## An introduction to zero-suppressed binary decision diagrams (2001)

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Venue: | in ‘Proceedings of the 12th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning |

Citations: | 10 - 0 self |

### BibTeX

@TECHREPORT{Mishchenko01anintroduction,

author = {Alan Mishchenko},

title = {An introduction to zero-suppressed binary decision diagrams},

institution = {in ‘Proceedings of the 12th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning},

year = {2001}

}

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### Abstract

### Citations

3180 | Graph-based algorithms for boolean function manipulation
- Bryant
- 1986
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Citation Context ...s) [21] provide an efficient way of solving problems expressed in terms of set theory. This tutorial paper presents ZDDs for a reader with a background in Boolean algebra and Binary Decision Diagrams =-=[4]-=-, without any prior experience with ZDDs. The case studies considered in the tutorial include the computation of the union of two sets, the generation of all primes of a Boolean function, and the comp... |

280 | Algebraic Decision Diagrams and Their Applications
- Bahar, Frohm, et al.
- 1993
(Show Context)
Citation Context ...ion, we discuss the generic structure of a recursive ZDD procedure. The presentation is also true for recursive procedures written using other types of decision diagrams, in particular, BDDs and ADDs =-=[2]-=-. Therefore, in this subsection we use the term “DD” instead of “ZDD”. Procedures written with DDs can be roughly divided into two classes: • Recursive procedures that rely on the DD structure to perf... |

152 |
Zero-suppressed BDDs for set manipulation in combinatorial problems
- Minato
(Show Context)
Citation Context ...uter Engineering Portland State University, Portland, OR 97207, USA alanmi@ee.pdx.edu; http://www.ee.pdx.edu/~alanmi/research.htm June 8, 2001 Abstract Zero-suppressed binary Decision Diagrams (ZDDs) =-=[21]-=- provide an efficient way of solving problems expressed in terms of set theory. This tutorial paper presents ZDDs for a reader with a background in Boolean algebra and Binary Decision Diagrams [4], wi... |

48 |
Binary Decision Diagrams and Applications for VLSI CAD
- Minato
- 1996
(Show Context)
Citation Context ...subcovers and the primary input variable and returns the composed cover. For a detailed analysis of ZDD in the representation of cube covers, it is recommended for the reader to review references [23]=-=[26]-=-[27] where some basic ZDD-based recursive operators are introduced and explained. 4 Basic ZDD procedures The basic functions dealing with ZDDs can be classified as follows: 4.1 Procedures working with... |

24 | Micheli: "Decision Diagrams and Pass Transistor Logic Synthesis
- Bertacco, Minato, et al.
- 1997
(Show Context)
Citation Context ...fficients [25]. • In exclusive SOP minimization [32][33][37]. • In symbolic traversal of FSMs and Petri Nets [38][41]. • In Davis-Putman resolution procedure [6]. • In pass-transistor logic synthesis =-=[3]-=-. • Finding all disjoint-support decompositions of completely specified logic functions [29]. • Unate decomposition of boolean functions [18].s9 Conclusions This tutorial introduces the reader into th... |

21 |
An approach to unified methodology of combinational switching circuits
- Cerny, Marin
- 1977
(Show Context)
Citation Context ...lar settings, the BDDs grow large making processing inefficient or impossible. In particular, this situation occurs when the applications work with sparse sets represented by characteristic functions =-=[5]-=-. A set is sparse if the number of elements in it is much smaller than the total number of elements that may appear in the set. Cube covers are an example of sparse sets, because a typical cube contai... |

21 |
Graph-Based Representations of Discrete Functions
- Minato
- 1995
(Show Context)
Citation Context ... to severalorder-of-magnitude speedups in some applications, for example [18]. 8 A complete list of published ZDD applications ZDDs have been introduced by S. Minato [21] in 1993 and presented in [26]=-=[28]-=-. Since that time several ZDD packages have been implemented [14][16][24][36]. ZDDs have been used to solve a number of problems arising in different areas of computer science and engineering: • To re... |

20 |
Fast generation of irredundant sum-of-products forms from binary decision diagrams
- Minato
(Show Context)
Citation Context ...erted into single-output functions, as described in [5]. 7.2 Computation of an irredundant SOP The algorithm for recursive computation of ISOP has been proposed in [30]. It has been implemented in [7]=-=[20]-=-. The pseudo-code is given in Fig. 10. Symbols “+”, “&”, and “-“ in the pseudo-code stand for the Boolean operations OR, AND, and SHARP. The procedure IrrSOP() to compute the irredundant sumof-product... |

