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10 - Binary Decision Diagrams

from Part IV - Graph Representations and Efficient Computation Models

Published online by Cambridge University Press:  05 June 2013

Beate Bollig
Affiliation:
Technische Universität Dortmund
Martin Sauerhoff
Affiliation:
Technische Universität Dortmund
Detlef Sieling
Affiliation:
Technische Universität Dortmund
Ingo Wegener
Affiliation:
Technische Universität Dortmund
Yves Crama
Affiliation:
Université de Liège, Belgium
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Summary

Introduction

Let f: DnR be a finite function: that is, D and R are finite sets. Such a function can be represented by the table of all (a, f (a)), a Є Dn, which always has an exponential size of ∣D∣n. Therefore, we are interested in representations that for many important functions are much more compact. The best-known representations are circuits and decision diagrams. Circuits are a hardware model reflecting the sequential and parallel time to compute f (a) froma (see Chapter 11). Decision diagrams (DDs), also called branching programs (BPs), are nonuniform programs for computing f (a) from a based on only two types of instructions represented by nodes in a graph (see also Figure 10.1):

  • Decision nodes: depending on the value of some input variable xi the next node is chosen.

  • Output nodes (also called sinks): a value from R is presented as output.

A decision diagram is a directed acyclic graph consisting of decision nodes and output nodes. Each node v represents a function fv defined in the following way. Let a = (a1, …, an) Є Dn. At decision nodes, choose the next node as described before. The value of fv(a) is defined as the value of the output node that is finally reached when starting at v. Hence, for each node each input a Є Dn activates a unique computation path that we follow during the computation of fv(a). An edge e = (v,w) of the diagram is called activated by a if the computation path starting at v runs via e.

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Publisher: Cambridge University Press
Print publication year: 2010

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