Problem Description

In this problem, we're given an array locations which contains distinct positive integers representing the position of different cities. The positions are essentially distances between some reference point and each city, but those distances are abstract in the problem; they don't need to represent real-world units.

Additionally, we have three integers: start, finish, and fuel. These represent the index of the starting city in the locations array, the index of the destination city, and the initial amount of fuel we have, respectively.

The rules of the game are:

• You can travel from any city i to any different city j.
• Traveling from city i to city j consumes |locations[i] - locations[j]| units of fuel (|x| denotes the absolute value of x). You cannot make the trip if you don't have enough fuel to reach city j.
• You can visit any city more than once, and the fuel left can NOT be negative at any point.

The goal is to determine how many different routes you can take from the start city to the finish city, given the constraints of fuel usage. Importantly, the number of routes can be very large, so we return the result modulo 10^9 + 7.

Intuition

To solve this problem, we can use depth-first search (DFS) combined with memoization to avoid redundant calculations. Here is the thought process we use to arrive at the solution:

1. The core of the solution is the function dfs(i, k) which calculates the number of possible routes from city i with k fuel remaining to reach the finish city.

2. Initially, if the current city i is the same as the destination finish, we record one possible route (as no fuel is needed to reach the destination if you are already there).

3. If we don't have enough fuel to directly reach the destination from our current city, there's no need to explore further from this state; the function returns 0.

4. For each city i, we iterate over all other cities j, and recursively call our dfs function to find out how many routes we can take from city j to finish, with the fuel reduced by the amount it would take to travel from i to j.

5. We use memoization to store already calculated states (representing unique combinations of the current city and the remaining fuel), so we do not repeat computations for those states. This dramatically improves performance.

6. Since we need to return the number of routes modulo 10^9 + 7, we take the modulo after every addition to keep the numbers within an integer range and comply with the problem constraints.

By summarizing this intuition, we get our dynamic top-down approach (DFS with memoization). It's important to note that the number of possible routes could be enormous, hence the need for modulo operation to avoid integer overflow.

For an alternative and more space-efficient approach, one could also use dynamic programming (DP) to solve this problem bottom-up. However, that approach is more complex and requires a deeper understanding of DP to implement correctly.

Learn more about Memoization and Dynamic Programming patterns.

Solution Approach

The solution to this problem leverages recursion and memoization to implement a depth-first search that efficiently counts all the possible routes from the starting city to the destination, considering the given fuel limit.

Here's how the implementation unfolds, step by step:

1. Memoization: To prevent the same calculations from being executed multiple times, we employ memoization. This is done by decorating our recursive function with @cache (available in Python 3.9+). The cache decorator automatically memorizes the return values of the function calls based on their input arguments.

2. Recursive Function dfs(i, k): This function takes the current city index i and the remaining fuel k as arguments. It returns the count of all possible routes from city i to the destination that can be achieved with k or less amount of fuel.

3. Base Cases:

   • If the remaining fuel k is less than the absolute difference in location between city i and the destination city finish (i.e., k < abs(locations[i] - locations[finish])), the function returns 0, indicating that the destination cannot be reached from this state.
   • If city i is the destination city (i.e., i == finish), the function initializes the route count ans with 1 since you're already at the destination.

4. Recursive Exploration:

   • We iterate through all other possible cities (represented by index j), and if j != i, we calculate the fuel required to travel from city i to city j.
   • We recursively call dfs(j, k - abs(locations[i] - locations[j])) to find out the number of routes from city j to the destination with the updated remaining fuel.
   • The result for each city j is added to ans modulo mod (which is 10**9 + 7) to ensure we maintain the result within the bounds of an integer and comply with the problem constraints about the modulo.

5. Returning the Result:

   • The function finally returns ans, which is the total number of routes from city i with fuel k to the destination.

6. Solution Execution:

   • The entry point for our calculation is dfs(start, fuel), this call initiates the depth-first search from the starting city with the given amount of fuel.

