3. Tree
A tree is a graph with the following properties:
The graph is connected (can go from anywhere to anywhere)
There are no cycles(acyclic)
Graphs that are not trees
Tree
4. Minimum Spanning Tree (MST)
4
• It is a tree (i.e., it is acyclic)
• It covers all the vertices V
• contains |V| - 1 edges
• A single graph can have many different spanning
trees.
Let G=(V,E) be an undirected connected graph.
A sub graph T=(V,E’) of G is a spanning tree of G iff T is
a tree.
6. A minimum cost spanning tree is a spanning tree which has a
minimum total cost.
A minimum spanning tree (MST) or minimum weight spanning
tree is then a spanning tree with weight less than or equal to the
weight of every other spanning tree.
Addition of even one single edge results in the spanning tree losing its
property of acyclicity and removal of one single edge results in its
losing the property of connectivity.
It is the shortest spanning tree .
The length of a tree is equal to the sum of the length of the arcs on the
tree.
7. Properties
Possible multiplicity
There may be several minimum spanning trees of the same weight
having a minimum number of edges
if all the edge weights of a given graph are the same, then every
spanning tree of that graph is minimum.
If there are n vertices in the graph, then each tree has n-1 edges.
Uniqueness
If each edge has a distinct weight then there will be only one, unique
minimum spanning tree.
8. Cycle Property:
Let T be a minimum spanning tree of a weighted graph G
Let e be an edge of G that is not in T and let C be the cycle formed
by e with T
For every edge f of C, weight(f) weight(e)
If weight(f) > weight(e) we can get a spanning tree of smaller weight
by replacing e with f
8
4
2 3
6
7
7
9
8
e
C
f
8
4
2 3
6
7
7
9
8
C
e
f
Replacing f with e
yields
a better spanning tree
9. Partition Property:
Consider a partition of the vertices
of G into subsets U and V
Let e be an edge of minimum weight
across the partition
There is a minimum spanning tree
of G containing edge e
Proof:
Let T be an MST of G
If T does not contain e, consider the
cycle C formed by e with T and let f
be an edge of C across the partition
By the cycle property,
weight(f) weight(e)
Thus, weight(f) = weight(e)
We obtain another MST by replacing
f with e
U V
7
4
2
8
5
7
3
9
8 e
f
7
4
2
8
5
7
3
9
8 e
f
Replacing f with e
yields
another MST
U V
10. Minimum-cost spanning trees
If we have a connected undirected graph with a weight (or cost)
associated with each edge
The cost of a spanning tree would be the sum of the costs of its edges
A minimum-cost spanning tree is a spanning tree that has the lowest
cost
A B
E D
F C
16
19
21 11
33 14
18
10
6
5
A connected, undirected
graph
A B
E D
F C
16
11
18
6
5
A minimum-cost spanning tree
11. Applications of minimum spanning trees
Consider an application where n stations are to be linked using a
communication network.
The laying of communication links between any two stations involves a
cost.
The problem is to obtain a network of communication links which
while preserving the connectivity between stations does it with
minimum cost.
The ideal solution to the problem would be to extract a sub graph
termed minimum cost spanning tree.
It preserves the connectedness of the graph yields minimum cost.
12. Applications cont’d
• Suppose you want to supply a set of houses with:
electric power
water
sewage lines
telephone lines
• To keep costs down, you could connect these houses with
a spanning tree ( for example, power lines)
•However, the houses are not all equal distances apart
• To reduce costs even further, you could connect the
houses with a minimum-cost spanning tree
13. Applications cont’d
• Constructing highways or railroads spanning several
cities
• Designing local access network
• Making electric wire connections on a control panel
• Laying pipelines connecting offshore drilling sites,
refineries, and consumer markets
14. Applications cont’d
The phone company task is to provide phone lines to a village with 10
houses, each labeled H1 through H10.
A single cable must connects each home. The cable must run through
houses H1, H2, and so forth, up through H10.
Each node is a house, and the edges are the means by which one house
can be wired up to another.
The weights of the edges dictate the distance between the homes.
Their task is to wire up all ten houses using the least amount of
telephone wiring possible.
