## Tuesday, August 03, 2010

### Dijkstra

Dijkstra's algorithm is a well known algorithm for solving the single-source shortest path problem in a graph. It is a "global" pathfinding algorithm, in that it will find the optimal solution to all the other nodes in the graph (provided the graph has no negative cost edges or cycles). Unfortunately the algorithm is not very easy to run in parallel.

The algorithm requires two lists, a visited list, and a pending list, and an array of the current distances to each node, plus the precursors.

In essence the algorithm begins at a starting node, and all other nodes are added to a 'pending' list. The distances are initialised to 'infinite'. We begin at the starting node that has been specified.

1. From this node we calculate the distance to each neighbouring node. If the distance is less than the current record, we update the distance measurement (cost of getting to current node plus cost to get to neighbour node), and update the precursor to the neighbour node to be the present node.

2. From all nodes, the node with the minimum distance on the pending list is selected as the next node to visit. This new node is then added to the visited list and removed from the pending list..

It's quite simple, but an animation will really help get the point across:

In code, the algorithm is as follows:
(This code stores the graph in the "dist" array, with dist[i][j] representing an edge cost between i and j. If it contains '0' then there is no connection).

The initial loop sets the distances to infinity. The first inner loop finds the minimum cost node, the second inner loop updates the distances based on the current node.
The minimum distance to each node from the starting node is stored in the 'd' array. The visited array is used to record which nodes have already been processed.
```void dijkstra(int s) {
int i, k, mini;
int visited[GRAPHSIZE];
//initialize all distances to infinite, and set the visited nodes to none.
for (i = 1; i <= n; ++i) {
d[i] = INFINITY; visited[i] = 0;
}
//set the distance to the starting node 's' to zero
d[s] = 0;
//outer loop - iterate through all nodes
for (k = 1; k <= n; ++k) {
//find the minimum cost node
//if the node has not been visited, and the minimum cost node is not initialised,
//then set the minimum cost node to the current node
//if the distance to the current node is less than the current minimum,
//then update the minimum
mini = -1;
for (i = 1; i <= n; ++i)
if (!visited[i]&&((mini==-1)||(d[i]<[mini])))
mini = i;
//now we have the current minimum node, set it to visited
visited[mini] = 1; //visit node
//for each node, if there is a valid edge between the current minimum node
//then calculate the cost to visit this neighbour node via the current node.
//if the cost is less than the current record of minimum costs, update the minimum cost.
for (i = 1; i <= n; ++i)
if (dist[mini][i]) //update distances
if (d[mini] + dist[mini][i] < d[i])
d[i] = d[mini] + dist[mini][i];
}
}```
In future blog posts I'll look at the A* algorithm, a parallel reduction for Dijkstra and some SIMD/GPU/CUDA optimisations for an 8-connected Dijkstra.