An evolutionary algorithm for graph planarisation by vertex deletion

A non-planar graph can only be planarised if it is structurally modified. This work presents a new heuristic algorithm that uses vertices deletion to modify a non-planar graph in order to obtain a planar subgraph. The proposed algorithm aims to delete a minimum number of vertices to achieve its goal. The vertex deletion number of a graph G = (V,E) is the smallest integer k ? 0 such that there is an induced planar subgraph of G obtained by the removal of k vertices of G. Considering that the corresponding decision problem is NPcomplete and an approximation algorithm for graph planarisation by vertices deletion does not exist, this work proposes an evolutionary algorithm that uses a constructive heuristic algorithm to planarise a graph. This constructive heuristic has time complexity of O(n+m), where m = |V| and n = |E|, and it is based on the PQ-trees data structure and on the vertex deletion operation. The algorithm performance is verified by means of case studies

High-Fidelity Wing Design Exploration with Gradient-Based Optimization

Numerical optimization has been applied to wing design problems for over 40 years. Over the decades, the scope and detail of optimization problems have advanced considerably. At the present time, the state-of-the-art in wing design optimization incorporates high-fidelity modeling of the steady-state aeroelastic response of the wing at both on-design and off-design operating conditions. Reynolds-averaged solutions of the Navier–Stokes equations coupled with linear finite element anal- ysis offer the highest fidelity modeling currently tenable in an optimization con- text. However, the complexity of implementing and cost of executing high-fidelity aerostructural optimization have limited the extent of research on the topic. The goal of this dissertation is to examine the general application of these tools to wing design problems and highlight several factors pertaining to their usefulness and versatility. Two types of wing design problems are considered in this dissertation: refin- ing and exploratory. Refining problems are more common in practice, especially for high-fidelity optimization, because they start from a good design and make small changes to improve it. Exploratory problems are intended to have liberal parametrizations predisposed to have significant differences between the original and final designs. The investigation of exploratory problems yields novel findings regarding multimodality in the design space and robustness of the framework. Multimodality in the design space can impact the usefulness and versatility of gradient-based optimization in wing design. Both aerodynamic and aerostructural wing design problems are shown to be amenable to gradient-based optimization despite the existence of multimodality in some cases. For example, a rectangular wing with constant cross-section is successfully converted, through gradient-based optimization, into a swept-back wing with transonic airfoils and a minimum-mass structure. These studies introduce new insights into the tradeoff between skin- friction and induced drag and its impact on multimodality and optimization. The results of these studies indicate that multimodality is dependent on model fidelity and geometric parametrization. It is shown that artificial multimodality can be eliminated by improving model fidelity and numerical accuracy of functions and derivatives, whereas physically significant multimodality can be controlled with the application of geometric constraints. The usefulness of numerical optimization in wing design hinges on the ability of the optimizer to competently balance fundamental tradeoffs. With comprehensive access to the relevant design parameters and physics models of the aerostructural system, an optimizer can converge to a better multidisciplinary design than is pos- sible with a traditional, sequential design process. This dissertation features the high-fidelity aerostructural optimization of an Embraer regional jet, in which si- multaneous optimization of airfoil shape, planform, and structural sizing variables yields a significantly improved wing over the baseline design. For a regional jet, it is shown that the inclusion of climb and descent segments in the fuel burn com- putation has a significant impact on the tradeoff between structural weight and aspect ratio. Another study addresses the tradeoff between cruise performance and low-speed, high-lift flight characteristics. A separation constraint at a low-speed, high-lift condition is introduced as an effective method of preserving low-speed performance while still achieving significant fuel burn reduction in cruise.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163242/1/nbons_1.pd

Design and Analysis of Algorithms: Course Notes

These are my lecture notes from CMSC 651: Design and Analysis of Algorithms}, a one semester course that I taught at University of Maryland in the Spring of 1993. The course covers core material in algorithm design, and also helps students prepare for research in the field of algorithms. The reader will find an unusual emphasis on graph theoretic algorithms, and for that I am to blame. The choice of topics was mine, and is biased by my personal taste. The material for the first few weeks was taken primarily from the (now not so new) textbook on Algorithms by Cormen, Leiserson and Rivest. A few papers were also covered, that I personally feel give some very important and useful techniques that should be in the toolbox of every algorithms researcher. (Also cross-referenced as UMIACS-TR-93-72

Control of semiconductor manufacturing: CMP & thickness variations

Master'sMASTER OF ENGINEERIN

Variability-Aware Design of Static Random Access Memory Bit-Cell

The increasing integration of functional blocks in today's integrated circuit designs necessitates a large embedded memory for data manipulation and storage. The most often used embedded memory is the Static Random Access Memory (SRAM), with a six transistor memory bit-cell. Currently, memories occupy more than 50% of the chip area and this percentage is only expected to increase in future. Therefore, for the silicon vendors, it is critical that the memory units yield well, to enable an overall high yield of the chip. The increasing memory density is accompanied by aggressive scaling of the transistor dimensions in the SRAM. Together, these two developments make SRAMs increasingly susceptible to process-parameter variations. As a result, in the current nanometer regime, statistical methods for the design of the SRAM array are pivotal to achieve satisfactory levels of silicon predictability. In this work, a method for the statistical design of the SRAM bit-cell is proposed. Not only does it provide a high yield, but also meets the specifications for the design constraints of stability, successful write, performance, leakage and area. The method consists of an optimization framework, which derives the optimal design parameters; i.e., the widths and lengths of the bit-cell transistors, which provide maximum immunity to the variations in the transistor's geometry and intrinsic threshold voltage fluctuations. The method is employed to obtain optimal designs in the 65nm, 45nm and 32nm technologies for different set of specifications. The optimality of the resultant designs is verified. The resultant optimal bit-cell designs in the 65nm, 45nm and 32nm technologies are analyzed to study the SRAM area and yield trade-offs associated with technology scaling. In order to achieve 50% scaling of the bit-cell area, at every technology node, two ways are proposed. The resultant designs are further investigated to understand, which mode of failure in the bit-cell becomes more dominant with technology scaling. In addition, the impact of voltage scaling on the bit-cell designs is also studied

