Developing a Simple ZDD Package

1
Developing a Simple ZDD Package

• Alan Mishchenko
• University of California, Berkeley

2
Outline

• Decision diagrams
• ZDD package
• Experiment
• Summary

3
Decision Diagrams

• DDs are a useful data-structure
• It is a good exercise to build a new DD package
• This presentation is based on real code without
  any simplification
• The code can be found at https//bitbucket.org/ala
  nmi/abc/src/4dc7e5aa7805931d35ba1c310f90f5baed74d8
  59/src/misc/extra/extraUtilPerm.c

4
Building a DD Package

• Define data structures
• Consider memory management
• Write a procedure for creating a new node
• Develop traversal procedures
• Debug and clean up the code
• Develop and test application code based on the
  new package

5
Basic Data Structures

• typedef struct Abc_ZddObj_ Abc_ZddObj
• struct Abc_ZddObj_
• unsigned Var 31
• unsigned Mark 1
• unsigned True
• unsigned False
• typedef struct Abc_ZddEnt_ Abc_ZddEnt
• struct Abc_ZddEnt_
• int Arg0
• int Arg1
• int Arg2
• int Res

6
ZDD Manager

• typedef struct Abc_ZddMan_ Abc_ZddMan
• struct Abc_ZddMan_
• int nVars
• int nObjs
• int nObjsAlloc
• int nPermSize
• unsigned nUniqueMask
• unsigned nCacheMask
• int pUnique
• int pNexts
• Abc_ZddEnt pCache
• Abc_ZddObj pObjs
• int nCacheLookups
• int nCacheMisses
• word nMemory
• int pV2TI
• int pV2TJ
• int pT2V

7
Creating New Node

• static inline unsigned Abc_ZddHash( int Arg0, int
  Arg1, int Arg2 )
• return 12582917 Arg0 4256249 Arg1
  741457 Arg2
• static inline int Abc_ZddUniqueCreate(
• Abc_ZddMan p, int Var, int True, int False
  )
• if ( True 0 )
• return False
• else
• int q p-gtpUnique (Abc_ZddHash(Var,
  True, False) p-gtnUniqueMask)
• for ( q q p-gtpNexts q )
• if ( p-gtpObjsq.Var Var
• p-gtpObjsq.True True
• p-gtpObjsq.False False )
• return q
• assert( p-gtnObjs lt p-gtnObjsAlloc )
• q p-gtnObjs
• p-gtpObjsq.Var Var

8
Traversal Procedures

• Set operations
• Union, difference, intersection
• Product operations
• Dot-product, cross-product
• Counting operations
• Count the number of nodes and paths
• Permutation operations
• Transposition, permutation product

9
ZDD Union

• int Abc_ZddUnion( Abc_ZddMan p, int a, int b )
• Abc_ZddObj A, B
• int r0, r1, r
• if ( a 0 ) return b
• if ( b 0 ) return a
• if ( a b ) return a
• if ( a gt b ) return Abc_ZddUnion( p, b, a )
• if ( (r Abc_ZddCacheLookup(p, a, b,
  ABC_ZDD_OPER_UNION)) gt 0 )
• return r
• A Abc_ZddNode( p, a )
• B Abc_ZddNode( p, b )
• if ( A-gtVar lt B-gtVar )
• r0 Abc_ZddUnion( p, A-gtFalse, b ),
• r1 A-gtTrue
• else if ( A-gtVar gt B-gtVar )
• r0 Abc_ZddUnion( p, a, B-gtFalse ),
• r1 B-gtTrue
• else

10
Experiment

• Shin-ichi Minato proposed PiDDs, a ZDD-based
  data-structure to represent and manipulate
  permutations
• Shin-ichi Minato, "PiDD A new decision diagram
  for efficient problem solving in permutation
  space, Proc. SAT11, pp. 90-104
• One of the applications cited in the paper, is
  enumeration of reachable states of a simplified
  Rubiks cube
• Traditional cube is 3x3x3 and has 4.31019
  states
• http//en.wikipedia.org/wiki/Rubiks_Cube
• Simplified cube is 2x2x2 and has only 3,674,160
  states
• http//en.wikipedia.org/wiki/Pocket_cube
• Minatos ZDD-based implementation takes 207 sec
  to enumerate states of the simplified cube on a
  2.4 GHz Core2Duo PC

11
Cubes State Enconding
12
Cubes Transition Relation
13
Experiment from Minatos Paper
14
Experiment ZDD-Based Enumeration

• UC Berkeley, ABC 1.01 (compiled May 14 2014
  043019)
• abc 01gt cubeenum -z
• Enumerating states of 2x2x2 cube.
• Iter 0 -gt 1 Nodes 0 Used
  2 Time 0.00 sec
• Iter 1 -gt 10 Nodes 63 Used
  577 Time 0.00 sec
• Iter 2 -gt 64 Nodes 443 Used
  4349 Time 0.03 sec
• Iter 3 -gt 385 Nodes 2018 Used
  26654 Time 0.12 sec
• Iter 4 -gt 2232 Nodes 7451 Used
  119442 Time 0.43 sec
• Iter 5 -gt 12224 Nodes 25178 Used
  490038 Time 1.07 sec
• Iter 6 -gt 62360 Nodes 83955 Used
  1919750 Time 1.77 sec
• Iter 7 -gt 289896 Nodes 290863 Used
  7182932 Time 3.16 sec
• Iter 8 -gt 1159968 Nodes 614845 Used
  25301123 Time 8.18 sec
• Iter 9 -gt 3047716 Nodes 585664 Used
  66228369 Time 20.72 sec
• Iter 10 -gt 3671516 Nodes 19430 Used
  102292452 Time 34.28 sec
• Iter 11 -gt 3674160 Nodes 511 Used
  103545878 Time 34.80 sec
• Iter 12 -gt 3674160 Nodes 511 Used
  103566266 Time 34.81 sec
• ZDD stats Var 276 Obj 103566266 Alloc
  134217728
• Hit 63996630 Miss 141768893 Mem
  4608.00 MB

15
Experiment Explicit Enumeration

• UC Berkeley, ABC 1.01 (compiled May 14 2014
  043019)
• abc 01gt cubeenum
• Enumerating states of 2x2x2 cube.
• Iter 0 -gt 1 Time 0.00 sec
• Iter 1 -gt 10 Time 0.00 sec
• Iter 2 -gt 64 Time 0.00 sec
• Iter 3 -gt 385 Time 0.00 sec
• Iter 4 -gt 2232 Time 0.01 sec
• Iter 5 -gt 12224 Time 0.03 sec
• Iter 6 -gt 62360 Time 0.09 sec
• Iter 7 -gt 289896 Time 0.18 sec
• Iter 8 -gt 1159968 Time 0.36 sec
• Iter 9 -gt 3047716 Time 1.02 sec
• Iter 10 -gt 3671516 Time 2.44 sec
• Iter 11 -gt 3674160 Time 2.93 sec
• Iter 12 -gt 3674160 Time 2.93 sec

16
Conclusion

• Discussed basics of decision diagrams
• Proposed a simple ZDD package
• Analyzed performance of ZDDs vs explicit method
  for a state enumeration problem