What is the Minimum Number of Refueling Stops problem and how can it be solved using a greedy algorithm?

What is the Minimum Number of Refueling Stops problem and how can it be solved using a greedy algorithm?

The Minimum Number of Refueling Stops problem is a problem in which we are given a target distance, a car with a certain amount of fuel capacity, and a list of gas stations with their distances from the starting point. The goal is to determine the minimum number of refueling stops required to reach the target distance, assuming that the car starts with a full tank of fuel and can refuel at any gas station.

This problem can be solved using a greedy algorithm. The algorithm works as follows:

1. Initialize the number of refueling stops to 0 and the current position to the starting point.
2. While the current position is less than the target distance:
a. Find the next gas station that is reachable from the current position and has the maximum amount of fuel.
b. If no reachable gas station is found, return -1 (indicating that it is not possible to reach the target distance).
c. Increment the number of refueling stops by 1.
d. Update the current position to the distance of the selected gas station.
3. Return the number of refueling stops.

The greedy algorithm selects the gas station with the maximum amount of fuel at each step, ensuring that the car can travel the maximum distance possible before refueling. This approach guarantees that the minimum number of refueling stops is achieved, as the car is always refueled at the most optimal gas station.