Description
There is a regular convex polygon with n vertices. The vertices are labeled from 0 to n - 1 in a clockwise direction, and each vertex has exactly one monkey. The following figure shows a convex polygon of 6 vertices.
Simultaneously, each monkey moves to a neighboring vertex. A collision happens if at least two monkeys reside on the same vertex after the movement or intersect on an edge.
Return the number of ways the monkeys can move so that at least one collision happens. Since the answer may be very large, return it modulo 109 + 7.
Example 1:
Input: n = 3
Output: 6
Explanation:
There are 8 total possible movements.
Two ways such that they collide at some point are:
- Monkey 1 moves in a clockwise direction; monkey 2 moves in an anticlockwise direction; monkey 3 moves in a clockwise direction. Monkeys 1 and 2 collide.
- Monkey 1 moves in an anticlockwise direction; monkey 2 moves in an anticlockwise direction; monkey 3 moves in a clockwise direction. Monkeys 1 and 3 collide.
Example 2:
Input: n = 4
Output: 14
Constraints:
3 <= n <= 109
Solutions
Solution 1: Mathematics (Fast Power)
According to the problem description, each monkey has two ways of moving, either clockwise or counterclockwise. Therefore, there are a total of 2n ways to move. The non-collision ways of moving are only two, that is, all monkeys move clockwise or all monkeys move counterclockwise. Therefore, the number of collision ways of moving is 2n - 2.
We can use fast power to calculate the value of 2n, then use 2n - 2 to calculate the number of collision ways of moving, and finally take the remainder of 109 + 7.
The time complexity is O(log n), where n is the number of monkeys. The space complexity is O(1).
class Solution: def monkeyMove(self, n: int) -> int: mod = 10**9 + 7 return (pow(2, n, mod) - 2) % mod(code-box)
