## Dijkstra’s Algorithm

### January 4, 2011

We follow the implementation given in Section 24.3 of the third edition, published in 2009, of *Introduction to Algorithms* by Thomas Cormen, Charles Leiserson, Ronald Rivest and Clifford Stein. (hereafter referred to as CLRS). In it, vertices are given a number of attributes; we begin with a simple vertex class to collect them all in one spot:

`class Vertex(object):`

```
``` def __init__(self, adj={}, dist=float("inf"), name=None, pred=None):

self.adj = adj

self.dist = dist

self.name = name

self.pred = pred

return

def __cmp__(self, other):

return cmp(self.dist, other.dist)

` def __repr__(self):`

return self.name

A vertex has a distance parameter, a name, and a predecessor. In CLRS, vertices also have an adjacency list and edge weights are recorded in an external structure. We instead opt to have each vertex’s `adj`

attribute be a Python dictionary, where each key is an adjacent vertex and each value is the weight of the edge between them. We also give vertices a `__cmp__`

method to compare vertices by distance and a `__repr__`

method is included to give human readable print out.

The implementation of Dijkstra’s Algorithm in CLRS requires a min-priority queue; interestingly, this queue determines the speed of the algorithm as a whole, because we need to constantly retrieve the vertex of minimum distance and update the key of a node in the queue whenever a vertex’s distance estimate changes. While CLRS recommends a Fibonacci heap, we’ll use pairing-heaps which we studied in a previous exercise. Pairing heaps have most of the efficiency of Fibonacci heaps, while being easier to implement.

We’ll represent a digraph with a Python dictionary of string –> vertex pairs. First off, we have two procedures to build our digraph `D`

out of edges:

`def add_edge(D, u, v, weight):`

if u not in D:

D[u] = Vertex(name=u, adj={})

if v not in D:

D[v] = Vertex(name=v, adj={})

D[u].adj[v] = weight

return

```
```

`def add_edges(D, edges):`

for edge in edges:

add_edge(D, *edge)

return

When adding an edge, we first ensure both vertices are in the graph. Then the head vertex’s adjacency list is updated with tail vertex and the weight of the edge between them. The `add_edges`

procedure is included for convenience’s sake, so we don’t have to build the digraph one edge at a time.

Next, our two helper functions: `init_single_source`

and `relax`

. The first readies the digraph for work on the single source shortest path problem, while the latter updates each vertex when a shorter path is found to it:

`def init_single_source(D, s):`

for v in D:

D[v].dist = float("inf")

D[v].pred = None

D[s].dist = 0

return

```
```

`def relax(D, u, v):`

d = D[u].dist + D[u].adj[v]

if D[v].dist > d:

D[v].dist = d

D[v].pred = u

return

The main procedure is `dijkstra`

. First, we add each vertex (`D.values()`

is a list of all the vertices in `D`

) to the priority queue. Next we iteratively step through all the vertices of the digraph (indexed by distance from the source), relaxing each outward edge. We also clean up the queue with `merge_pairs`

so that the queue is again ordered by distance after relaxing edges.

`def dijkstra(D, s):`

init_single_source(D, s)

Q = make_heap()

for v in D.values():

Q = insert(v, Q)

while Q:

u, Q = find_min(Q), delete_min(Q)

for v in u.adj:

relax(D, u.name, v)

Q = merge_pairs(Q)

return

Finally, we build the shortest path by traversing backwards through the predecessor subtree created by the algorithm. We record the path with a list, appending on the right, and then reverse the list to get the path in order:

`def find_shortest_path(D, s, f):`

dijkstra(D, s)

path, v = [f], D[f].pred

while v is not None:

path.append(v)

v = D[v].pred

path.reverse()

return path

Testing on the digraph given in our picture:

`if __name__ == "__main__":`

D = {}

add_edges(D, [('a', 'c', 2), ('a', 'd', 6),

('b', 'a', 3), ('b', 'd', 8),

('c', 'd', 7), ('c', 'e', 5),

('d', 'e', 10)])

print find_shortest_path(D, 'a', 'e')

```
```

`# Output: ['a', 'c', 'e']`

The code including the priority-queue is available on http://codepad.org/lRUh1tq8.

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My Haskell solution (see http://bonsaicode.wordpress.com/2011/01/04/programming-praxis-dijkstra%E2%80%99s-algorithm/ for a version with comments):

[…] Programming Praxis: Dijkstra’s Algorithm […]

My commented version: http://www.gleocadie.net/?p=641&lang=en

My try in Rexx

slightly optimized REXX (no double init, no sort but select for next neighbour)