LeetCode Distilled Series: 46. Finding all Permutations

Given a list of unique letters, find all of its distinct permutations.

Input:

['a', 'b', 'c']

Output:

[['a', 'b', 'c'], ['a', 'c', 'b'], ['b', 'a', 'c'], ['b', 'c', 'a'], ['c', 'a', 'b'], ['c', 'b', 'a']]

Explanation

Intuition

This is a classic combinatorial search problem, let’s apply the 3-step system from backtracking template.

Algo Monster

1. Identify states

What state do we need to know whether we have reached a solution (and using it to construct a solution if the problem asks for it).

1. We need a state to keep track of the list of letters we have chosen for the current permutation

What state do we need to decide which child nodes should be visited next and which ones should be pruned.

1. We have to know what are the letters left that we can still use (since each letter can only be used once).

2. Draw the State-space Tree

3. DFS on the State-space tree

Using the backtracking template as basis, we add the two states we identified in step 1:

1. A list to represent permutation constructed so far, path
2. A list to record which letters are already used, used, used[i] == true means ith letter in the origin list has been used.

def permutations(l):           
    def dfs(path, used, res):
        if len(path) == len(l):
            res.append(path[:]) # note [:] make a deep copy since otherwise we'd be append the same list over and over
            return
        
        for i, letter in enumerate(l):
            # skip used letters
            if used[i]:
                continue
            # add letter to permutation, mark letter as used
            path.append(letter)
            used[i] = True
            dfs(path, used, res)
            # remove letter from permutation, mark letter as unused
            path.pop()
            used[i] = False
            
    res = []
    dfs([], [False] * len(l), res)
    return res

Note on deep copy

In the exit logic of the above solution, we append a deep copy of path res.append(path[:]). This creates a new list with elements being the same as current elements of path. Consider what happens if we don't make the deep copy:

>>> res = []
>>> path = []
>>> for i in range(3):
...     path.append(i)
...     res.append(path)
...
>>> res
[[0, 1, 2], [0, 1, 2], [0, 1, 2]]

We get the same copy three times! Because append(path) actually appends a reference (memory address), we actually appended the same list three times. path.append(i) mutates the list and affects all references to that list. To fix this, we create a new list before we append.