OBDD-Based Representation of Interval Graphs

A graph $G = (V,E)$ can be described by the characteristic function of the edge set $\chi_E$ which maps a pair of binary encoded nodes to 1 iff the nodes are adjacent. Using \emph{Ordered Binary Decision Diagrams} (OBDDs) to store $\chi_E$ can lead to a (hopefully) compact representation. Given the OBDD as an input, symbolic/implicit OBDD-based graph algorithms can solve optimization problems by mainly using functional operations, e.g. quantification or binary synthesis. While the OBDD representation size can not be small in general, it can be provable small for special graph classes and then also lead to fast algorithms. In this paper, we show that the OBDD size of unit interval graphs is $O(\ | V \ | /\log \ | V \ |)$ and the OBDD size of interval graphs is $O(\ | V \ | \log \ | V \ |)$which both improve a known result from Nunkesser and Woelfel (2009). Furthermore, we can show that using our variable order and node labeling for interval graphs the worst-case OBDD size is$\Omega(\ | V \ | \log \ | V \ |)$. We use the structure of the adjacency matrices to prove these bounds. This method may be of independent interest and can be applied to other graph classes. We also develop a maximum matching algorithm on unit interval graphs using$O(\log \ | V \ |)$operations and a coloring algorithm for unit and general intervals graphs using$O(\log^2 \ | V \ |)$ operations and evaluate the algorithms empirically.Comment: 29 pages, accepted for 39th International Workshop on Graph-Theoretic Concepts 201

Polynomial systems : graphical structure, geometry, and applications

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 199-208).Solving systems of polynomial equations is a foundational problem in computational mathematics, that has several applications in the sciences and engineering. A closely related problem, also prevalent in applications, is that of optimizing polynomial functions subject to polynomial constraints. In this thesis we propose novel methods for both of these tasks. By taking advantage of the graphical and geometrical structure of the problem, our methods can achieve higher efficiency, and we can also prove better guarantees. Various problems in areas such as robotics, power systems, computer vision, cryptography, and chemical reaction networks, can be modeled by systems of polynomial equations, and in many cases the resulting systems have a simple sparsity structure. In the first part of this thesis we represent this sparsity structure with a graph, and study the algorithmic and complexity consequences of this graphical abstraction. Our main contribution is the introduction of a novel data structure, chordal networks, that always preserves the underlying graphical structure of the system. Remarkably, many interesting families of polynomial systems admit compact chordal network representations (of size linear in the number of variables), even though the number of components is exponentially large. Our methods outperform existing techniques by orders of magnitude in applications from algebraic statistics and vector addition systems. We then turn our attention to the study of graphical structure in the computation of matrix permanents, a classical problem from computer science. We provide a novel algorithm that requires Õ(n 2[superscript w]) arithmetic operations, where [superscript w] is the treewidth of its bipartite adjacency graph. We also investigate the complexity of some related problems, including mixed discriminants, hyperdeterminants, and mixed volumes. Although seemingly unrelated to polynomial systems, our results have natural implications on the complexity of solving sparse systems. The second part of this thesis focuses on the problem of minimizing a polynomial function subject to polynomial equality constraints. This problem captures many important applications, including Max-Cut, tensor low rank approximation, the triangulation problem, and rotation synchronization. Although these problems are nonconvex, tractable semidefinite programming (SDP) relaxations have been proposed. We introduce a methodology to derive more efficient (smaller) relaxations, by leveraging the geometrical structure of the underlying variety. The main idea behind our method is to describe the variety with a generic set of samples, instead of relying on an algebraic description. Our methods are particularly appealing for varieties that are easy to sample from, such as SO(n), Grassmannians, or rank k tensors. For arbitrary varieties we can take advantage of the tools from numerical algebraic geometry. Optimization problems from applications usually involve parameters (e.g., the data), and there is often a natural value of the parameters for which SDP relaxations solve the (polynomial) problem exactly. The final contribution of this thesis is to establish sufficient conditions (and quantitative bounds) under which SDP relaxations will continue to be exact as the parameter moves in a neighborhood of the original one. Our results can be used to show that several statistical estimation problems are solved exactly by SDP relaxations in the low noise regime. In particular, we prove this for the triangulation problem, rotation synchronization, rank one tensor approximation, and weighted orthogonal Procrustes.by Diego Cifuentes.Ph. D

