Minimum Speed to Arrive on Time

class Solution {
public:
    double calculateTime(vector<int>& distances, int velocity) {
        double totalTime = 0.0;
        for (int index = 0; index < distances.size(); index++) {
            double segmentTime = (double) distances[index] / (double) velocity;
            totalTime += (index == distances.size() - 1 ? segmentTime : ceil(segmentTime));
        }
        return totalTime;
    }
    
    int minimumSpeed(vector<int>& distances, double hours) {
        int low = 1;
        int high = 10000000;
        int optimalSpeed = -1;
        
        while (low <= high) {
            int middle = (low + high) / 2;
            if (calculateTime(distances, middle) <= hours) {
                optimalSpeed = middle;
                high = middle - 1;
            } else {
                low = middle + 1;
            }
        }
        return optimalSpeed;
    }
};

The provided C++ code defines a solution for determining the minimum speed necessary to cover given distances within a specific time limit. The solution is encapsulated within a class called Solution that includes two primary functions: calculateTime and minimumSpeed.

• The calculateTime function accepts an array of distances and a velocity. It computes the total travel time needed to traverse each segment at the given velocity. For all segments except the last, the time is rounded up to the nearest whole number, accomplishing this rounding with the ceil function. The time for the last segment is added as is, without rounding.

• The minimumSpeed function uses a binary search approach to find the smallest integer speed between 1 and 10,000,000 that allows the total travel time computed by calculateTime to be less than or equal to the given time limit (hours). The function initializes two pointers, low and high, to bound the possible speeds and adjusts these bounds based on the computed travel times. If a speed meets the time constraint, it is noted as optimalSpeed, and the search continues to see if a lower feasible speed exists by adjusting the high pointer. If the computed time exceeds the allowed time, the search space is adjusted by moving the low pointer.

The logic in minimumSpeed effectively narrows the search space, determining the minimum necessary speed more efficiently than a linear search, leveraging the properties of sorted data inherent in the speed values being sequentially incremental. This binary search ensures that the function will return the lowest possible speed that meets the requirement, or -1 if no valid speed can be found (although the given range should typically suffice for realistic inputs).

class Solution {
    double calculateTime(int[] distances, int velocity) {
        double totalTime = 0.0;
        for (int idx = 0; idx < distances.length; idx++) {
            double segmentTime = (double) distances[idx] / (double) velocity;
            totalTime += (idx == distances.length - 1 ? segmentTime : Math.ceil(segmentTime));
        }
        return totalTime;
    }

    public int findMinimumSpeed(int[] distances, double hours) {
        int low = 1;
        int high = 10000000;
        int optimalSpeed = -1;

        while (low <= high) {
            int mid = (low + high) / 2;
            if (calculateTime(distances, mid) <= hours) {
                optimalSpeed = mid;
                high = mid - 1;
            } else {
                low = mid + 1;
            }
        }
        return optimalSpeed;
    }
}

The Java solution provided tackles the problem of determining the minimum speed required to travel given distances within a specified number of hours. The process leverages a binary search algorithm to efficiently find the optimal speed.

• The calculateTime method computes the total travel time for an array of distances at a given speed. For each segment, except the last one, it rounds up the segment's travel time to the nearest whole number using Math.ceil. This adjustment is crucial for segments where rounding might affect the ability to meet the total hours.

• The findMinimumSpeed function initiates the search for the minimum speed necessary to meet the deadline as defined by the hours parameter. The function defines a search range for the speed (from 1 to 10,000,000). Using a binary search approach:

  • The midpoint value of the current speed range (mid) is used as the trial speed.
  • The calculateTime method determines if the total travel time with this speed meets the deadline.
  • Based on this evaluation, adjust the search boundaries (low and high) to either half the search range if the current speed is sufficient (i.e., reduce high), or increase it if more speed is needed (i.e., increase low).
  • This binary refinement continues until the lowest possible speed meeting the time constraint is found.

The output from findMinimumSpeed is the minimal speed required such that travel across all distances can be completed within or under the given hours, effectively balancing speed and time constraints using an optimized approach.