Dijkstra's algorithm

Dijkstra's algorithm is a method to find the shortest paths between nodes in a graph. It is faster than many other ways to do this, but it needs all of the distances between nodes in the graph to be zero or more.
Algorithm
Keep doing these steps:
Find the thing that you have not yet drawn a mark next to that has the smallest number written next to it
Draw a mark next to this thing
Do these steps for each other thing connected to this place:
Add the number written next to this thing and the distance of the connection together
If the connected thing does not have a number written next to it, write the new number and the name of this thing next to the connected thing
If the connected thing has a number written next to it and the new number is smaller than the written number:
Draw a line over what is already written
Write the new number and the name of this thing instead
Stop when you have drawn a mark next to every thing in the list
When you have done all of these steps you can use the list to find the shortest way from the first thing to any other thing. First write down the other thing. Then keep doing these steps:
Find the last thing you wrote down in the list
Write down the thing written next to it
Stop when you find the first thing
The connections between the things you have written down are the shortest way from the first thing to the other thing.
Pseudocode
In this pseudocode algorithm, the code , searches for the vertex in the vertex set that has the least value. returns the length of the edge connecting the two neighbor-nodes and . The variable on line 18 is the length of the path from the root node to the neighbor node if it were to go through . If this path is shorter than the current shortest path recorded for , that current path is replaced with this path. The array contains a pointer to the next node on the source graph to get the shortest route to the source.
1 function Dijkstra(Graph, source):
2
3 create vertex set Q
4
5 for each vertex v in Graph:
6 dist[v] - INFINITY
7 prev[v] - UNDEFINED
8 add v to Q
9 dist[source] - 0
10
11 while Q is not empty:
12 u - vertex in Q with min dist[u]
13
14 remove u from Q
15
16 for each neighbor v of u: // only v that are still in Q
17 alt - dist[u] + length(u, v)
18 if alt < dist[v]:
19 dist[v] - alt
20 prev[v] - u
21
22 return dist[], prev[]
If we only want the shortest path between vertices and , we can stop the search after line 15 if (if the current node is the target node).
Now we can read the shortest path from to by going backwards:
1 S - empty sequence
2 u - target
3 if prev[u] is defined or u = source: // Do something only if the vertex is reachable
4 while u is defined: // Construct the shortest path with a stack S
5 insert u at the beginning of S // Push the vertex onto the stack
6 u - prev[u] // Traverse from target to source
Now sequence is the list of vertices that make up one of the shortest paths from to . If a path from to does not exist, sequence will be empty.
References
Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). "Section 24.3: Dijkstra's algorithm". Introduction to Algorithms (Second ed.). MIT Press and McGraw-Hill. pp. 595-601. .
https://github.com/xtaci/algorithms/blob/master/include/dijkstra.h
Other websites
Dijkstra's Algorithm in C++
Algorithms
Graph theory