def prims_algorithm_mst(graph):
"""
Prim's algorithm, this algorithm
has the same time complexity as Djikstra's
single source shorted path algorithm.
This particular algorithm is
O((V ** 2) + (V * E))
but with a priority queue or vEB tree this can be reduced
to
O((V * log(V)) + (V * E))
a time complexity equivalent to Djikstra's
"""
if not isinstance(graph, Graph):
raise TypeError('this function expects an instance of Graph')
queue = PriorityQueue(graph)
root = None
nodes = {u: TreeNode(u) for u in graph.vertices}
while not queue.is_empty():
u = queue.pop_min()
if root is None:
root = nodes[u]
for v in graph.adj[u]:
if queue.contains(v):
queue.update(v, graph.weights[(u, v)])
nodes[v].parent = nodes[u]
for n in nodes:
node = nodes[n]
if node.parent is not None:
node.parent.children.append(node)
return root
