How to generate a permutation?

问题

My question is: given a list L of length n, and an integer i such that 0 <= i < n!, how can you write a function perm(L, n) to produce the ith permutation of L in O(n) time? What I mean by ith permutation is just the ith permutation in some implementation defined ordering that must have the properties:

For any i and any 2 lists A and B, perm(A, i) and perm(B, i) must both map the jth element of A and B to an element in the same position for both A and B.
For any inputs (A, i), (A, j) perm(A, i)==perm(A, j) if and only if i==j.

NOTE: this is not homework. In fact, I solved this 2 years ago, but I've completely forgotten how, and it's killing me. Also, here is a broken attempt I made at a solution:

def perm(s, i):
  n = len(s)
  perm = [0]*n
  itCount = 0
  for elem in s:
    perm[i%n + itCount] = elem
    i = i / n
    n -= 1
    itCount+=1
  return perm

ALSO NOTE: the O(n) requirement is very important. Otherwise you could just generate the n! sized list of all permutations and just return its ith element.

回答1:

def perm(sequence, index):
    sequence = list(sequence)
    result = []
    for x in xrange(len(sequence)):
        idx = index % len(sequence)
        index /= len(sequence)
        result.append( sequence[idx] )
        # constant time non-order preserving removal
        sequence[idx] = sequence[-1]
        del sequence[-1]
    return result

Based on the algorithm for shuffling, but we take the least significant part of the number each time to decide which element to take instead of a random number. Alternatively consider it like the problem of converting to some arbitrary base except that the base name shrinks for each additional digit.

回答2:

Could you use factoradics? You can find an illustration via this MSDN article.

Update: I wrote an extension of the MSDN algorithm that finds i'th permutation of n things taken r at a time, even if n != r.

回答3:

A computational minimalistic approach (written in C-style pseudocode):

function perm(list,i){
    for(a=list.length;a;a--){
        list.switch(a-1,i mod a);
        i=i/a;
    }
    return list;
}

Note that implementations relying on removing elements from the original list tend to run in O(n^2) time, at best O(n*log(n)) given a special tree style list implementation designed for quickly inserting and removing list elements.

The above code rather than shrinking the original list and keeping it in order just moves an element from the end to the vacant location, still makes a perfect 1:1 mapping between index and permutation, just a slightly more scrambled one, but in pure O(n) time.

回答4:

So, I think I finally solved it. Before I read any answers, I'll post my own here.

def perm(L, i):
  n = len(L)
  if (n == 1):
    return L
  else:
    split = i%n
    return [L[split]] + perm(L[:split] + L[split+1:], i/n)

回答5:

There are n! permutations. The first character can be chosen from L in n ways. Each of those choices leave (n-1)! permutations among them. So this idea is enough for establishing an order. In general, you will figure out what part you are in, pick the appropriate element and then recurse / loop on the smaller L.

The argument that this works correctly is by induction on the length of the sequence. (sketch) For a length of 1, it is trivial. For a length of n, you use the above observation to split the problem into n parts, each with a question on an L' with length (n-1). By induction, all the L's are constructed correctly (and in linear time). Then it is clear we can use the IH to construct a solution for length n.

来源：https://stackoverflow.com/questions/4120743/how-to-generate-a-permutation

标签

algorithm

permutation