codeforces#P1365C. Rotation Matching
Rotation Matching
Description
After the mysterious disappearance of Ashish, his two favourite disciples Ishika and Hriday, were each left with one half of a secret message. These messages can each be represented by a permutation of size $n$. Let's call them $a$ and $b$.
Note that a permutation of $n$ elements is a sequence of numbers $a_1, a_2, \ldots, a_n$, in which every number from $1$ to $n$ appears exactly once.
The message can be decoded by an arrangement of sequence $a$ and $b$, such that the number of matching pairs of elements between them is maximum. A pair of elements $a_i$ and $b_j$ is said to match if:
- $i = j$, that is, they are at the same index.
- $a_i = b_j$
His two disciples are allowed to perform the following operation any number of times:
- choose a number $k$ and cyclically shift one of the permutations to the left or right $k$ times.
A single cyclic shift to the left on any permutation $c$ is an operation that sets $c_1:=c_2, c_2:=c_3, \ldots, c_n:=c_1$ simultaneously. Likewise, a single cyclic shift to the right on any permutation $c$ is an operation that sets $c_1:=c_n, c_2:=c_1, \ldots, c_n:=c_{n-1}$ simultaneously.
Help Ishika and Hriday find the maximum number of pairs of elements that match after performing the operation any (possibly zero) number of times.
The first line of the input contains a single integer $n$ $(1 \le n \le 2 \cdot 10^5)$ — the size of the arrays.
The second line contains $n$ integers $a_1$, $a_2$, ..., $a_n$ $(1 \le a_i \le n)$ — the elements of the first permutation.
The third line contains $n$ integers $b_1$, $b_2$, ..., $b_n$ $(1 \le b_i \le n)$ — the elements of the second permutation.
Print the maximum number of matching pairs of elements after performing the above operations some (possibly zero) times.
Input
The first line of the input contains a single integer $n$ $(1 \le n \le 2 \cdot 10^5)$ — the size of the arrays.
The second line contains $n$ integers $a_1$, $a_2$, ..., $a_n$ $(1 \le a_i \le n)$ — the elements of the first permutation.
The third line contains $n$ integers $b_1$, $b_2$, ..., $b_n$ $(1 \le b_i \le n)$ — the elements of the second permutation.
Output
Print the maximum number of matching pairs of elements after performing the above operations some (possibly zero) times.
Samples
5
1 2 3 4 5
2 3 4 5 1
5
5
5 4 3 2 1
1 2 3 4 5
1
4
1 3 2 4
4 2 3 1
2
Note
For the first case: $b$ can be shifted to the right by $k = 1$. The resulting permutations will be $\{1, 2, 3, 4, 5\}$ and $\{1, 2, 3, 4, 5\}$.
For the second case: The operation is not required. For all possible rotations of $a$ and $b$, the number of matching pairs won't exceed $1$.
For the third case: $b$ can be shifted to the left by $k = 1$. The resulting permutations will be $\{1, 3, 2, 4\}$ and $\{2, 3, 1, 4\}$. Positions $2$ and $4$ have matching pairs of elements. For all possible rotations of $a$ and $b$, the number of matching pairs won't exceed $2$.