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Minimum Spanning Tree
Algorithms
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Prim's Algorithm for Minimum Spanning Tree (MST), Slides of Design and Analysis of Algorithms
The implementation and explanation of prim's algorithm for finding the minimum spanning tree (mst) of a weighted connected graph. The algorithm is presented step by step with visual examples.
Typology: Slides
2011/2012
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Minimum Spanning Tree
Algorithms
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Greedy Algorithms
^
Main Concept: Make the
best
or
greedy
choice at any
given step.
^
Choices are made in sequence such that»^
Each individual choice is best according to some limited “short-term” criterion, that is not too expensive to evaluate» Once a choice is made, it cannot be undone!^
Even if it becomes evident later that it was a poor choice Sometimes life is like that
^
The goal is to»^
take a sequence of locally optimal actions, hoping to yield aglobally optimal solution.
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Minimum Spanning Tree (MST)
^
A^
spanning tree
for a connected, undirected
graph,
G
=(
V , E ) is
1.^
a connected subgraph of
G^
that forms an
2.^
undirected tree incident with each vertex.
^
In a weighted graph
G
=(
V ,
E , W ),
»^
the weight of a subgraph is the sum of the weights ofthe edges in the subgraph.
^
A^
minimum spanning tree
(MST) for a weighted
graph is»^
a spanning tree with the minimum weight.
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Minimum Spanning Tree
^
Problem: given a connected, undirected, weightedgraph:^14
find a
spanning tree
using edges that
minimize the total weight
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Minimum Spanning Tree
^
Answer:
H^
B^
C
G^
E^
D
A F
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Another Example
^
Given a weighted graph
G
V ,
E
, W
), find a MST
of G^3
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Prime’s Algorithm(High-Level Pseudocode) ^
Prime(G)//Input: A weighted connected graph G = <V, E>//Output: E
--- the set of edges composing MST of GT^
VT^
= {v
ET^
=^
for
i = 1
to^
|V| - 1
do
find a minimum-weight edge e* = (u, v) among all the edges (u, v) such that u is in V
and v is in V-VT
T
VT^
= V
T
{v*}
ET^
= E
T
{e*}
return
ET
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Prime’s Algorithm(High-Level Pseudocode) ^
Prime(G)//Input: A weighted connected graph G = <V, E>//Output: E
--- the set of edges composing MST of GT^
VT^
= {v
ET^
=^
for
i = 1
to^
|V| - 1
do
find a minimum-weight edge e* = (u, v) among all the edges (u, v) such that u is in V
and v is in V-VT
T
VT^
= V
T
{v*}
ET^
= E
T
{e*}
return
ET
H^
B^
C
G^
E^
D
A F
14
6 10 3
4 5
2
(^915)
8
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Prim’s Algorithm
MST-Prim(G,
w,
r)
Q^
=^
V[G];
for
each
u
Q
key[u]
key[r]
p[r]
NULL;
while
(Q
not
empty)
u^
=^
ExtractMin(Q); for
each
v
Adj[
u ]
if (v
Q
and
w(
u,v
)^
<^
key[
v ])
p[v]
u; key[v]
Grow a single tree by repeatedlyadding the least cost edge thatconnects a vertex in the existing treeto a vertex not in the existing tree^ w(u,v);
Intermediary solution is a subtree
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Prim’s Algorithm
MST-Prim(G,
w,
r)
Q^
=^
V[G];
for
each
u
Q
key[u]
key[r]
p[r]
NULL;
while
(Q
not
empty)
u^
=^
ExtractMin(Q); for
each
v
Adj[
u ]
if (v
Q
and
w(
u,v
)^
<^
key[
v ])
p[v]
u; key[v]
14^ w(u,v);
6 10 3
4 5
2
(^915)
8 Run on example graph
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Prim’s Algorithm
MST-Prim(G,
w,
r)
Q^
=^
V[G];
for
each
u
Q
key[u]
key[r]
p[r]
NULL;
while
(Q
not
empty)
u^
=^
ExtractMin(Q); for
each
v
Adj[
u ]
if (v
Q
and
w(
u,v
)^
<^
key[
v ])
p[v]
u; key[v]
^ w(u,v);
^
^
14
6 10 3
4 5
2
(^915)
8 Pick a start vertex r
r
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Prim’s Algorithm
MST-Prim(G,
w,
r)
Q^
=^
V[G];
for
each
u
Q
key[u]
key[r]
p[r]
NULL;
while
(Q
not
empty)
u^
=^
ExtractMin(Q); for
each
v
Adj[
u ]
if (v
Q
and
w(
u,v
)^
<^
key[
v ])
p[v]
u; key[v]
^ w(u,v);
^
^
14
6 10 3
4 5
2
(^915)
8
Red vertices have been removed from Q
u
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Prim’s Algorithm
MST-Prim(G,
w,
r)
Q^
=^
V[G];
for
each
u
Q
key[u]
key[r]
p[r]
NULL;
while
(Q
not
empty)
u^
=^
ExtractMin(Q); for
each
v
Adj[
u ]
if (v
Q
and
w(
u,v
)^
<^
key[
v ])
p[v]
u; key[v]
14 w(u,v);
^
^
^3
14
6 10 3
4 5
2
(^915)
8
u
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Prim’s Algorithm
MST-Prim(G,
w,
r)
Q^
=^
V[G];
for
each
u
Q
key[u]
key[r]
p[r]
NULL;
while
(Q
not
empty)
u^
=^
ExtractMin(Q); for
each
v
Adj[
u ]
if (v
Q
and
w(
u,v
)^
<^
key[
v ])
p[v]
u; key[v]
14 w(u,v);
^
^
^3
14
6 10 3
4 5
2
(^915)
8
u
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