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3928. Minimum Cost to Buy Apples II

Description

You are given an integer n and an integer array prices of length n, where prices[i] is the price of apples at shop i.

You are also given a 2D integer array roads, where roads[i] = [ui, vi, costi, taxi] represents a bidirectional road:

  • ui and vi are the shops connected by the road.
  • costi is the cost to travel the road without carrying apples.
  • taxi is the multiplier applied to costi when traveling with apples.

For each shop i, you can either:

  • Buy apples locally at shop i for prices[i].
  • Travel empty to any shop j using any number of roads, buy apples for prices[j], and return to shop i while carrying apples, paying cost * tax on each road used for the return trip.

The forward path, where you travel empty, and the return path may be different.

Return an integer array ans of length n, where ans[i] is the minimum total cost to buy apples starting from shop i.

 

Example 1:

Input: n = 2, prices = [8,3], roads = [[0,1,1,2]]

Output: [6,3]

Explanation:

Shop i prices[i] Shop j prices[j] costi taxi Travel cost Return cost Total Minimum
0 8 1 3 1 2 1 1 * 2 = 2 1 + 2 + 3 = 6 min(8, 6) = 6
1 3 0 8 1 2 1 1 * 2 = 2 1 + 2 + 8 = 11 min(3, 11) = 3

Thus, the answer is [6, 3].

Example 2:

Input: n = 3, prices = [9,4,6], roads = [[0,1,1,3],[1,2,4,2]]

Output: [8,4,6]

Explanation:

​​​​​​​

Shop i prices[i] Shop j prices[j] costi taxi Travel cost Return cost Total Minimum
0 9 1 4 1 3 1 1 * 3 = 3 1 + 3 + 4 = 8 min(9, 8) = 8
1 4 2 6 4 2 4 4 * 2 = 8 4 + 8 + 6 = 18 min(4, 18) = 4
2 6 1 4 4 2 4 4 * 2 = 8 4 + 8 + 4 = 16 min(6, 16) = 6

Thus, the answer is [8, 4, 6].

Example 3:

Input: n = 3, prices = [10,11,1], roads = [[0,2,1,3],[1,2,3,4],[0,1,5,2]]

Output: [5,11,1]

Explanation:

​​​​​​​​​​​​​​

Shop i prices[i] Shop j prices[j] costi taxi Travel cost Return cost Total Minimum
0 10 2 1 1 3 1 1 * 3 = 3 1 + 3 + 1 = 5 min(10, 5) = 5
1 11 2 1 3 4 3 3 * 4 = 12 3 + 12 + 1 = 16 min(11, 16) = 11
2 1 0 10 1 3 1 1 * 3 = 3 1 + 3 + 10 = 14 min(1, 14) = 1

Thus, the answer is [5, 11, 1].

 

Constraints:

  • 1 <= n <= 1000
  • prices.length == n
  • 1 <= prices[i] <= 109
  • 0 <= roads.length <= min(n × (n - 1) / 2, 2000)
  • roads[i] = [ui, vi, costi, taxi]
  • 0 <= ui, vi <= n - 1
  • ui != vi
  • 1 <= costi <= 109
  • ​​​​​​​1 <= tax​​​​​​​i <= 100​​​​​​​
  • There are no repeated edges.

Solutions

Solution 1

Thinking

A separate shortest-path search from every shop toward every other shop is about \(O(n^2\log n)\), which is tight for \(n\le 1000\). Outbound travel is empty-handed while the return multiplies by \(\textit{tax}\), so the two directions do not share one distance table.

Compute empty-handed distances and loaded distances separately — the latter is shortest paths on edges \(cost\cdot tax\). For shop \(i\) the answer is \(\min_j(\mathrm{dist}_{\mathrm{empty}}(i,j)+\textit{prices}[j]+\mathrm{dist}_{\mathrm{load}}(j,i))\).

This directory has no implemented solution yet; the walkthrough stops at that two-weight reduction.

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