By Peter Prevos
Euler Problem 24 asks to develop lexicographic permutations which are ordered arrangements of objects in lexicographic order. Tushar Roy of Coding Made Simple has shared a great introduction on how to generate lexicographic permutations.
Euler Problem 24 Definition
A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are:
012 021 102 120 201 210
What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?
Brute Force Solution
The digits 0 to 9 have permutations (including combinations that start with 0). Most of these permutations are, however, not in lexicographic order. A brute-force way to solve the problem is to determine the next lexicographic permutation of a number string and repeat this one million times.
nextPerm 1 && a[i - 1] >= a[i]) i
This code takes the following steps:
- Find largest index such that .
- If no such index exists, then this is already the last permutation.
- Find largest index such that and .
- Swap and .
- Reverse the suffix starting at .
A more efficient solution is to use combinatorics, thanks to MathBlog. The last nine digits can be ordered in ways. So the first permutations start with a 0. By extending this thought, it follows that the millionth permutation must start with a 2.
From this rule, it follows that the 725761st permutation is 2013456789. We now need 274239 more lexicographic permutations:
We can repeat this logic to find the next digit. The last 8 digits can be ordered in 40320 ways. The second digit is the 6th digit in the remaining numbers, which is 7 (2013456789).
This process is repeated until all digits have been used.
R blogger Tony's Bubble Universe created a generalised function to solve this problem a few years ago.
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