421 research outputs found
Logic based Benders' decomposition for orthogonal stock cutting problems
We consider the problem of packing a set of rectangular items into a strip of fixed width, without overlapping, using minimum height. Items must be packed with their edges parallel to those of the strip, but rotation by 90\ub0 is allowed. The problem is usually solved through branch-and-bound algorithms. We propose an alternative method, based on Benders' decomposition. The master problem is solved through a new ILP model based on the arc flow formulation, while constraint programming is used to solve the slave problem. The resulting method is hybridized with a state-of-the-art branch-and-bound algorithm. Computational experiments on classical benchmarks from the literature show the effectiveness of the proposed approach. We additionally show that the algorithm can be successfully used to solve relevant related problems, like rectangle packing and pallet loading
Timing-Driven Macro Placement
Placement is an important step in the process of finding physical layouts for electronic computer chips. The basic task during placement is to arrange the building blocks of the chip, the circuits, disjointly within a given chip area. Furthermore, such positions should result in short circuit interconnections which can be routed easily and which ensure all signals arrive in time. This dissertation mostly focuses on macros, the largest circuits on a chip. In order to optimize timing characteristics during macro placement, we propose a new optimistic timing model based on geometric distance constraints. This model can be computed and evaluated efficiently in order to predict timing traits accurately in practice. Packing rectangles disjointly remains strongly NP-hard under slack maximization in our timing model. Despite of this we develop an exact, linear time algorithm for special cases. The proposed timing model is incorporated into BonnMacro, the macro placement component of the BonnTools physical design optimization suite developed at the Research Institute for Discrete Mathematics. Using efficient formulations as mixed-integer programs we can legalize macros locally while optimizing timing. This results in the first timing-aware macro placement tool. In addition, we provide multiple enhancements for the partitioning-based standard circuit placement algorithm BonnPlace. We find a model of partitioning as minimum-cost flow problem that is provably as small as possible using which we can avoid running time intensive instances. Moreover we propose the new global placement flow Self-Stabilizing BonnPlace. This approach combines BonnPlace with a force-directed placement framework. It provides the flexibility to optimize the two involved objectives, routability and timing, directly during placement. The performance of our placement tools is confirmed on a large variety of academic benchmarks as well as real-world designs provided by our industrial partner IBM. We reduce running time of partitioning significantly and demonstrate that Self-Stabilizing BonnPlace finds easily routable placements for challenging designs – even when simultaneously optimizing timing objectives. BonnMacro and Self-Stabilizing BonnPlace can be combined to the first timing-driven mixed-size placement flow. This combination often finds placements with competitive timing traits and even outperforms solutions that have been determined manually by experienced designers
Approximating Dynamic Time Warping and Edit Distance for a Pair of Point Sequences
We give the first subquadratic-time approximation schemes for dynamic time
warping (DTW) and edit distance (ED) of several natural families of point
sequences in , for any fixed . In particular, our
algorithms compute -approximations of DTW and ED in time
near-linear for point sequences drawn from k-packed or k-bounded curves, and
subquadratic for backbone sequences. Roughly speaking, a curve is
-packed if the length of its intersection with any ball of radius
is at most , and a curve is -bounded if the sub-curve
between two curve points does not go too far from the two points compared to
the distance between the two points. In backbone sequences, consecutive points
are spaced at approximately equal distances apart, and no two points lie very
close together. Recent results suggest that a subquadratic algorithm for DTW or
ED is unlikely for an arbitrary pair of point sequences even for . Our
algorithms work by constructing a small set of rectangular regions that cover
the entries of the dynamic programming table commonly used for these distance
measures. The weights of entries inside each rectangle are roughly the same, so
we are able to use efficient procedures to approximately compute the cheapest
paths through these rectangles
Improved Layout Structure with Complexity Measures for the Manufacturer’s Pallet Loading Problem (MPLP) Using a Block Approach
Purpose: The purpose of this paper is to study the Manufacturers pallet-loading problem (MPLP), by loading identical small boxes onto a rectangle pallet to maximise the pallet utilization percentage while reducing the Complexity of loading.
Design/methodology/approach: In this research a Block approach is proposed using a Mixed integer linear programming (MILP) model that generates layouts of an improved structure, which is very effective due to its properties in grouping boxes in a certain orientation along the X and Y axis. Also, a novel complexity index is introduced to compare the complexity for different pallet loading, which have the same pallet size but different box arrangements.
Findings: The proposed algorithm has been tested against available data-sets in literature and the complexity measure and graphical layout results clearly demonstrate the superiority of the proposed approach compared with literature Manufacturers pallet-loading problem layouts.
