Lesson: Dijkstra’s Algorithm
Dijkstra’s algorithm is used to solve the single source reachability problem on
graphs using the edge-list representation. It is also a nice example of the use of
various different data structures. Imagine a graph such as the following:
We want to find the shortest distance to various cities starting from Pierre. The
graph is represented by a map keyed by a city name. The value stored for each
key is also a map, with a city as key and a weight as value. The map
representation of the above graph is as follows:
Dijkstra’s algorithm uses an internal priority queue of distance/city pairs. This
queue is organized so that the value with smallest distance is at the top of the
queue. Initially the queue contains the starting city and distance zero. The map
of reachable cities is initially zero. As a city if pulled from the queue, if it is already
known to be reachable it is ignored. Otherwise it is placed into the queue of
reachable cities, and the neighbors of the new city are placed into the queue,
adding the distance to the city and the distance to the new neighbor.
The following table shows the values of the priority queue at various stages:
Pierre: 0
Pierre: 0
Pendleton: 2
Pendleton: 2
Phoenix: 6, Pueblo: 10
Phoenix: 6
Pueblo: 9, Peoria 10, Pueblo: 10, Pittsburgh: 16
Pueblo: 9
Peoria: 10, Pueblo: 10, Pierre: 12, Pittsburgh: 16
Peoria: 10
Pueblo: 10, Pierre: 12, Pueblo: 13 Pittsburgh: 15, Pittsburgh: 16
Pittsburgh: 15
Pittsburgh: 16, Pensacola: 19
Pensacola: 19
Phoenix: 24
Notice how duplicates are removed only when pulled from the queue.
