gistlib - a trail is an array of coordinates. determine the shortest path along some trails to a goal in swift

One approach to solve this problem is to use the concept of graphs and apply a shortest path algorithm like Dijkstra's algorithm to find the shortest path from the starting point to the goal.

First, we need to represent the trails as a graph. We can do this by treating each coordinate as a node and each trail segment between two coordinates as a weighted edge between those nodes. The weight of an edge can be calculated as the Euclidean distance between the two nodes.

Here's a sample code to create a graph from an array of trails:


main.swift
typealias Node = Int

struct Edge {
    let to: Node
    let weight: Double
}

var graph = [[Edge]]()
var coordinates = [CGPoint]() // array of coordinates representing the trail

// Populate the graph with edges based on the trail coordinates
for i in 0..<coordinates.count {
    var edges = [Edge]()
    for j in i+1..<coordinates.count {
        let weight = Double(coordinates[i].distance(to: coordinates[j]))
        edges.append(Edge(to: j, weight: weight))
    }
    graph.append(edges)
}
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Once we have the graph, we can use Dijkstra's algorithm to find the shortest path from the starting point to the goal. Here's a sample code to apply Dijkstra's algorithm:


main.swift
func dijkstra(start: Node, graph: [[Edge]]) -> [Double] {
    var shortestDistances = [Double](repeating: Double.infinity, count: graph.count)
    shortestDistances[start] = 0.0
    
    var visited = Set<Node>()
    var queue = [start]
    
    while let currentNode = queue.popLast() {
        visited.insert(currentNode)
        
        for edge in graph[currentNode] {
            if !visited.contains(edge.to) {
                let tentativeDistance = shortestDistances[currentNode] + edge.weight
                if tentativeDistance < shortestDistances[edge.to] {
                    shortestDistances[edge.to] = tentativeDistance
                    queue.append(edge.to)
                }
            }
        }
    }
    
    return shortestDistances
}
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After running Dijkstra's algorithm, we can get the shortest distance to all nodes from the starting point. We can then backtrack from the goal to the starting point to get the shortest path. Here's a sample code to backtrack from the goal to the starting point:


main.swift
func backtrack(from: Node, to: Node, shortestDistances: [Double], graph: [[Edge]] ) -> [Node] {
    var path = [to]
    var currentNode = to
    
    while currentNode != from {
        for edge in graph[currentNode] {
            if shortestDistances[edge.to] + edge.weight == shortestDistances[currentNode] {
                path.append(edge.to)
                currentNode = edge.to
                break
            }
        }
    }
    
    return path.reversed()
}
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With these functions, we can find the shortest path from the starting point to the goal as follows:


main.swift
let start = 0 // index of the starting coordinate
let goal = 10 // index of the goal coordinate

let shortestDistances = dijkstra(start: start, graph: graph)
let shortestPath = backtrack(from: start, to: goal, shortestDistances: shortestDistances, graph: graph)
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The shortestPath array will contain the indices of the nodes representing the shortest path from the starting point to the goal. We can convert these indices to actual coordinates to get the actual trail path.

a trail is an array of coordinates. determine the shortest path along some trails to a goal in swift

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