<< problem 123 - Prime square remainders | Palindromic sums - problem 125 >> |
Problem 124: Ordered radicals
(see projecteuler.net/problem=124)
The radical of n, rad(n), is the product of the distinct prime factors of n.
For example, 504 = 23 * 32 * 7, so rad(504) = 2 * 3 * 7 = 42.
If we calculate rad(n) for 1 <= n <= 10, then sort them on rad(n), and sorting on n if the radical values are equal, we get:
Unsorted Sorted
nrad(n) nrad(n)k
11 111
22 222
33 423
42 824
55 335
66 936
77 557
82 668
93 779
1010 101010
Let E(k) be the kth element in the sorted n column; for example, E(4) = 8 and E(6) = 9.
If rad(n) is sorted for 1 <= n <= 100000, find E(10000).
My Algorithm
Similar to a prime sieve I find all multiples of each prime.
There is one struct Radical
for every number n
, its product
starts with 1
and is multiplied by a prime factors of n
.
A simple comparison operator<()
will be used to sort all radicals.
A full sort isn't needed - there is the nice nth_element
algorithm in C++'s STL which ensures that a single element is at the correct position -
without a strict guarantee that other elements are perfectly sorted, too. This partial sort is often much faster than a full sort.
Modifications by HackerRank
I don't properly re-use radicals for consecutive test cases and can't handle huge search spaces. I couldn't care less ...
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "2 10 4 10 6" | ./124
Output:
Note: the original problem's input 100000 10000
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 <algorithm>
// store a single radical
struct Radical
{
// current number
unsigned int n;
// product of all prime factors
unsigned int product;
// compare two object, prefer lower product (and lower n if products are equal)
bool operator<(const Radical& other) const
{
if (product != other.product)
return product < other.product;
return n < other.n;
}
};
// all relevant redicals
std::vector<Radical> rads;
// return a certain radical (1-based index)
unsigned int getNth(unsigned int index)
{
index--; // 1-based instead of 0-based
// partial sort
std::nth_element(rads.begin(), rads.begin() + index, rads.end());
return rads[index].n;
}
int main()
{
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int limit = 100000;
std::cin >> limit;
// initialize all radicals
rads.clear();
rads.reserve(limit + 1);
for (unsigned int i = 0; i <= limit; i++)
{
Radical current;
current.n = i;
current.product = 1;
rads.push_back(current);
}
// some big value to push zero to the back
rads[0].product = 999999999;
// compute radicals using a "sieve"
for (auto current : rads)
{
// process only primes
if (current.product != 1)
continue;
// adjust all of their multiples
for (unsigned int j = current.n; j <= limit; j += current.n)
rads[j].product *= current.n;
}
// get n-th element
unsigned int pos = 10000;
std::cin >> pos;
std::cout << getNth(pos) << std::endl;
}
return 0;
}
This solution contains 10 empty lines, 13 comments and 3 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
May 19, 2017 submitted solution
May 22, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler124
My code solves 4 out of 10 test cases (score: 30%)
I failed 4 test cases due to wrong answers and 2 because of timeouts
Difficulty
Project Euler ranks this problem at 25% (out of 100%).
Hackerrank describes this problem as advanced.
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=124 - 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-124-sorted-radical-function/ (written by Kristian Edlund)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p124.java (written by Nayuki)
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 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 123 - Prime square remainders | Palindromic sums - problem 125 >> |