def dijkstra(graph=None, source_node=1):
""" Return shortest path lengths computed by Dijkstra's algorithm. """
if graph is None:
return None
shortest_path_lengths = {} #dict for keeping track of shortest path legnths
# min heap to keep track of crossing cuts
crossing_cuts = Heap([(float("inf"), int(node)) for node in graph.nodes()])
crossing_cuts.update_key(source_node, 0)
# main algorithm body
while crossing_cuts:
# get smallest crossing cut cut
shortest_cut = crossing_cuts.pop()
head_node, path_length = shortest_cut.name, shortest_cut.key
shortest_path_lengths[head_node] = path_length
# update heap
unadded_tail_nodes = set(graph[head_node]) - set(shortest_path_lengths)
for tail_node in unadded_tail_nodes:
old_key = crossing_cuts[tail_node].key
new_key = shortest_path_lengths[head_node] + graph[head_node][tail_node]
if new_key < old_key:
crossing_cuts.update_key(tail_node, new_key)
return shortest_path_lengths
def prim(graph):
"""
Implements Prim's Minimum Spanning Tree Algorithm in O(VlogE) time with a min heap.
"""
mst = defaultdict(dict)
initial_node = random.choice(graph.keys())
# Initialize minimum heap to contain the nodes of the graph, whose keys
# are the edge values
crossing_cuts = Heap([((float("inf"), float("inf")), int(node))
for node in graph.keys() if node != initial_node])
# Populate min heap with initial node's crossing cuts
for node_name in graph[initial_node].keys():
new_key = (graph[initial_node][node_name], initial_node)
crossing_cuts.update_key(node_name, new_key)
# Pop nodes from minimum heap until minimum spanning tree is formed
while crossing_cuts:
smallest_node = crossing_cuts.pop()
(node_out, (weight, node_in)) = smallest_node.name, smallest_node.key
mst[node_in][node_out] = weight
mst[node_out][node_in] = weight
for node_name in (set(graph[node_out].keys()) - set(mst.keys())):
new_key = (graph[node_out][node_name], node_out)
if node_name in crossing_cuts.node_names():
old_key = crossing_cuts[node_name].key
if new_key[0] < old_key[0]:
crossing_cuts.update_key(node_name, new_key)
return mst
