Count All Valid Pickup and Delivery Options

Problem 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 pickup(i). Since the answer may be too large, return it modulo (10^9 + 7).

Key Insights

• Each order consists of a pickup and a delivery, and the delivery for each order must occur after its respective pickup.
• The problem can be approached using combinatorial mathematics, specifically Catalan numbers.
• The total valid sequences of n pickups and n deliveries can be represented using dynamic programming or combinatorial formulas.
• The result must be computed modulo (10^9 + 7) to handle large numbers.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

To solve the problem, we can use a dynamic programming approach. We'll define a dynamic programming array dp where dp[i] represents the number of valid pickup and delivery sequences for i orders.

The relationship can be derived from the fact that for each order, we can place the pickup in various positions, and the delivery must follow it. The total number of valid sequences for i orders can be calculated as:

[ dp[i] = dp[i-1] \times (2 \times i) \times (2 \times i - 1) ]

This formula derives from the principle of choosing positions for the pickups and deliveries while maintaining their order.

The base case is dp[0] = 1 because there is one way to arrange zero orders (doing nothing).

Code Solutions