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2187. Minimum Time to Complete Trips

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; that is, the next trip can start immediately after completing the current trip. Also, each bus operates independently; that is, the trips of one bus do not influence the trips of any other bus.

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.

 

Example 1:

Input: time = [1,2,3], totalTrips = 5
Output: 3
Explanation:
- At time t = 1, the number of trips completed by each bus are [1,0,0]. 
  The total number of trips completed is 1 + 0 + 0 = 1.
- At time t = 2, the number of trips completed by each bus are [2,1,0]. 
  The total number of trips completed is 2 + 1 + 0 = 3.
- At time t = 3, the number of trips completed by each bus are [3,1,1]. 
  The total number of trips completed is 3 + 1 + 1 = 5.
So the minimum time needed for all buses to complete at least 5 trips is 3.

Example 2:

Input: time = [2], totalTrips = 1
Output: 2
Explanation:
There is only one bus, and it will complete its first trip at t = 2.
So the minimum time needed to complete 1 trip is 2.

 

Constraints:

  • 1 <= time.length <= 105
  • 1 <= time[i], totalTrips <= 107

Solutions

We notice that if we can complete at least $totalTrips$ trips in $t$ time, then we can also complete at least $totalTrips$ trips in $t' > t$ time. Therefore, we can use the method of binary search to find the smallest $t$.

We define the left boundary of the binary search as $l = 1$, and the right boundary as $r = \min(time) \times totalTrips$. For each binary search, we calculate the middle value $\text{mid} = \frac{l + r}{2}$, and then calculate the number of trips that can be completed in $\text{mid}$ time. If this number is greater than or equal to $totalTrips$, then we reduce the right boundary to $\text{mid}$, otherwise we increase the left boundary to $\text{mid} + 1$.

Finally, return the left boundary.

The time complexity is $O(n \times \log(m \times k))$, where $n$ and $k$ are the length of the array $time$ and $totalTrips$ respectively, and $m$ is the minimum value in the array $time$. The space complexity is $O(1)$.

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class Solution:
    def minimumTime(self, time: List[int], totalTrips: int) -> int:
        mx = min(time) * totalTrips
        return bisect_left(
            range(mx), totalTrips, key=lambda x: sum(x // v for v in time)
        )

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class Solution {
    public long minimumTime(int[] time, int totalTrips) {
        int mi = time[0];
        for (int v : time) {
            mi = Math.min(mi, v);
        }
        long left = 1, right = (long) mi * totalTrips;
        while (left < right) {
            long cnt = 0;
            long mid = (left + right) >> 1;
            for (int v : time) {
                cnt += mid / v;
            }
            if (cnt >= totalTrips) {
                right = mid;
            } else {
                left = mid + 1;
            }
        }
        return left;
    }
}

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class Solution {
public:
    long long minimumTime(vector<int>& time, int totalTrips) {
        int mi = *min_element(time.begin(), time.end());
        long long left = 1, right = 1LL * mi * totalTrips;
        while (left < right) {
            long long cnt = 0;
            long long mid = (left + right) >> 1;
            for (int v : time) {
                cnt += mid / v;
            }
            if (cnt >= totalTrips) {
                right = mid;
            } else {
                left = mid + 1;
            }
        }
        return left;
    }
};

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func minimumTime(time []int, totalTrips int) int64 {
    mx := slices.Min(time) * totalTrips
    return int64(sort.Search(mx, func(x int) bool {
        cnt := 0
        for _, v := range time {
            cnt += x / v
        }
        return cnt >= totalTrips
    }))
}

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function minimumTime(time: number[], totalTrips: number): number {
    let left = 1n;
    let right = BigInt(Math.min(...time)) * BigInt(totalTrips);
    while (left < right) {
        const mid = (left + right) >> 1n;
        const cnt = time.reduce((acc, v) => acc + mid / BigInt(v), 0n);
        if (cnt >= BigInt(totalTrips)) {
            right = mid;
        } else {
            left = mid + 1n;
        }
    }
    return Number(left);
}

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