On the size of (generalized) OBDDs for threshold functions

Ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for Boolean functions. Among the many areas of application are hardware verification, model checking, and symbolic graph algorithms. Threshold functions are the basic functions for discrete neural networks and are used as building blocks in the design of some symbolic graph algorithms. In this paper the first exponential lower bound on the size of a more general model than OBDDs and the first nontrivial asymptotically optimal bound on the OBDD size for a threshold function are presented.     

Exact OBDD Bounds for Some Fundamental Functions

Ordered binary decision diagrams (OBDDs) are nowadays one of the most common dynamic data structures or representation types for Boolean functions. Among the many areas of application are verification, model checking, computer aided design, relational algebra, and symbolic graph algorithms. Although many exponential lower bounds on the OBDD size of Boolean functions are known, there are only few functions where the OBDD size is asymptotically known exactly. In this paper the exact OBDD sizes of the fundamental functions multiplexer and addition of n-bit numbers are determined.     

A note on the size of OBDDs for the graph of integer multiplication

Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations and ordered binary decision diagrams (OBDDs) are the most common dynamic data structure for Boolean functions. Among the many areas of application are verification, model checking, computer-aided design, relational algebra, and symbolic graph algorithms. Analyzing the limits of symbolic graph algorithms for the all-pairs-shortest paths problem which work on OBDD-represented graph instances the so-called graph of integer multiplication has been investigated by Sawitzki [D. Sawitzki, Lower bounds on the OBDD size of graphs of some popular functions, in: Proc. of SOFSEM, LNCS, vol. 3381, 2005, pp. 298-309]. Using simple arguments his lower bound of ²n/768−1 on the size of OBDDs representing the graph of integer multiplication is improved up to ²n/24.     

基于有序二叉决策图的装配序列符号表示方法

通过建立装配状态的二进制编码和装配操作的布尔特征函数,给出了装配序列描述的有序二叉决策图(OBDD)方法;建立了从装配序列的与或图模型到OBDD模型的转换规则;并对装配序列表示的与或图模型和OBDD模型进行了存储效率比较.实验结果表明:OBDD方法具有较好的存储性能,可以改善复杂装配体的装配序列表示的存储效率,适合于复杂装配体的可行装配序列的描述.     

基于OBDD的含圈与或图搜索算法研究*

与或图搜索是人工智能领域一项一定范围内通用的问题求解技术。基于传统数据结构的与或图表示技术极大地限制了与或图搜索算法可求解问题的规模。本文在Mahanti等提出的含圈与或图理论框架基础上,给出了基于OBDD的含圈与或图符号表示方法,并提出了一种求解含圈与或图最小代价解图的符号搜索算法。实验结果表明：该算法在处理大规模含圈与或图时具有明显优势。     

Symbolic Graphs: Linear Solutions to Connectivity Related Problems

The importance of symbolic data structures such as Ordered Binary Decision Diagrams (OBDD) is rapidly growing in many areas of Computer Science where the large dimensions of the input models is a challenging feature: OBDD based graph representations allowed to define truly new standards in the achievable dimensions for the Model Checking verification technique. However, OBDD representations pose strict constraints in the algorithm design issue. For example, Depth-First Search (DFS) is not feasible in a symbolic framework and, consequently, many state-of-the-art DFS based algorithms (e.g., connectivity procedures) cannot be easily rearranged to work on symbolically represented graphs. We devise here a symbolic algorithmic strategy, based on the new notion of spine-set, that is general enough to be the engine of linear symbolic step algorithms for both strongly connected components and biconnected components. Our procedures improve on previously designed connectivity symbolic algorithms. Moreover, by an application to the so-called “bad cycle detection problem”, our technique can be used to efficiently solve the emptiness problem for various kinds of ω-automata. This work is a revised and extended version of [22,23]. It is partially supported by the projects PRIN 2005015491 and BIOCHECK.     

On the OBDD complexity of the most significant bit of integer multiplication

Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations. Ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for boolean functions. Among the many areas of application are verification, model checking, computer-aided design, relational algebra, and symbolic graph algorithms. In this paper it is shown that the OBDD complexity of the most significant bit of integer multiplication is exponential answering an open question posed by Wegener (2000) [18].     

On the Minimization of (Complete) Ordered Binary Decision Diagrams

Ordered binary decision diagrams (OBDDs) are a popular data structure for Boolean functions. Some applications work with a restricted variant called complete OBDDs which is strongly related to nonuniform deterministic finite automata. One of its complexity measures is the width which has been investigated in several areas in computer science like machine learning, property testing, and the design and analysis of implicit graph algorithms. For a given function the size and the width of a (complete) OBDD is very sensitive to the choice of the variable ordering but the computation of an optimal variable ordering for the OBDD size is known to be NP-hard. Since optimal variable orderings with respect to the OBDD size are not necessarily optimal for the complete model or the OBDD width and hardly anything about the relation between optimal variable orderings with respect to the size and the width is known, this relationship is investigated. Here, using a new reduction idea it is shown that the size minimization problem for complete OBDDs and the width minimization problem are NP-hard.     

