Binary decision diagram

In computer science, a binary decision diagram (BDD) or branching program is a data structure that is used to represent a Boolean function. On a more abstract level, BDDs can be considered as a compressed representation of sets or relations. Unlike other compressed representations, operations are performed directly on the compressed representation, i.e. without decompression. Similar data structures include negation normal form (NNF), Zhegalkin polynomials, and propositional directed acyclic graphs (PDAG). Definition A Boolean function can be represented as a rooted, directed, acyclic graph, which consists of several (decision) nodes and two terminal nodes. The two terminal nodes are labeled 0 (FALSE) and 1 (TRUE). Each (decision) node u is labeled by a Boolean variable x_i and has two child nodes called low child and high child. The edge from node u to a low (or high) child represents an assignment of the value FALSE (or TRUE, respe

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Efficient Data Structures For Decision Diagrams

Stéphane Rabie

Dynamic Programming Optimization (DPOP) is an algorithm proposed to solve distributed constraint optimization problems. In order to represent the multi-values functions manipulated in this algorithm, a data structure called Hypercube was implemented. A more efficient data structure, the Utility Diagram, was then proposed as an alternative to the Hypercube. DPOP also required the implementation of several operations (such as join, project, slice, split and reorder) on these two data structures. This project is a follow-up of Nacereddine Ouaret’s master thesis, which consisted in implementing all of these data structures and theirs associated operations. As DPOP may have to work on very large decision diagrams, and perform a lot of successive operations on them, having implementations which are efficient in term of speed and memory is critical. The aim of this project was therefore to seek for new ways to improve the already implemented functions for hypercubes and utility diagrams, both in term of execution time and memory consumption. This report will thus first present a quick overview of hypercubes, utility diagrams, and their associated operations (a more complete description of these objects, as well as the details about their original implementation, can be found in Nacereddine Ouaret’s report on Efficient Data Structures for Decision Diagrams). The second part will then cover various improvements made to their implementation during the course of this project. Finally, a variant for the methods used by hypercube, more economical in term of memory as it reuses existing hypercubes rather than creating new ones, will be presented.

2009

Efficient Data Structures for Decision Diagrams

Nacereddine Ouaret

Dynamic Programming OPtimization (DPOP) is an algorithm proposed for solving distributed constraint optimization problems. In this algorithm, Hypercubes are used as the format of the messages exchanged between the agents. Then, Hybrid-DPOP (H-DPOP), a hybrid algorithm based on DPOP was proposed as an improved solution. Its main advantage over DPOP lies under the use of Multi-valued Decision Diagrams (MDDs) instead of Hypercubes. Furthermore, MDDs give a more compact representation of the messages. The MDDs were implemented and presented in by Kumar even though it was presented as Constrained Decision Diagrams (CDD) which is a generalization of MDDs. The first part of this project aims at implementing Hypercubes. An Implementation already exists. Therefore, the goal is to propose an implementation which is more efficient in term of speed and memory usage for the already implemented functions (join, project and slice). The solution implemented during this project should also provide new functions (split and re-order) and the possibility of saving Hypercubes in the XML format. The second part of the project consists in proposing and implementing a data structure that is more efficient than the MDDs implemented in H-DPOP. This data structure should also provide more functions, and also provide the possibility of saving data in the XML format.

2008

Variants of Homomorphism Polynomials Complete for Algebraic Complexity Classes

Aditya Vardhan Varre

We present polynomial families complete for the well-studied algebraic complexity classes VF, VBP, VP, and VNP. The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. (2018). We consider three different variants of graph homomorphisms, namely injective homomorphisms, directed homomorphisms, and injective directed homomorphisms, and obtain polynomial families complete for VF, VBP, VP, and VNP under each one of these. The polynomial families have the following properties: The polynomial families complete for VF, VBP, and VP are model independent, i.e., they do not use a particular instance of a formula, algebraic branching programs, or circuit for characterising VF, VBP, or VP, respectively. All the polynomial families are hard under p-projections.

ASSOC COMPUTING MACHINERY2021

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