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Lecture 24 Network models : Minimal Spanning Tree problem 24.1 Minimal Spanning Tree Problem A tree is defined to be an undirected, acyclic and connected graph. A spanning tree is a subgraph of G (undirected, connected graph), is a tree and contains all the vertices of G. A minimum spanning tree is a spanning tree but has weights or lengths associated with edges and the total weight is at the minimum. Prim’s Algorithm It starts at any vertex (say A) in a graph and finds the least cost vertex (say B) connected to the start vertex. Now either from A or B, it will find the next least costly vertex connection, without creating cycle (say C) Now either from A, B or C find the next least costly vertex connection, without creating a cycle and so on. Eventually all the vertices will be connected without any cycles and a minimum spanning tree will be the result. Example 1 Suppose it is desired to establish a cable communication network that links major cities, which is shown in the figure. Determine how the cities are connected such that the total cable mileage is minimized. Solution C = {LA} C = {LA, SE} C = {LA, SE, DE} C = {LA, SE, DE, DA} C = {LA, SE, DE, DA, EH} C = {LA, SE, DE, DA, EH, NY} C = {LA, SE, DE, DA, EH, NY, DC} C' = {SE, DE, DA, EH, NY, DC} C' = {DE, DA, EH, NY, DC} C' = {DA, EH, NY, DC} C' = {EH, NY, DC} C' = {NY, DC} C' = {DC} C' = { } The resultant network is 1 Thus the total cable mileage is 1100 + 1300 + 780 + 900 + 800 + 200 = 5080 Example 2 For the following graph obtain the minimum spanning tree. The numbers on the branches represent the cost. Solution C = {A} C = {A, D} C = {A, D, B} C = {A, D, B, C} C = {A, D, B, C, G} C = {A, D, B, C, G, F} C = {A, D, B, C, G, F, E} C' = {B, C, D, E, F, G} C' = {B, C, E, F, G} C' = {C, E, F, G} C' = {E, F, G} C' = {E, F} C' = {E} C' = { } The resultant network is 2 Cost = 2 + 1 + 4 + 3 + 3 + 5 = 18 units 3