Minimum Cost to Buy Apples - Problem
You're a traveling merchant looking to buy apples from various cities and return home! ๐
Given n cities numbered from 1 to n, you have a network of bidirectional roads connecting them. Each road has a travel cost, and each city sells apples at different prices. Here's the twist: after buying an apple, all road costs get multiplied by factor k for your return journey!
Your goal: For each possible starting city, find the minimum total cost to:
- Travel to any city to buy exactly one apple
- Return to your starting city (with increased road costs)
Input:
ncities androadsarray whereroads[i] = [a, b, cost]appleCostarray whereappleCost[i-1]is the cost of buying an apple from city i- Factor
kthat multiplies return journey costs
Output: Array where answer[i-1] is the minimum total cost starting from city i
Input & Output
example_1.py โ Basic Triangle
$
Input:
n = 3, roads = [[1,2,5],[2,3,2],[1,3,3]], appleCost = [1,2,3], k = 2
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Output:
[8, 8, 10]
๐ก Note:
From city 1: Best is to buy from city 1 (cost 0 + 1 + 0*2 = 1) but we need to consider the return journey cost structure. Actually going to city 2 costs 5+2+5*2 = 17, going to city 3 costs 3+3+3*2 = 12. So minimum is buying from city 1 itself: 0+1+0 = 1. Wait, let me recalculate properly...
example_2.py โ Linear Path
$
Input:
n = 2, roads = [[1,2,10]], appleCost = [5,1], k = 3
โบ
Output:
[35, 31]
๐ก Note:
From city 1: Buy from city 1 (0+5+0=5) or buy from city 2 (10+1+10*3=41). Min = 5. From city 2: Buy from city 2 (0+1+0=1) or buy from city 1 (10+5+10*3=45). Min = 1.
example_3.py โ Single City
$
Input:
n = 1, roads = [], appleCost = [10], k = 5
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Output:
[10]
๐ก Note:
Only one city, must buy apple there with no travel cost.
Constraints
- 1 โค n โค 1000
- 0 โค roads.length โค min(n ร (n - 1) / 2, 2000)
- roads[i].length == 3
- 1 โค ai, bi โค n
- ai โ bi
- 1 โค costi โค 1000
- appleCost.length == n
- 1 โค appleCost[i] โค 1000
- 1 โค k โค 10
- There are no repeated edges
Visualization
Tap to expand
Understanding the Visualization
1
Map the Network
Build a graph of cities connected by roads with travel costs
2
Calculate Outbound Costs
From each starting city, find shortest paths to all apple markets
3
Calculate Return Costs
From each apple market, find shortest paths back with traffic multiplier k
4
Find Optimal Strategy
For each starting city, choose the apple market that minimizes total journey cost
Key Takeaway
๐ฏ Key Insight: The problem combines shortest path finding with economic optimization. Use Dijkstra's algorithm twice: once for normal costs and once with multiplied return costs.
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Explanation
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