<< problem 9 - Special Pythagorean triplet | Largest product in a grid - problem 11 >> |
Problem 10: Summation of primes
(see projecteuler.net/problem=10)
The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
Find the sum of all the primes below two million.
My Algorithm
The prime sieve is more or less unchanged from problem 7.
Then I create a lookup table sums
which contains for each prime number p
the sum of all prime numbers <=p.
The test cases may contain numbers which are not prime, too.
Therefore I use upper_bound
to find the smallest entry which >=p.
And since we that entry "is one step too far", I go back to the previous entry and print it.
Modifications by HackerRank
My 2-step design was heavily influenced by Hackerrank's large number of test cases:
the "expensive" precomputation is done once and the test cases are computationally very "cheap".
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "1 10" | ./10
Output:
Note: the original problem's input 2000000
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.
#include <iostream>
#include <vector>
#include <map>
int main()
{
// prime numbers beyond this are not relevant for the problem
const unsigned int MaxPrime = 2000000;
// precompute all relevant prime numbers
std::vector<unsigned int> primes;
// the only even prime
primes.push_back(2);
// now check all odd numbers for primality
for (unsigned int i = 3; i <= MaxPrime; i += 2)
{
bool isPrime = true;
for (auto p : primes)
{
// no larger prime factor possible ?
if (p*p > i)
break;
// no prime number ?
if (i % p == 0)
{
isPrime = false;
break;
}
}
// yes, we have a new prime
if (isPrime)
primes.push_back(i);
}
// prime numbers were found in ascending order,
// let's add their value and store in a map such that
// [prime number] => [sum of all prime numbers up to the current]
// note: long long is required to avoid overflows
std::map<unsigned int, unsigned long long> sums;
unsigned long long sum = 0;
for (auto p : primes)
{
sum += p;
sums[p] = sum;
}
// the test cases are more or less "smart" lookups
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int x;
std::cin >> x;
// find the closest prime number which is bigger than the input
auto i = sums.upper_bound(x);
// go back to the closest prime number which is smaller than the input
i--;
// show the sum associated to that prime number
std::cout << i->second << std::endl;
}
return 0;
}
This solution contains 8 empty lines, 15 comments and 3 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.22 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 12 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
February 23, 2017 submitted solution
March 29, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler010
My code solves 8 out of 8 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 5% (out of 100%).
Hackerrank describes this problem as medium.
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.
Similar problems at Project Euler
Problem 7: 10001st prime
Note: I'm not even close to solving all problems at Project Euler. Chances are that similar problems do exist and I just haven't looked at them.
Links
projecteuler.net/thread=10 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
C# www.mathblog.dk/sum-of-all-primes-below-2000000-problem-10/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/10-19/problem10.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p010.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/10 Summation of primes.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem010.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p010.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p010.hs (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler010.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 | |
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 !!!
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