The bidimensionality theory and its algorithmic applications

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 201-219).Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixed-parameter algorithms and approximation algorithms for NP- hard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k x k grid graph (and similar graphs) grows with k, typically as Q(k²), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex- removal parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). Bidimensional problems have many structural properties; for example, any graph embeddable in a surface of bounded genus has treewidth bounded above by the square root of the problem's solution value. These properties lead to efficient-often subexponential-fixed-parameter algorithms, as well as polynomial-time approximation schemes, for many minor-closed graph classes. One type of minor-closed graph class of particular relevance has bounded local treewidth, in the sense that the treewidth of a graph is bounded above in terms of the diameter; indeed, we show that such a bound is always at most linear. The bidimensionality theory unifies and improves several previous results.(cont.) The theory is based on algorithmic and combinatorial extensions to parts of the Robertson-Seymour Graph Minor Theory, in particular initiating a parallel theory of graph contractions. The foundation of this work is the topological theory of drawings of graphs on surfaces and our results regarding the relation (the linearity) of the size of the largest grid minor in terms of treewidth in bounded-genus graphs and more generally in graphs excluding a fixed graph H as a minor. In this thesis, we also develop the algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into L₁ (and even Euclidean embeddings) are insufficient, but that the additional structure provided by many embedding theorems does suffice for our purposes. We obtain an O[sq. root( log n)] approximation for min-ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be [theta][sq. root(log n)]. We also prove various approximate max-flow/min-vertex- cut theorems, which in particular give a constant-factor approximation for min-ratio vertex cuts in any excluded-minor family of graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best-known ratio was O(log n). These results have a number of applications. We exhibit an O[sq. root (log n)] pseudo-approximation for finding balanced vertex separators in general graphs.(cont.) Furthermore, we obtain improved approximation ratios for treewidth: In any graph of treewidth k, we show how to find a tree decomposition of width at most O(k[sq. root(log k)]), whereas previous algorithms yielded O(k log k). For graphs excluding a fixed graph as a minor, we give a constant-factor approximation for the treewidth; this via the bidimensionality theory can be used to obtain the first polynomial-time approximation schemes for problems like minimum feedback vertex set and minimum connected dominating set in such graphs.by MohammadTaghi Hajiaghayi.Ph.D

Plane and simple : using planar subgraphs for efficient algorithms

In this thesis, we showcase how planar subgraphs with special structural properties can be used to fi nd efficient algorithms for two NP-hard problems in combinatorial optimization. In the fi rst part, we develop algorithms for the computation of Tutte paths and show how these special subgraphs can be used to efficiently compute long cycles and other relaxations of Hamiltonicity if we restrict the input to planar graphs. We give an O(n^2) time algorithm for the computation of Tutte paths in circuit graphs and generalize it to the computation of Tutte paths between any two given vertices and a prescribed intermediate edge in 2-connected planar graphs. In the second part, we study the Maximum Planar Subgraph Problem (MPS) and show how dense planar subgraphs can be used to develop new approximation algorithms for this problem. All new algorithms and arguments we present are based on a novel approach that focuses on maximizing the number of triangular faces in the computed subgraph. For this, we define a new optimization problem called Maximum Planar Triangles (MPT). We show that this problem is NP-hard and quantify how good an approximation algorithm for MPT performs as an approximation for MPS. We give a greedy 1/11-approximation algorithm for Mpt and show that the approximation ratio can be improved to 1/6 by using locally optimal triangular cactus subgraphs.In dieser Dissertation zeigen wir, wie planare Teilgraphen mit speziellen Eigenschaften verwendet werden können, um effiziente Algorithmen für zwei NP-schwere Probleme in der kombinatorischen Optimierung zu fi nden. Im ersten Teil entwickeln wir Algorithmen zur Berechnung von Tutte-Wegen und zeigen, wie diese verwendet werden können, um lange Kreise und andere Lockerungen der Hamilton-Charakteristik zu finden, wenn wir uns auf Graphen in der Ebene beschränken. Wir beschreiben zunächst einen O(n^2)-Algorithmus in Circuit-Graphen und verallgemeinern diesen anschließend für die Berechnung von Tutte-Wegen in 2-zusammenhängenden planaren Graphen. Im zweiten Teil untersuchen wir das Maximum Planar Subgraph Problem (MPS) und zeigen, wie besonders dichte planare Teilgraphen verwendet werden können, um neue Approximationsalgorithmen zu entwickeln. Unsere Ergebnisse basieren auf einem neuartigen Ansatz, bei dem die Anzahl der dreieckigen Gebiete im berechneten Teilgraphen maximiert wird. Dazu de finieren wir ein neues Optimierungsproblem namens Maximum Planar Triangles (MPT). Wir zeigen, dass dieses Problem NP-schwer ist und quantifi zieren, wie gut ein Approximationsalgorithmus für MPT als Approximation für MPS funktioniert. Wir geben einen 1/11-Approximationsalgorithmus für MPT und zeigen, wie dies durch die Verwendung von lokal optimaler Kaktus-Teilgraphen auf 1/6 verbessert werden kann