Developing Developing

a Simple ZDD Packagea Simple ZDD Package

Alan MishchenkoAlan Mishchenko

University of California, BerkeleyUniversity of California, Berkeley

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OutlineOutline Decision diagramsDecision diagrams ZDD packageZDD package ExperimentExperiment SummarySummary

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Decision DiagramsDecision Diagrams

DDs are a useful data-structureDDs are a useful data-structure It is a good exercise to build a new DD packageIt is a good exercise to build a new DD package This presentation is based on real code without This presentation is based on real code without

any simplificationany simplification The code can be found atThe code can be found at

https://bitbucket.org/alanmi/abc/src/4dc7e5aa7805931d35ba1c310fhttps://bitbucket.org/alanmi/abc/src/4dc7e5aa7805931d35ba1c310f90f5baed74d859/src/misc/extra/extraUtilPerm.c90f5baed74d859/src/misc/extra/extraUtilPerm.c

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Building a DD PackageBuilding a DD Package

Define data structuresDefine data structuresConsider memory managementConsider memory management

Write a procedure for creating a new nodeWrite a procedure for creating a new nodeDevelop traversal proceduresDevelop traversal proceduresDebug and clean up the codeDebug and clean up the codeDevelop and test application code based Develop and test application code based

on the new packageon the new package

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Basic Data StructuresBasic Data Structurestypedef struct Abc_ZddObj_ Abc_ZddObj;typedef struct Abc_ZddObj_ Abc_ZddObj;struct Abc_ZddObj_ struct Abc_ZddObj_ {{ unsigned Var : 31;unsigned Var : 31; unsigned Mark : 1;unsigned Mark : 1; unsigned True;unsigned True; unsigned False;unsigned False;}; };

typedef struct Abc_ZddEnt_ Abc_ZddEnt;typedef struct Abc_ZddEnt_ Abc_ZddEnt;struct Abc_ZddEnt_ struct Abc_ZddEnt_ {{ int Arg0;int Arg0; int Arg1;int Arg1; int Arg2;int Arg2; int Res;int Res;};};

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ZDD ManagerZDD Managertypedef struct Abc_ZddMan_ Abc_ZddMan;typedef struct Abc_ZddMan_ Abc_ZddMan;struct Abc_ZddMan_ struct Abc_ZddMan_ {{ int nVars;int nVars; int nObjs;int nObjs; int nObjsAlloc;int nObjsAlloc; int nPermSize;int nPermSize; unsigned nUniqueMask;unsigned nUniqueMask; unsigned nCacheMask;unsigned nCacheMask; int * pUnique;int * pUnique; int * pNexts;int * pNexts; Abc_ZddEnt * pCache;Abc_ZddEnt * pCache; Abc_ZddObj * pObjs;Abc_ZddObj * pObjs; int nCacheLookups;int nCacheLookups; int nCacheMisses;int nCacheMisses; word nMemory;word nMemory; int * pV2TI;int * pV2TI; int * pV2TJ;int * pV2TJ; int * pT2V;int * pT2V;};};

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Creating New NodeCreating New Nodestatic inline unsigned Abc_ZddHash( int static inline unsigned Abc_ZddHash( int Arg0, int , int Arg1, int , int Arg2 ) ) { return 12582917 { return 12582917 * Arg0 + 4256249 4256249 * Arg1 + 741457 741457 * Arg2; } ; }

static inline int Abc_ZddUniqueCreate( static inline int Abc_ZddUniqueCreate( Abc_ZddMan * p, int Var, int True, int False )Abc_ZddMan * p, int Var, int True, int False ){{ if ( True == 0 ) if ( True == 0 ) return False;return False; elseelse {{ int *q = p->pUnique + (Abc_ZddHash(Var, True, False) & p->nUniqueMask);int *q = p->pUnique + (Abc_ZddHash(Var, True, False) & p->nUniqueMask); for ( ; *q; q = p->pNexts + *q )for ( ; *q; q = p->pNexts + *q ) if ( p->pObjs[*q].Var == Var && if ( p->pObjs[*q].Var == Var && p->pObjs[*q].True == True && p->pObjs[*q].True == True && p->pObjs[*q].False == False )p->pObjs[*q].False == False ) return *q;return *q; assert( p->nObjs < p->nObjsAlloc ); assert( p->nObjs < p->nObjsAlloc ); *q = p->nObjs++;*q = p->nObjs++; p->pObjs[*q].Var = Var;p->pObjs[*q].Var = Var; p->pObjs[*q].True = True;p->pObjs[*q].True = True; p->pObjs[*q].False = False;p->pObjs[*q].False = False; return *q;return *q; }}}}

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Traversal ProceduresTraversal Procedures

