# Dijkstra's Algorithm

## Contents

Definition of Dijkstra's Algortithm

"Dijkstra's algorithm, named after Dutch computer scientist Edsger Dijkstra who conceived it in 1959, is a graph search algorithm in order to solve the single-source shortest path problem for a graph with non negative edge path costs, It computes length of the shortest path from the source to each of the remaining vertices in the graph. This algorithm is often used in routing."

Relevant concepts:

1. short path: The problem of finding the shortest path in a graph from one vertex to another. "Shortest" may be least number of edges, least total weight, etc.
2. source: A vertex of a directed graph with no incoming edges. More formally, a vertex with in-degree 0.
3. vertex: An item in a graph. It is also known as a node.
4. edge: A connection between two vertices of a graph.
5. weight: In graph area, it means the distance between two nodes.

Despcription of Algorithm

"The algorithm works by keeping for each vertex v the cost d[v] of the shortest path found so far between s and v. Initially, this value is 0 for the source vertex s (d[s]=0), and infinity for all other vertices, representing the fact that we do not know any path leading to those vertices (d[v]=? for every v in V, except s). When the algorithm finishes, d[v] will be the cost of the shortest path from s to v -- or infinity, if no such path exists. The basic operation of Dijkstra's algorithm is edge relaxation: if there is an edge from u to v, then the shortest known path from s to u (d[u]) can be extended to a path from s to v by adding edge (u,v) at the end. This path will have length d[u]+w(u,v). If this is less than the current d[v], we can replace the current value of d[v] with the new value. Edge relaxation is applied until all values d[v] represent the cost of the shortest path from s to v. The algorithm is organized so that each edge (u,v) is relaxed only once, when d[u] has reached its final value."

"The algorithm maintains two sets of vertices S and Q. Set S contains all vertices for which we know that the value d[v] is already the cost of the shortest path and set Q contains all other vertices. Set S starts empty, and in each step one vertex is moved from Q to S. This vertex is chosen as the vertex with lowest value of d[u]. When a vertex u is moved to S, the algorithm relaxes every outgoing edge (u,v)."

Example

An example can explain the basic idea of alogrithm above and let user understand easily. The example will briefly explain each step that is taken and how the distance is calculated.

Consider the following example: Figure1: Weighted-directed graph

The above weighted graph has 5 vertices from A to E. The value between the two vertices is known as the weight between two vertices. For example the weight between A and C is 1. Using the above graph the Dijkstra’s algorithm is used to determine the shortest path from the source A to the remaining vertices in the graph.

Step 1
sDist[A]=0, the initial value to the source itself; sDist[B]= ∞, sDist[C]= ∞, sDist[D]= ∞, sDist[E]= ∞, the nodes not processed yet. Figure2: shortest path to vertices B, C from A

Step 2
Adj[A]={B,C}; computing the value of the adjacent vertices of the graph, sDist[B]=4; sDist[C]=1; Figure3: Shortest path from B, D using C as intermediate vertex

Step 3
Computation from vertex C, Adj[C] = {B, D}; sDist[B] > sDist[C]+ EdgeCost[C,B], 4 > 1+2 (True), Therefore, sDist[B]=3; sDist[D]=3; Figure4: Shortest path to E using B as intermediate vertex

Step 4
Adj[B]={E}; sDist[E]=sDist[B]+EdgeCost[B,E]=3+3=6; Adj[D]={E}; sDist[E]=sDist[D]+EdgeCost[D,E]=3+3=6. This is same as the initial value that was computed so sDist[E] value is not changed. Figure5: the path obtained using Dijkstra’s Algorithm

Step 5
Adj[E]=0; means there is no outgoing edges from E, and no more vertices, algorithm terminated. Hence the path is presented in figure5.

Reference

Copyright: Yan Huang and Jiachang Yang