We don't consider dynamic programming (DP) approach here, but it's important to note that it could also be utilized in a situation where memory consumption needs to be optimized. The DP approach would iteratively build up a table f[i][k] representing the number of paths from city i with k remaining fuel to finish. The final answer would then be read from f[start][fuel].

Both DFS with memoization and the DP approach use time complexity O(n^2 * m) and space complexity O(n * m) where n is the number of cities and m is the amount of fuel.

Example Walkthrough

Let's go through an example to illustrate the solution approach using the depth-first search (DFS) with memoization.

Suppose we have the following input:

• locations array: [4, 3, 1]
• start: 0 (the starting city is at position 4)
• finish: 2 (the destination city is at position 1)
• fuel: 4 units

Using the solution approach:

1. Initialization: We start by initializing our memoization cache (not shown in this example). Then we begin our algorithm at city index 0 (which is at location 4) with 4 fuel.

2. Recursive Function dfs(i, k): Our first call will be dfs(0, 4).

3. Base Cases: We check our base cases:

   • i is not equal to finish, so we do not add 1 to our route count.
   • We have enough fuel (more than the distance from our current city to finish), so we do not return 0.

4. Recursive Exploration:

   • From city index 0, we calculate the fuel required to get to other cities:
     • To city index 1 (locations[1] == 3): fuel cost is |4 - 3| = 1. We then recurse with dfs(1, 4 - 1), which is dfs(1, 3).
     • To city index 2 (locations[2] == 1): fuel cost is |4 - 1| = 3. We then recurse with dfs(2, 4 - 3), which is dfs(2, 1).

5. Exploring dfs(1, 3):

   • Check base case: Not at destination, and have enough fuel.
   • Recursive exploration from city index 1:
     • To city index 0 (already visited, not destination): dfs(0, 2).
     • To city index 2 (destination): direct fuel cost is |3 - 1| = 2. Since we have 3 fuel, we can reach the destination, so dfs(2, 1) from here adds 1 to our routes.

6. Exploring dfs(2, 1):

   • Check base case: We're at the destination, so this recursive path adds 1 to our route count.

7. Exploring dfs(0, 2):

   • Check base case: Not at destination, and have enough fuel.
   • Recursive exploration from city index 0:
     • To city index 1: dfs(1, 1).
     • To city index 2 (destination): direct fuel cost is 3, but we only have 2 fuel, so this path does not add to our route count.

8. Exploring dfs(1, 1):

   • Check base case: Not at destination, and we don't have enough fuel left to go anywhere else, so this does not add to our route count.

9. Adding up the routes:

   • From the original dfs(0, 4), we got routes from exploring both dfs(1, 3) and dfs(2, 1), which in total gives us 2 routes (the exact paths are 0->2 and 0->1->2).

10. Returning the Result:

    • The total number of possible routes from our starting city with the given fuel to reach the destination is 2 when using DFS with memoization. The answer is calculated as (1 from dfs(2, 1)) + (1 from dfs(1, 3) -> dfs(2, 1)). Thus, we conclude dfs(0, 4) = 2.

This example demonstrates how we recursively explore different paths, using memoization to avoid redundant calculations, and how we handle the returns when we reach the destination or when we lack the fuel to continue. Each recursive call returns the number of ways to get from the current city to the destination with the given amount of remaining fuel. Finally, we sum all possible routes to get the total count.

Solution Implementation

Time and Space Complexity

The time complexity of the code is O(n^2 * m) where n is the number of locations and m is the amount of fuel provided. This is because, for every recursive call to dfs, there's a potential for iterating over all n locations (except the current one), and this can happen at most m times due to the iterative decrement of fuel at each state. Multiplying these together, we end up with n^2 * m considering every state (i, k) for each location i and fuel k.

The space complexity of the code is O(n * m) due to the memoization used in the dfs function. This memoization stores the results of each unique call to dfs(i, k). Since i can take n different values and k can take up to m different values, the cache could potentially store n * m distinct states.

Learn more about how to find time and space complexity quickly using problem constraints.