PROBLEM LINK:

Practice
Contest

Author: Asif Haque
Tester:Jingbo Shang
Editorialist:Jingbo Shang

DIFFICULTY:

Medium

PREREQUISITES:

Fast Matrix Power with Mod

PROBLEM:

Given a m-order linear recurrence, calculate the n-th number (mod 10^9+7).

EXPLANATION:

There are two key points in this problem: Write the linear recurrence in matrix representation. That is using a m×m matrix to state one step of recurrence. Fast calculate the matrix power operation. That is using O(m^3 logn) time to get A^n mod 10^9+7

Let’s take a example of Fibonacci Numbers. fib[n] = fib[n - 1] + fib[n - 2] This is a 2-order linear recurrence. And we can write this as the following matrix representation: (fib[n-1],fib[n]) * (0, 1; 1, 1) = (fib[n], fib[n+1]) Here (0, 1; 1, 1) is a 2*2 matrix, denote as A.

After that, we need to calculate A^n mod 10^9+7. We can prepare A^1,A^2,A^4…A^logn in O(m^3 logn) time. And then, multiple corresponding matrices together to get A^n. Based on the binary expression of n,it is obvious that these logn matrices will be multiplied at most once. Therefore, we can get the answer in O(m^3 logn).

AUTHOR'S AND TESTER'S SOLUTIONS:

Author's solution can be found here.
Tester's solution can be found here.