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"""Prim's Algorithm.
Determines the minimum spanning tree(MST) of a graph using the Prim's Algorithm.
Details: https://en.wikipedia.org/wiki/Prim%27s_algorithm
"""
import
heapq
as
hq
import
math
from
collections
.
abc
import
Iterator
class
Vertex
:
"""Class Vertex."""
def
__init__
(
self
,
id_
):
"""
Arguments:
id - input an id to identify the vertex
Attributes:
neighbors - a list of the vertices it is linked to
edges - a dict to store the edges's weight
"""
self
.
id
=
str
(
id_
)
self
.
key
=
None
self
.
pi
=
None
self
.
neighbors
=
[]
self
.
edges
=
{}
# {vertex:distance}
def
__lt__
(
self
,
other
):
"""Comparison rule to < operator."""
return
self
.
key
<
other
.
key
def
__repr__
(
self
):
"""Return the vertex id."""
return
self
.
id
def
add_neighbor
(
self
,
vertex
):
"""Add a pointer to a vertex at neighbor's list."""
self
.
neighbors
.
append
(
vertex
)
def
add_edge
(
self
,
vertex
,
weight
):
"""Destination vertex and weight."""
self
.
edges
[
vertex
.
id
]
=
weight
def
connect
(
graph
,
a
,
b
,
edge
):
# add the neighbors:
graph
[
a
-
1
].
add_neighbor
(
graph
[
b
-
1
])
graph
[
b
-
1
].
add_neighbor
(
graph
[
a
-
1
])
# add the edges:
graph
[
a
-
1
].
add_edge
(
graph
[
b
-
1
],
edge
)
graph
[
b
-
1
].
add_edge
(
graph
[
a
-
1
],
edge
)
def
prim
(
graph
:
list
,
root
:
Vertex
)
->
list
:
"""Prim's Algorithm.
Runtime:
O(mn) with `m` edges and `n` vertices
Return:
List with the edges of a Minimum Spanning Tree
Usage:
prim(graph, graph[0])
"""
a
=
[]
for
u
in
graph
:
u
.
key
=
math
.
inf
u
.
pi
=
None
root
.
key
=
0
q
=
graph
[:]
while
q
:
u
=
min
(
q
)
q
.
remove
(
u
)
for
v
in
u
.
neighbors
:
if
(
v
in
q
)
and
(
u
.
edges
[
v
.
id
]
<
v
.
key
):
v
.
pi
=
u
v
.
key
=
u
.
edges
[
v
.
id
]
for
i
in
range
(
1
,
len
(
graph
)):
a
.
append
((
int
(
graph
[
i
].
id
)
+
1
,
int
(
graph
[
i
].
pi
.
id
)
+
1
))
return
a
def
prim_heap
(
graph
:
list
,
root
:
Vertex
)
->
Iterator
[
tuple
]:
"""Prim's Algorithm with min heap.
Runtime:
O((m + n)log n) with `m` edges and `n` vertices
Yield:
Edges of a Minimum Spanning Tree
Usage:
prim(graph, graph[0])
"""
for
u
in
graph
:
u
.
key
=
math
.
inf
u
.
pi
=
None
root
.
key
=
0
h
=
list
(
graph
)
hq
.
heapify
(
h
)
while
h
:
u
=
hq
.
heappop
(
h
)
for
v
in
u
.
neighbors
:
if
(
v
in
h
)
and
(
u
.
edges
[
v
.
id
]
<
v
.
key
):
v
.
pi
=
u
v
.
key
=
u
.
edges
[
v
.
id
]
hq
.
heapify
(
h
)
for
i
in
range
(
1
,
len
(
graph
)):
yield
(
int
(
graph
[
i
].
id
)
+
1
,
int
(
graph
[
i
].
pi
.
id
)
+
1
)
def
test_vector
()
->
None
:
"""
# Creates a list to store x vertices.
>>> x = 5
>>> G = [Vertex(n) for n in range(x)]
>>> connect(G, 1, 2, 15)
>>> connect(G, 1, 3, 12)
>>> connect(G, 2, 4, 13)
>>> connect(G, 2, 5, 5)
>>> connect(G, 3, 2, 6)
>>> connect(G, 3, 4, 6)
>>> connect(G, 0, 0, 0) # Generate the minimum spanning tree:
>>> G_heap = G[:]
>>> MST = prim(G, G[0])
>>> MST_heap = prim_heap(G, G[0])
>>> for i in MST:
... print(i)
(2, 3)
(3, 1)
(4, 3)
(5, 2)
>>> for i in MST_heap:
... print(i)
(2, 3)
(3, 1)
(4, 3)
(5, 2)
"""
if
__name__
==
"__main__"
:
import
doctest
doctest
.
testmod
()
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