Tirer parti de la structure des données incertaines

The management of data uncertainty can lead to intractability, in the case of probabilistic databases, or even undecidability, in the case of open-world reasoning under logical rules. My thesis studies how to mitigate these problems by restricting the structure of uncertain data and rules. My first contribution investigates conditions on probabilistic relational instances that ensure the tractability of query evaluation and lineage computation. I show that these tasks are tractable when we bound the treewidth of instances, for various probabilistic frameworks and provenance representations. Conversely, I show intractability under mild assumptions for any other condition on instances. The second contribution concerns query evaluation on incomplete data under logical rules, and under the finiteness assumption usually made in database theory. I show that this task is decidable for unary inclusion dependencies and functional dependencies. This establishes the first positive result for finite open-world query answering on an arbitrary-arity language featuring both referential constraints and number restrictions.La gestion des données incertaines peut devenir infaisable, dans le cas des bases de données probabilistes, ou même indécidable, dans le cas du raisonnement en monde ouvert sous des contraintes logiques. Cette thèse étudie comment pallier ces problèmes en limitant la structure des données incertaines et des règles. La première contribution présentée s'intéresse aux conditions qui permettent d'assurer la faisabilité de l'évaluation de requêtes et du calcul de lignage sur les instances relationnelles probabilistes. Nous montrons que ces tâches sont faisables, pour diverses représentations de la provenance et des probabilités, quand la largeur d'arbre des instances est bornée. Réciproquement, sous des hypothèses faibles, nous pouvons montrer leur infaisabilité pour toute autre condition imposée sur les instances. La seconde contribution concerne l'évaluation de requêtes sur des données incomplètes et sous des contraintes logiques, sous l'hypothèse de finitude généralement supposée en théorie des bases de données. Nous montrons la décidabilité de cette tâche pour les dépendances d'inclusion unaires et les dépendances fonctionnelles. Ceci constitue le premier résultat positif, sous l'hypothèse de la finitude, pour la réponse aux requêtes en monde ouvert avec un langage d'arité arbitraire qui propose à la fois des contraintes d'intégrité référentielle et des contraintes de cardinalité

Entwurf funktionaler Implementierungen von Graphalgorithmen

Classic graph algorithms are usually presented and analysed in imperative programming languages. Imperative programming languages are well-suited for the description of a program flow, in which the order in which the operations are performed is important. One common example of such a description is the successive, typically destructive modification of objects. This kind of iteration often occurs in the context of graph algorithms that deal with a certain kind of optimisation. In functional programming, the order of execution is abstracted and problem solutions are described as compositions of intermediate solutions. Additionally, functional programming languages are referentially transparent and thus destructive updates of objects are discouraged. The development of purely functional graph algorithms begins with the decomposition of a given problem into simpler problems. In many cases the solutions of these partial problems can be used to solve different problems as well. What is more, this compositionality allows exchanging functions for more efficient or more comprehensible versions with little effort. An algebraic approach with a focus on relation algebra as defined by Tarski is used as an intermediate step in this dissertation. One advantage of this approach is the formality of the resulting specifications. Despite their formality, the resulting expressions are still readable, because the algebraic operations have intuitive interpretations. Another advantage is that the specification is executable, once the necessary operations are implemented. This dissertation presents the basics of the algebraic approach in the functional programming language Haskell. Using this foundation, some exemplary graph-theoretic problems are solved in the presented framework. Finally, optimisations of the presented implementations are discussed and pointers are provided to further problems that can be solved using the above methods.Klassische Graphalgorithmen werden üblicherweise in imperativen Programmiersprachen beschrieben und analysiert. Imperative Programmiersprachen eignen sich gut, um Programmabläufe zu beschreiben, in welchen die Reihenfolge der Operationen wichtig ist. Dies betrifft insbesondere die schrittweise, in der Regel destruktive Veränderung von Objekten, wie sie häufig im Falle von Optimierungsproblemen auf Graphen vorkommt. In der funktionalen Programmierung abstrahiert man von einer festen Berechnungsreihenfolge und beschreibt Problemlösungen als Kompositionen von Teillösungen. Ferner sind funktionale Programmiersprachen referentiell transparent, sodass destruktive Veränderungen nur bedingt möglich sind. Die Entwicklung rein funktionaler Graphalgorithmen setzt bei der Zerlegung der bestehenden Probleme in einfachere Probleme an. Oftmals können Lösungen dieser Teilprobleme auch in anderen Situationen eingesetzt werden. Darüber hinaus erlaubt es diese Kompositionalität, einzelne Funktionen mit wenig Aufwand durch effizientere oder verständlichere Fassungen auszutauschen. Als Zwischenschritt in der Entwicklung wird in dieser Dissertation ein algebraischer Ansatz basierend auf der Relationenalgebra im Sinne von Tarski verwendet. Ein Vorteil dieses Ansatzes ist die Formalität der entstehenden Spezifikationen. Trotz ihrer Formalität bleiben die entstehenden Ausdrücke oft leserlich, weil die algebraischen Operationen anschauliche Interpretationen zulassen. Ein weiterer Vorteil ist, dass Spezifikationen ausführbar werden, sobald bestimmte Basisoperationen implementiert sind. In dieser Dissertation werden Grundlagen einer Implementierung des algebraischen Ansatzes in der funktionalen Programmiersprache Haskell behandelt. Ausgehend hiervon werden exemplarisch einige Probleme der Graphentheorie gelöst. Schließlich werden Optimierungen der vorgestellten Implementierungen und weitere Probleme, welche mit den obigen Methoden lösbar sind, diskutiert

Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing

Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing

Logic perturbation based circuit partitioning and optimum FPGA switch-box designs.

