Minimum Cost to Buy Apples - Problem

You're a traveling merchant looking to buy apples from various cities and return home! ๐ŸŽ

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:

  1. Travel to any city to buy exactly one apple
  2. Return to your starting city (with increased road costs)

Input:

  • n cities and roads array where roads[i] = [a, b, cost]
  • appleCost array where appleCost[i-1] is the cost of buying an apple from city i
  • Factor k that 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
โ€บ Output: [8, 8, 10]
๐Ÿ’ก 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
โ€บ Output: [35, 31]
๐Ÿ’ก 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
โ€บ Output: [10]
๐Ÿ’ก Note: Only one city, must buy apple there with no travel cost.

Constraints

  • 1 โ‰ค n โ‰ค 1000
  • 0 โ‰ค roads.length โ‰ค min(n ร— (n - 1) / 2, 2000)
  • roads[i].length == 3
  • 1 โ‰ค ai, bi โ‰ค n
  • ai โ‰  bi
  • 1 โ‰ค costi โ‰ค 1000
  • appleCost.length == n
  • 1 โ‰ค appleCost[i] โ‰ค 1000
  • 1 โ‰ค k โ‰ค 10
  • There are no repeated edges

Visualization

Tap to expand
๐ŸŽ Apple Shopping StrategyHome๐ŸŽ$2๐ŸŽ$1Cost: 3Cost: 5Return: 3ร—2=6Return: 5ร—2=10๐Ÿ’ฐ Cost AnalysisRoute 1 (Red Apple):Go: 3 + Buy: $2 + Return: 6 = $11Route 2 (Orange Apple):Go: 5 + Buy: $1 + Return: 10 = $16โœ… Best Choice: Route 1 ($11)๐Ÿ—๏ธ LegendOutbound JourneyReturn Journey (kร—cost)Starting CityApple Market
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
๐ŸŽฏ 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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