https://github.com/appleboiy/competitive-contest-355
Contest for course CS355 Competitive Programming
https://github.com/appleboiy/competitive-contest-355
Last synced: 2 months ago
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Contest for course CS355 Competitive Programming
- Host: GitHub
- URL: https://github.com/appleboiy/competitive-contest-355
- Owner: AppleBoiy
- Created: 2024-01-22T08:11:52.000Z (over 1 year ago)
- Default Branch: master
- Last Pushed: 2024-03-31T16:47:22.000Z (about 1 year ago)
- Last Synced: 2025-02-12T05:45:05.098Z (4 months ago)
- Language: C++
- Homepage: https://appleboiy.github.io/competitive-contest-355-docs/
- Size: 24.7 MB
- Stars: 1
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# Fibonacci Modulo Problem
## Problem Statement
You are given two integers, **n** and **m**, and you need to find the **nth Fibonacci number modulo m**.
### Approach
The naive approach to find the Fibonacci number at position **n** and then taking the modulo **m** might be inefficient
for large values of **n**. However, you can optimize the solution using the Pisano Period.
### Pisano Period
The Pisano period, also known as the Fibonacci period or cycle, refers to the periodic and repetitive nature of the
remainders when the Fibonacci sequence is divided by a positive integer called the "modulus."
[read more](https://en.wikipedia.org/wiki/Pisano_period)
#### Fibonacci Sequence
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting
with 0 and 1.
Example Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
#### Pisano Period
When the Fibonacci sequence is divided by a modulus (a positive integer), the remainders exhibit a periodic pattern
known as the Pisano period. The length of this period depends on the modulus chosen.
The Pisano period is useful in various applications, especially in problems related to number theory and modular
arithmetic.
#### Formula
Given a modulus 'm,' the Pisano period is denoted as π(m). The period starts with the pair (0, 1) and repeats after a
certain number of terms.
#### Example
For example, if the modulus is 5:
Fibonacci sequence modulo 5
0, 1, 1, 2, 3, 0, 3, 3, 1, 4,
0, 4, 4, 3, 2, 0, 2, 2, 4, 1,
--- Pisano period starts here ---
0, 1, 1, 2, 3, 0, 3, 3, 1, 4,
0, 4, 4, 3, 2, 0, 2, 2, 4, 1,
0, 1, 1, 2, 3, 0, 3, 3, 1, 4
... and so on
In this case, the Pisano period (π(5)) is 20, as the sequence starts repeating after 20 terms.
### Code Explanation
Below is the C++ code implementing this approach:
```cpp
#include
#define ll long long
ll fib_fast(ll n, ll m) {
if (n <= 1)
return n;
// Pisano period
ll remainder = 0;
ll a = 0, b = 1, c;
for (ll i = 0; i < n - 1; i++) {
c = (a + b) % m;
a = b;
b = c;
if (a == 0 && b == 1) {
remainder = i + 1;
break;
}
}
ll new_n = n % remainder;
if (new_n <= 1)
return new_n;
a = 0, b = 1, c = a + b;
for (ll i = 0; i < new_n - 1; i++) {
c = (a + b) % m;
a = b;
b = c;
}
return c % m;
}
int main() {
long long n, m;
std::cin >> n >> m;
std::cout << fib_fast(n, m) << '\n';
}
```
### Input
The program takes two inputs, **n** and **m**, representing the position of the Fibonacci number and the modulo value.
### Output
The program outputs the **nth Fibonacci number modulo m**.
### Time Complexity
The time complexity of this solution is **O(remainder + new_n)**, where **remainder** is the Pisano Period and **new_n**
is the position of the Fibonacci number within the Pisano Period. This is a significant optimization over the naive
approach.