17 |
Two-level logic minimization: An overview”, Integration
- Coudert
- 1994
(Show Context)
Citation Context ... procedures to perform conversions between them. Taken together, BDDs and 2 ZDDs provide a powerful framework to solve problems in logic synthesis, such as two-level sum-of-product (SOP) minimization =-=[8]-=-, three-level minimization, factorization [27][35], and decomposition [18]. The use of ZDDs is not limited to logic synthesis. They have been used, independently of BDDs, in a number of applications, ... |

16 |
Doing Two-Level Logic Minimization 100 Times Faster
- Coudert
- 1995
(Show Context)
Citation Context ...ssing media for Boolean or multivalued functions and (2) a representation facilitating the analysis of data leading to new implicit algorithms, which tend to be more efficient than the classical ones =-=[9]-=-. One of the reasons why decision diagrams, and in particular BDDs, became useful for the CAD tool developers, is that they provide canonical representation of discrete objects. Canonical means that u... |

16 |
BDDs vs. zero-suppressed BDDs: for CTL symbolic model checking of Petri Nets
- Yoneda, Hatori, et al.
- 1996
(Show Context)
Citation Context ...nder length constraint [17]. • To represent and manipulate polynomials with integer coefficients [25]. • In exclusive SOP minimization [32][33][37]. • In symbolic traversal of FSMs and Petri Nets [38]=-=[41]-=-. • In Davis-Putman resolution procedure [6]. • In pass-transistor logic synthesis [3]. • Finding all disjoint-support decompositions of completely specified logic functions [29]. • Unate decompositio... |

15 |
Finding All Simple Disjunctive Decompositions Using Irredundant Sum-of-Products Forms
- Minato, Micheli
- 1998
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Citation Context ...imentation shows that this implementation of ISOP cover is more efficient than the one proposed in the CUDD package. The ISOP computed by the above algorithm has some remarkable propertied studied in =-=[29]-=-. The actual cover depends on the ordering of variables in the DD manager. As shown in [7][20], in many cases the quality of the ISOP computed is close to the quality of the exact minimum cover (typic... |

15 | Synthesis of Finite State Machines: Logic Optimization - Villa, Brayton - 1997 |

14 | ZRes: The Old Davis-Putnam Procedure Meets ZBDDs
- Chatalic, Simon
(Show Context)
Citation Context ...nd manipulate polynomials with integer coefficients [25]. • In exclusive SOP minimization [32][33][37]. • In symbolic traversal of FSMs and Petri Nets [38][41]. • In Davis-Putman resolution procedure =-=[6]-=-. • In pass-transistor logic synthesis [3]. • Finding all disjoint-support decompositions of completely specified logic functions [29]. • Unate decomposition of boolean functions [18].s9 Conclusions T... |

12 |
Solving Graph Optimization Problems with ZBDDs
- Coudert
- 1997
(Show Context)
Citation Context ... cube covers [22][27][35]. • To solve unate covering problem arising in multi-layer planar routing [10]. • To find dichotomy-based constraint encoding [11][13]. • To solve graph optimization problems =-=[12]-=-. • To represent and manipulate regular expressions under length constraint [17]. • To represent and manipulate polynomials with integer coefficients [25]. • In exclusive SOP minimization [32][33][37]... |

10 | Calculation of Unate Cube Set Algebra Using Zero-Suppressed BDDs
- Minato
(Show Context)
Citation Context ...ree subcovers and the primary input variable and returns the composed cover. For a detailed analysis of ZDD in the representation of cube covers, it is recommended for the reader to review references =-=[23]-=-[26][27] where some basic ZDD-based recursive operators are introduced and explained. 4 Basic ZDD procedures The basic functions dealing with ZDDs can be classified as follows: 4.1 Procedures working ... |

10 | Implicit manipulation of polynomials using zerosuppressed BDDs,” unpublished manuscript
- Minato
- 1994
(Show Context)
Citation Context ...][13]. • To solve graph optimization problems [12]. • To represent and manipulate regular expressions under length constraint [17]. • To represent and manipulate polynomials with integer coefficients =-=[25]-=-. • In exclusive SOP minimization [32][33][37]. • In symbolic traversal of FSMs and Petri Nets [38][41]. • In Davis-Putman resolution procedure [6]. • In pass-transistor logic synthesis [3]. • Finding... |

