<< problem 190 - Maximising a weighted product | Squarefree Numbers - problem 193 >> |
Problem 191: Prize Strings
(see projecteuler.net/problem=191)
A particular school offers cash rewards to children with good attendance and punctuality.
If they are absent for three consecutive days or late on more than one occasion then they forfeit their prize.
During an n-day period a trinary string is formed for each child consisting of L's (late), O's (on time), and A's (absent).
Although there are eighty-one trinary strings for a 4-day period that can be formed, exactly forty-three strings would lead to a prize:
OOOO OOOA OOOL OOAO OOAA OOAL OOLO OOLA OAOO OAOA
OAOL OAAO OAAL OALO OALA OLOO OLOA OLAO OLAA AOOO
AOOA AOOL AOAO AOAA AOAL AOLO AOLA AAOO AAOA AAOL
AALO AALA ALOO ALOA ALAO ALAA LOOO LOOA LOAO LOAA
LAOO LAOA LAAO
How many "prize" strings exist over a 30-day period?
My Algorithm
Another dynamic programming problem ... my function count
has 3 parameters:
day
stands for the number of days to be evaluated, initially 30absent
counts the consecutive absent dayslate
represents the total number of days where the pupil was late
1.
absent
has to be reset to zero everytime the pupil shows up (it doesn't matter whether on time or late)2. A massive amount of situations produce the same parameter set. That means, some parameter sets are evaluated several thousand times.
A simple memoization scheme brings down the computation time from several seconds to less than 10 milliseconds:
There are at most 30*2*3 = 1800 different parameter sets:
- 30 days
- child was 0 or 1 days late so far
- he/she was 0, 1 or 2 days absent (only counting the most recent days)
days * 2 * 3 + absent * 2 + late
is unique for each parameter set and used as a index for a small cache
.
Note
My cache can hold the results for up to 80 days.
The result fits in unsigned int
for 30 days, too.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 4 | ./191
Output:
Note: the original problem's input 30
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++ and can be compiled with G++, Clang++, Visual C++. You can download it, too.
#include <iostream>
// memoize intermediate results
unsigned long long cache[80*2*3] = { 0 };
// recursively evaluate a day:
// days - number of days left
// absent - number of consecutive absent day (only consider up to three past days)
// late - total number of late days
unsigned long long count(unsigned int days, unsigned int absent = 0, unsigned int late = 0)
{
// too many consecutive absent days ?
if (absent == 3)
return 0;
// too late to often ?
if (late > 1)
return 0;
// all days passed ? => collect prize
if (days == 0)
return 1;
// unique ID of current parameter set
unsigned int hash = days * 2 * 3 + absent * 2 + late;
// already computed ?
if (cache[hash] != 0)
return cache[hash];
unsigned long long result;
// assume pupil is today neither late nor absent
result = count(days - 1, 0, late);
// assume pupil is absent today
result += count(days - 1, absent + 1, late);
// assume pupil is late today
result += count(days - 1, 0, late + 1);
// store result
cache[hash] = result;
return result;
}
int main()
{
unsigned int days;
std::cin >> days;
std::cout << count(days) << std::endl;
return 0;
}
This solution contains 7 empty lines, 14 comments and 1 preprocessor command.
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 25, 2017 submitted solution
May 25, 2017 added comments
Difficulty
Project Euler ranks this problem at 45% (out of 100%).
Links
projecteuler.net/thread=191 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/LaurentMazare/ProjectEuler/blob/master/e191.py (written by Laurent Mazare)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p191.py (written by Nayuki)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/191.cc (written by Meng-Gen Tsai)
C github.com/zmwangx/Project-Euler/blob/master/191/191.c (written by Zhiming Wang)
Java github.com/HaochenLiu/My-Project-Euler/blob/master/191.java (written by Haochen Liu)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p134.java (written by Nayuki)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p191.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem191.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem191.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p191.mathematica (written by Nayuki)
Mathematica github.com/steve98654/ProjectEuler/blob/master/191.nb
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p191.hs (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 | |
[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 190 - Maximising a weighted product | Squarefree Numbers - problem 193 >> |