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Extended find of unique tuples in a relation represented by a BDD

由 断了今生、忘了曾经 提交于 2019-12-05 10:15:01

There exist simple and efficient solution if variables are ordered in such a way that x1 and x2 are at the top of BDD.

Consider BDD for second example.

You can traverse (in breadth-first order) first two layers of it to get four sub-BDDs. One for each possible combination of x1,x2. Three of those sub-BDDs a rooted at y1 and fourth is empty (constant False).

Now you can count number of elements in each sub-BDD (Algorithm C from Knuth's Volume 4 Fascicle 1, Bitwise Tricks & Techniques; Binary Decision Diagrams).

If number of elements in sub-BDD is greater than 1 then drop it (shortcut from parent node directly to False), otherwise leave it as it is.

It is possible to run this algorithm in single pass by memoizing partial results while counting elements.

Here is my implementation. I have studied author's proposed solution and it seems to me that it is the best if not the only simple BDD-based solution for arbitrary ordering. However, there may be some improvements if the algorithm is implemented in my way- PLEASE CHECK. I am using my own wrapper over BDD package but you should not have any troubles to understand it.

EDITED: I have simplified the solution, function Bdd_GetVariableChar() is not used anymore.

/* TESTING SOLUTION FOR QUESTION ON STACK OVERFLOW */
/* bdd_termFalse,bdd_termTrue: Boolean constants */
/* Bdd_isTerminal(f): check if f is Boolean constant */
/* Bdd_Low(f),Bdd_High(f): 'else' and 'then' subfunction */
/* Bdd_Top(f): literal function representing topvar of f */
/* Bdd_IsSmaller(f,g): check if topvar of f is above topvar of g */
/* existentialAbstraction(f,cube): \exist v.f for all v in cube */

Bdd_Edge specialAbstraction(Bdd_Edge f, Bdd_Edge cube) {
  if (Bdd_isTerminal(cube)) return f;
  if (Bdd_isTerminal(f)) return bdd_termFalse;
  if (Bdd_IsSmaller(f,cube)) {
    Bdd_Edge E,T;
    E = specialAbstraction(Bdd_Low(f),cube);
    T = specialAbstraction(Bdd_High(f),cube);
    return Bdd_ITE(Bdd_Top(f),T,E);
  } else if (Bdd_IsSmaller(cube,f)) {
    return bdd_termFalse;
  } else {
    Bdd_Edge E,T;
    cube = Bdd_High(cube);
    E = Bdd_Low(f);
    T = Bdd_High(f);
    if (Bdd_isEqv(E,bdd_termFalse)) {
      return specialAbstraction(T,cube);
    } else if (Bdd_isEqv(T,bdd_termFalse)) {
      return specialAbstraction(E,cube);
    } else {
      Bdd_Edge EX,TX,R;
      EX = existentialAbstraction(E,cube);
      TX = existentialAbstraction(T,cube);
      if (Bdd_isEqv(EX,TX)) return bdd_termFalse;
      R = Bdd_ITE(Bdd_ITE(EX,bdd_termFalse,T),
                  bdd_termTrue,
                  Bdd_ITE(TX,bdd_termFalse,E));
      return specialAbstraction(R,cube);
    }
  }
}

And, yes, if variable ordering is fixed with x above y, the algorithm can really be much more efficient - you can remove all the calculations from the most complex 'else' block and just return 0.

Here are some testing runs:

CUBE (JUST IN CASE YOU ARE NOT FAMILIAR WITH BDD ALGORITHMS)
  +  y1 y2 y3 y4 y5

ORIGINAL (ORDERED WITH X ABOVE Y)
  +  *x1 *x2 x3 *x4 x5 y1 *y2 y3 y4 y5
  +  *x1 x2 *x3 *x4 *x5 y1 y2 *y3 y4 y5
  +  *x1 x2 *x3 *x4 x5 *y1 y2 *y3 y4 y5
  +  *x1 x2 *x3 x4 *x5 y1 *y2 y3 *y4 *y5
  +  *x1 x2 x3 *x4 x5 *y1 *y2 *y3 *y4 y5
  +  *x1 x2 x3 *x4 x5 *y1 y2 y3 *y4 *y5
  +  x1 *x2 *x3 *x4 *x5 y1 y2 y3 y4 *y5
  +  x1 x2 *x3 x4 x5 *y1 *y2 *y4 *y5
  +  x1 x2 x3 *x4 *x5 *y1 *y2 *y3 y4 *y5