10 | Recursive Operators for Prime Implicant and Irredundant Normal Form Determination - Morreale - 1970 |

9 | Implicit Prime Cover Computation: An Overview
- Coudert, Madre, et al.
(Show Context)
Citation Context ...onverted into single-output functions, as described in [5]. 7.2 Computation of an irredundant SOP The algorithm for recursive computation of ISOP has been proposed in [30]. It has been implemented in =-=[7]-=-[20]. The pseudo-code is given in Fig. 10. Symbols “+”, “&”, and “-“ in the pseudo-code stand for the Boolean operations OR, AND, and SHARP. The procedure IrrSOP() to compute the irredundant sumof-pro... |

9 |
Fast Weak-Division Method for Implicit Cube Representation
- Minato
- 1993
(Show Context)
Citation Context ...eas of computer science and engineering: • To represent sets in various problems [23][34]. • To represent cubes and essential primes in two-level SOP minimization [8] and factorization of cube covers =-=[22]-=-[27][35]. • To solve unate covering problem arising in multi-layer planar routing [10]. • To find dichotomy-based constraint encoding [11][13]. • To solve graph optimization problems [12]. • To repres... |

7 |
EXTRA Library of the DD procedures. http://www.ee.pdx.edu/~alanmi/research/extra.htm
- Mishchenko
(Show Context)
Citation Context ...ckage. The appendix to the paper contains a list of 35+ ZDD procedures included in the decision diagram package CUDD Release 2.3.1 [36] and 50+ additional ZDD procedures included in the EXTRA library =-=[30]-=- available as a public-domain extension of CUDD. Table of Contents 1 Introduction 2 2 Definitions 3 3 Comparing BDDs and ZDDs 3 3.1 Boolean functions 3 3.2 Sets of subsets 4 3.3 Cube covers 4 4 Basic ... |

5 |
An Efficient Method for Generating Kernels on Implicit Cube Set Representations
- Yamashita
(Show Context)
Citation Context ...Taken together, BDDs and 2 ZDDs provide a powerful framework to solve problems in logic synthesis, such as two-level sum-of-product (SOP) minimization [8], three-level minimization, factorization [27]=-=[35]-=-, and decomposition [18]. The use of ZDDs is not limited to logic synthesis. They have been used, independently of BDDs, in a number of applications, ranging from the graph-theory problems to handling... |

4 | Unate decomposition of Boolean functions
- Jacob, Mishchenko
(Show Context)
Citation Context ... 2 ZDDs provide a powerful framework to solve problems in logic synthesis, such as two-level sum-of-product (SOP) minimization [8], three-level minimization, factorization [27][35], and decomposition =-=[18]-=-. The use of ZDDs is not limited to logic synthesis. They have been used, independently of BDDs, in a number of applications, ranging from the graph-theory problems to handling polynomials and regular... |

4 |
Fast Factorization Method for Implicit Cube Cover Representation
- Minato
- 1996
(Show Context)
Citation Context ...em. Taken together, BDDs and 2 ZDDs provide a powerful framework to solve problems in logic synthesis, such as two-level sum-of-product (SOP) minimization [8], three-level minimization, factorization =-=[27]-=-[35], and decomposition [18]. The use of ZDDs is not limited to logic synthesis. They have been used, independently of BDDs, in a number of applications, ranging from the graph-theory problems to hand... |

3 | On the Properties of Combination Set Operations
- Okuno, Minato, et al.
- 1998
(Show Context)
Citation Context ...s have been implemented [14][16][24][36]. ZDDs have been used to solve a number of problems arising in different areas of computer science and engineering: • To represent sets in various problems [23]=-=[34]-=-. • To represent cubes and essential primes in two-level SOP minimization [8] and factorization of cube covers [22][27][35]. • To solve unate covering problem arising in multi-layer planar routing [10... |

2 |
A New Paradigm of Dichotomy-based Constraint Encoding
- Coudert
(Show Context)
Citation Context ...vel SOP minimization [8] and factorization of cube covers [22][27][35]. • To solve unate covering problem arising in multi-layer planar routing [10]. • To find dichotomy-based constraint encoding [11]=-=[13]-=-. • To solve graph optimization problems [12]. • To represent and manipulate regular expressions under length constraint [17]. • To represent and manipulate polynomials with integer coefficients [25].... |

