1359. Count All Valid Pickup and Delivery Options
Description
Given n
orders, each order consists of a pickup and a delivery service.
Count all valid pickup/delivery possible sequences such that delivery(i) is always after of pickup(i).
Since the answer may be too large, return it modulo 10^9 + 7.
Example 1:
Input: n = 1 Output: 1 Explanation: Unique order (P1, D1), Delivery 1 always is after of Pickup 1.
Example 2:
Input: n = 2 Output: 6 Explanation: All possible orders: (P1,P2,D1,D2), (P1,P2,D2,D1), (P1,D1,P2,D2), (P2,P1,D1,D2), (P2,P1,D2,D1) and (P2,D2,P1,D1). This is an invalid order (P1,D2,P2,D1) because Pickup 2 is after of Delivery 2.
Example 3:
Input: n = 3 Output: 90
Constraints:
1 <= n <= 500
Solutions
Solution 1: Dynamic Programming
We define $f[i]$ as the number of all valid pickup/delivery sequences for $i$ orders. Initially, $f[1] = 1$.
We can choose any of these $i$ orders as the last delivery order $D_i$, then its pickup order $P_i$ can be at any position in the previous $2 \times i  1$, and the number of pickup/delivery sequences for the remaining $i  1$ orders is $f[i  1]$, so $f[i]$ can be expressed as:
$$ f[i] = i \times (2 \times i  1) \times f[i  1] $$
The final answer is $f[n]$.
We notice that the value of $f[i]$ is only related to $f[i  1]$, so we can use a variable instead of an array to reduce the space complexity.
The time complexity is $O(n)$, where $n$ is the number of orders. The space complexity is $O(1)$.









