<< 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?
My 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.
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 toecho "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)
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.
Benchmark
The correct solution to the original Project Euler problem was found in less than 0.01 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++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 solves 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 rarely 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:
C# www.mathblog.dk/project-euler-24-millionth-lexicographic-permutation/ (written by Kristian Edlund)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p024.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/24 Lexicographic permutations.js (written by David Ernst)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p024.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p024.hs (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler024.scala (written by Michael Bayne)
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own. Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
Heatmap
Please click on a problem's number to open my solution to that problem:
green | solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too | |
yellow | solutions score less than 100% at Hackerrank (but still solve the original problem easily) | |
gray | problems are already solved but I haven't published my solution yet | |
blue | solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much | |
orange | problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte | |
red | problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too | |
black | problems are solved but access to the solution is blocked for a few days until the next problem is published | |
[new] | the flashing problem is the one I solved most recently |
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I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
Look at my progress and performance pages to get more details.
Copyright
I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.
All of my solutions can be used for any purpose and I am in no way liable for any damages caused.
You can even remove my name and claim it's yours. But then you shall burn in hell.
The problems and most of the problems' images were created by Project Euler.
Thanks for all their endless effort !!!
<< problem 23 - Non-abundant sums | 1000-digit Fibonacci number - problem 25 >> |