Excursions in Modern Mathematics Sixth Edition

1. Excursions in Modern MathematicsSixth Edition Peter Tannenbaum

2. Chapter 7The Mathematics of Networks The Cost of Being Connected

3. The Mathematics of NetworksOutline/learning Objectives • To identify and use a graph to model minimum network problems. • To classify which graphs are trees. • To implement Kruskal’s algorithm to find a minimal spanning tree. • To understand Torricelli’s construction for finding a Steiner point. • To recognize when the shortest network connecting three points uses a Steiner point. • To understand basic properties of the shortest network connecting a set of (more than three) points.

4. The Mathematics of Networks 7.1 Trees

5. Trees What is a Network? • Network is any connected graph (or tree). • Nodes or Terminals are the vertices. • Links are the edges. The theme of this chapter is finding optimal networks that connect a set of points.

6. Trees • Tree A network with no circuits. • Spanning Tree A subgraph that connects all the vertices of the network and has no circuits. • Minimum Spanning Tree (MST) Among all spanning trees of a weighted network, one with the least total weight.

7. Trees Interesting notes about trees: • One of the most useful structures in discrete mathematics. • Invaluable to computer scientists. • Computer programmers depend on trees when creating, sorting, and searching programs.

8. Trees • Family Tree

9. Trees • Sorting quadrilaterals

10. Trees • C2H6 (ethane)

11. Trees • Possible outcomes A rootedtree that happens to also be a binarytree.

12. Trees • Expression Tree (4 + 6) * 8 – 4/2

13. Trees

14. Trees Property 1: • In a tree, there is one and only one path joining any two vertices. • If there is one and only one path joining any two vertices of a graph, then the graph must be a tree.

15. Trees Property 2: • In a tree, every edge is a bridge. • If every edge of a graph is a bridge, then the graph must be a tree.

16. Trees Property 3: • A tree with N vertices has N – 1 edges. • If a network has N vertices and N – 1 edges, then it must be a tree.

17. Trees Notice that a disconnected graph (not a network) can have N vertices and N – 1 edges.

18. The Mathematics of Networks 7.2 Spanning Trees

19. Spanning Trees Property 4: • If a network has N vertices and M edges, then MN – 1. [R = M – (N – 1) as the redundancy of the network.] • If M = N – 1, the network is a tree; if M  N – 1, the network has circuits and is not a tree. (In other words, a tree is a network with zero redundancy and a network with positive redundancy is not a tree.

20. Spanning Trees The network in (a) has N = 8 vertices and M = 8 edges. The redundancy of the network is R = 1, so to find a spanning tree we will have to “discard” one edge.

21. Spanning Trees Five of these edges are bridges of the network, and they will have to be part of any spanning tree. The other three edges (BC, CG, and GB) form a circuit of length 3, and

22. Spanning Trees if we exclude any of the three edges we will have a spanning tree. Thus, the network has three different spanning trees (b), (c), and (d).

23. Spanning Trees • How many spanning trees are in this figure?

24. Spanning Trees • How many spanning trees in this figure?

25. Spanning Trees • Can you create a minimum spanning tree?

26. The Mathematics of Networks 7.3 Kruskal’s Algorithm

27. Kruskal’s Algorithm There are several well-known algorithms for finding minimum spanning trees. In this section we will discuss one of the the nicest of these, called Kruskal’s algorithm.

28. Kruskal’s Algorithm Kruskal’s Algorithm • Almost exactly like the cheapest-link algorithm. • Choose the cheapest link that doesn’t close the circuit. • Traces the Minimum Spanning Tree. (MST) • It always gives the optimal solution and is efficient.

29. Kruskal’s Algorithm What is the minimum spanning tree (MST) of the network shown in (b)?

30. Kruskal’s Algorithm We will use Kruskal’s algorithm to find the MST of the network. Step 1. Among all the possible links, we choose the cheapest one, in this case GF (at a cost of $42 million). This link is going to be a part of the MST, and we mark it in red as shown in (a).

