Minimum Time to Complete Trips

Problem Description

You are given an array time where time[i] denotes the time taken by the ith bus to complete one trip. Each bus can make multiple trips successively, and they operate independently. You are also given an integer totalTrips, which denotes the number of trips all buses should make in total. Return the minimum time required for all buses to complete at least totalTrips trips.

Key Insights

• Each bus can complete multiple trips, and the number of trips completed by a bus in a given time can be calculated as time[i].
• The problem can be solved using a binary search approach to efficiently find the minimum time.
• The total number of trips completed by all buses at a given time t can be calculated by summing t // time[i] for each bus.

Space and Time Complexity

Time Complexity: O(n log(max_time)), where n is the number of buses and max_time is the maximum possible time required to complete all trips.
Space Complexity: O(1), since we are using a constant amount of space.

Solution

To solve the problem, we can use binary search to determine the minimum time needed to complete at least totalTrips. The algorithm works as follows:

1. Define the search space for time between 1 and min(time) * totalTrips.
2. Use binary search:
   • Calculate the midpoint time.
   • Count the total number of trips completed by all buses at that midpoint time.
   • If the count meets or exceeds totalTrips, adjust the search space to search for a smaller time; otherwise, search for a larger time.
3. The answer will be the minimum time that satisfies the trip requirement.

Code Solutions