Cheung Chak Chung.Thesis (M.Phil.)--Chinese University of Hong Kong, 2001.Includes bibliographical references (leaves 101-114).Abstracts in English and Chinese.Abstract --- p.iAcknowledgments --- p.iiiVita --- p.vTable of Contents --- p.viList of Figures --- p.xList of Tables --- p.xivChapter 1 --- Introduction --- p.1Chapter 1.1 --- Motivation --- p.1Chapter 1.2 --- Aims and Contribution --- p.4Chapter 1.3 --- Thesis Overview --- p.5Chapter 2 --- VLSI Design Cycle --- p.6Chapter 2.1 --- Logic Synthesis --- p.7Chapter 2.1.1 --- Logic Minimization --- p.8Chapter 2.1.2 --- Technology Mapping --- p.8Chapter 2.1.3 --- Testability --- p.8Chapter 2.2 --- Physical Design Synthesis --- p.8Chapter 2.2.1 --- Partitioning --- p.9Chapter 2.2.2 --- Floorplanning & Placement --- p.10Chapter 2.2.3 --- Routing --- p.11Chapter 2.2.4 --- "Compaction, Extraction & Verification" --- p.12Chapter 2.2.5 --- Physical Design of FPGAs --- p.12Chapter 3 --- Alternative Wiring --- p.13Chapter 3.1 --- Introduction --- p.13Chapter 3.2 --- Notation and Definitions --- p.15Chapter 3.3 --- Application of Rewiring --- p.17Chapter 3.3.1 --- Logic Optimization --- p.17Chapter 3.3.2 --- Timing Optimization --- p.17Chapter 3.3.3 --- Circuit Partitioning and Routing --- p.18Chapter 3.4 --- Logic Optimization Analysis --- p.19Chapter 3.4.1 --- Global Flow Optimization --- p.19Chapter 3.4.2 --- OBDD Representation --- p.20Chapter 3.4.3 --- Automatic Test Pattern Generation (ATPG) --- p.22Chapter 3.4.4 --- Graph Based Alternative Wiring (GBAW) --- p.23Chapter 3.5 --- Augmented GBAW --- p.26Chapter 3.6 --- Logic Optimization by using GBAW --- p.28Chapter 3.7 --- Conclusions --- p.31Chapter 4 --- Multi-way Partitioning using Rewiring Techniques --- p.33Chapter 4.1 --- Introduction --- p.33Chapter 4.2 --- Circuit Partitioning Algorithm Analysis --- p.38Chapter 4.2.1 --- The Kernighan-Lin (KL) Algorithm --- p.39Chapter 4.2.2 --- The Fiduccia-Mattheyses (FM) Algorithm --- p.42Chapter 4.2.3 --- Geometric Representation Algorithm --- p.46Chapter 4.2.4 --- The Multi-level Partitioning Algorithm --- p.49Chapter 4.2.5 --- Hypergraph METIS - hMETIS --- p.51Chapter 4.3 --- The GBAW Partitioning Algorithm --- p.53Chapter 4.4 --- Experimental Results --- p.56Chapter 4.5 --- Conclusions --- p.58Chapter 5 --- Optimum FPGA Switch-Box Designs - HUSB --- p.62Chapter 5.1 --- Introduction --- p.62Chapter 5.2 --- Background and Definitions --- p.65Chapter 5.2.1 --- Routing Architectures --- p.65Chapter 5.2.2 --- Global Routing --- p.67Chapter 5.2.3 --- Detailed Routing --- p.67Chapter 5.3 --- FPGA Router Comparison --- p.69Chapter 5.3.1 --- CGE --- p.69Chapter 5.3.2 --- SEGA --- p.70Chapter 5.3.3 --- TRACER --- p.71Chapter 5.3.4 --- VPR --- p.72Chapter 5.4 --- Switch Box Design --- p.73Chapter 5.4.1 --- Disjoint type switch box (XC4000-type) --- p.73Chapter 5.4.2 --- Anti-symmetric switch box --- p.74Chapter 5.4.3 --- Universal Switch box --- p.74Chapter 5.4.4 --- Switch box Analysis --- p.75Chapter 5.5 --- Terminology --- p.77Chapter 5.6 --- "Hyper-universal (4, W)-design analysis" --- p.82Chapter 5.6.1 --- "H3 is an optimum (4, 3)-design" --- p.84Chapter 5.6.2 --- "H4 is an optimum (4,4)-design" --- p.88Chapter 5.6.3 --- "Hi is a hyper-universal (4, i)-design for i = 5,6,7" --- p.90Chapter 5.7 --- Experimental Results --- p.92Chapter 5.8 --- Conclusions --- p.95Chapter 6 --- Conclusions --- p.99Chapter 6.1 --- Thesis Summary --- p.99Chapter 6.2 --- Future work --- p.100Chapter 6.2.1 --- Alternative Wiring --- p.100Chapter 6.2.2 --- Partitioning Quality --- p.100Chapter 6.2.3 --- Routing Devices Studies --- p.100Bibliography --- p.101Chapter A --- 5xpl - Berkeley Logic Interchange Format (BLIF) --- p.115Chapter B --- Proof of some 2-local patterns --- p.122Chapter C --- Illustrations of FM algorithm --- p.124Chapter D --- HUSB Structures --- p.127Chapter E --- Primitive minimal 4-way global routing Structures --- p.13