279 lines
7.7 KiB
Python
279 lines
7.7 KiB
Python
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"""Utilities for enumeration of finite and countably infinite sets.
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"""
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from __future__ import absolute_import, division, print_function
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###
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# Countable iteration
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# Simplifies some calculations
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class Aleph0(int):
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_singleton = None
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def __new__(type):
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if type._singleton is None:
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type._singleton = int.__new__(type)
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return type._singleton
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def __repr__(self): return '<aleph0>'
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def __str__(self): return 'inf'
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def __cmp__(self, b):
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return 1
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def __sub__(self, b):
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raise ValueError("Cannot subtract aleph0")
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__rsub__ = __sub__
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def __add__(self, b):
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return self
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__radd__ = __add__
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def __mul__(self, b):
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if b == 0: return b
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return self
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__rmul__ = __mul__
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def __floordiv__(self, b):
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if b == 0: raise ZeroDivisionError
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return self
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__rfloordiv__ = __floordiv__
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__truediv__ = __floordiv__
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__rtuediv__ = __floordiv__
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__div__ = __floordiv__
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__rdiv__ = __floordiv__
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def __pow__(self, b):
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if b == 0: return 1
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return self
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aleph0 = Aleph0()
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def base(line):
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return line*(line+1)//2
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def pairToN(pair):
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x,y = pair
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line,index = x+y,y
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return base(line)+index
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def getNthPairInfo(N):
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# Avoid various singularities
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if N==0:
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return (0,0)
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# Gallop to find bounds for line
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line = 1
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next = 2
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while base(next)<=N:
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line = next
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next = line << 1
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# Binary search for starting line
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lo = line
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hi = line<<1
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while lo + 1 != hi:
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#assert base(lo) <= N < base(hi)
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mid = (lo + hi)>>1
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if base(mid)<=N:
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lo = mid
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else:
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hi = mid
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line = lo
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return line, N - base(line)
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def getNthPair(N):
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line,index = getNthPairInfo(N)
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return (line - index, index)
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def getNthPairBounded(N,W=aleph0,H=aleph0,useDivmod=False):
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"""getNthPairBounded(N, W, H) -> (x, y)
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Return the N-th pair such that 0 <= x < W and 0 <= y < H."""
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if W <= 0 or H <= 0:
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raise ValueError("Invalid bounds")
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elif N >= W*H:
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raise ValueError("Invalid input (out of bounds)")
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# Simple case...
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if W is aleph0 and H is aleph0:
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return getNthPair(N)
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# Otherwise simplify by assuming W < H
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if H < W:
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x,y = getNthPairBounded(N,H,W,useDivmod=useDivmod)
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return y,x
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if useDivmod:
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return N%W,N//W
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else:
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# Conceptually we want to slide a diagonal line across a
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# rectangle. This gives more interesting results for large
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# bounds than using divmod.
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# If in lower left, just return as usual
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cornerSize = base(W)
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if N < cornerSize:
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return getNthPair(N)
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# Otherwise if in upper right, subtract from corner
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if H is not aleph0:
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M = W*H - N - 1
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if M < cornerSize:
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x,y = getNthPair(M)
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return (W-1-x,H-1-y)
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# Otherwise, compile line and index from number of times we
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# wrap.
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N = N - cornerSize
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index,offset = N%W,N//W
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# p = (W-1, 1+offset) + (-1,1)*index
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return (W-1-index, 1+offset+index)
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def getNthPairBoundedChecked(N,W=aleph0,H=aleph0,useDivmod=False,GNP=getNthPairBounded):
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x,y = GNP(N,W,H,useDivmod)
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assert 0 <= x < W and 0 <= y < H
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return x,y
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def getNthNTuple(N, W, H=aleph0, useLeftToRight=False):
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"""getNthNTuple(N, W, H) -> (x_0, x_1, ..., x_W)
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Return the N-th W-tuple, where for 0 <= x_i < H."""
