7 The Mathematics of Networks

2. 7 The Mathematics of Networks 7.1 Trees 7.2 Spanning Trees 7.3 Kruskal’s Algorithm 7.4 The Shortest Network Connecting Three Points 7.5 Shortest Networks for Four or More Points

3. Shortest Network Minimum spanning trees represent the optimal way to connect a set of pointsbased on one key assumption–that all the connections should be along the linksof the network. But what if, in a manner of speaking, we don’t have to follow the road? What if weare free to create new vertices and links “outside” the original network? To clarifythe distinction, let’s look at a new type of cable network problem.

4. Example 7.8 The Outback Cable Network The Amazonia Telephone Company has relocated to Australia and now calls itselfthe Outback Cable Network. The company has a new contract to create an underground fiber-optic network connecting the towns of Alcie Springs, Booker Creek,and Camoorea–three isolated towns located in the heart of the Australian outback. By freakish coincidence (or good planning) the towns are equidistant, forming an equilateral triangle 500 miles on each side.

5. Example 7.8 The Outback Cable Network The three towns areconnected by three unpaved straight roads. What is the cheapest underground fiber-optic cable network connectingthe three towns?

6. Example 7.8 The Outback Cable Network The Australian outback is flat scrub with few majorroads to speak of–there is little or no disadvantage to going offroad. We will assume that in the outback cable lines can be laid anywhere (alongthe road or offroad) and that there is a fixed cost of $100,000 per mile. Here,cheapest means “shortest,” so the name of the game is to design a network that isas short as possible. We shall call such a network the shortest network (SN).

7. Example 7.8 The Outback Cable Network The MST can always be found using Kruskal’s algorithm, and it givesus a ceiling on the length of the shortest network. In this example the MST consistsof two (any two) of the three sides of the equilateral triangle, and itslength is 1000 miles.

8. Example 7.8 The Outback Cable Network The T-network is clearly shorter. The length of the height AJof the triangle can be computed using properties of 30-60-90 triangles and isapproximately 433 miles. It follows that the length of this network is approximately 933 miles.

9. Example 7.8 The Outback Cable Network The Y-network shown is shorter than theT-network. There is a “Y”-junction at S, with threeequal branches connecting S to each of A, B, and C. This network is approximately 866 miles long. A key feature of this network is the way the three branches cometogether at the junction point S, forming equal 120ºangles.

10. Example 7.8 The Outback Cable Network It is not hard to convinceoneself that the Y-network is the shortest network connecting thethree towns. An informal argument goes like this: Since the original layout of thetowns is completely symmetric (it looks the same from each of the three towns),we would expect that the shortest network should also be completely symmetric.The only network that looks the same from each of the three towns is a “perfect”Y where all three branches of the Y have equal length.

11. Shortest Network Our discussion of shortest networks will rely heavily on a careful analysis ofthe junction points of the network we create. In some cases, the junction point is one of the vertices of the original network–we will refer to such a junction point as a native junction point.

12. Shortest Network In othercases, the junction points arepoints that were not in the original network–they are new points introduced inthe process of creating the network.

13. Shortest Network We call such points interior junction points ofthe network. In addition, the junction point S has the special property that it makes for a “perfect Y” junction–three branches coming together atequal 120º angles. We will call such a junction point a Steiner point of the network.

14. JUNCTION POINTS ■A junction point of a network is a point where two or more branches of thenetwork come together. ■A native junction point is a junction point that is also one of the originalvertices of the graph.

15. JUNCTION POINTS ■A junction point that is not a native junction point is called an interiorjunction point of the network. ■An interior junction point formed by three branches coming together atequal 120ºangles is called a Steiner point of the network.

16. Example 7.9 The Third Trans-Pacific Cable Network This is a true story. In 1989 a consortium of several of the world’s biggest telephonecompanies (among them AT&T, MCI, Sprint, and British Telephone) completedthe Third Trans-Pacific Cable (TPC-3) line, a network of submarine fiber-optic lineslinking Japan and Guam to the United States (via Hawaii). The approximatestraight-line distances (in miles) between the three terminals of the network(Chikura, Japan; Tanguisson Point, Guam; and Oahu, Hawaii) are shown next.