Originality/value: This study aids real life manufactures operations when less complex operations are essential to reduce the complexity of pallet loading
Efficient bulk-loading methods for temporal and multidimensional index structures
Nahezu alle naturwissenschaftlichen Bereiche profitieren von neuesten Analyse- und Verarbeitungsmethoden fĂĽr groĂźe Datenmengen. Diese Verfahren setzten eine effiziente Verarbeitung von geo- und zeitbezogenen Daten voraus, da die Zeit und die Position wichtige Attribute vieler Daten
sind. Die effiziente Anfrageverarbeitung wird insbesondere durch den Einsatz von Indexstrukturen
ermöglicht. Im Fokus dieser Arbeit liegen zwei Indexstrukturen: Multiversion B-Baum
(MVBT) und R-Baum. Die erste Struktur wird fĂĽr die Verwaltung von zeitbehafteten Daten,
die zweite fĂĽr die Indexierung von mehrdimensionalen Rechteckdaten eingesetzt.
Ständig- und schnellwachsendes Datenvolumen stellt eine große Herausforderung an die Informatik
dar. Der Aufbau und das Aktualisieren von Indexen mit herkömmlichen Methoden (Datensatz
fĂĽr Datensatz) ist nicht mehr effizient. Um zeitnahe und kosteneffiziente Datenverarbeitung
zu ermöglichen, werden Verfahren zum schnellen Laden von Indexstrukturen dringend benötigt.
Im ersten Teil der Arbeit widmen wir uns der Frage, ob es ein Verfahren fĂĽr das Laden von MVBT
existiert, das die gleiche I/O-Komplexität wie das externe Sortieren besitz. Bis jetzt blieb diese
Frage unbeantwortet. In dieser Arbeit haben wir eine neue Kostruktionsmethode entwickelt und
haben gezeigt, dass diese gleiche Zeitkomplexität wie das externe Sortieren besitzt. Dabei haben
wir zwei algorithmische Techniken eingesetzt: Gewichts-Balancierung und Puffer-Bäume. Unsere
Experimenten zeigen, dass das Resultat nicht nur theoretischer Bedeutung ist.
Im zweiten Teil der Arbeit beschäftigen wir uns mit der Frage, ob und wie statistische Informationen
über Geo-Anfragen ausgenutzt werden können, um die Anfrageperformanz von R-Bäumen zu
verbessern. Unsere neue Methode verwendet Informationen wie Seitenverhältnis und Seitenlängen
eines repräsentativen Anfragerechtecks, um einen guten R-Baum bezüglich eines häufig eingesetzten
Kostenmodells aufzubauen. Falls diese Informationen nicht verfĂĽgbar sind, optimieren
wir R-Bäume bezüglich der Summe der Volumina von minimal umgebenden Rechtecken der Blattknoten.
Da das Problem des Aufbaus von optimalen R-Bäumen bezüglich dieses Kostenmaßes
NP-hart ist, führen wir zunächst das Problem auf ein eindimensionales Partitionierungsproblem
zurück, indem wir die Daten bezüglich optimierte raumfüllende Kurven sortieren. Dann lösen
wir dieses Problem durch Einsatz vom dynamischen Programmieren. Die I/O-Komplexität des
Verfahrens ist gleich der von externem Sortieren, da die I/O-Laufzeit der Methode durch die
Laufzeit des Sortierens dominiert wird.
Im letzten Teil der Arbeit haben wir die entwickelten Partitionierungsvefahren fĂĽr den Aufbau
von Geo-Histogrammen eingesetzt, da diese ähnlich zu R-Bäumen eine disjunkte Partitionierung
des Raums erzeugen. Ergebnisse von intensiven Experimenten zeigen, dass sich unter Verwendung
von neuen Partitionierungstechniken sowohl R-Bäume mit besserer Anfrageperformanz als
auch Geo-Histogrammen mit besserer Schätzqualität im Vergleich zu Konkurrenzverfahren generieren
lassen
A Quasiphysical and Dynamic Adjustment Approach for Packing the Orthogonal Unequal Rectangles in a Circle with a Mass Balance: Satellite Payload Packing
Packing orthogonal unequal rectangles in a circle with a mass balance (BCOURP) is a typical combinational optimization problem with the NP-hard nature. This paper proposes an effective quasiphysical and dynamic adjustment approach (QPDAA). Two embedded degree functions between two orthogonal rectangles and between an orthogonal rectangle and the container are defined, respectively, and the extruded potential energy function and extruded resultant force formula are constructed based on them. By an elimination of the extruded resultant force, the dynamic rectangle adjustment, and an iteration of the translation, the potential energy and static imbalance of the system can be quickly decreased to minima. The continuity and monotony of two embedded degree functions are proved to ensure the compactness of the optimal solution. Numerical experiments show that the proposed QPDAA is superior to existing approaches in performance
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