Symbolic OBDD representations for mechanical assembly sequences

Assembly sequence planning is one typical combinatorial optimization problem, where the size of parts involved is a significant and often prohibitive difficulty. The compact storage and efficient evaluation of all the feasible assembly sequences is one crucial concern. Ordered binary decision diagram (OBDD) is a canonical form to represent and manipulate the Boolean functions efficiently, and appears to give improved results for large-scale combinatorial optimization problems. In this paper, subassemblies, assembly states and assembly tasks are represented as Boolean characteristic functions, and the symbolic OBDD representation of assembly sequences is proposed. In this framework, the procedures to transform directed graph and AND/OR graph into OBDDs are presented. The great advantage of OBDD-based scheme is that the storage space of OBDD-based representation of all the feasible assembly sequences does not increase with the part count of assembly dramatically so quickly as that of both directed graph and AND/OR graph do. We undertake many experimental tests using Visual C++ and CUDD package. It was shown that the OBDD scheme represented all the feasible assembly sequences correctly and completely, and outperforms either directed graph or AND/OR graph in storage efficiency.     

Randomized OBDDs for the most significant bit of multiplication need exponential space

Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations and ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for Boolean functions. Only in 2008 the question whether the deterministic OBDD complexity of the most significant bit of integer multiplication is exponential has been answered affirmatively. Since probabilistic methods have turned out to be useful in almost all areas of computer science, one may ask whether randomization can help to represent the most significant bit of integer multiplication in smaller size. Here, it is proved that the randomized OBDD complexity is also exponential.     

A uniform framework for weighted decision diagrams and its implementation

This paper introduces a generic framework for OBDD variants with weighted edges. It covers many boolean and multi-valued OBDD-variants that have been studied in the literature and applied to the symbolic representation and manipulation of discrete functions. Our framework supports reasoning about reducedness and canonicity of new OBDD-variants and provides a platform for the implementation and comparison of OBDD-variants. Furthermore, we introduce a new multi-valued OBDD-variant, called normalized algebraic decision diagram, which supports min/max-operations and turns out to be very useful for, e.g., integer linear programming and model checking probabilistic systems. Finally, we explain the main features of an implementation of a newly developed BDD-package, called JINC, which relies on our generic notion of weighted decision diagrams, and realizes various synthesis algorithms, reordering techniques and transformation algorithms for boolean and multi-terminal OBDDs, with or without edge-attributes, and their zero-suppressed variants.     

Slow emergence of cooperation for win-stay lose-shift on trees

We consider a group of agents on a graph who repeatedly play the prisoner’s dilemma game against their neighbors. The players adapt their actions to the past behavior of their opponents by applying the win-stay lose-shift strategy. On a finite connected graph, it is easy to see that the system learns to cooperate by converging to the all-cooperate state in a finite time. We analyze the rate of convergence in terms of the size and structure of the graph. Dyer et al. (2002) showed that the system converges rapidly on the cycle, but that it takes a time exponential in the size of the graph to converge to cooperation on the complete graph. We show that the emergence of cooperation is exponentially slow in some expander graphs. More surprisingly, we show that it is also exponentially slow in bounded-degree trees, where many other dynamics are known to converge rapidly. Editors: Amy Greenwald and Michael Littman.     

Learning to rank on graphs

Graph representations of data are increasingly common. Such representations arise in a variety of applications, including computational biology, social network analysis, web applications, and many others. There has been much work in recent years on developing learning algorithms for such graph data; in particular, graph learning algorithms have been developed for both classification and regression on graphs. Here we consider graph learning problems in which the goal is not to predict labels of objects in a graph, but rather to rank the objects relative to one another; for example, one may want to rank genes in a biological network by relevance to a disease, or customers in a social network by their likelihood of being interested in a certain product. We develop algorithms for such problems of learning to rank on graphs. Our algorithms build on the graph regularization ideas developed in the context of other graph learning problems, and learn a ranking function in a reproducing kernel Hilbert space (RKHS) derived from the graph. This allows us to show attractive stability and generalization properties. Experiments on several graph ranking tasks in computational biology and in cheminformatics demonstrate the benefits of our framework.     

