---
# System prepended metadata

title: 1575. Count All Possible Routes
tags: [leetcode]

---

## 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 109 + 7.

- Example 1:

>Input: locations = [2,3,6,8,4], start = 1, finish = 3, fuel = 5
Output: 4
>>Explanation: The following are all possible routes, each uses 5 units of fuel:
1 -> 3
1 -> 2 -> 3
1 -> 4 -> 3
1 -> 4 -> 2 -> 3

- Example 2:

>Input: locations = [4,3,1], start = 1, finish = 0, fuel = 6
Output: 5
>>Explanation: The following are all possible routes:
1 -> 0, used fuel = 1
1 -> 2 -> 0, used fuel = 5
1 -> 2 -> 1 -> 0, used fuel = 5
1 -> 0 -> 1 -> 0, used fuel = 3
1 -> 0 -> 1 -> 0 -> 1 -> 0, used fuel = 5

- Example 3:

>Input: locations = [5,2,1], start = 0, finish = 2, fuel = 3
Output: 0
>>Explanation: It is impossible to get from 0 to 2 using only 3 units of fuel since the shortest route needs 4 units of fuel.

- Constraints:

>2 <= locations.length <= 100
1 <= locations[i] <= 109
All integers in locations are distinct.
0 <= start, finish < locations.length
1 <= fuel <= 200

## Solution
- The problem can be done by using dynamic programming, with the index showing the `starting from location ith and the left fuel jth`
- We can construct the array by the given constraint value
```cpp=
vector<vector<int>> dp(locations.size(), vector<int>(201, -1));
```
- So, the return value should be
```cpp=
return search(locations, dp, start, finish, fuel);
```
- For the searching, return the value if the array has already gotten the value
```cpp=
if (dp[start][fuel] != -1) return dp[start][fuel];
```
- For zero fuel, the value can be returned at once, it is only pausible if it is the finish index
```cpp=
if (fuel == 0) return dp[start][0] = (start == finish);
```
- Because we can iterate back and forth, the answer for the start should be `1` if it is ending point (meaning that we do not need to do anything)
```cpp=
int cur, ans = (start == finish);
```
- Then iterate through all the points, find the pausible nexthop, which means that the fuel still remains. Do not construct a self loop
```cpp=
if (i == start) continue;
cur = fuel - abs(locations[start] - locations[i]);
if (cur >= 0) ans = (ans + search(locations, dp, i, finish, cur)) % mod;
```
- Then return the answer back and store it at the same time
```cpp=
return dp[start][fuel] = ans;
```