17. Example 7.9 The Third Trans-Pacific Cable Network

18. Example 7.9 The Third Trans-Pacific Cable Network By and large, laying submarine cable has a fixed cost per mile (somewhere between$50,000 and $70,000), so cheapest means “shortest” and the problem is once againto find the shortest network connecting the three terminals.Given what we learned in Example 7.8, a reasonable guess is that the shortestnetwork is going to require an interior junction point, somewhere inside the Japan-Guam-Hawaii triangle. But where?

19. Example 7.9 The Third Trans-Pacific Cable Network The junctionpoint S is located in such a way that three branches of the network coming out of Sform equal 120º angles–in other words, S is a Steiner point of the network.

20. Example 7.9 The Third Trans-Pacific Cable Network The length of the shortest network is5180 miles, but this is a theoretical length only, based on straight-line distances.With submarine cable one has to add as much as 10% to the straight-linelengths because of the uneven nature of the ocean floor. The actual length ofsubmarine cable used in TPC-3 is about 5690 miles.

21. Kruskal’s Algorithm From Examples 7.8 and 7.9 we are tempted to conclude that the key to findingthe shortest network connecting three points (cities) A, B, and C is to find a Steinerpoint S inside triangle ABC and make this point the junction point of the network(i.e.,the network consists of the three segments AS, BS, and CS forming equal 120ºangles at S).This is true as long as we can find a Steiner point inside the triangle,but ,as we will see in the next example, not every triangle has a Steiner point!

22. Example 7.10 A High-Speed Rail Network Off and on, there is talk of building a high-speed rail connection between LosAngeles and Las Vegas. To make it more interesting, let’s add a third city–SaltLake City–to the mix.

23. Example 7.10 A High-Speed Rail Network The straight-line distances between the three cities areshown in Fig. 7-21(a). The mathematical question again is, What is the shortestnetwork connecting these three cities?

24. Example 7.10 A High-Speed Rail Network The first observation is a simple but important general property of triangles illustrated:For any triangle ABCand interior point S, angle ASC must be bigger than angle ABC. It follows that forangle ASC to measure 120º, the measure of angle ABC must be less than 120º.

25. Example 7.10 A High-Speed Rail Network Unfortunately (or fortunately–depends on how you look at it), the angleABC in this example measures about 155º; therefore, there is no Steiner junctionpoint inside the triangle. Without a Steiner junction point, how do we find theshortest network?

26. Example 7.10 A High-Speed Rail Network The answer turns out to be surprisingly simple: In this situationthe shortest network consists of the two shortest sides of the triangle, as shown. Please notice that this shortest network happens to be the minimumspanning tree as well!

27. THE SHORTEST NETWORK CONNECTING THREE POINTS ■If one of the angles of the triangle is 120º or more, the shortest networklinking the three vertices consists of the two shortest sides of the triangle. In this situation, the shortest network coincides with the minimum spanning tree.

28. THE SHORTEST NETWORK CONNECTING THREE POINTS ■If all three angles of the triangle are less than 120º, the shortest network isobtained by finding a Steiner point S inside the triangle and joining S to eachof the vertices.

29. Torricelli’s Construction Finding the shortest network connecting three points is a problem with a longand an interesting history going back some 400 years. In the early 1600s ItalianEvangelista Torricelli came up with a remarkably simple and elegant method forlocating a Steiner junction point inside a triangle. All you need to carry out Torricelli’s construction is a straightedge and a compass; all you need to understandwhy it works is a few facts from high school geometry. Here it goes:

30. TORRICELLI’S CONSTRUCTION Suppose A, B, and C form a triangle such that all three angles of thetriangle are less than 120º.

31. TORRICELLI’S CONSTRUCTION Step 1 Choose any of the three sides of the triangle (say BC) and construct an equilateral triangle BCX so that X and A are onopposite sides of BC.

32. TORRICELLI’S CONSTRUCTION Step 2 Circumscribe a circle around equilateral triangle BCX.

33. TORRICELLI’S CONSTRUCTION Step 3 Join X to A with a straight line. The point of intersection of the line segment XA with the circle is the Steiner point!