic phi-functions which allow us to derive phi-functions for all 2D objects that are formed by linear segments and circular arcs. Our phi-functions support translations and rotations of objects. In order to deal with restrictions on minimal or maximal distances between objects, we also propose adjusted phi-functions. Our phi-functions are expressed by simple linear and quadratic formulas without radicals. The use of radical-free phi-functions allows us to increase efficiency of optimization algorithms. We include several model examples

Minimizing waste in the 2-dimensional cutting stock problem

The 2-dimensional cutting stock problem is an important problem in the garment manufacturing industry. The problem is to arrange a given set of 2-dimensional patterns onto a rectangular bolt of cloth such that the efficiency is maximised. This arrangement is called a marker. Efficiency is measured by pattern area I marker area. Efficiency varies depending on the shape and number of patterns being cut, but an improvement in efficiency can result in significant savings. Markers are usually created by humans with the aid of CAD software. Many researchers have attempted to create automatic marker making software but have failed to produce marker efficiencies as high as human generated ones. This thesis presents a mathematical model which optimally solves the 2-dimensional cutting stock problem. However, the model can only be solved in a practical amount of time for small markers. Subsequently, two compaction algorithms based on mathematical modelling have been developed to improve the efficiency of human generated markers. The models developed in this thesis make use of a geometrical calculation known as the no-fit polygon. The no-fit polygon is a tool for determining whether polygons A and B overlap. It also gives all feasible positions for polygons B with respect to polygon A, such that the two polygons do not overlap. For the case when both polygons A and B are non-convex, current calculation methods are either time consuming or unreliable. This thesis presents a method which is both computationally efficient and robust for calculating the no-fit polygon when polygons A and B are non-convex. When tested on a set of industrial markers, the compaction algorithms improved the marker efficiencies by over 1.5% on average

Smallest Number of Sensors for k-Covering

This paper presents some theoretical results on the smaller number Nk(a, b) of sensors to achieve k coverage for the rectangular area [0, a] × [0, b]. The first properties show the numbers Nk(a, b) are sub-additive and increasing on each variable. Based on these results, some lower and upper bounds for Nk(a, b) are introduced. The main result of the article proves that the minimal density of sensors to achieve k-coverage is (k) ≤ k/2 improving a previous result of Ammari and Das [2]. Finally, the numbers N1(a, b) are tabled for some small values of a, b

Decomposing and packing polygons / Dania el-Khechen.

In this thesis, we study three different problems in the field of computational geometry: the partitioning of a simple polygon into two congruent components, the partitioning of squares and rectangles into equal area components while minimizing the perimeter of the cuts, and the packing of the maximum number of squares in an orthogonal polygon. To solve the first problem, we present three polynomial time algorithms which given a simple polygon P partitions it, if possible, into two congruent and possibly nonsimple components P 1 and P 2 : an O ( n 2 log n ) time algorithm for properly congruent components and an O ( n 3 ) time algorithm for mirror congruent components. In our analysis of the second problem, we experimentally find new bounds on the optimal partitions of squares and rectangles into equal area components. The visualization of the best determined solutions allows us to conjecture some characteristics of a class of optimal solutions. Finally, for the third problem, we present three linear time algorithms for packing the maximum number of unit squares in three subclasses of orthogonal polygons: the staircase polygons, the pyramids and Manhattan skyline polygons. We also study a special case of the problem where the given orthogonal polygon has vertices with integer coordinates and the squares to pack are (2 {604} 2) squares. We model the latter problem with a binary integer program and we develop a system that produces and visualizes optimal solutions. The observation of such solutions aided us in proving some characteristics of a class of optimal solutions

Feasible Form Parameter Design of Complex Ship Hull Form Geometry

This thesis introduces a new methodology for robust form parameter design of complex hull form geometry via constraint programming, automatic differentiation, interval arithmetic, and truncated hierarchical B- splines. To date, there has been no clearly stated methodology for assuring consistency of general (equality and inequality) constraints across an entire geometric form parameter ship hull design space. In contrast, the method to be given here can be used to produce guaranteed narrowing of the design space, such that infeasible portions are eliminated. Furthermore, we can guarantee that any set of form parameters generated by our method will be self consistent. It is for this reason that we use the title feasible form parameter design. In form parameter design, a design space is represented by a tuple of design parameters which are extended in each design space dimension. In this representation, a single feasible design is a consistent set of real valued parameters, one for every component of the design space tuple. Using the methodology to be given here, we pick out designs which consist of consistent parameters, narrowed to any desired precision up to that of the machine, even for equality constraints. Furthermore, the method is developed to enable the generation of complex hull forms using an extension of the basic rules idea to allow for automated generation of rules networks, plus the use of the truncated hierarchical B-splines, a wavelet-adaptive extension of standard B-splines and hierarchical B-splines. The adaptive resolution methods are employed in order to allow an automated program the freedom to generate complex B-spline representations of the geometry in a robust manner across multiple levels of detail. Thus two complementary objectives are pursued: ensuring feasible starting sets of form parameters, and enabling the generation of complex hull form geometry

