In this lecture, we will cover the ‘Minimum Cost Pathfinding’ problem, which is frequently presented in coding tests, and examine step-by-step how to solve it using Swift. We will guide you on understanding the algorithm as well as efficiently solving the problem using Swift’s features.
Problem Description
This problem entails finding a path that minimizes the cost of traveling from one city to another.
Each city is represented as a node in a graph, and cities are connected by roads.
Each road has a different cost associated with traveling on it.
The goal is to find the minimum cost to reach the destination city from the starting city.
Here is an example of the problem:
Input:
The number of cities N, and the number of roads M
Following this, M lines will provide information about each road.
The road information is given as (A, B, C), indicating that traveling from city A to city B requires a cost of C.
Lastly, the starting city and destination city are provided.
Output:
The minimum cost from the starting city to the destination city
Problem Solving Strategy
To solve this problem, we will use Dijkstra’s algorithm.
Dijkstra’s algorithm is a method for finding the shortest path in a weighted graph and is particularly effective for handling non-negative costs.
The main idea of this algorithm is to select the nearest unvisited node based on the shortest path found so far.
Here is how to solve the problem step by step using Dijkstra’s algorithm:
- Construct the graph in the form of an adjacency list.
- Initialize an array to store the minimum cost to each city.
- Begin at the starting city and update the information for the adjacent cities.
- Use a priority queue to determine the next city to visit.
- Repeat until the destination city is reached.
Swift Code Implementation
Now, let’s implement the code in Swift according to the above strategy.
Here is an example code implementing Dijkstra’s algorithm:
import Foundation
struct Edge {
let to: Int
let cost: Int
}
func dijkstra(start: Int, end: Int, graph: [[Edge]]) -> Int {
var minCosts = Array(repeating: Int.max, count: graph.count)
var priorityQueue = [(cost: Int, vertex: Int)]()
minCosts[start] = 0
priorityQueue.append((0, start))
while !priorityQueue.isEmpty {
let current = priorityQueue.removeFirst()
if current.vertex == end {
return current.cost
}
for edge in graph[current.vertex] {
let newCost = current.cost + edge.cost
if newCost < minCosts[edge.to] {
minCosts[edge.to] = newCost
priorityQueue.append((newCost, edge.to))
}
}
priorityQueue.sort { $0.cost < $1.cost }
}
return minCosts[end] == Int.max ? -1 : minCosts[end]
}
// Example of the graph
let N = 5 // Number of cities
let M = 7 // Number of roads
var graph = [[Edge]](repeating: [], count: N)
let roads: [(Int, Int, Int)] = [
(0, 1, 10),
(0, 2, 5),
(1, 2, 2),
(1, 3, 1),
(2, 1, 3),
(2, 3, 9),
(3, 4, 2)
]
for road in roads {
graph[road.0].append(Edge(to: road.1, cost: road.2))
graph[road.1].append(Edge(to: road.0, cost: road.2)) // Assuming bidirectional roads
}
// Starting city and destination city
let start = 0
let end = 4
let minCost = dijkstra(start: start, end: end, graph: graph)
print("Minimum cost: \(minCost)") // Output: Minimum cost: 12
Code Explanation
The Swift code above implements an algorithm that finds the minimum cost to travel from one city to another using several functions.
The dijkstra function, starting from each city, receives connection information about the cities through the graph parameter and returns the minimum cost via the starting and destination cities.
1. Edge Structure:
Stores connection information between cities. to is the index of the connected city, and cost is the cost of that movement.
2. dijkstra Function:
This is the main logic for calculating the minimum cost. It explores the cities adjacent to the starting city and continuously updates the minimum cost.
A priority queue is used to determine the next move to the closest city.
3. Graph Structure:
The graph is constructed in an adjacency list format and contains information about the roads connected to each city.
4. Result: Outputs the minimum cost from the given starting city to the destination city.
Result Analysis
In the code above, based on the input cities and road information, the minimum cost to travel from starting city 0 to destination city 4 is calculated as 12.
This indicates that an appropriate connection and cost management were achieved.
Therefore, we can conclude that the optimized path provided through Dijkstra's algorithm can indeed be found.
Conclusion
In this lecture, we applied Dijkstra's algorithm to solve the minimum cost problem using Swift.
We learned how to understand the algorithm and implement it in actual code.
Understanding and utilizing the algorithm well will provide a foundation that can be useful in various coding tests and real-world applications.
Additionally, to tackle more complex problems, it is advisable to learn various algorithms and continuously practice whenever encountering problems.
In the next lecture, we will cover other algorithms to solve more problems and further enhance our skills.