How do you solve Count All Valid Pickup and Delivery Options on LeetCode?

Count All Valid Pickup and Delivery Options (LeetCode #1359) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n) time complexity and O(n) space complexity. Key insight: The order of deliveries must always respect the pickup order, leading to a combinatorial structure.

What is the time complexity of Count All Valid Pickup and Delivery Options?

The optimal solution for Count All Valid Pickup and Delivery Options runs in O(n) time and uses O(n) space. The time complexity is O(n) because we compute the valid sequences in a single pass through the DP array, and space complexity is O(n) due to the DP array.

What is the brute force solution for Count All Valid Pickup and Delivery Options?

The brute force approach for Count All Valid Pickup and Delivery Options is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute-force approach generates all possible sequences of pickups and deliveries, then filters out invalid ones. This is straightforward but inefficient, as it involves checking every permutation. The optimal Optimal Solution approach improves this to O(n).

What algorithm or data structure is used to solve Count All Valid Pickup and Delivery Options?

Count All Valid Pickup and Delivery Options uses the following concepts: Math, Dynamic Programming, Combinatorics. The recommended patterns to study are: Dynamic Programming, Combinatorics.

What are the interview tips for Count All Valid Pickup and Delivery Options?

Always consider edge cases, such as the smallest input size. Explain your thought process clearly; it helps the interviewer understand your reasoning. Practice recognizing patterns in problems, as many can be solved using similar techniques.

What are common mistakes when solving Count All Valid Pickup and Delivery Options?

Failing to account for the modulo operation when calculating large numbers. Not recognizing the combinatorial nature of the problem, leading to inefficient brute-force solutions.

#1359

Count All Valid Pickup and Delivery Options

Hard

MathDynamic ProgrammingCombinatoricsDynamic ProgrammingCombinatorics

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(n)
Space	O(1)	O(n)

💡

Intuition

Time O(n)Space O(n)

The optimal solution uses dynamic programming to build the count of valid sequences incrementally. Each new order adds combinations based on previous counts, leveraging combinatorial principles.

⚙️

Algorithm

3 steps

1Step 1: Initialize a DP array where dp[i] represents the number of valid sequences for i orders.
2Step 2: Use the formula dp[i] = dp[i-1] * (2 * i - 1) * i to calculate the number of valid sequences for each i.
3Step 3: Return dp[n] modulo 10^9 + 7.

solution.py7 lines

1def countOrders(n):
2    MOD = 10**9 + 7
3    dp = [0] * (n + 1)
4    dp[0] = 1
5    for i in range(1, n + 1):
6        dp[i] = dp[i - 1] * (2 * i - 1) * i % MOD
7    return dp[n]

ℹ

Complexity note: The time complexity is O(n) because we compute the valid sequences in a single pass through the DP array, and space complexity is O(n) due to the DP array.

1The order of deliveries must always respect the pickup order, leading to a combinatorial structure.
2Dynamic programming allows us to build solutions incrementally, reducing the need for exhaustive checks.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.