Networking models for quantitative methods of research

1. MINIMUM SPANNING A Networking Model Report for DPA 702 QUANTITATIVE METHODS OF RESEARCH By: ALONA M. SALVACebu Technological University

3. MST Algorithms

4. Kruskal

5. Reverse Delete

6. Prim

7. Brief History

9. Networking Basics A network is an arrangement of paths connected at various points, through which items move. Introduction to Management Science, 9thed, Bernard W. Taylor III, 2006 Practical Optimization: A Gentle Introduction, John W. Chinneck, 2010 http://www.sce.carleton.ca/faculty/chinneck/po.html

10. Definitions & Networking Basics Some further definitions associated with graphs and networks: chain: a sequence of arcs connecting two nodes i and j. For example, in Figure 8.1(a), we might connect nodes A and E via the chains ABCE or ADCE. path: a sequence of directed arcs connecting two nodes. In this case, you must respect the arc directions. For example, in Figure 8.1(b), a path from A to E might be ABDE, but the chain ABCE is not a path because it traverses arc BC in the wrong direction. cycle: a chain that connects a node to itself without any retracing. Practical Optimization: A Gentle Introduction, John W. Chinneck, 2010 http://www.sce.carleton.ca/faculty/chinneck/po.html

11. Definitions & Networking Basics connected graph (or connected network): has just one part. In other words you can reach any node in the graph or network via a chain from any other node. Practical Optimization: A Gentle Introduction, John W. Chinneck, 2010 http://www.sce.carleton.ca/faculty/chinneck/po.html

12. tree: a connected graph having no cycles. Some examples are shown in Figure 8.2(a). Practical Optimization: A Gentle Introduction, John W. Chinneck, 2010 http://www.sce.carleton.ca/faculty/chinneck/po.html

13. Definitions spanning tree: normally a tree selected from among the arcs in a graph or network so that all of the nodes in the tree are connected. Practical Optimization: A Gentle Introduction, John W. Chinneck, 2010 http://www.sce.carleton.ca/faculty/chinneck/po.html

14. Definitions flow capacity: an upper (and sometimes lower) limit on the amount of flow in an arc in a network. For example, the maximum flow rate of water in a pipe, or the maximum simultaneous number of calls on a telephone connection. source (or source node): a node which introduces flow into a network. This happens at the boundary between the network under study and the external world. sink (or sink node): a node which removes flow from a network. This happens at the boundary between the network under study and the external world. Practical Optimization: A Gentle Introduction, John W. Chinneck, 2010 http://www.sce.carleton.ca/faculty/chinneck/po.html

15. The Minimum Spanning Tree Problem Given a graph in which the arcs are labeled with the distances between the nodes that they connect, find a spanning tree which has the minimum total length. NO CYCLES ALLOWED!! ALGORITHMS – a problem solving procedure following a set of rules Cycle- the nodes are connected by more than one edge. Practical Optimization: A Gentle Introduction, John W. Chinneck, 2010 http://www.sce.carleton.ca/faculty/chinneck/po.html

16. Steps of the minimal spanning tree solution method: Select any starting node (conventionally, node 1 is selected). Select the node closest to the starting node to join the spanning tree. Select the closest node not presently in the spanning tree. Repeat step 3 until all nodes have joined the spanning tree. Introduction to Management Science, 9thed, Bernard W. Taylor III, 2006

17. Practical Optimization: A Gentle Introduction, John W. Chinneck, 2010 http://www.sce.carleton.ca/faculty/chinneck/po.html

19. Reverse Delete

20. PrimIntroduction to Management Science, 9thed, Bernard W. Taylor III, 2006

21. KRUSKAL’S ALGORITHM Pick out the smallest edges Repeat Step 1 as long as the edges selected does not create a cycle When all nodes have been connected, you are done http://www.slideshare.net/zhaokatherine/minimum-spanning-tree

22. REVERSE-DELETE ALGORITHM This is the opposite of Kruskal’s algorithm Start with all edges Delete the longest edge Continue deleting longest edges as long as all nodes are connected and no cycles are formed http://www.slideshare.net/zhaokatherine/minimum-spanning-tree

23. PRIM’S ALGORITHM Pick out a nod Pick out the shortest edge that is connected to your tree so far, as long as it doesn’t create a cycle Continue this until all nodes are covered http://www.slideshare.net/zhaokatherine/minimum-spanning-tree

26. 19 i i a b h c d e f g a b h c d e f g 8 7 8 7 9 9 4 4 2 2 11 11 14 14 4 4 6 7 6 7 10 10 8 8 2 1 2 1

27. 20 i i a b h c d e f g a b h c d e f g 8 7 8 7 9 9 4 4 2 2 11 11 14 14 4 4 6 7 6 7 10 10 8 8 2 1 2 1

28. 21 i i a b h c d e f g a b h c d e f g 8 7 8 7 9 9 4 4 2 2 11 11 14 14 4 4 6 7 6 7 10 10 8 8 2 1 2 1

29. 22 i i a b h c d e f g a b h c d e f g 8 7 8 7 9 9 4 4 2 2 11 11 14 14 4 4 6 7 6 7 10 10 8 8 2 1 2 1

30. 23 Prim's algorithm(basic part) MST_PRIM(G,w,r) A={} S:={r} (r is an arbitrary node in V) 3. Q=V-{r}; 4. while Q is not empty do { 5 take an edge (u, v) such that (1) u S and v  Q (v S ) and (u, v) is the shortest edge satisfying (1) 6 add (u, v) to A, add v to S and delete v from Q }

31. 24 Prim's algorithm MST_PRIM(G,w,r) 1 for each u in Q do key[u]:=∞ parent[u]:=NIL 4 key[r]:=0; parent[r]=NIL; 5 QV[Q] 6 while Q!={} do 7 u:=EXTRACT_MIN(Q); if parent[u]Nil print (u, parent[u]) 8 for each v in Adj[u] do 9 if v in Q and w(u,v)<key[v] 10 then parent[v]:=u 11 key[v]:=w(u,v)

33. Q is a priority queue, holding all vertices that are not in the tree now.

34. key[v] is the minimum weight of any edge connecting v to a vertex in the tree.

35. parent[v] names the parent of v in the tree.

36. When the algorithm terminates, Q is empty; the minimum spanning tree A for G is thus A={(v,parent[v]):v∈V-{r}}.

38. 27 i i a b h c d e f g a b h c d e f g 8 7 8 7 9 9 4 4 2 2 11 11 14 14 4 4 6 7 6 7 10 10 8 8 2 1 2 1

39. 28 i i a b h c d e f g a b h c d e f g 8 7 8 7 9 9 4 4 2 2 11 11 14 14 4 4 6 7 6 7 10 10 8 8 2 1 2 1

40. 29 i i a b h c d e f g a b h c d e f g 8 7 8 7 9 9 4 4 2 2 11 11 14 14 4 4 6 7 6 7 10 10 8 8 2 1 2 1

41. 30 i a b h c d e f g 8 7 9 4 2 11 14 4 6 7 10 8 2 1 Bottleneck spanning tree: A spanning tree of G whose largest edge weight is minimum over all spanning trees of G. The value of the bottleneck spanning tree is the weight of the maximum-weight edge in T. Theorem:A minimum spanning tree is also a bottleneck spanning tree. (Challenge problem)

44. Why we like MST’s MST’s are fun to work with It helps us find the shortest route “The shortest distance between two people is a sMILE.”