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Proceedings of the 1993 Conference on Intelligent Computer-Aided Training and Virtual Environment Technology

The volume 2 proceedings from the 1993 Conference on Intelligent Computer-Aided Training and Virtual Environment Technology are presented. Topics discussed include intelligent computer assisted training (ICAT) systems architectures, ICAT educational and medical applications, virtual environment (VE) training and assessment, human factors engineering and VE, ICAT theory and natural language processing, ICAT military applications, VE engineering applications, ICAT knowledge acquisition processes and applications, and ICAT aerospace applications

Layoutautomatisierung im analogen IC-Entwurf mit formalisiertem und nicht-formalisiertem Expertenwissen

After more than three decades of electronic design automation, most layouts for analog integrated circuits are still handcrafted in a laborious manual fashion today. Obverse to the highly automated synthesis tools in the digital domain (coping with the quantitative difficulty of packing more and more components onto a single chip – a desire well known as More Moore), analog layout automation struggles with the many diverse and heavily correlated functional requirements that turn the analog design problem into a More than Moore challenge. Facing this qualitative complexity, seasoned layout engineers rely on their comprehensive expert knowledge to consider all design constraints that uncompromisingly need to be satisfied. This usually involves both formally specified and nonformally communicated pieces of expert knowledge, which entails an explicit and implicit consideration of design constraints, respectively. Existing automation approaches can be basically divided into optimization algorithms (where constraint consideration occurs explicitly) and procedural generators (where constraints can only be taken into account implicitly). As investigated in this thesis, these two automation strategies follow two fundamentally different paradigms denoted as top-down automation and bottom-up automation. The major trait of top-down automation is that it requires a thorough formalization of the problem to enable a self-intelligent solution finding, whereas a bottom-up automatism –controlled by parameters– merely reproduces solutions that have been preconceived by a layout expert in advance. Since the strengths of one paradigm may compensate the weaknesses of the other, it is assumed that a combination of both paradigms –called bottom-up meets top-down– has much more potential to tackle the analog design problem in its entirety than either optimization-based or generator-based approaches alone. Against this background, the thesis at hand presents Self-organized Wiring and Arrangement of Responsive Modules (SWARM), an interdisciplinary methodology addressing the design problem with a decentralized multi-agent system. Its basic principle, similar to the roundup of a sheep herd, is to let responsive mobile layout modules (implemented as context-aware procedural generators) interact with each other inside a user-defined layout zone. Each module is allowed to autonomously move, rotate and deform itself, while a supervising control organ successively tightens the layout zone to steer the interaction towards increasingly compact (and constraint compliant) layout arrangements. Considering various principles of self-organization and incorporating ideas from existing decentralized systems, SWARM is able to evoke the phenomenon of emergence: although each module only has a limited viewpoint and selfishly pursues its personal objectives, remarkable overall solutions can emerge on the global scale. Several examples exhibit this emergent behavior in SWARM, and it is particularly interesting that even optimal solutions can arise from the module interaction. Further examples demonstrate SWARM’s suitability for floorplanning purposes and its application to practical place-and-route problems. The latter illustrates how the interacting modules take care of their respective design requirements implicitly (i.e., bottom-up) while simultaneously paying respect to high level constraints (such as the layout outline imposed top-down by the supervising control organ). Experimental results show that SWARM can outperform optimization algorithms and procedural generators both in terms of layout quality and design