Set operationsSet operationsUnion, difference, intersection Union, difference, intersection

Product operationsProduct operationsDot-product, cross-productDot-product, cross-product

Counting operationsCounting operationsCount the number of nodes and pathsCount the number of nodes and paths

Permutation operationsPermutation operationsTransposition, permutation productTransposition, permutation product

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ZDD UnionZDD Unionint Abc_ZddUnion( Abc_ZddMan * p, int a, int b )int Abc_ZddUnion( Abc_ZddMan * p, int a, int b ){{ Abc_ZddObj * A, * B; Abc_ZddObj * A, * B; int r0, r1, r;int r0, r1, r; if ( a == 0 ) return b;if ( a == 0 ) return b; if ( b == 0 ) return a;if ( b == 0 ) return a; if ( a == b ) return a;if ( a == b ) return a; if ( a > b ) return Abc_ZddUnion( p, b, a );if ( a > b ) return Abc_ZddUnion( p, b, a ); if ( (r = Abc_ZddCacheLookup(p, a, b, ABC_ZDD_OPER_UNION)) >= 0 )if ( (r = Abc_ZddCacheLookup(p, a, b, ABC_ZDD_OPER_UNION)) >= 0 ) return r;return r; A = Abc_ZddNode( p, a );A = Abc_ZddNode( p, a ); B = Abc_ZddNode( p, b );B = Abc_ZddNode( p, b ); if ( A->Var < B->Var )if ( A->Var < B->Var ) r0 = Abc_ZddUnion( p, A->False, b ), r0 = Abc_ZddUnion( p, A->False, b ), r1 = A->True;r1 = A->True; else if ( A->Var > B->Var )else if ( A->Var > B->Var ) r0 = Abc_ZddUnion( p, a, B->False ), r0 = Abc_ZddUnion( p, a, B->False ), r1 = B->True;r1 = B->True; elseelse r0 = Abc_ZddUnion( p, A->False, B->False ), r0 = Abc_ZddUnion( p, A->False, B->False ), r1 = Abc_ZddUnion( p, A->True, B->True );r1 = Abc_ZddUnion( p, A->True, B->True ); r = Abc_ZddUniqueCreate( p, Abc_MinInt(A->Var, B->Var), r1, r0 );r = Abc_ZddUniqueCreate( p, Abc_MinInt(A->Var, B->Var), r1, r0 ); return Abc_ZddCacheInsert( p, a, b, ABC_ZDD_OPER_UNION, r );return Abc_ZddCacheInsert( p, a, b, ABC_ZDD_OPER_UNION, r );}}

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ExperimentExperiment

Shin-ichi Minato proposed PiDDs, a ZDD-based data-Shin-ichi Minato proposed PiDDs, a ZDD-based data-structure to represent and manipulate permutationsstructure to represent and manipulate permutations Shin-ichi Minato, "PiDD: A new decision diagram for efficient Shin-ichi Minato, "PiDD: A new decision diagram for efficient

problem solving in permutation space,“ Proc. SAT’11, pp. 90-104problem solving in permutation space,“ Proc. SAT’11, pp. 90-104

One of the applications cited in the paper, is enumeration One of the applications cited in the paper, is enumeration of reachable states of a simplified Rubik’s cubeof reachable states of a simplified Rubik’s cube Traditional cube is 3x3x3 and has 4.3*10^19 statesTraditional cube is 3x3x3 and has 4.3*10^19 states

http://en.wikipedia.org/wiki/Rubiks_Cubehttp://en.wikipedia.org/wiki/Rubiks_Cube Simplified cube is 2x2x2 and has only 3,674,160 statesSimplified cube is 2x2x2 and has only 3,674,160 states

http://en.wikipedia.org/wiki/Pocket_cubehttp://en.wikipedia.org/wiki/Pocket_cube Minato’s ZDD-based implementation takes 207 sec to enumerate Minato’s ZDD-based implementation takes 207 sec to enumerate

states of the simplified cube on a 2.4 GHz Core2Duo PCstates of the simplified cube on a 2.4 GHz Core2Duo PC

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Cube’s State EncondingCube’s State Enconding

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Cube’s Transition RelationCube’s Transition Relation

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Experiment from Minato’s PaperExperiment from Minato’s Paper

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Experiment: ZDD-Based EnumerationExperiment: ZDD-Based Enumeration

