Discrrete mathematics for computer science prim

Minimum Spanning Trees


Subgraph
• A graph G is a subgraph of graph H if
– The vertices of G are a subset of the
vertices of H, and
– The edges of G are a subset of the
edges of H.


Spanning Tree
• A spanning tree of a graph G is a
subgraph of G that is a tree and includes
all the vertices of G.
• Every connected graph has a spanning
tree
– Start with G. If G is a tree then it is a ST of G.
If not, G has a cycle. Remove any edge of the
cycle. The result is a connected subgraph of
G. Repeat. Process must terminate since the
number of edges is finite.


Spanning Trees


Spanning Trees

a spanning tree

Spanning Trees

another spanning tree
(can have many)


Weighted Graphs
• A weighted graph is a graph with a
mapping of edges to R+ assigning a
weight to each edge
• A minimum spanning tree of a
weighted graph is a spanning tree of
minimum total weight (sum of the
weights of the edges in the tree)


Minimum Spanning Trees
3

4
1

4
6

3

7
1

2


Minimum Spanning Trees
3

4
1

4
6

3

7
1

1. Choose any vertex

2


Minimum Spanning Trees
3

4
1

4

6

3

7
1

2

2. Choose minimum weight
edge connected to that vertex


Minimum Spanning Trees
3

4
1

4
6

3

7
1

2

3. Choose minimum weight

edge connected to chosen
vertices


Minimum Spanning Trees
3

4
1

4
6

3

7
1

2

4. Choose minimum weight
edge connected to chosen
vertices


Minimum Spanning Trees
3

4
1

4
6

3

7
1

2

5. Choose minimum weight
edge connected to chosen
vertices without creating a


Minimum Spanning Trees
3

4
1

4
6

3

7
1

2

6. Choose minimum weight
edge connected to chosen
vertices without creating a


Prim’s algorithm
• To find a MST of a connected weighted
graph G, we build a tree one edge at a
time
– Start with T being any node of G
– Pick a minimum weight edge that does not
create a cycle and that connects a node in
T to a node not in T. Add that edge to T.
– Repeat until all nodes of G have been
added to T.

• A “Greedy algorithm”


Proof that Prim’s algorithm works
• Invariant: At every stage, T is a
subgraph of a MST
• True initially since T is just a single node
• The algorithm runs until all nodes have
been added to T, so in the end the
invariant says that T is a MST.
• So we just need to show that each stage
of the algorithm preserves the invariant.

Proof that the algorithm maintains
the invariant
• Proof by contradiction. Suppose there is
a stage at which the invariant is true
before and false after.
• Let e be the edge added when the
invariant is first violated. Suppose e
joins node v in T to node w not in T.
• So T is a subgraph of a MST but T+e is
not.
• Let T’ be a MST of which T is a subgraph.
• Edge e is not in T’.


Proof that the algorithm maintains
the invariant
• Since T’ is a spanning tree, there is a path p from v to
w in T’. Path p does not go through e since e is not
part of T’. So p+e is a cycle consisting of two different
paths from u to v.
• Path p must include an edge f which has one end in T
and the other not in T.
• Edge f must have weight ≥ the weight of e, otherwise
the algorithm would have chosen to add edge f to T
instead of e.
• Removing f from T’ and adding e results in a spanning
tree of weight ≤ the weight of T’ and including T as a
subgraph, contradicting the assumption that T+e was

not a subgraph of a MST. ✔


Finis