<< problem 248 - Numbers for which Euler's totient function ... | 250250 - problem 250 >> |
Problem 249: Prime Subset Sums
(see projecteuler.net/problem=249)
Let S = {2, 3, 5, ..., 4999} be the set of prime numbers less than 5000.
Find the number of subsets of S, the sum of whose elements is a prime number.
Enter the rightmost 16 digits as your answer.
My Algorithm
The efficient prime sieve was copied from my toolbox.
Initially, there is one empty set with sum zero.
Then I iterate over all prime numbers up to 5000 and add each prime to each set.
For example, 2 is added to the empty set so that its sum is 2.
There was no set with sum=2 but the number of sets with sum=0 is added to number of sets with sum=2.
Then 3 is added to each set: to the empty set with sum=0 and the set with sum=2.
Hence there is now one more set with sum=0+3=3 and one with sum=2+3=5.
When the next prime number is processed, it's 5, it's added to the one empty sets.
There was already one set with sum=5, now there is a second one !
(and at least one new set with sum=2+5=7, sum=3+5=8 and sum=5+5=10).
To avoid overflows, the number of sets is repeatedly truncated to the last 16 digits (mod 10^16).
I achieved a significant performance improvement by processing the sums in descending order because then
I only need one array instead of two (otherwise collisions produce a wrong result).
The last step is to count only those sums that are prime numbers.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 100 | ./249
Output:
Note: the original problem's input 5000
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. Or just jump to my GitHub repository.
#include <iostream>
#include <vector>
const auto Modulo = 10000000000000000ULL;
// odd prime numbers are marked as "true" in a bitvector
std::vector<bool> sieve;
// return true, if x is a prime number
bool isPrime(unsigned int x)
{
// handle even numbers
if ((x & 1) == 0)
return x == 2;
// lookup for odd numbers
return sieve[x >> 1];
}
// find all prime numbers from 2 to size
void fillSieve(unsigned int size)
{
// store only odd numbers
const unsigned int half = size >> 1;
// allocate memory
sieve.resize(half, true);
// 1 is not a prime number
sieve[0] = false;
// process all relevant prime factors
for (unsigned int i = 1; 2*i*i < half; i++)
// do we have a prime factor ?
if (sieve[i])
{
// mark all its multiples as false
unsigned int current = 3*i+1;
while (current < half)
{
sieve[current] = false;
current += 2*i+1;
}
}
}
int main()
{
unsigned int limit = 5000;
std::cin >> limit;
// sum of all prime numbers below 5000
fillSieve(limit);
unsigned int maxSum = 2;
for (unsigned int i = 3; i <= limit; i += 2)
if (isPrime(i))
maxSum += i;
// one empty set, its sum is zero
std::vector<unsigned long long> count(maxSum + 1, 0);
unsigned int largest = 0;
count[largest] = 1;
// add each prime to each set
for (unsigned int i = 2; i < limit; i++)
if (isPrime(i))
{
largest += i;
// look at all potential sums of sets
for (auto j = largest; j >= i; j--)
{
count[j] += count[j - i];
count[j] %= Modulo;
}
}
// enlarge prime sieve
fillSieve(maxSum); // maxSum is about 1600000
// sum of all count where index is a prime number
unsigned long long result = 0;
for (unsigned int i = 0; i < count.size(); i++)
if (isPrime(i))
{
result += count[i];
result %= Modulo;
}
// show result
std::cout << result << std::endl;
return 0;
}
This solution contains 13 empty lines, 18 comments and 2 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.7 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 14 MByte.
(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
July 14, 2017 submitted solution
July 14, 2017 added comments
Difficulty
Project Euler ranks this problem at 60% (out of 100%).
Links
projecteuler.net/thread=249 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/Meng-Gen/ProjectEuler/blob/master/249.py (written by Meng-Gen Tsai)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p249.py (written by Nayuki)
Python github.com/steve98654/ProjectEuler/blob/master/249.py
C++ github.com/roosephu/project-euler/blob/master/249.cpp (written by Yuping Luo)
C++ github.com/smacke/project-euler/blob/master/cpp/249.cpp (written by Stephen Macke)
C github.com/LaurentMazare/ProjectEuler/blob/master/e249.c (written by Laurent Mazare)
Java github.com/dcrousso/ProjectEuler/blob/master/PE249.java (written by Devin Rousso)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p249.java (written by Nayuki)
Perl github.com/shlomif/project-euler/blob/master/project-euler/249/euler-249-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:
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 |
I stopped working on Project Euler problems around the time they released 617.
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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 248 - Numbers for which Euler's totient function ... | 250250 - problem 250 >> |