Quantum Algorithm for Finding the Optimal Variable Ordering for Binary Decision Diagrams

An ordered binary decision diagram (OBDD) is a directed acyclic graph that represents a Boolean function. Since OBDDs have many nice properties as data structures, they have been extensively studied for decades in both theoretical and practical fields, such as VLSI (Very Large Scale Integration) design, formal verification, machine learning, and combinatorial problems. Arguably, the most crucial problem in using OBDDs is that they may vary exponentially in size depending on their variable ordering (i.e., the order in which the variables are to be read) when they represent the same function. Indeed, it is NP hard to find an optimal variable ordering that minimizes an OBDD for a given function. Friedman and Supowit provided a clever deterministic algorithm with time/space complexity O^?(3?), where n is the number of variables of the function, which is much better than the trivial brute-force bound O^?(n!2?). This paper shows that a further speedup is possible with quantum computers by presenting a quantum algorithm that produces a minimum OBDD together with the corresponding variable ordering in O^?(2.77286?) time and space with an exponentially small error probability. Moreover, this algorithm can be adapted to constructing other minimum decision diagrams such as zero-suppressed BDDs

Compressing Binary Decision Diagrams

The paper introduces a new technique for compressing Binary Decision Diagrams in those cases where random access is not required. Using this technique, compression and decompression can be done in linear time in the size of the BDD and compression will in many cases reduce the size of the BDD to 1-2 bits per node. Empirical results for our compression technique are presented, including comparisons with previously introduced techniques, showing that the new technique dominate on all tested instances.Comment: Full (tech-report) version of ECAI 2008 short pape

Decision diagrams in machine learning: an empirical study on real-life credit-risk data.

Decision trees are a widely used knowledge representation in machine learning. However, one of their main drawbacks is the inherent replication of isomorphic subtrees, as a result of which the produced classifiers might become too large to be comprehensible by the human experts that have to validate them. Alternatively, decision diagrams, a generalization of decision trees taking on the form of a rooted, acyclic digraph instead of a tree, have occasionally been suggested as a potentially more compact representation. Their application in machine learning has nonetheless been criticized, because the theoretical size advantages of subgraph sharing did not always directly materialize in the relatively scarce reported experiments on real-world data. Therefore, in this paper, starting from a series of rule sets extracted from three real-life credit-scoring data sets, we will empirically assess to what extent decision diagrams are able to provide a compact visual description. Furthermore, we will investigate the practical impact of finding a good attribute ordering on the achieved size savings.Advantages; Classifiers; Credit scoring; Data; Decision; Decision diagrams; Decision trees; Empirical study; Knowledge; Learning; Real life; Representation; Size; Studies;

A memory efficient algorithm for network reliability

We combine the Augmented Ordered Binary Decision Diagram (OBDD-A) with the use of boundary sets to create a method for computing the exact K-terminal or all-terminal reliability of an undirected network with failed edges and perfect vertices. We present the results of implementing this algorithm and show that the execution time is comparable with the state of the art and the space requirement is greatly reduced. Indeed the space remains constant when networks increase in size but maintain their structure and maximum boundary set size; with the same amount of memory used for computing a 312 and a 31000 grid network

On augmented OBDD and performability for sensor networks

The expected hop count (EHC) or performability of a wireless sensor network (WSN) with probabilistic node failures provides the expected number of operational nodes a message traverses from a set of sensors to reach its target station. This paper proposes a novel approach for computing the EHC of a practical communication model for WSN, k-of-all-sources to any-terminal (k-of-S,t). Techniques based on factoring and Boolean techniques solve the EHC when k=1 for |S| greater than/equal to 1 However, they fail to scale with large WSN and are not useful for computing the EHC with k>1. To overcome these problems, we propose an Augmented Ordered Binary Decision Diagram (OBDD-A) approach, which obtains the EHC for all cases of (k-of-S,t). We use randomly generated wireless networks and grid networks having up to 4.6x1020 (s,t)-minpaths to generate results. Results show that OBDD-A can obtain the EHC for networks that are unsolvable with existing approaches

Technical Communications of ICLP

Abstract Dynamic programming (DP) on tree decompositions is a well studied approach for solving hard problems efficiently. State-of-the-art implementations usually rely on tables for storing information, and algorithms specify how the tuples are manipulated during traversal of the decomposition. However, a major bottleneck of such table-based algorithms is relatively high memory consumption. The goal of the doctoral thesis herein discussed is to mitigate performance and memory shortcomings of such algorithms. The idea is to replace tables with an efficient data structure that no longer requires to enumerate intermediate results explicitly during the computation. To this end, Binary Decision Diagrams (BDDs) and related concepts are studied with respect to their applicability in this setting. Besides native support for efficient storage, from a conceptual point of view BDDs give rise to an alternative approach of how DP algorithms are specified. Instead of tuple-based manipulation operations, the algorithms are specified on a logical level, where sets of models can be conjointly updated. The goal of the thesis is to provide a general tool-set for problems that can be solved efficiently via DP on tree decompositions

МИНИМИЗАЦИЯ МНОГОУРОВНЕВЫХ ПРЕДСТАВЛЕНИЙ СИСТЕМ БУЛЕВЫХ ФУНКЦИЙ НА ОСНОВЕ РАЗЛОЖЕНИЯ ШЕННОНА

A locally optimal algorithm is proposed to form a permutation of variables, which are used to obtain successive Shannon decompositions of a system of disjunctive normal forms of completely specified Boolean functions. The goal of it is a multilevel representation of functions which is called a reduced ordered Binary Decision Diagram. The results of experimental comparison of the program implementing the proposed algorithm and the program implementing the algorithm for enumeration of random permutations are given. The results show the advantage of the proposed algorithm when it is used for synthesis of logical circuits on the basis of library elements.Предлагается приближенный алгоритм формирования перестановки переменных, по каждой из которых последовательно строятся разложения Шеннона системы дизъюнктивных нормальных форм полностью определенных булевых функций с целью получения многоуровневого представления функций, называемого в литературе сокращенной упорядоченной диаграммой двоичного выбора либо диаграммой двоичных решений. Приводятся результаты экспериментального сравнения программы, реализующей предложенный алгоритм, с программой, реализующей алгоритм перебора случайных перестановок

Discrete Function Representations Utilizing Decision Diagrams and Spectral Techniques

All discrete function representations become exponential in size in the worst case. Binary decision diagrams have become a common method of representing discrete functions in computer-aided design applications. For many functions, binary decision diagrams do provide compact representations. This work presents a way to represent large decision diagrams as multiple smaller partial binary decision diagrams. In the Boolean domain, each truth table entry consisting of a Boolean value only provides local information about a function at that point in the Boolean space. Partial binary decision diagrams thus result in the loss of information for a portion of the Boolean space. If the function were represented in the spectral domain however, each integer-valued coefficient would contain some global information about the function. This work also explores spectral representations of discrete functions, including the implementation of a method for transforming circuits from netlist representations directly into spectral decision diagrams

Exact BDD Minimization for Path-Related Objective Functions

Abstract. In this paper we investigate the exact optimization of BDDs with respect to path-related objective functions. We aim at a deeper understanding of the computational effort of exact methods targeting the new objective functions. This is achieved by an approach based on Dynamic Programming which generalizes the framework of Friedman and Supowit. A prime reason for the computational complexity can be identified using this framework. For the first time, experimental results give the minimal expected path length of BDDs for benchmark functions. They have been obtained by an exact Branch&Bound method which can be derived from the general framework. The exact solutions are used to evaluate a heuristic approach. Apart from a few exceptions, the results prove the high quality of the heuristic solutions