Ninja And Stops

Posted: 16 Mar, 2021
Difficulty: Hard

PROBLEM STATEMENT

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Ninja wants to travel from his house to a given destination, which is ‘X’ miles from his house. Along the way, he needs to fill gas in his vehicle. He knows there are ‘Y’ stations in his way. He also knows the distance between the station and his house, and how many liters of gas that particular station has.

Ninja starts his journey, with an infinite capacity of the tank filled with ‘Z’ liters of starting fuel. Suppose his vehicle uses 1 liter of gas for every mile, and ninja can stop at any gas station, transfer all the available gas at that station and then move ahead.

Now, you need to find out what is the minimum number of stops Ninja must make to reach his desired destination.

Note:
Note that if Ninja reaches a particular stop with no fuel, it can still fill his tank at that stop and continue his journey ahead. Similarly, if he reaches his destination with no fuel, it is still considered to have arrived.
For example :
Given X = 10, Y = 4, ARR[Y] = {[1, 6], [2, 3], [3, 3], [6, 4]} and Z = 1
So the path followed in this case would look like this:

Ninja starts with 1L of gas. 
Drives to the first gas station at position 1, using 1L of gas, then refueling with 6L of gas.
Then, drive to position 6, using 5L of gas, then refueling 4L in the current 1L of gas, making it a total of 5L of gas.
Finally, drive to the destination consuming 4L of gas.
So, Ninja made 2 refueling stops before reaching the destination. So, you need to print 2. 
Input Format:
The first line contains an integer ‘T’ which denotes the number of test cases or queries to be run. Then the test cases are as follows.

The first line of each test case contains three space-separated integers ‘X’, ‘Y’ and ‘Z’, denoting distance in miles, number of gas stations, and starting fuel of the vehicle.

The next ‘Y’ lines of each test contain an array of ‘Y’ pairs where each pair denotes the distance from the house and available fuel for a refill.
Output Format:
For each test case, you need to return a single integer denoting the minimum stops made to reach the destination.

Print the output of each test case in a separate line.
Note:
You don’t need to print anything; It has already been taken care of. Just implement the given function.
Constraints:
1 <= T <= 10
1 <= X, Z  <= 10^7
0 <= size of Y <= 10^5
1 <= Y1, Y2 <= 10^7

Time limit: 1 sec
Approach 1

The simple idea that we use in this approach will be to check all the available paths. For this we create a recursive function let’s say MINIMUM_STOP_HELPER() that will return the desired path. The function will take fuel left, distance travelled, next gas station, and the array of all the stations as its parameters.

 

The base conditions for this recursive function will be:

 

  • If you have enough fuel
  • If you have reached the target
  • If no stations left and you did not reach the target
  • If you cannot reach the next fuel station with the current fuel.

 

Algorithm:

 

  • Create the recursive function.
  • Base Cases:
    • If you have enough fuel to reach destination from start
      • Return 0
    • If you have reached the target
      • Return 0
    • If no stations left and you did not reach the target
      • Return -1
    • If you cannot reach the next fuel station with the current fuel.
      • Return -1
  • For the main function:
    • Find minimum of the two cases:
      • If you refill at the next station.
      • If you do not refill at the next station.
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