#2188

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum Time to Finish the Race

Minimize the time to complete a race with tire swaps using dynamic programming and state transitions.

Problem Statement

You are given a 2D array, tires, where each tire type has a formula that determines how long it takes to finish each successive lap. The formula is fi * ri(x-1) seconds for the xth lap, with fi being the base time and ri being the increase in time for successive laps. You are also given an integer changeTime, representing how long it takes to switch between tire types, and an integer numLaps, which represents the total laps in the race.

You can choose any tire at the beginning and swap tires after each lap, waiting changeTime seconds. The goal is to calculate the minimum total time to finish the race with the given constraints. Your solution should be optimized using dynamic programming to account for the transitions between tire types, ensuring the most efficient way to complete the race.

Examples

Example 1

Input: tires = [[2,3],[3,4]], changeTime = 5, numLaps = 4

Output: 21

Lap 1: Start with tire 0 and finish the lap in 2 seconds. Lap 2: Continue with tire 0 and finish the lap in 2 * 3 = 6 seconds. Lap 3: Change tires to a new tire 0 for 5 seconds and then finish the lap in another 2 seconds. Lap 4: Continue with tire 0 and finish the lap in 2 * 3 = 6 seconds. Total time = 2 + 6 + 5 + 2 + 6 = 21 seconds. The minimum time to complete the race is 21 seconds.

Example 2

Input: tires = [[1,10],[2,2],[3,4]], changeTime = 6, numLaps = 5

Output: 25

Lap 1: Start with tire 1 and finish the lap in 2 seconds. Lap 2: Continue with tire 1 and finish the lap in 2 * 2 = 4 seconds. Lap 3: Change tires to a new tire 1 for 6 seconds and then finish the lap in another 2 seconds. Lap 4: Continue with tire 1 and finish the lap in 2 * 2 = 4 seconds. Lap 5: Change tires to tire 0 for 6 seconds then finish the lap in another 1 second. Total time = 2 + 4 + 6 + 2 + 4 + 6 + 1 = 25 seconds. The minimum time to complete the race is 25 seconds.

Constraints

1 <= tires.length <= 105
tires[i].length == 2
1 <= fi, changeTime <= 105
2 <= ri <= 105
1 <= numLaps <= 1000

Solution Approach

State Transition Dynamic Programming

This problem can be solved using dynamic programming (DP) where each state represents the minimal time required to finish a certain number of laps with a specific tire. You will track the minimum time required for each tire at every lap and explore all possible transitions between tire types, considering both lap time and change time.

Time Calculation for Successive Laps

For each tire, the time to complete successive laps increases exponentially based on the tire's formula. This means the solution needs to account for both the increasing time due to the formula and the additional time incurred by changing tires. Optimize the transitions between tire types to ensure that tire swaps occur only when necessary.

Optimizing Tire Transitions

Instead of brute-forcing every possible tire choice, optimize tire transitions by storing the minimal times for each lap in the DP table. Focus on the maximum number of laps that can be completed with a single tire before it becomes inefficient to continue without switching.

Complexity Analysis

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

The time complexity depends on how you manage the transitions between tires and track the minimal times for each tire choice. In the worst case, the solution requires iterating over each tire for each lap, resulting in a time complexity of O(n * m), where n is the number of tires and m is the number of laps. Space complexity also depends on the number of laps and tire types stored in the DP table, with a space complexity of O(n * m).

What Interviewers Usually Probe

Look for the ability to optimize state transitions using dynamic programming.
Check if the candidate can handle exponential growth in lap times and tire changes effectively.
Evaluate the candidate's approach to minimizing tire swaps and understanding the impact on time.

Common Pitfalls or Variants

Common pitfalls

Forgetting to account for the exponential increase in time for successive laps with each tire.
Improperly handling tire swaps, either by not considering them at all or by switching tires unnecessarily.
Overcomplicating the problem by trying to brute force all possible combinations instead of using dynamic programming.

Follow-up variants

Introduce additional tire types with different increase rates and see how the solution scales.
Change the structure of the problem by limiting the number of tire swaps or changing the number of laps.
Modify the problem to include constraints on when tire changes can occur, such as only changing after a set number of laps.

FAQ

How does dynamic programming help in this problem?

Dynamic programming helps by breaking down the problem into smaller subproblems, tracking the minimum time to finish the race with each tire choice and lap. This allows for efficient computation without redundant recalculations.

What is the impact of tire swaps on the total time?

Tire swaps add a fixed change time, but they can result in significant savings if a tire's efficiency decreases after several laps. Minimizing unnecessary swaps is key to optimizing the total time.

How do I handle the exponential increase in lap times?

The exponential increase in lap times with each tire requires careful tracking of when to stop using a tire. You must weigh the benefits of switching tires against the added change time and the diminishing returns from using a tire for too long.

What are the edge cases I should consider for this problem?

Consider cases where only one tire type is available, where the change time is very high, or where there are very few laps. These edge cases will test the efficiency and correctness of your solution.

Can this problem be solved without dynamic programming?

While a brute force approach could work, dynamic programming provides a much more efficient way to solve this problem by storing intermediate results and avoiding redundant calculations.

terminal

Solution

Solution 1

#### Python3

class Solution:
    def minimumFinishTime(
        self, tires: List[List[int]], changeTime: int, numLaps: int
    ) -> int:
        cost = [inf] * 18
        for f, r in tires:
            i, s, t = 1, 0, f
            while t <= changeTime + f:
                s += t
                cost[i] = min(cost[i], s)
                t *= r
                i += 1
        f = [inf] * (numLaps + 1)
        f[0] = -changeTime
        for i in range(1, numLaps + 1):
            for j in range(1, min(18, i + 1)):
                f[i] = min(f[i], f[i - j] + cost[j])
            f[i] += changeTime
        return f[numLaps]

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