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# Spanning trees & applications

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### Spanning trees & applications

1. 1. CONTENTS Tree Minimum spanning tree  Definition  Properties  Example  Applications
2. 2. TreeA tree is a graph with the following properties: The graph is connected (can go from anywhere to anywhere) There are no cycles(acyclic) Tree Graphs that are not trees
3. 3. Minimum Spanning Tree (MST)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 isa tree.• 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. 4
4. 4. Connected undirected graph Spanning trees
5. 5.  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.
6. 6. PropertiesPossible 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.
7. 7. 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 f 8 f 8 Replacing f with e 4 4 yields C C 9 9 2 6 2 6 a better spanning tree 3 3 e e 8 7 8 7 7 7
8. 8. Partition Property:  Consider a partition of the vertices of G into subsets U and V U V f 7  Let e be an edge of minimum weight across the partition 4 9  There is a minimum spanning tree 5 2 of G containing edge e 8 8 e 3Proof: 7  Let T be an MST of G Replacing f with e  If T does not contain e, consider the yields cycle C formed by e with T and let f another MST be an edge of C across the partition U V f 7  By the cycle property, 4 weight(f)  weight(e) 9  Thus, weight(f) = weight(e) 2 5 8  We obtain another MST by replacing 8 e 3 f with e 7
9. 9. 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 16 16 A B A B 21 11 6 11 6 19 5 5 F C F C 33 14 10 18 E 18 D E D A connected, undirected A minimum-cost spanning tree graph
10. 10. 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.
11. 11. 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 witha spanning tree ( for example, power lines) •However, the houses are not all equal distances apart• To reduce costs even further, you could connect thehouses with a minimum-cost spanning tree
12. 12. 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
13. 13. 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.
14. 14. Graphical representation of hooking up a 10-home village withphone lines
15. 15.  The two valid spanning trees from the above graph. The edges forming the spanning tree are bolded.
16. 16. Problem: Laying Telephone Wire Central office
17. 17. Wiring: Naïve Approach Central office Expensive!
18. 18. Wiring: Better Approach Central officeMinimize the total length of wire connecting the customers
19. 19. Thank you