<< problem 114 - Counting block combinations I | Red, green or blue tiles - problem 116 >> |
Problem 115: Counting block combinations II
(see projecteuler.net/problem=115)
NOTE: This is a more difficult version of Problem 114.
A row measuring n units in length has red blocks with a minimum length of m units placed on it,
such that any two red blocks (which are allowed to be different lengths) are separated by at least one black square.
Let the fill-count function, F(m, n), represent the number of ways that a row can be filled.
For example, F(3, 29) = 673135 and F(3, 30) = 1089155.
That is, for m = 3, it can be seen that n = 30 is the smallest value for which the fill-count function first exceeds one million.
In the same way, for m = 10, it can be verified that F(10, 56) = 880711 and F(10, 57) = 1148904, so n = 57 is the least value for which the fill-count function first exceeds one million.
For m = 50, find the least value of n for which the fill-count function first exceeds one million.
My Algorithm
... almost the same as problem 114 !
Only the main
function had to adjusted to find in a simple loop the smallest value to exceed one million.
Modifications by HackerRank
My approach needs a bit of memory. Hackerrank has inputs up to 10^18 which clearly exceed the RAM size of a desktop PC.
[TODO] find closed formula
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 3 1000000" | ./115
Output:
Note: the original problem's input 50 1000000
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>
#define ORIGINAL
// memoized solutions
const long long Unknown = -1;
std::vector<long long> solutions;
// find result for row with a certain length
unsigned long long count(unsigned long long space, unsigned int minBlockLength)
{
// finished ?
if (space == 0)
return 1;
// already know the answer ?
if (solutions[space] != Unknown)
return solutions[space];
// one option is to leave the next cell black
auto result = count(space - 1, minBlockLength);
// insert red blocks at the current position with all possible spaces
for (auto block = minBlockLength; block <= space; block++)
{
// how much is left after inserting ?
auto next = space - block;
// must be followed by a black cell
if (next > 0)
next--;
// count all combinations
result += count(next, minBlockLength);
}
// memoize result
solutions[space] = result;
return result;
}
int main()
{
unsigned int tests;
std::cin >> tests;
while (tests--)
{
// minimum length of each red block
unsigned int minBlockLength = 3;
// what number should be exceeded ?
unsigned long long limit = 1000000;
std::cin >> minBlockLength >> limit;
// cached results
solutions.clear();
solutions.resize(limit + 1, Unknown); // not enough for Hackerrank
for (unsigned int totalLength = 1; ; totalLength++)
{
auto combinations = count(totalLength, minBlockLength);
if (combinations > limit)
{
std::cout << totalLength << std::endl;
break;
}
}
}
return 0;
}
This solution contains 12 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.
Peak memory usage was about 10 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 13, 2017 submitted solution
May 13, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler115
My code solves 1 out of 15 test cases (score: 0%)
I failed 10 test cases due to wrong answers and 4 because of timeouts
Difficulty
Project Euler ranks this problem at 35% (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=115 - 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-115-separated-blocks/ (written by Kristian Edlund)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p115.java (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler115.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 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 114 - Counting block combinations I | Red, green or blue tiles - problem 116 >> |