<< problem 321 - Swapping Counters | Building a tower - problem 324 >> |
Problem 323: Bitwise-OR operations on random integers
(see projecteuler.net/problem=323)
Let y_0, y_1, y_2,... be a sequence of random unsigned 32 bit integers (i.e. 0 <= y_i < 2^32, every value equally likely).
For the sequence x_i the following recursion is given:
x_0 = 0 and
x_i = x_{i-1} | y_{i-1}, for i > 0. ( | is the bitwise-OR operator)
It can be seen that eventually there will be an index N such that x_i = 2^{32}-1 (a bit-pattern of all ones) for all i >= N.
Find the expected value of N.
Give your answer rounded to 10 digits after the decimal point.
My Algorithm
I evaluate n rounds and find their likelihood of producing the final value.
The "chance of survival" of an unset bit decreases after each round so that the result eventually stabilizes.
I stop once the delta falls below my threshold 10^-11 (one digit more than needed to prevent rounding issues).
A single is zero after a certain number of rounds with probability:
(1) 0.5^{round}
And it became 1 after all those rounds:
(2) 1 - 0.5^{round}
The chance that all 32 bits are 1:
(3) (1 - 0.5^{round})^32
Therefore the chance that at least one out of 32 bits is still zero:
(4) 1 - (1 - 0.5^{round})^32
I keep adding all values of equation (4) until the value of (4) drops below 10^-11.
It takes a little more than 40 iterations to find the correct result.
Note
Actually, my first step was writing a simple Monte-Carlo simulation to get an approximation of the result.
Of course it's impossible to find all 10 digits after the decimal point (unless you invest a huge amount of time).
The code can still be found in montecarlo
(and its helper function myrand
).
It's worth mentioning that the results of my Monte-Carlo simulation differed significantly from their true values.
I spent more than an hour trying to find a bug in my "mathematical" approach because I trusted the Monte-Carlo result.
As it turned out, myrand
was flawed because the simple LCG algorithm had bad parameters.
When I choose different parameters (from en.wikipedia.org/wiki/Linear_congruential_generator) then everything turned out to be fine.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 8 | ./323
Output:
Note: the original problem's input 32
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 <iomanip>
#include <cmath>
// 32 bits
unsigned int maxBits = 32;
// ---------- first approach: find a good estimation with Monte-Carlo simulation ----------
// a simple pseudo-random number generator
// (produces the same result no matter what compiler you have - unlike rand() from math.h)
unsigned int myrand()
{
static unsigned long long seed = 0;
seed = 6364136223846793005ULL * seed + 1;
return seed >> 30;
}
// randomized simulation
double montecarlo(unsigned int iterations)
{
// all 32 bits are set
const unsigned int allBits = (1ULL << maxBits) - 1;
unsigned int numSteps = 0;
for (unsigned int i = 0; i < iterations; i++)
{
// start with zero
unsigned int current = 0;
// and repeat until all bits are set
do
{
// and some random bits
current |= myrand() & allBits;
numSteps++;
} while (current != allBits);
}
// return average number of steps until "completion"
return numSteps / double(iterations);
}
// ---------- final approach: solve "smarter" ! ----------
int main()
{
std::cin >> maxBits;
// number of accurate digits
const unsigned int digits = 10;
// 0.00000000001 => one more zero ensures that the last digit is correct even after rounding
const double Epsilon = pow(10.0, -(double)(digits + 1));
// at least one round
double result = 0;
unsigned int round = 0;
// until change is smaller than Epsilon
while (true)
{
// chance of "survival" for a single zero after all those rounds
auto hasZero = pow(0.5, round);
// were all 32 bits flipped from 0 to 1 ?
auto isDone = pow(1 - hasZero, maxBits);
// remaining
auto delta = 1 - isDone;
// enough digits ?
if (delta < Epsilon)
break;
// next iteration
result += delta;
round++; // stop after 42 rounds
}
// display result
std::cout << std::fixed << std::setprecision(digits)
<< result << std::endl;
// run Monte-Carlo simulation
//while (true)
// std::cout << montecarlo(100000000) << std::endl;
return 0;
}
This solution contains 17 empty lines, 24 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
July 31, 2017 submitted solution
July 31, 2017 added comments
Difficulty
Project Euler ranks this problem at 20% (out of 100%).
Links
projecteuler.net/thread=323 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/Meng-Gen/ProjectEuler/blob/master/323.py (written by Meng-Gen Tsai)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p323.py (written by Nayuki)
Python github.com/smacke/project-euler/blob/master/python/323.py (written by Stephen Macke)
C++ github.com/roosephu/project-euler/blob/master/323.cpp (written by Yuping Luo)
C++ github.com/steve98654/ProjectEuler/blob/master/323.cpp
C github.com/LaurentMazare/ProjectEuler/blob/master/e323.c (written by Laurent Mazare)
Java github.com/dcrousso/ProjectEuler/blob/master/PE323.java (written by Devin Rousso)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p323.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem323.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem323.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p323.mathematica (written by Nayuki)
Mathematica github.com/steve98654/ProjectEuler/blob/master/323.nb
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p323.hs (written by Nayuki)
Perl github.com/shlomif/project-euler/blob/master/project-euler/323/euler-323-v1.pl (written by Shlomi Fish)
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 |
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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 321 - Swapping Counters | Building a tower - problem 324 >> |