<< problem 179 - Consecutive positive divisors | RSA encryption - problem 182 >> |
Problem 181: Investigating in how many ways objects of two different colours can be grouped
(see projecteuler.net/problem=181)
Having three black objects B and one white object W they can be grouped in 7 ways like this:
(BBBW), (B,BBW), (B,B,BW), (B,B,B,W), (B,BB,W), (BBB,W), (BB,BW)
In how many ways can sixty black objects B and forty white objects W be thus grouped?
My Algorithm
My solution is somehow similar to the coin-change algorithm:
- each group can be treated as a sequence of black and white objects, BWBW is the same as BBWW
- all groups can be sorted by their size and, if multiple groups have the same size, by their lexicographical order
maxBlack = 3
and maxWhite = 1
these groups exist:(B,B,B,W), (B,B,BW), (B,W,BB), (B, BBW), (W, BBB), (BB, BW), (BBBW)
These are exactly the same as in the problem statement but in a different order.
Two outer loops iterate over all possible sequences of black and white objects.
The inner loops place them at every possible positions (until the number of available objects is exhausted).
The result will be found in
current[60][40]
.
Alternative Approaches
You can solve this problem with Dynamic Programming, too.
I wrote a simple prototype but it turned out to be much slower (8 seconds vs. 0.04 seconds).
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "3 1" | ./181
Output:
Note: the original problem's input 60 40
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.
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>
// limits
const unsigned int MaxBlack = 160;
const unsigned int MaxWhite = 160;
const unsigned int Modulo = 1000000007; // Hackerrank only
int main()
{
#define ORIGINAL
#ifndef ORIGINAL
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
#endif
{
unsigned int maxBlack = MaxBlack;
unsigned int maxWhite = MaxWhite;
std::cin >> maxBlack >> maxWhite;
// static array size: actually it would be sufficient to use maxBlack instead of MaxBlack
// (and maxWhite instead of MaxWhite)
unsigned long long previous[MaxBlack + 1][MaxWhite + 1];
unsigned long long current [MaxBlack + 1][MaxWhite + 1];
// initialize
for (unsigned int i = 0; i <= maxBlack; i++)
for (unsigned int j = 0; j <= maxWhite; j++)
previous[i][j] = 0;
previous[0][0] = 1;
// all possible subsets
for (unsigned int useBlack = 0; useBlack <= maxBlack; useBlack++)
for (unsigned int useWhite = 0; useWhite <= maxWhite; useWhite++)
{
// skip empty subset
if (useBlack == 0 && useWhite == 0)
continue;
// put subset at every possible position
for (unsigned int i = 0; i <= maxBlack; i++)
for (unsigned int j = 0; j <= maxWhite; j++)
{
current[i][j] = 0;
// place it repeatedly
unsigned int k = 0;
while (i >= k * useBlack && j >= k * useWhite)
{
current[i][j] += previous[i - k * useBlack][j - k * useWhite];
k++;
}
#ifndef ORIGINAL
current[i][j] %= Modulo;
#endif
}
// copy for next iteration
for (unsigned int i = 0; i <= maxBlack; i++)
for (unsigned int j = 0; j <= maxWhite; j++)
previous[i][j] = current[i][j];
}
// print result
std::cout << current[maxBlack][maxWhite] << std::endl;
}
return 0;
}
This solution contains 10 empty lines, 10 comments and 6 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 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
June 28, 2017 submitted solution
June 28, 2017 added comments
July 28, 2017 modified to solve Hackerrank, too
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler181
My code solves 2 out of 3 test cases (score: 9.09%)
I failed 0 test cases due to wrong answers and 1 because of timeouts
Difficulty
Project Euler ranks this problem at 70% (out of 100%).
Hackerrank describes this problem as easy.
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=181 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/HaochenLiu/My-Project-Euler/blob/master/181.py (written by Haochen Liu)
Python github.com/Meng-Gen/ProjectEuler/blob/master/181.py (written by Meng-Gen Tsai)
Python github.com/smacke/project-euler/blob/master/python/181.py (written by Stephen Macke)
C++ github.com/roosephu/project-euler/blob/master/181.cpp (written by Yuping Luo)
Go github.com/frrad/project-euler/blob/master/golang/Problem181.go (written by Frederick Robinson)
Mathematica github.com/steve98654/ProjectEuler/blob/master/181.nb
GP-Pari github.com/frrad/project-euler/blob/master/gp-pari/Problem181.gp (written by Frederick Robinson)
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 179 - Consecutive positive divisors | RSA encryption - problem 182 >> |