Handshakes That Don't Cross — LeetCode 1259 Python Solution

The problem

You are given an even number of people numPeople that stand around a circle and each person shakes hands with someone else so that there are numPeople / 2 handshakes total. Return the number of ways these handshakes could occur such that none of the handshakes cross.

Example

Input

numPeople = 4

Output

2

Explanation

There are two ways to do it, the first way is [(1,2),(3,4)] and the second one is [(2,3),(4,1)].

Python solution

class Solution:
    def numberOfWays(self, numPeople: int) -> int:
        @cache
        def dfs(i: int) -> int:
            if i < 2:
                return 1
            ans = 0
            for l in range(0, i, 2):
                r = i - l - 2
                ans += dfs(l) * dfs(r)
                ans %= mod
            return ans

        mod = 10**9 + 7
        return dfs(numPeople)

Complexity

Pattern: Dynamic Programming

Define a state, write the transition, and stop recomputing the same subproblem. LeetCode 1259. Handshakes That Don't Cross is filed here on both counts: the reference solution below belongs to the algorithm family this hub collects, and LeetCode tags it Dynamic Programming.

The dynamic programming guide has the Python template for the pattern and the 481 LeetCode problems that use it.

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