<< problem 23 - Non-abundant sums | 1000-digit Fibonacci number - problem 25 >> |

# Problem 24: Lexicographic permutations

(see projecteuler.net/problem=24)

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?

# Algorithm

The original problem can be solved with STL's handy `std::next_permutation`

.

Even though we call it 999999 times it is still very fast (< 10 ms).

*Note:* the first permutation has index 0, hence 999999 instead of 1000000 iterations.

## Alternative Approaches

Hackerrank's modified problem has a much larger "search space" and causes timeout.

Therefore I implemented an alternative algorithm based on the "Factorial number system" (see en.wikipedia.org/wiki/Factorial_number_system):

499999 in our decimal system is:

4 * 10^4 + 9 * 10^4 + 9 * 10^3 + 9 * 10^2 + 9 * 10^1 + 9 * 10^0

The factorial number system replaces 10^x by x!:

1 * 9! + 3 * 8! + 3 * 7! + 1 * 6! + 2 * 5! + 3 * 4! + 1 * 3! + 0 * 2! + 1 * 1! + 0 * 0!

= 1 * 362880 + 3 * 40320 + 3 * 5040 + 1 * 720 + 2 * 120 + 3 * 24 + 1 * 6 + 0 * 2 + 1 * 1 + 0 * 1

= 499999

The coefficients 1, 3, 3, 1, 2, 3, 1, 0, 1, 0 define which indices of our original, unpermutated string we have to choose.

But there is a twist: whenever we select an element, we have to remove it from the original. And everything's 0-based.

That means:

`0123456789`

→ choose element 1: `1`

→ `1`

`023456789 `

→ choose element 3: `4`

→ `14`

`02356789 `

→ choose element 3: `5`

→ `145`

`0236789 `

→ choose element 1: `2`

→ `1452`

`036789 `

→ choose element 2: `6`

→ `14526`

`03789 `

→ choose element 3: `8`

→ `145268`

`0379 `

→ choose element 1: `3`

→ `1452683`

`079 `

→ choose element 0: `0`

→ `14526830`

`79 `

→ choose element 1: `9`

→ `145268309`

`7 `

→ choose element 0: `7`

→ `1452683097`

## Modifications by HackerRank

The string `abcdefghijklm`

is used instead of `0123456789`

.

In the end, we have 13! instead of 10! potential permutations (1715x more permutations).

## Note

The "live test" is based on the Hackerrank problem.

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

The code contains `#ifdef`

s to switch between the original problem and the Hackerrank version.

Enable `#ifdef ORIGINAL`

to produce the result for the original problem (default setting for most problems).

#include <iostream>
#include <string>
#include <algorithm>
int main()
{
//#define ORIGINAL

#ifdef ORIGINAL
unsigned int numPermutation = 1000000;
std::string current = "0123456789";
while (--numPermutation)
std::next_permutation(current.begin(), current.end());
std::cout << current << std::endl;
#else
const std::string abc = "abcdefghijklm";
unsigned int tests;
std::cin >> tests;
while (tests--)
{
// to find the permutation we treat the input number as something written
// in a "factorial" system:
// x = pos0 * 12! + pos1 * 11! + pos2 * 10! + ... + pos12 * 1!
// (we have 13 letters, therefore the position of the first letter is in 12! radix)
// precomputed 0! .. 12!
const unsigned long long factorials[13+1] =
{ 1,1,2,6,24,120,720,5040,40320,362880,3628800,39916800,479001600,6227020800 };
// 13! which exceed 32 bits
unsigned long long x;
std::cin >> x;
// reduce to a single cycle (repeats after 13! iterations)
x %= factorials[abc.size()];
// that factorial system is zero-based ...
x--;
// strip off single letters (until empty)
auto remain = abc;
// our wanted permutation
std::string result;
while (!remain.empty())
{
// get next digit in that strange number system :-)
auto currentFactorial = factorials[remain.size() - 1];
auto pos = x / currentFactorial;
// store the associated letter
result += remain[pos];
// and remove it from the still unprocessed data
remain.erase(pos, 1);
// eliminate the processed digit
x %= currentFactorial;
}
std::cout << result << std::endl;
}
#endif
return 0;
}

This solution contains 11 empty lines, 15 comments and 6 preprocessor commands.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This live test is based on the Hackerrank problem.

This is equivalent to`echo "1 500000" | ./24`

Output:

*Note:* the original problem's input `1000000`

__cannot__ be entered

because just copying results is a soft skill reserved for idiots.

*(this interactive test is still under development, computations will be aborted after one second)*

# Benchmark

The correct solution to the original Project Euler problem was found in **less than 0.01** seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

(compiled for x86_64 / Linux, GCC flags: `-O3 -march=native -fno-exceptions -fno-rtti -std=c++11 -DORIGINAL`

)

See here for a comparison of all solutions.

**Note:** interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without `-DORIGINAL`

.

# Changelog

February 23, 2017 submitted solution

April 4, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler024

My code solved **10** out of **10** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **5%** (out of 100%).

Hackerrank describes this problem as **easy**.

*Note:*

Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.

In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is never an option.

# Links

projecteuler.net/thread=24 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-24-millionth-lexicographic-permutation/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p024.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p024.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p024.mathematica (written by Nayuki)

Javascript: github.com/dsernst/ProjectEuler/blob/master/24 Lexicographic permutations.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler024.scala (written by Michael Bayne)

# Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.

yellow problems score less than 100% at Hackerrank (but still solve the original problem).

gray problems are already solved but I haven't published my solution yet.

blue problems are already solved and there wasn't a Hackerrank version of it (at the time I solved it) or I didn't care about it because it differed too much.

*Please click on a problem's number to open my solution to that problem:*

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