基于重量分析的OBDD变量排序算法

有序的二叉判决图（ＯＢＤＤ）是布尔表达式的一种有效表示方法，但它的体积对变量排序具有较强的依赖性。本文提出一种电路结构图，并在此基础上定义了原始输入重量和节点重量等参数，并建立了用重量分析来指导的ＯＢＤＤ变量排序算法。由于从考虑变量对输出函数的影响出发与从考虑ＯＢＤＤ节点共享性出发对变量排序的要求不同，本文分别设计了两类算法。实验结果表明，本文对大多数标准电路变量排序的效果都优于国际上的同类算法，     

OBDD在组合逻辑电路测试中的应用研究

传统的组合逻辑电路测试方法在搜索过程中都不可避免地要进行反向回溯,由于反向回溯的次数过多,往往会降低算法的效率,文中利用OBDD来表示电路中每个节点所代表的逻辑函数,把传统算法中的反向回溯过程转换为OBDD图的问题,从而加快了故障测试的速度,同时,OBDD在测试矢量集的生成以及必要值的确定中也显示出一定的优越性。     

Graphillion: software library for very large sets of labeled graphs

Several graph libraries have been developed in the past few decades, and they were basically designed to work with a few graphs. However, there are many problems in which we have to consider all subgraphs satisfying certain constraints on a given graph. Since the number of subgraphs can increase exponentially with the graph size, explicitly representing these sets is infeasible. Hence, libraries concerned with efficiently representing a single graph instance are not suitable for such problems. In this paper, we develop Graphillion, a software library for very large sets of (vertex-)labeled graphs, based on zero-suppressed binary decision diagrams. Graphillion is not based on a traditional representation of graphs. Instead, a graph set is simply regarded as a “set of edge sets” ignoring vertices, which allows us to employ powerful tools of a “family of sets” (a set of sets) and permits large graph sets to be handled efficiently. We also utilize advanced graph enumeration algorithms, which enable the simple family tools to understand the graph structure. Graphillion is implemented as a Python library to encourage easy development of its applications, without introducing significant performance overheads. In experiments, we consider two case studies, a puzzle solver and a power network optimizer, in which several operations and heavy optimization have to be performed over very large sets of constrained graphs (i.e., cycles or forests with complicated conditions). The results show that Graphillion allows us to manage a huge number of graphs with very low development effort.     

一种基于符号关系图的快速符号数据聚类算法

由于在实际应用中有大量的符号数据生成,符号数据聚类成为了聚类分析的一个重要研究领域。目前,已有许多符号数据聚类算法被提出,但将它们应用于大数据环境时,仍然存在计算成本高、运行速度慢等问题。文中提出了一种基于符号关系图的快速符号数据聚类算法。该算法使用符号关系图替代原始数据,缩小数据集的规模,有效地解决了这一问题。大量的实验分析显示新算法相比其他算法是有效的。     

Trust-aware recommendation based on heterogeneous multi-relational graphs fusion

Users’ trust relations have a significant influence on their choice towards different products. However, few recommendation or prediction algorithms both consider users’ social trust relations and item-related knowledge, which makes them difficult to cope with cold start and the data sparsity problems. In this paper, we propose a novel trust-ware recommendation method based on heterogeneous multi-relational graphs fusion, termed as T-MRGF. In contrast with other traditional methods, it fuses the user-related and item-related graphs with the user–item interaction graph and fully utilizes the high-level connections existing in heterogeneous graphs. Specifically, we first establish the user–user trust relation graph, user–item interaction graph and item–item knowledge graph, and the user feature and item feature, which have been obtained from the user–item graph, are used as the input of the user-related graph and the item-related graph respectively. The fusion is achieved through the cascade of feature vectors before and after feature propagation. In this way, the heterogeneous multi-relational graphs are fused for the feature propagation, which largely refines the user and item representation for model prediction. Simulation results show that the proposed method significantly improve the recommendation performance compared to the state-of-the-art KG-based algorithms both in accuracy and training efficiency.     

Study of the diagnosability of automated production systems based on functional graphs

Functional graphs are a convenient representation that we have introduced to model automated production systems. They are useful for the monitoring and the supervision of manufacturing processes or other industrial processes, such as chemical processes. An approach based on relational theory and graph theory is presented in this paper. This approach allows to characterize formally structural properties of a functional graph and to map it into a set of relations translating all the complete paths existing in the initial graph. Two kinds of functional graphs are analyzed and algorithms exploiting their structures are presented. We introduce the concept of diagnosability as a system property that reflects the possibility to observe the behavior of a system with respect to faults. The diagnosability is defined and analyzed by means of computable states and mathematical relations. Propositions explaining causality relations between functions of a functional graph are given.     

基于符号迁移图的互模拟验证算法

符号迁移图是传值进程的一种直观而简洁的语义表示模型，该模型由Hennessy和Lin首先提出，随后又被Lin推广至带赋值的符号迁移图，本文不但定义了符号迁移图各种版本（基／符号）的强操作语义和强互模拟，提出了相互的强互模拟算法，而且通过引入符号观察图和符号同余图，给出了其弱互模拟等价和观察同余的验证算法，给出并证明了了τ－循环和τ－边消去定理，在应用任何弱互模拟观察同余验证算法之前，均可利用这些定理对所给符号迁移图进行化简。