<< problem 24 - Lexicographic permutations | Reciprocal cycles - problem 26 >> |
Problem 25: 1000-digit Fibonacci number
(see projecteuler.net/problem=25)
The Fibonacci sequence is defined by the recurrence relation:
F_n = F_{n-1} + F_{n-2}, where F_1 = 1 and F_2 = 1.
Hence the first 12 terms will be:
F_1 = 1
F_2 = 1
F_3 = 2
F_4 = 3
F_5 = 5
F_6 = 8
F_7 = 13
F_8 = 21
F_9 = 34
F_{10} = 55
F_{11} = 89
F_{12} = 144
The 12th term, F_{12}, is the first term to contain three digits.
What is the index of the first term in the Fibonacci sequence to contain 1000 digits?
My Algorithm
I precompute all Fibonacci number with up to 5000 digits (a design decision influenced by Hackerrank's modified problem) and keep those results in cache
.
Unfortunately, there is a small problem with C++ ...
F_{47}=2971215073 is the largest Fibonacci number that fits in a 32-bit integer and
F_{94}=19740274219868223167 is too big for a 64-bit integer.
My program stores such large number as a std::vector
where index 0 contains the least significant digit ("in reverse order").
E.g. F_{23}=28657 is represented as { 7, 5, 6, 8, 2 }
The function add
returns the sum of two large numbers a
and b
where b>=a
.
The algorithm behind this function is exactly the same you were taught in primary school.
Alternative Approaches
The main problem was adding two very large numbers. When programming in Python, Java, etc. you get these things for free.
Modifications by HackerRank
The large amount of test cases was the main cause for dividing my solution into two parts;
1. precompute all relevant Fibonacci numbers (done once - "expensive")
2. look up the result (performed many, many times - "cheap")
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" | ./25
Output:
Note: the original problem's input 1000
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 <vector>
#include <iostream>
// store single digits because numbers get too big for C++
typedef std::vector<unsigned int> Digits;
// Hackerrank's upper limit
const unsigned int MaxDigits = 5000;
// add two long number where b >= a
Digits add(const Digits& a, const Digits& b)
{
Digits result = b;
unsigned int carry = 0;
for (unsigned int i = 0; i < result.size(); i++)
{
// "a" might have less digits than "b"
if (i < a.size())
result[i] += a[i];
// don't forget about the carry ...
result[i] += carry;
// handle overflow
if (result[i] >= 10)
{
carry = 1;
result[i] -= 10;
}
else
carry = 0;
}
// largest digit not overflowing ?
if (carry != 0)
result.push_back(carry);
return result;
}
int main()
{
// precompute number of steps we needed for each number of digits
// [number of digits] => [index of smallest Fibonacci number]
std::vector<unsigned int> cache = { 0, 1 }; // F_0 is undefined
cache.reserve(MaxDigits);
// f(1) = 1
Digits a = { 1 };
// f(2) = 1
Digits b = { 1 };
// we have predefined F_1 and F_2
unsigned int fiboIndex = 2;
while (cache.size() <= MaxDigits)
{
// next Fibonacci number
fiboIndex++;
auto next = add(a, b);
a = std::move(b);
b = std::move(next);
// digits of current Fibonacci number
auto numDigits = b.size();
// digits of the previously largest Fibonacci number
auto largestKnown = cache.size() - 1; // don't count the 0th element
// one more digit than before ?
if (largestKnown < numDigits)
cache.push_back(fiboIndex);
}
// simply look up the result
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int numDigits;
std::cin >> numDigits;
std::cout << cache[numDigits] << std::endl;
}
return 0;
}
This solution contains 15 empty lines, 17 comments and 2 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.4 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
February 24, 2017 submitted solution
April 4, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler025
My code solves 4 out of 4 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 5% (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=25 - 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-25-fibonacci-sequence-1000-digits/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/20-29/problem25.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p025.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/25 1000-digit Fibonacci number.js (written by David Ernst)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p025.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p025.hs (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler025.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/025-1000-digit-Fibonacci-number.pl (written by Gustaf Erikson)
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 24 - Lexicographic permutations | Reciprocal cycles - problem 26 >> |