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if useLeftToRight:
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elts = [None]*W
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for i in range(W):
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elts[i],N = getNthPairBounded(N, H)
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return tuple(elts)
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else:
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if W==0:
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return ()
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elif W==1:
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return (N,)
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elif W==2:
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return getNthPairBounded(N, H, H)
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else:
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LW,RW = W//2, W - (W//2)
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L,R = getNthPairBounded(N, H**LW, H**RW)
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return (getNthNTuple(L,LW,H=H,useLeftToRight=useLeftToRight) +
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getNthNTuple(R,RW,H=H,useLeftToRight=useLeftToRight))
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def getNthNTupleChecked(N, W, H=aleph0, useLeftToRight=False, GNT=getNthNTuple):
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t = GNT(N,W,H,useLeftToRight)
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assert len(t) == W
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for i in t:
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assert i < H
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return t
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def getNthTuple(N, maxSize=aleph0, maxElement=aleph0, useDivmod=False, useLeftToRight=False):
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"""getNthTuple(N, maxSize, maxElement) -> x
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Return the N-th tuple where len(x) < maxSize and for y in x, 0 <=
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y < maxElement."""
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# All zero sized tuples are isomorphic, don't ya know.
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if N == 0:
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return ()
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N -= 1
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if maxElement is not aleph0:
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if maxSize is aleph0:
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raise NotImplementedError('Max element size without max size unhandled')
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bounds = [maxElement**i for i in range(1, maxSize+1)]
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S,M = getNthPairVariableBounds(N, bounds)
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else:
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S,M = getNthPairBounded(N, maxSize, useDivmod=useDivmod)
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return getNthNTuple(M, S+1, maxElement, useLeftToRight=useLeftToRight)
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def getNthTupleChecked(N, maxSize=aleph0, maxElement=aleph0,
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useDivmod=False, useLeftToRight=False, GNT=getNthTuple):
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# FIXME: maxsize is inclusive
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t = GNT(N,maxSize,maxElement,useDivmod,useLeftToRight)
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assert len(t) <= maxSize
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for i in t:
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assert i < maxElement
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return t
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def getNthPairVariableBounds(N, bounds):
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"""getNthPairVariableBounds(N, bounds) -> (x, y)
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Given a finite list of bounds (which may be finite or aleph0),
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return the N-th pair such that 0 <= x < len(bounds) and 0 <= y <
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bounds[x]."""
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if not bounds:
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raise ValueError("Invalid bounds")
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if not (0 <= N < sum(bounds)):
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raise ValueError("Invalid input (out of bounds)")
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level = 0
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active = list(range(len(bounds)))
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active.sort(key=lambda i: bounds[i])
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prevLevel = 0
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for i,index in enumerate(active):
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level = bounds[index]
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W = len(active) - i
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if level is aleph0:
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H = aleph0
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else:
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H = level - prevLevel
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levelSize = W*H
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if N<levelSize: # Found the level
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idelta,delta = getNthPairBounded(N, W, H)
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return active[i+idelta],prevLevel+delta
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else:
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N -= levelSize
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prevLevel = level
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else:
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raise RuntimError("Unexpected loop completion")
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def getNthPairVariableBoundsChecked(N, bounds, GNVP=getNthPairVariableBounds):
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x,y = GNVP(N,bounds)
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assert 0 <= x < len(bounds) and 0 <= y < bounds[x]
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return (x,y)
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###
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def testPairs():
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W = 3
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H = 6
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a = [[' ' for x in range(10)] for y in range(10)]
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b = [[' ' for x in range(10)] for y in range(10)]
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for i in range(min(W*H,40)):
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x,y = getNthPairBounded(i,W,H)
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x2,y2 = getNthPairBounded(i,W,H,useDivmod=True)
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print(i,(x,y),(x2,y2))
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a[y][x] = '%2d'%i
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b[y2][x2] = '%2d'%i
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print('-- a --')
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for ln in a[::-1]:
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if ''.join(ln).strip():
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print(' '.join(ln))
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print('-- b --')
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for ln in b[::-1]:
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if ''.join(ln).strip():
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print(' '.join(ln))
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def testPairsVB():
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bounds = [2,2,4,aleph0,5,aleph0]
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a = [[' ' for x in range(15)] for y in range(15)]
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b = [[' ' for x in range(15)] for y in range(15)]
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for i in range(min(sum(bounds),40)):
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x,y = getNthPairVariableBounds(i, bounds)
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print(i,(x,y))
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a[y][x] = '%2d'%i
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print('-- a --')
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for ln in a[::-1]:
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if ''.join(ln).strip():
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print(' '.join(ln))
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###
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# Toggle to use checked versions of enumeration routines.
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if False:
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getNthPairVariableBounds = getNthPairVariableBoundsChecked
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getNthPairBounded = getNthPairBoundedChecked
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getNthNTuple = getNthNTupleChecked
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getNthTuple = getNthTupleChecked
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if __name__ == '__main__':
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testPairs()
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testPairsVB()
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