2013-03-13

Encoding permutations as integers via the Lehmer code (JavaScript)

This blog post explores the Lehmer code, a way of mapping integers to permutations. It can be used to compute a random permutation (by computing a random integer and mapping it to a permutation) and more.

Permutations

A permutation of an array is an array that contains the same elements, but possibly in a different order. For example, given the array
    [ 'a', 'b', 'c' ]
All of its permutations are:
    [ 'a', 'b', 'c' ]
    [ 'a', 'c', 'b' ]
    [ 'b', 'a', 'c' ]
    [ 'b', 'c', 'a' ]
    [ 'c', 'a', 'b' ]
    [ 'c', 'b', 'a' ]

Computing a permutation: a naive solution

In order to motivate the Lehmer code, let’s first implement a naive algorithm for computing a permutation of an array. Given the following array arr.
    var arr = [ 'a', 'b', 'c' ];
A simple way of computing a random permutation of arr is:
  • Compute a random number i, 0 ≤ i < 3. arr[i] is the first element of the permutation. Remove element i from arr.
  • Compute a random number i, 0 ≤ i < 2. arr[i] is the second element of the permutation. Remove element i from arr.
  • The remaining element of arr is the last element of the permutation.
In order to implement the above algorithm, we need a function to compute a random integer in a range starting at 0, up to and excluding an upper bound upper. The following function performs that duty.
    /**
     * @returns a number 0 <= n < upper
     */
    function getRandomInteger(upper) {
        return Math.floor(Math.random() * upper);
    }
Furthermore, we don’t want to change the input array arr, which means that we need a function that non-destructively removes an element:
    /**
     * @returns a fresh copy of arr, with the element at index removed
     */
    function removeElement(arr, index) {
        return arr.slice(0, index).concat(arr.slice(index+1));
    }
The algorithm itself looks as follows:
    function createPermutation(arr) {
        if (arr.length === 0) {
            return [];
        }
        var index = getRandomInteger(arr.length);
        return [arr[index]].concat(
            createPermutation(
                removeElement(arr, index)));
    }
Interaction:
    > createPermutation([ 'a', 'b', 'c' ])
    [ 'a', 'c', 'b' ]
Note: createPermutation() could be implemented more efficiently, but the current implementation expresses our intentions very clearly.

Encoding permutations as integers

An alternative to the above algorithm is to find a way to map single integers to permutations. We can then simply compute a random integer and map it to a permutation.

The naive algorithm in two steps

As a first step towards this new approach, lets first split up the previous algorithm into two steps:
  1. Compute the indices for the (continually shrinking) array arr.
  2. Turn the indices into a permutation.
The following function performs step 1. We don’t need to know anything about arr, except for its length len. The first index of the returned array is in the range [0,len), the second in the range [0,len−1), etc.
    function createLehmerCode(len) {
        var result = [];
        for(var i=len; i>0; i--) {
            result.push(getRandomInteger(i));
        }
        return result;
    }
Interaction:
    > createLehmerCode(3)
    [ 0, 1, 0 ]
The above way of encoding a permutation as a sequence of numbers is called a Lehmer code. Such a code can easily be converted into a permutation, via the following function (step 2):
    function codeToPermutation(elements, code) {
        return code.map(function (index) {
            var elem = elements[index];
            elements = removeElement(elements, index);
            return elem;
        });
    }
Interaction:
    > codeToPermutation(['a','b','c'], [0,0,0])
    [ 'a', 'b', 'c' ]
    > codeToPermutation(['a','b','c'], [2,1,0])
    [ 'c', 'b', 'a' ]

Mapping integers to Lehmer codes

The next step is to replace createLehmerCode() by a function that maps an integer to a Lehmer code. Afterwards, we compute that integer randomly and not the code itself, any more. To that end, lets look at ways of encoding sequences of digits (e.g. indices) as single integers. If each of the digits has the same range, we can use a positional system whose radix is the (excluded) upper bound of the range.

The decimal system. For example, if each digit is in the range 0–9 then we can use the fixed radix 10 and the decimal system. That is, each digit has the same radix. “Radix” is just another way of saying “upper bound of a digit”. The following table reminds us of the decimal system.

Digit position 2 1 0
Digit range 0–9 0–9 0–9
Multiplier 100 = 102 10 = 101 1 = 100
Radix 10 10 10

The value of a position is the value of the digit multiplied by the multiplier. The value of a complete decimal number is the sum of the values of the positions.

Note that each multiplier is one plus the sum of all previous highest digits multiplied by their multipliers. For example:

100 = 1 + (9x10 + 9x1)

The factoradic system. Encoding the digits of a Lehmer code into an integer is more complex, because each digit has a different radix. We want the mapping to be bijective (a one-to-one mapping without “holes”). The factoradic system is what we need, as explained via the following table. The digit ranges reflect the rules of the Lehmer code.

Digit position 3 2 1 0
Digit range 0–3 0–2 0–1 0
Multiplier 6 = 3! 2 = 2! 1 = 1! 1 = 0!
Radix 4 3 2 1

The last digit is always zero and often omitted. Again, a multiplier is one plus the highest value that you can achive via previous positions. For example:

6 = 1 + (2x2 + 1x1 + 0x1)
The following function performs the mapping from integers to Lehmer codes.
    function integerToCode(int, permSize) {
        if (permSize <= 1) {
            return [0];
        }
        var multiplier = factorial(permSize-1);
        var digit = Math.floor(int / multiplier);
        return [digit].concat(
            integerToCode(
                int % multiplier,
                permSize-1));
    }
We start with the highest position. Its digit can be determined by dividing int by the position’s multiplier. Afterwards the remainder of that division becomes the new int and we continue with the next position.

integerToCode() uses the following function to compute the factorial of a number:

    function factorial(n) {
        if (n <= 0) {
            return 1;
        } else {
            return n * factorial(n-1);
        }
    }

Putting everything together

We now have all the pieces to compute a permutation as originally planned:
    function createPermutation(arr) {
        var int = getRandomInteger(factorial(arr.length));
        var code = integerToCode(int);
        return codeToPermutation(arr, code);
    }
The range of the random integer representing a permutation is [0,arr.length). That is an indication that the mapping between integers and permutations is really bijective, because arr has arr.length! permutations,

Practically useful?

Is the Lehmer code practically useful? It is if you need to encode permutations as integers. There are two additional use cases for it: Computing a random permutation and enumerating all permutations. For both use cases, Lehmer codes give you convenient solutions, but not efficient ones. If you want efficiency, consider the following two algorithms:

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