UC Berkeley, ABC 1.01 (compiled May 14 2014 04:30:19)UC Berkeley, ABC 1.01 (compiled May 14 2014 04:30:19)abc 01> cubeenum -zabc 01> cubeenum -zEnumerating states of 2x2x2 cube.Enumerating states of 2x2x2 cube.Iter 0 -> 1 Nodes = 0 Used = 2 Time = 0.00 secIter 0 -> 1 Nodes = 0 Used = 2 Time = 0.00 secIter 1 -> 10 Nodes = 63 Used = 577 Time = 0.00 secIter 1 -> 10 Nodes = 63 Used = 577 Time = 0.00 secIter 2 -> 64 Nodes = 443 Used = 4349 Time = 0.03 secIter 2 -> 64 Nodes = 443 Used = 4349 Time = 0.03 secIter 3 -> 385 Nodes = 2018 Used = 26654 Time = 0.12 secIter 3 -> 385 Nodes = 2018 Used = 26654 Time = 0.12 secIter 4 -> 2232 Nodes = 7451 Used = 119442 Time = 0.43 secIter 4 -> 2232 Nodes = 7451 Used = 119442 Time = 0.43 secIter 5 -> 12224 Nodes = 25178 Used = 490038 Time = 1.07 secIter 5 -> 12224 Nodes = 25178 Used = 490038 Time = 1.07 secIter 6 -> 62360 Nodes = 83955 Used = 1919750 Time = 1.77 secIter 6 -> 62360 Nodes = 83955 Used = 1919750 Time = 1.77 secIter 7 -> 289896 Nodes = 290863 Used = 7182932 Time = 3.16 secIter 7 -> 289896 Nodes = 290863 Used = 7182932 Time = 3.16 secIter 8 -> 1159968 Nodes = 614845 Used = 25301123 Time = 8.18 secIter 8 -> 1159968 Nodes = 614845 Used = 25301123 Time = 8.18 secIter 9 -> 3047716 Nodes = 585664 Used = 66228369 Time = 20.72 secIter 9 -> 3047716 Nodes = 585664 Used = 66228369 Time = 20.72 secIter 10 -> 3671516 Nodes = 19430 Used = 102292452 Time = 34.28 secIter 10 -> 3671516 Nodes = 19430 Used = 102292452 Time = 34.28 secIter 11 -> 3674160 Nodes = 511 Used = 103545878 Time = 34.80 secIter 11 -> 3674160 Nodes = 511 Used = 103545878 Time = 34.80 secIter 12 -> 3674160 Nodes = 511 Used = 103566266 Time = 34.81 secIter 12 -> 3674160 Nodes = 511 Used = 103566266 Time = 34.81 secZDD stats: Var = 276 Obj = 103566266 Alloc = 134217728 ZDD stats: Var = 276 Obj = 103566266 Alloc = 134217728 Hit = 63996630 Miss = 141768893 Mem = 4608.00 MBHit = 63996630 Miss = 141768893 Mem = 4608.00 MB

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Experiment: Explicit EnumerationExperiment: Explicit Enumeration

UC Berkeley, ABC 1.01 (compiled May 14 2014 04:30:19)UC Berkeley, ABC 1.01 (compiled May 14 2014 04:30:19)abc 01> cubeenumabc 01> cubeenumEnumerating states of 2x2x2 cube.Enumerating states of 2x2x2 cube.Iter 0 -> 1 Time = 0.00 secIter 0 -> 1 Time = 0.00 secIter 1 -> 10 Time = 0.00 secIter 1 -> 10 Time = 0.00 secIter 2 -> 64 Time = 0.00 secIter 2 -> 64 Time = 0.00 secIter 3 -> 385 Time = 0.00 secIter 3 -> 385 Time = 0.00 secIter 4 -> 2232 Time = 0.01 secIter 4 -> 2232 Time = 0.01 secIter 5 -> 12224 Time = 0.03 secIter 5 -> 12224 Time = 0.03 secIter 6 -> 62360 Time = 0.09 secIter 6 -> 62360 Time = 0.09 secIter 7 -> 289896 Time = 0.18 secIter 7 -> 289896 Time = 0.18 secIter 8 -> 1159968 Time = 0.36 secIter 8 -> 1159968 Time = 0.36 secIter 9 -> 3047716 Time = 1.02 secIter 9 -> 3047716 Time = 1.02 secIter 10 -> 3671516 Time = 2.44 secIter 10 -> 3671516 Time = 2.44 secIter 11 -> 3674160 Time = 2.93 secIter 11 -> 3674160 Time = 2.93 secIter 12 -> 3674160 Time = 2.93 secIter 12 -> 3674160 Time = 2.93 sec

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ConclusionConclusion Discussed basics of decision diagramsDiscussed basics of decision diagrams Proposed a simple ZDD packageProposed a simple ZDD package Analyzed performance of ZDDs vs explicit Analyzed performance of ZDDs vs explicit

method for a state enumeration problemmethod for a state enumeration problem