2473. Minimum Cost to Buy Apples π
Description
You are given a positive integer n
representing n
cities numbered from 1
to n
. You are also given a 2D array roads
, where roads[i] = [ai, bi, costi]
indicates that there is a bidirectional road between cities ai
and bi
with a cost of traveling equal to costi
.
You can buy apples in any city you want, but some cities have different costs to buy apples. You are given the 1-based array appleCost
where appleCost[i]
is the cost of buying one apple from city i
.
You start at some city, traverse through various roads, and eventually buy exactly one apple from any city. After you buy that apple, you have to return back to the city you started at, but now the cost of all the roads will be multiplied by a given factor k
.
Given the integer k
, return a 1-based array answer
of size n
where answer[i]
is the minimum total cost to buy an apple if you start at city i
.
Example 1:
Input: n = 4, roads = [[1,2,4],[2,3,2],[2,4,5],[3,4,1],[1,3,4]], appleCost = [56,42,102,301], k = 2 Output: [54,42,48,51] Explanation: The minimum cost for each starting city is the following: - Starting at city 1: You take the path 1 -> 2, buy an apple at city 2, and finally take the path 2 -> 1. The total cost is 4 + 42 + 4 * 2 = 54. - Starting at city 2: You directly buy an apple at city 2. The total cost is 42. - Starting at city 3: You take the path 3 -> 2, buy an apple at city 2, and finally take the path 2 -> 3. The total cost is 2 + 42 + 2 * 2 = 48. - Starting at city 4: You take the path 4 -> 3 -> 2 then you buy at city 2, and finally take the path 2 -> 3 -> 4. The total cost is 1 + 2 + 42 + 1 * 2 + 2 * 2 = 51.
Example 2:
Input: n = 3, roads = [[1,2,5],[2,3,1],[3,1,2]], appleCost = [2,3,1], k = 3 Output: [2,3,1] Explanation: It is always optimal to buy the apple in the starting city.
Constraints:
2 <= n <= 1000
1 <= roads.length <= 2000
1 <= ai, bi <= n
ai != bi
1 <= costi <= 105
appleCost.length == n
1 <= appleCost[i] <= 105
1 <= k <= 100
- There are no repeated edges.
Solutions
Solution 1: Heap-optimized Dijkstra's Algorithm
We enumerate the starting point, and for each starting point, we use Dijkstra's algorithm to find the shortest distance to all other points, and update the minimum value accordingly.
The time complexity is $O(n \times m \times \log m)$, where $n$ and $m$ are the number of cities and roads, respectively.
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