productivity. From an academic point of view, SWARM’s grand achievement is to tap fertile virgin soil for future works on novel bottom-up meets top-down automatisms. These may one day be the key to close the automation gap in analog layout design.Nach mehr als drei Jahrzehnten Entwurfsautomatisierung werden die meisten Layouts für analoge integrierte Schaltkreise heute immer noch in aufwändiger Handarbeit entworfen. Gegenüber den hochautomatisierten Synthesewerkzeugen im Digitalbereich (die sich mit dem quantitativen Problem auseinandersetzen, mehr und mehr Komponenten auf einem einzelnen Chip unterzubringen – bestens bekannt als More Moore) kämpft die analoge Layoutautomatisierung mit den vielen verschiedenen und stark korrelierten funktionalen Anforderungen, die das analoge Entwurfsproblem zu einer More than Moore Herausforderung machen. Angesichts dieser qualitativen Komplexität bedarf es des umfassenden Expertenwissens erfahrener Layouter um sämtliche Entwurfsconstraints, die zwingend eingehalten werden müssen, zu berücksichtigen. Meist beinhaltet dies formal spezifiziertes als auch nicht-formal übermitteltes Expertenwissen, was eine explizite bzw. implizite Constraint Berücksichtigung nach sich zieht. Existierende Automatisierungsansätze können grundsätzlich unterteilt werden in Optimierungsalgorithmen (wo die Constraint Berücksichtigung explizit erfolgt) und prozedurale Generatoren (die Constraints nur implizit berücksichtigen können). Wie in dieser Arbeit eruiert wird, folgen diese beiden Automatisierungsstrategien zwei grundlegend unterschiedlichen Paradigmen, bezeichnet als top-down Automatisierung und bottom-up Automatisierung. Wesentliches Merkmal der top-down Automatisierung ist die Notwendigkeit einer umfassenden Problemformalisierung um eine eigenintelligente Lösungsfindung zu ermöglichen, während ein bottom-up Automatismus –parametergesteuert– lediglich Lösungen reproduziert, die vorab von einem Layoutexperten vorgedacht wurden. Da die Stärken des einen Paradigmas die Schwächen des anderen ausgleichen können, ist anzunehmen, dass eine Kombination beider Paradigmen –genannt bottom-up meets top down– weitaus mehr Potenzial hat, das analoge Entwurfsproblem in seiner Gesamtheit zu lösen als optimierungsbasierte oder generatorbasierte Ansätze für sich allein. Vor diesem Hintergrund stellt die vorliegende Arbeit Self-organized Wiring and Arrangement of Responsive Modules (SWARM) vor, eine interdisziplinäre Methodik, die das Entwurfsproblem mit einem dezentralisierten Multi-Agenten-System angeht. Das Grundprinzip besteht darin, ähnlich dem Zusammentreiben einer Schafherde, reaktionsfähige mobile Layoutmodule (realisiert als kontextbewusste prozedurale Generatoren) in einer benutzerdefinierten Layoutzone interagieren zu lassen. Jedes Modul darf sich selbständig bewegen, drehen und verformen, wobei ein übergeordnetes Kontrollorgan die Zone schrittweise verkleinert, um die Interaktion auf zunehmend kompakte (und constraintkonforme) Layoutanordnungen hinzulenken. Durch die Berücksichtigung diverser Selbstorganisationsgrundsätze und die Einarbeitung von Ideen bestehender dezentralisierter Systeme ist SWARM in der Lage, das Phänomen der Emergenz hervorzurufen: obwohl jedes Modul nur eine begrenzte Sichtweise hat und egoistisch seine eigenen Ziele verfolgt, können sich auf globaler Ebene bemerkenswerte Gesamtlösungen herausbilden. Mehrere Beispiele veranschaulichen dieses emergente Verhalten in SWARM, wobei besonders interessant ist, dass sogar optimale Lösungen aus der Modulinteraktion entstehen können. Weitere Beispiele demonstrieren SWARMs Eignung zwecks Floorplanning sowie die Anwendung auf praktische Place-and-Route Probleme. Letzteres verdeutlicht, wie die interagierenden Module ihre jeweiligen Entwurfsanforderungen implizit (also: bottom-up) beachten, während sie gleichzeitig High-Level-Constraints berücksichtigen (z.B. die Layoutkontur, die top-down vom übergeordneten Kontrollorgan auferlegt wird). Experimentelle Ergebnisse zeigen, dass Optimierungsalgorithmen und prozedurale Generatoren von SWARM sowohl bezüglich Layoutqualität als auch Entwurfsproduktivität übertroffen werden können. Aus akademischer Sicht besteht SWARMs große Errungenschaft in der Erschließung fruchtbaren Neulands für zukünftige Arbeiten an neuartigen bottom-up meets top-down Automatismen. Diese könnten eines Tages der Schlüssel sein, um die Automatisierungslücke im analogen Layoutentwurf zu schließen