<< problem 287 - Quadtree encoding (a simple compression algorithm) | Digital Signature - problem 290 >> |
Problem 288: An enormous factorial
(see projecteuler.net/problem=288)
For any prime p the number N(p,q) is defined by N(p,q) = sum_{n=0 ... q}{T_n * p^n}
with T_n generated by the following random number generator:
S_0 = 290797
S_{n+1} = S_n^2 mod 50515093
T_n = S_n mod p
Let Nfac(p,q) be the factorial of N(p,q).
Let NF(p,q) be the number of factors p in Nfac(p,q).
You are given that NF(3,10000) mod 3^20 = 624955285.
Find NF(61, 10^7) mod 61^10
My Algorithm
I started with the easy part:
The function countFactors()
returns the exponent of a given prime in a factorial's prime factorization.
It relies on Legendre's formula which I already used for other problems, too ( see en.wikipedia.org/wiki/Legendre's_formula).
The pseudo-random number generator wasn't hard as well. I wrote that stuff when I saw the problem for the first time but then I was stuck and continued with other problems.
Upen revisiting this problem I made the crucial observation that the result has to be printed mod 61^10 → and 61 is the same prime used for NF(61, 10^7).
(the same hold true for the example NF(3, 10000) mod 3^20).
Therefore the result depends on three parameters:
- a prime p = 61
- a positive number q = 10^7
- an exponent of the modulus exponent = 10
It can't exceed 61^10 because of the mod 61^10 part in the final formula. The variable
maxPower
will be 1, 61, 3721, ... until it reaches 61^10 and remains at the level.There are only 61 possible values for T_n = S_n mod 61. That means that the number of different T_n * p^n is very small (in fact it's 71).
Caching those results of
countFactors(t * p^n)
let's the runtime of the program drop from 2.6 to 0.35 seconds.
Note
T_n is at most 60, so max(T_n * 61^10) = 60 * 61^10 approx 4.28 * 10^19 is the largest value I will encounter in my computations.
Unfortunately this value is beyond the range of 64 bit integers (it requires 66 bits) and I needed to switch to GCC's 128 bit extension.
Therefore this code doesn't compile with Visual C++ etc.
I didn't like this problem because I spent too much time "discovering" that I need 128 bit arithmetic.
Interactive test
This feature is not available for the current problem.
My code
… was written in C++11 and can be compiled with G++. You can download it, too.
#include <iostream>
#include <vector>
#include <unordered_map>
// I had overflows with 64 bit arithmetic, so I switched to GCC's 128 bit extension
typedef __int128 Number;
// compute the exponent of prime in the factorization of f!
unsigned long long countFactors(Number f, unsigned int prime)
{
// Legendre's formula, https://en.wikipedia.org/wiki/Legendre%27s_formula
unsigned long long result = 0;
Number power = prime;
while (power <= f)
{
result += f / power;
power *= prime;
}
return result;
}
// variables of the problem statement:
// p = prime = 61
// q = iterations = 10^7
// moduloExponent = 10 (because of 61^10)
unsigned long long solve(unsigned int prime, unsigned int iterations, unsigned int exponent)
{
// compute prime^moduloExponent => 61^10
unsigned long long modulo = 1;
for (unsigned int i = 1; i <= exponent; i++)
modulo *= prime;
// will be 61^i => 61^0, 61^1, 61^2, ..., 61^10 and keep it at 61^10
Number maxPower = 1;
// seed of pseudo-random number generator
unsigned long long s = 290797;
unsigned long long result = 0;
for (unsigned int i = 0; i <= iterations; i++)
{
// T(i, 61) * 61^i
auto t = s % prime;
// => but 61^i limited to 61^10
auto product = t * maxPower;
// I observed only 71 different products (in 1000000 iterations !), let's cache them
static std::unordered_map<Number, unsigned long long> cache;
auto lookup = cache.find(product);
if (lookup == cache.end())
{
// new, unknown parameters, must call countFactors()
auto current = countFactors(product, prime);
// cache the value
cache[product] = current;
// add it to the result, too
result += current;
}
else
{
// add looked up value
result += lookup->second;
}
// keep under 61^10
result %= modulo;
// 61^i => 61^(i+1), but limit to 61^10
if (maxPower < modulo)
maxPower *= prime;
// next random number
s *= s;
s %= 50515093;
}
return result;
}
int main()
{
unsigned int prime = 61;
unsigned int iterations = 10000000;
unsigned int exponent = 10;
std::cin >> prime >> iterations >> exponent;
std::cout << solve(prime, iterations, exponent) << std::endl;
return 0;
}
This solution contains 13 empty lines, 20 comments and 3 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.3 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
October 19, 2017 submitted solution
October 19, 2017 added comments
Difficulty
Project Euler ranks this problem at 35% (out of 100%).
Links
projecteuler.net/thread=288 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python blog.dreamshire.com/project-euler-288/ (written by Mike Molony)
Python github.com/evilmucedin/project-euler/blob/master/euler288/288.py (written by Den Raskovalov)
Python github.com/LaurentMazare/ProjectEuler/blob/master/e288.py (written by Laurent Mazare)
Python github.com/Meng-Gen/ProjectEuler/blob/master/288.py (written by Meng-Gen Tsai)
Python github.com/smacke/project-euler/blob/master/python/288.py (written by Stephen Macke)
Python github.com/steve98654/ProjectEuler/blob/master/288.py
Java github.com/thrap/project-euler/blob/master/src/Java/Problem288.java (written by Magnus Solheim Thrap)
Perl github.com/shlomif/project-euler/blob/master/project-euler/288/euler-288-v1.pl (written by Shlomi Fish)
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:
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gray | problems are already solved but I haven't published my solution yet | |
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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 | |
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[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 287 - Quadtree encoding (a simple compression algorithm) | Digital Signature - problem 290 >> |