<< problem 186 - Connectedness of a network | The hyperexponentiation of a number - problem 188 >> |
Problem 187: Semiprimes
(see projecteuler.net/problem=187)
A composite is a number containing at least two prime factors. For example, 15 = 3 * 5; 9 = 3 * 3; 12 = 2 * 2 * 3.
There are ten composites below thirty containing precisely two, not necessarily distinct, prime factors: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26.
How many composite integers, n < 10^8, have precisely two, not necessarily distinct, prime factors?
My Algorithm
My fast prime sieve generates all prime numbers below 5 * 10^7 in about 120ms.
Then all primes numbers are pairwise multiplied. I count the number of all products below 10^8.
Alternative Approaches
A prime counting function can solve this almost instantly (see my solution for problem 501).
Or even simpler: prime[j]
is the smallest prime to be multiplied with prime[i]
.
Then limit / prime[i]
will be the maximum number for prime[j]
. A simple binary search in primes
can count how many primes are relevant.
(that's prime counting, too, and equivalent to the speed-up trick in problem 501 for small n
).
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 30 | ./187
Output:
Note: the original problem's input 100000000
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 <vector>
#include <algorithm>
// ---------- standard prime sieve from my toolbox ----------
// 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;
}
}
}
// ---------- and now problem-specific code ----------
int main()
{
// generate all primes
unsigned int limit = 100000000;
unsigned int largestPrime = limit / 2 + 100; // at least one more prime than strictly needed
fillSieve(largestPrime);
// extract all prime numbers from sieve
std::vector<unsigned int> primes = { 2 };
primes.reserve(3002000); // avoid frequent re-allocations
for (unsigned int i = 3; i < largestPrime; i += 2)
if (isPrime(i))
primes.push_back(i);
#define ORIGINAL
#ifndef ORIGINAL
// Hackerrank has several test cases
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
#endif
{
std::cin >> limit;
// compute all products of primes i and j where i <= j
unsigned int count = 0;
for (unsigned int i = 0; primes[i] * primes[i] < limit; i++)
for (unsigned int j = i; primes[i] * primes[j] < limit; j++)
// found one more solutions ...
count++;
std::cout << count << std::endl;
}
return 0;
}
This solution contains 14 empty lines, 18 comments and 6 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.18 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 17 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
May 23, 2017 submitted solution
May 23, 2017 added comments
August 18, 2017 modified to solve Hackerrank, too
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler187
My code solves 26 out of 26 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 20% (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.
Links
projecteuler.net/thread=187 - the best forum on the subject (note: you have to submit the correct solution first)
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 a new problem is published | |
the flashing problem is the one I solved most recently |
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I scored 13,486 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 186 - Connectedness of a network | The hyperexponentiation of a number - problem 188 >> |