#1359

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Count All Valid Pickup and Delivery Options

Count all valid pickup and delivery sequences for n orders where deliveries occur after pickups using dynamic programming.

math dynamic programming combinatorics

Problem Statement

You are given n orders. Each order consists of a pickup and a delivery service, where pickup(i) corresponds to delivery(i). The task is to determine how many valid sequences can be made from these orders. A sequence is valid if and only if delivery(i) always occurs after pickup(i) for every order.

Since the result can be large, return the total count modulo 10^9 + 7. The problem can be efficiently solved using a dynamic programming approach that incorporates combinatorics, especially utilizing the permutation and combination theory for state transitions as you add each new (pickup, delivery) pair.

Examples

Example 1

Input: n = 1

Output: 1

Unique order (P1, D1), Delivery 1 always is after of Pickup 1.

Example 2

Input: n = 2

Output: 6

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

Example details omitted.

Constraints

1 <= n <= 500

Solution Approach

Dynamic Programming Approach

Use dynamic programming to build the solution incrementally. For each new (pickup, delivery) pair, update the DP table by considering the valid transitions based on the prior sequence. The key transition involves calculating the number of ways to insert a new (pickup, delivery) pair into a valid sequence.

Combinatorial Optimization

Apply combinatorial techniques, specifically permutations and combinations, to optimize the DP solution. The number of valid sequences is influenced by the choices made in inserting each (pickup, delivery) pair into the sequence. This helps reduce the complexity of the solution by considering combinations instead of explicitly generating every sequence.

Modulo Operations

Due to the large possible result, incorporate modulo 10^9 + 7 in every calculation to avoid overflow and ensure the result fits within the required constraints. This ensures the correctness of the answer while keeping computations manageable.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity is O(n^2), considering the need to compute the valid state transitions for each of the n pairs. The space complexity is also O(n^2) for storing the DP table of size n x n, which tracks valid transitions for each (pickup, delivery) pair.

What Interviewers Usually Probe

Candidate should demonstrate a strong understanding of dynamic programming and combinatorics.
Watch for candidates applying permutations and combinations to efficiently calculate the number of valid sequences.
Test if the candidate can explain and handle modulo operations during the computation process.

Common Pitfalls or Variants

Common pitfalls

Overcomplicating the problem by generating all sequences explicitly instead of using dynamic programming.
Forgetting to account for the modulo 10^9 + 7 operation, leading to incorrect results due to overflow.
Misunderstanding the state transition process and failing to correctly compute valid transitions between pickup and delivery pairs.

Follow-up variants

Extending the problem to account for more than one delivery per pickup, increasing the complexity.
Changing the constraints, such as limiting n to smaller values or increasing the range of possible values for n.
Modifying the problem to work with different numbers of pickups and deliveries, not tied to a one-to-one correspondence.

FAQ

What is the core pattern of the Count All Valid Pickup and Delivery Options problem?

The core pattern involves dynamic programming with state transitions and combinatorics to compute valid sequences while considering the constraints of each pickup and delivery pair.

How does dynamic programming help in solving this problem?

Dynamic programming is used to build up solutions for smaller subproblems, ensuring that all valid sequences are counted by transitioning states as new (pickup, delivery) pairs are added.

Why is modulo 10^9 + 7 important in this problem?

Modulo 10^9 + 7 is crucial to ensure that the result remains within the problem's constraints, avoiding overflow and keeping computations manageable.

What combinatorial concepts are used in this problem?

Permutation and combination theory is used to calculate the number of valid ways to insert each new (pickup, delivery) pair into an existing sequence.

How can GhostInterview assist in preparing for this dynamic programming problem?

GhostInterview provides step-by-step guidance and feedback on implementing dynamic programming approaches, helping candidates handle transitions, optimizations, and modulo operations effectively.

terminal

Solution

Solution 1: Dynamic Programming

We define $f[i]$ as the number of all valid pickup/delivery sequences for $i$ orders. Initially, $f[1] = 1$.

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class Solution:
    def countOrders(self, n: int) -> int:
        mod = 10**9 + 7
        f = 1
        for i in range(2, n + 1):
            f = (f * i * (2 * i - 1)) % mod
        return f

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