Permutations by swapping: Difference between revisions
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Here, bfsjt0 N gives the initial permutation of order N, and bfsjtn^:M bfsjt0M gives the Mth Steinhaus–Johnson–Trotter permutation of order N. (bf stands for "brute force".) |
Here, bfsjt0 N gives the initial permutation of order N, and bfsjtn^:M bfsjt0M gives the Mth Steinhaus–Johnson–Trotter permutation of order N. (bf stands for "brute force".) |
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To convert from the Steinhaus–Johnson–Trotter representation of a permutation to J's representation, use <:@|, or to find J's |
To convert from the Steinhaus–Johnson–Trotter representation of a permutation to J's representation, use <:@|, or to find J's anagram index of a Steinhaus–Johnson–Trotter representation of a permutation, use A.<:@| |
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Example use: |
Example use: |
Revision as of 20:01, 27 July 2012
Generate permutations of n items in which successive permutations differ from each other by the swapping of any two items. Also generate the sign of the permutation which is +1 when the permutation is generated from an even number of swaps from the initial state, and -1 for odd. Show the permutations and signs of three items, in order of generation here.
Such data are of use in generating the determinant of a square matrix and any functions created should bear this in mind.
Note: The Steinhaus–Johnson–Trotter algorithm generates successive permutations where adjacent items are swapped, but from this discussion adjacency is not a requirement.
- References
- Cf.
J
J's built in mechanism for generating permutations (which is designed around the idea of selecting a permutation uniquely by an integer) does not seem seem to have an obvious mapping to Steinhaus–Johnson–Trotter. Perhaps someone with a sufficiently deep view of the subject of permutations can find a direct mapping?
Meanwhile, here's an inductive approach, using negative integers to look left and positive integers to look right:
<lang J>bfsjt0=: _1 - i. lookingat=: 0 >. <:@# <. i.@# + * next=: | >./@:* | > | {~ lookingat bfsjtn=: (((] <@, ] + *@{~) | i. next) C. ] * _1 ^ next < |)^:(*@next)</lang>
Here, bfsjt0 N gives the initial permutation of order N, and bfsjtn^:M bfsjt0M gives the Mth Steinhaus–Johnson–Trotter permutation of order N. (bf stands for "brute force".)
To convert from the Steinhaus–Johnson–Trotter representation of a permutation to J's representation, use <:@|, or to find J's anagram index of a Steinhaus–Johnson–Trotter representation of a permutation, use A.<:@|
Example use:
<lang J> bfsjtn^:(i.!3) bfjt0 3 _1 _2 _3 _1 _3 _2 _3 _1 _2
3 _2 _1
_2 3 _1 _2 _1 3
<:@| bfsjtn^:(i.!3) bfjt0 3
0 1 2 0 2 1 2 0 1 2 1 0 1 2 0 1 0 2
A. <:@| bfsjtn^:(i.!3) bfjt0 3
0 1 4 5 3 2</lang>
Python
Python: iterative
When saved in a file called spermutations.py it is used in the Python example to the Matrix arithmetic task.
<lang python>debug = False
def spermutations(n):
'permutations by swapping. Yields: perm, sign' sign = 1 p = [[i, 0 if i==0 else -1] # [num, direction] for i in range(n)]
if debug: print ' #', p yield tuple(pp[0] for pp in p), sign while any(pp[1] for pp in p): # moving i1, largest = max(((i, pp) for i, pp in enumerate(p) if pp[1]), key=lambda x: x[1]) n1, d1 = largest sign *= -1 if d1 == -1: # Swap down i2 = i1-1 p[i1], p[i2] = p[i2], p[i1] # If this causes the chosen element to reach the First or last # position within the permutation, or if the next element in the # same direction is larger than the chosen element: if i2 == 0 or p[i2-1][0] > n1: # The direction of the chosen element is set to zero p[i2][1] = 0 elif d1 == 1: # Swap up i2 = i1+1 p[i1], p[i2] = p[i2], p[i1] # If this causes the chosen element to reach the first or Last # position within the permutation, or if the next element in the # same direction is larger than the chosen element: if i2 == n-1 or p[i2+1][0] > n1: # The direction of the chosen element is set to zero p[i2][1] = 0 if debug: print ' #', p yield tuple(pp[0] for pp in p), sign
for i3, pp in enumerate(p): n3, d3 = pp if n3 > n1: pp[1] = 1 if i3 < i2 else -1 if debug: print ' # Set Moving'
if __name__ == '__main__':
from itertools import permutations
for n in (3, 4): print '\nPermutations and sign of %i items' % n sp = set() for i in spermutations(n): sp.add(i[0]) print('Perm: %r Sign: %2i' % i) if debug: raw_input('?') # Test p = set(permutations(range(n))) assert sp == p, 'Two methods of generating permutations do not agree'</lang>
- Sample output
Permutations and sign of 3 items Perm: (0, 1, 2) Sign: 1 Perm: (0, 2, 1) Sign: -1 Perm: (2, 0, 1) Sign: 1 Perm: (2, 1, 0) Sign: -1 Perm: (1, 2, 0) Sign: 1 Perm: (1, 0, 2) Sign: -1 Permutations and sign of 4 items Perm: (0, 1, 2, 3) Sign: 1 Perm: (0, 1, 3, 2) Sign: -1 Perm: (0, 3, 1, 2) Sign: 1 Perm: (3, 0, 1, 2) Sign: -1 Perm: (3, 0, 2, 1) Sign: 1 Perm: (0, 3, 2, 1) Sign: -1 Perm: (0, 2, 3, 1) Sign: 1 Perm: (0, 2, 1, 3) Sign: -1 Perm: (2, 0, 1, 3) Sign: 1 Perm: (2, 0, 3, 1) Sign: -1 Perm: (2, 3, 0, 1) Sign: 1 Perm: (3, 2, 0, 1) Sign: -1 Perm: (3, 2, 1, 0) Sign: 1 Perm: (2, 3, 1, 0) Sign: -1 Perm: (2, 1, 3, 0) Sign: 1 Perm: (2, 1, 0, 3) Sign: -1 Perm: (1, 2, 0, 3) Sign: 1 Perm: (1, 2, 3, 0) Sign: -1 Perm: (1, 3, 2, 0) Sign: 1 Perm: (3, 1, 2, 0) Sign: -1 Perm: (3, 1, 0, 2) Sign: 1 Perm: (1, 3, 0, 2) Sign: -1 Perm: (1, 0, 3, 2) Sign: 1 Perm: (1, 0, 2, 3) Sign: -1
Python: recursive
After spotting the pattern of highest number being inserted into each perm of lower numbers from right to left, then left to right, I developed this recursive function: <lang python>def spermutations(n):
return [(tuple(item), -1 if i %2 else 1) for i, item in enumerate(sperm(n))]
def sperm(items):
if items <= 0: return [[]] else: dir = 1 new_items = [] for item in sperm(items - 1): if dir == 1: # step down new_items += [item[:i] + [items-1] + item[i:] for i in range(len(item), -1, -1)] else: # step up new_items += [item[:i] + [items-1] + item[i:] for i in range(len(item) + 1)] dir *= -1 return new_items</lang>
- Sample output
The output is the same as before except it is a list of all results rather than yielding each result from a generator function.