1928. Minimum Cost to Reach Destination in Time
Description
There is a country of n
cities numbered from 0
to n - 1
where all the cities are connected by bi-directional roads. The roads are represented as a 2D integer array edges
where edges[i] = [xi, yi, timei]
denotes a road between cities xi
and yi
that takes timei
minutes to travel. There may be multiple roads of differing travel times connecting the same two cities, but no road connects a city to itself.
Each time you pass through a city, you must pay a passing fee. This is represented as a 0-indexed integer array passingFees
of length n
where passingFees[j]
is the amount of dollars you must pay when you pass through city j
.
In the beginning, you are at city 0
and want to reach city n - 1
in maxTime
minutes or less. The cost of your journey is the summation of passing fees for each city that you passed through at some moment of your journey (including the source and destination cities).
Given maxTime
, edges
, and passingFees
, return the minimum cost to complete your journey, or -1
if you cannot complete it within maxTime
minutes.
Example 1:
Input: maxTime = 30, edges = [[0,1,10],[1,2,10],[2,5,10],[0,3,1],[3,4,10],[4,5,15]], passingFees = [5,1,2,20,20,3] Output: 11 Explanation: The path to take is 0 -> 1 -> 2 -> 5, which takes 30 minutes and has $11 worth of passing fees.
Example 2:
Input: maxTime = 29, edges = [[0,1,10],[1,2,10],[2,5,10],[0,3,1],[3,4,10],[4,5,15]], passingFees = [5,1,2,20,20,3] Output: 48 Explanation: The path to take is 0 -> 3 -> 4 -> 5, which takes 26 minutes and has $48 worth of passing fees. You cannot take path 0 -> 1 -> 2 -> 5 since it would take too long.
Example 3:
Input: maxTime = 25, edges = [[0,1,10],[1,2,10],[2,5,10],[0,3,1],[3,4,10],[4,5,15]], passingFees = [5,1,2,20,20,3] Output: -1 Explanation: There is no way to reach city 5 from city 0 within 25 minutes.
Constraints:
1 <= maxTime <= 1000
n == passingFees.length
2 <= n <= 1000
n - 1 <= edges.length <= 1000
0 <= xi, yi <= n - 1
1 <= timei <= 1000
1 <= passingFees[j] <= 1000
- The graph may contain multiple edges between two nodes.
- The graph does not contain self loops.
Solutions
Solution 1: Dynamic Programming
We define $f[i][j]$ to represent the minimum cost to reach city $j$ from city $0$ after $i$ minutes. Initially, $f[0][0] = \textit{passingFees}[0]$, and the rest $f[0][j] = +\infty$.
Next, within the time range $[1, \textit{maxTime}]$, we traverse all edges. For each edge $(x, y, t)$, if $t \leq i$, then we:
- Can first spend $i - t$ minutes to reach city $y$ from city $0$, then spend $t$ minutes to reach city $x$ from city $y$, and add the passing fee to reach city $x$, i.e., $f[i][x] = \min(f[i][x], f[i - t][y] + \textit{passingFees}[x])$;
- Can also first spend $i - t$ minutes to reach city $x$ from city $0$, then spend $t$ minutes to reach city $y$ from city $x$, and add the passing fee to reach city $y$, i.e., $f[i][y] = \min(f[i][y], f[i - t][x] + \textit{passingFees}[y])$.
The final answer is $\min{f[i][n - 1]}$, where $i \in [0, \textit{maxTime}]$. If the answer is $+\infty$, return $-1$.
The time complexity is $O(\textit{maxTime} \times (m + n))$, where $m$ and $n$ are the number of edges and cities, respectively. The space complexity is $O(\textit{maxTime} \times n)$.
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