Dijkstra Algorithm
The Dijkstra Algorithm, named after its creator, Edsger W. Dijkstra, is a widely-used graph traversal algorithm that finds the shortest path between a starting node and all other nodes in a weighted graph. Often used in routing and navigation applications, Dijkstra's algorithm works by iteratively selecting the node with the smallest known distance from the starting node, and updating the distances of its neighboring nodes based on the edge weights connecting them. This process is repeated until all nodes have been visited or the shortest path to the destination node has been determined.
In order to efficiently keep track of distances and visited nodes, the algorithm typically uses a priority queue data structure. Initially, the starting node is assigned a distance of zero, while all other nodes are assigned an infinite distance. As the algorithm progresses, it extracts nodes with the smallest distance from the priority queue, marking them as visited and updating the distances of their neighbors. This ensures that nodes are visited in the order of their proximity to the starting node, guaranteeing the discovery of the shortest path. Once the destination node has been visited or all nodes have been processed, the path can be reconstructed by tracing back from the destination node to the starting node via the shortest path tree formed during the algorithm's execution.
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package com.algorithmexamples.graphs.directed.weighted
import com.algorithmexamples.datastructures.IndexedPriorityQueue
import com.algorithmexamples.datastructures.Stack
import com.algorithmexamples.graphs.NoSuchPathException
class Dijkstra(graph: DWGraph, val from: Int) {
/**
* distTo[v] = distance of shortest s->v path
*/
private val distTo: DoubleArray = DoubleArray(graph.V, { if (it == from) 0.0 else Double.POSITIVE_INFINITY })
/**
* edgeTo[v] = last edge on shortest s->v path
*/
private val edgeTo: Array<DWGraph.Edge?> = arrayOfNulls(graph.V)
/**
* priority queue of vertices
*/
private val pq: IndexedPriorityQueue<Double> = IndexedPriorityQueue(graph.V)
init {
if (graph.edges().any { it.weight < 0 }) {
throw IllegalArgumentException("there is a negative weight edge")
}
// relax vertices in order of distance from s
pq.insert(from, distTo[from])
while (!pq.isEmpty()) {
val v = pq.poll().first
for (e in graph.adjacentEdges(v)) {
relax(e)
}
}
}
// relax edge e and update pq if changed
private fun relax(e: DWGraph.Edge) {
val v = e.from
val w = e.to
if (distTo[w] > distTo[v] + e.weight) {
distTo[w] = distTo[v] + e.weight
edgeTo[w] = e
if (pq.contains(w)) {
pq.decreaseKey(w, distTo[w])
} else {
pq.insert(w, distTo[w])
}
}
}
/**
* Returns the length of a shortest path from the source vertex `s` to vertex `v`.
* @param v the destination vertex
* @return the length of a shortest path from the source vertex `s` to vertex `v`;
* `Double.POSITIVE_INFINITY` if no such path
*/
fun distTo(v: Int): Double {
return distTo[v]
}
/**
* Returns true if there is a path from the source vertex `s` to vertex `v`.
* @param v the destination vertex
* @return `true` if there is a path from the source vertex
* `s` to vertex `v`; `false` otherwise
*/
fun hasPathTo(v: Int): Boolean {
return distTo[v] < java.lang.Double.POSITIVE_INFINITY
}
/**
* Returns a shortest path from the source vertex `s` to vertex `v`.
* @param v the destination vertex
* @return a shortest path from the source vertex `s` to vertex `v`
* as an iterable of edges, and `null` if no such path
*/
fun pathTo(v: Int): Iterable<DWGraph.Edge> {
if (!hasPathTo(v)) throw NoSuchPathException("There is no path from [$from] to [$v]")
val path = Stack<DWGraph.Edge>()
var e = edgeTo[v]
while (e != null) {
path.push(e)
e = edgeTo[e.from]
}
return path
}
}