2 |
An Exact Minimization of AND-EXOR Expressions Using Encoded MRCF
- Ochi
- 1996
(Show Context)
Citation Context ...roblems [12]. • To represent and manipulate regular expressions under length constraint [17]. • To represent and manipulate polynomials with integer coefficients [25]. • In exclusive SOP minimization =-=[32]-=-[33][37]. • In symbolic traversal of FSMs and Petri Nets [38][41]. • In Davis-Putman resolution procedure [6]. • In pass-transistor logic synthesis [3]. • Finding all disjoint-support decompositions o... |

1 |
Exact Multi-Layer Topological Planar Routing
- Coudert, Shi
(Show Context)
Citation Context ...34]. • To represent cubes and essential primes in two-level SOP minimization [8] and factorization of cube covers [22][27][35]. • To solve unate covering problem arising in multi-layer planar routing =-=[10]-=-. • To find dichotomy-based constraint encoding [11][13]. • To solve graph optimization problems [12]. • To represent and manipulate regular expressions under length constraint [17]. • To represent an... |

1 |
Exact Dichotomy-based Constraint Encoding
- Coudert, Shi
(Show Context)
Citation Context ...o-level SOP minimization [8] and factorization of cube covers [22][27][35]. • To solve unate covering problem arising in multi-layer planar routing [10]. • To find dichotomy-based constraint encoding =-=[11]-=-[13]. • To solve graph optimization problems [12]. • To represent and manipulate regular expressions under length constraint [17]. • To represent and manipulate polynomials with integer coefficients [... |

1 |
Manipulation of Regular Expressions Under Length Constraints Using Zero-Suppressed BDDs
- Ishihara, Minato
(Show Context)
Citation Context ...yer planar routing [10]. • To find dichotomy-based constraint encoding [11][13]. • To solve graph optimization problems [12]. • To represent and manipulate regular expressions under length constraint =-=[17]-=-. • To represent and manipulate polynomials with integer coefficients [25]. • In exclusive SOP minimization [32][33][37]. • In symbolic traversal of FSMs and Petri Nets [38][41]. • In Davis-Putman res... |

1 |
R.Brayton, A.Sangiovanni-Vincentelli. Synthesis of Finite State Machines: Functional Optimization
- Kam
- 1997
(Show Context)
Citation Context ...ows in the unate covering problem represented by BDDs. So far, this problem has been solved either explicitly, without DDs, or by applying formulas with quantifiers to the BDDs of dominance relations =-=[19]-=-. 5 Manipulation of sets In this section, we apply the principles of traversal procedures discussed above to a particular example of computation of the union of two sets of subsets. For example, given... |

1 |
A Zero-Suppressed BDD package with Pruning and Its Applications to GRM Minimization
- Ochi
- 1996
(Show Context)
Citation Context ...ems [12]. • To represent and manipulate regular expressions under length constraint [17]. • To represent and manipulate polynomials with integer coefficients [25]. • In exclusive SOP minimization [32]=-=[33]-=-[37]. • In symbolic traversal of FSMs and Petri Nets [38][41]. • In Davis-Putman resolution procedure [6]. • In pass-transistor logic synthesis [3]. • Finding all disjoint-support decompositions of co... |

1 |
A New Approach to Exact ESOP Minimization
- Steinbach, Mishchenko
- 2001
(Show Context)
Citation Context ...[12]. • To represent and manipulate regular expressions under length constraint [17]. • To represent and manipulate polynomials with integer coefficients [25]. • In exclusive SOP minimization [32][33]=-=[37]-=-. • In symbolic traversal of FSMs and Petri Nets [38][41]. • In Davis-Putman resolution procedure [6]. • In pass-transistor logic synthesis [3]. • Finding all disjoint-support decompositions of comple... |

1 | Partial Order Reduction in Symbolic State Space Traversal Using ZBDDs
- Tomisaka, Yoneda
(Show Context)
Citation Context ...ns under length constraint [17]. • To represent and manipulate polynomials with integer coefficients [25]. • In exclusive SOP minimization [32][33][37]. • In symbolic traversal of FSMs and Petri Nets =-=[38]-=-[41]. • In Davis-Putman resolution procedure [6]. • In pass-transistor logic synthesis [3]. • Finding all disjoint-support decompositions of completely specified logic functions [29]. • Unate decompos... |