ABSTRACTION
  +  *x1 *x2 x3 *x4 x5
  +  *x1 x2 *x3 *x4
  +  *x1 x2 *x3 x4 *x5
  +  x1 *x2 *x3 *x4 *x5
  +  x1 x2 x3 *x4 *x5

ORIGINAL (ORDERED WITH Y ABOVE X)
  +  *y1 *y2 *y3 *y4 *y5 x1 x2 *x3 x4 x5
  +  *y1 *y2 *y3 *y4 y5 *x1 x2 x3 *x4 x5
  +  *y1 *y2 *y3 y4 *y5 x1 x2 x3 *x4 *x5
  +  *y1 *y2 y3 *y4 *y5 x1 x2 *x3 x4 x5
  +  *y1 y2 *y3 y4 y5 *x1 x2 *x3 *x4 x5
  +  *y1 y2 y3 *y4 *y5 *x1 x2 x3 *x4 x5
  +  y1 *y2 y3 *y4 *y5 *x1 x2 *x3 x4 *x5
  +  y1 *y2 y3 y4 y5 *x1 *x2 x3 *x4 x5
  +  y1 y2 *y3 y4 y5 *x1 x2 *x3 *x4 *x5
  +  y1 y2 y3 y4 *y5 x1 *x2 *x3 *x4 *x5

ABSTRACTION
  +  *x1 *x2 x3 *x4 x5
  +  *x1 x2 *x3 *x4
  +  *x1 x2 *x3 x4 *x5
  +  x1 *x2 *x3 *x4 *x5
  +  x1 x2 x3 *x4 *x5

ORIGINAL (MIXED ORDER)
  +  *x1 *x2 y1 *y2 y3 y4 y5 x3 *x4 x5
  +  *x1 x2 *y1 *y2 *y3 *y4 y5 x3 *x4 x5
  +  *x1 x2 *y1 y2 *y3 y4 y5 *x3 *x4 x5
  +  *x1 x2 *y1 y2 y3 *y4 *y5 x3 *x4 x5
  +  *x1 x2 y1 *y2 y3 *y4 *y5 *x3 x4 *x5
  +  *x1 x2 y1 y2 *y3 y4 y5 *x3 *x4 *x5
  +  x1 *x2 y1 y2 y3 y4 *y5 *x3 *x4 *x5
  +  x1 x2 *y1 *y2 *y3 *y4 *y5 *x3 x4 x5
  +  x1 x2 *y1 *y2 *y3 y4 *y5 x3 *x4 *x5
  +  x1 x2 *y1 *y2 y3 *y4 *y5 *x3 x4 x5

ABSTRACTION
  +  *x1 *x2 x3 *x4 x5
  +  *x1 x2 *x3 *x4
  +  *x1 x2 *x3 x4 *x5
  +  x1 *x2 *x3 *x4 *x5
  +  x1 x2 x3 *x4 *x5

One approach involves directly translating the definition of uniqueness:

R(x,y) and forall z . ~R(x,z) or y = z

An implementation may look like this:

def inspect(function, name, nvars):
    sys.stdout.write(name) # avoid print's trailing space or newline
    function.summary(nvars) # minterm count needs number of variables
    function.printCover()

import sys
from cudd import Cudd
m = Cudd()

nx = 2
ny = 3
x = [m.bddVar() for i in range(nx)]
y = [m.bddVar() for i in range(ny)]

R = (~x[0] & x[1] & (~y[0] & y[1] | y[1] & y[2]) |
     x[0] & ~x[1] & (y[0] ^ y[1]) & y[2] |
     ~x[0] & ~x[1] & y[0] & ~y[1] & ~y[2])

# This approach is independent of variable order.  We are free to enable
# reordering or call it explicitly.
m.reduceHeap()

inspect(R, 'R', nx+ny)

# Create auxiliary variables and selector function.
z = [m.bddVar() for i in range(ny)]
zcube = reduce(lambda a, b: a & b, z)
P = ~m.xeqy(y,z)

# A pair is in L iff:
#   - it is in R
#   - there is no other pair in R with the same x and different y
L = R & ~(R.swapVariables(y,z).andAbstract(P,zcube))

inspect(L, 'L', nx+ny)

Result of running that code:

R: 10 nodes 1 leaves 6 minterms
01-11 1
10101 1
10011 1
00100 1
0101- 1

L: 6 nodes 1 leaves 1 minterms
00100--- 1

The first two variables encode the first element of the pair; the next three variables encode the second element of the pair; the last three variables are the auxiliary variables.

The code applies DeMorgan's to the formula above to make use of andAbstract.

标签
algorithm
binary-decision-diagram
cudd
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