31. Kruskal’s Algorithm Kruskal’s algorithm Step 2. The next cheapest link available is BD at $45 million. We choose it for the MST and mark it in red. Step 3. The cheapest link available is AD at $49 million. Again, we choose it for the MST and mark it in red.

32. Kruskal’s Algorithm Kruskal’s algorithm Step 4. For the next cheapest link there is a tie between AB and DG, both at $51 million. But we can rule out AB– it would create a circuit in the MST, and we can’t have that!) The link DG, on the other hand, is just fine, so we mark in red and make ti part of the MST.

33. Kruskal’s Algorithm Kruskal’s algorithm Step 5. The next cheapest link available is CD at $53 million. No problems here, so again, we mark it in red and make it part of the MST. Step 6. The next cheapest link available is BC at $55 million, but this link would create a circuit, so we cross it out.

34. Kruskal’s Algorithm Kruskal’s algorithm Step 6 (cont.). The next possible choice is CF at $56 million, but once again, this choice creates a circuit so we must cross it out. The next possible choice is CE at $59 million, and this is one we do choose. We mark it in red and make it part of the MST.

35. Kruskal’s Algorithm Kruskal’s algorithm Step … Wait a second– we are finished! We can tell we are done– six links is exactly what is needed for an MST on seven vertices (N – 1). Figure (c) shows the MST in red. The total cost of the network is $299 million.

36. Kruskal’s Algorithm • Find the minimum spanning tree for the below figure.

37. Kruskal’s Algorithm • Find the minimum spanning tree for the below figure:

38. The Mathematics of Networks 7.4 The Shortest Network Connecting Three Points

39. The Shortest Network Connecting Three Points What is the cheapest underground fiber-optic cable network connecting the three towns?

40. The Shortest Network Connecting Three Points Here, cheapest means “shortest”, so the name of the game to design a network that is as short as possible. We shall call such a network the shortest network (SN).

41. The Shortest Network Connecting Three Points The search for the shortest network often starts with a look at the minimum spanning tree. The MST can always be found using Kruskal’s algorithm and it gives us a ceiling on the length of the shortest network.

42. The Shortest Network Connecting Three Points In this example the MST consists of two (any two) of the three sides of the equilateral triangle (a), and its length is 1000 miles.

43. The Shortest Network Connecting Three Points It is not hard to find a network connecting the three towns shorter than the MST. The T- network (b) is clearly shorter. The length of the segment CJ is approximately 433 miles. The length of this network is 933 miles.

44. The Shortest Network Connecting Three Points We can do better. The Y- network shown in (c) is even shorter than the T- network. In this network there is a “Y”- junction at S, with three equal branches connecting S to each of A, B, and C.

45. The Shortest Network Connecting Three Points This network is approximately 866 miles long. A key feature is the way the three branches come together at the junction point S, forming equal 120 angles.

46. The Shortest Network Connecting Three Points Before we move on, we need to discuss briefly the notion of a junction point on a network. ( A junction pointin a network is any point where two or more segments of the network come together.

47. The Shortest Network Connecting Three Points The MST in (a) has a junction point at A, then network in (b), has a junction point at J, and the shortest network in (c) has a junction point at S.

48. The Shortest Network Connecting Three Points There are three important terms that we will use in connection with junction points: • In a network connecting a set of vertices, a junction point is said to be native junction point if it is located at one of the vertices. • A nonnative junction point located somewhere other than at one of the original vertices is called an interior junction point of the network.

49. The Shortest Network Connecting Three Points • An interior junction point consisting of three line segments coming together forming equal 120 angles (a “perfect” Y- junction if you will) is called a Steiner pointof the network.

50. The Shortest Network Connecting Three Points • The Shortest Network Connecting Three Points • If one of the angles of the triangle is 120 or more, the shortest network linking the three vertices consists of the two shortest sides of the triangle (a).