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.

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

My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

The code contains #ifdefs 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.

Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Number of test cases (1-5):

Input data (separated by spaces or newlines):
Note: Enter the minimum length of the red blocks and which number of combinations has to be exceeded

This is equivalent to
echo "1 3 1000000" | ./115

Output:

(please click 'Go !')

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)

Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 seconds on a 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=c++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 never 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:

Python: 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)

Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.
yellow problems score less than 100% at Hackerrank (but still solve the original problem).
gray problems are already solved but I haven't published my solution yet.
blue problems are solved and there wasn't a Hackerrank version of it at the time I solved it or I didn't care about it because it differed too much.

Please click on a problem's number to open my solution to that problem:

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The 163 solved problems had an average difficulty of 22.2% at Project Euler and I scored 11,907 points (out of 13200) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
more about me can be found on my homepage, especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !