Count All Possible Routes

Problem Description

You are given an array of distinct positive integers locations where locations[i] represents the position of city i. You are also given integers start, finish and fuel representing the starting city, ending city, and the initial amount of fuel you have, respectively.

At each step, if you are at city i, you can pick any city j such that j != i and 0 <= j < locations.length and move to city j. Moving from city i to city j reduces the amount of fuel you have by |locations[i] - locations[j]|. Please notice that |x| denotes the absolute value of x.

Notice that fuel cannot become negative at any point in time, and that you are allowed to visit any city more than once (including start and finish).

Return the count of all possible routes from start to finish. Since the answer may be too large, return it modulo 10^9 + 7.

Key Insights

• You can move between cities as long as you have enough fuel.
• The problem can be approached using Dynamic Programming or Memoization to avoid redundant calculations.
• Each route can be represented as a recursive function that explores all possible moves from a given city.
• The absolute difference in locations determines the fuel cost for each move.

Space and Time Complexity

Time Complexity: O(n * fuel^2), where n is the number of cities and fuel is the maximum fuel available. Space Complexity: O(n * fuel), for storing the memoization table.

Solution

The solution can be approached using a recursive function combined with memoization to count possible routes from the start city to the finish city while keeping track of the remaining fuel. The algorithm explores each possible next city and deducts the fuel cost accordingly.

1. Use a recursive function countRoutes(currentCity, remainingFuel):

   • Base Case: If you reach the finish city, increment the route count.
   • For each city, calculate the fuel cost and check if you can move there.
   • Use memoization to store results for each city and fuel state to avoid recomputation.

2. The recursive function will be called with the starting city and the initial fuel.

Code Solutions