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 Networks for Four Points 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 7.11 Four-City Networks Imagine four cities (A, B, C, and D) that need to be connected by an undergroundfiber-optic cable network. Suppose that the cities sit on the vertices of a square500 miles on each side, as shown. What does the optimal networkconnecting these cities look like? It depends on the situation.

5. Example 7.11 Four-City Networks If we don’t want to create any interior junction points in the network (eitherbecause we don’t want to venture off the prescribed paths–as in the jungle scenario–or because the cost of creating a new junction is too high), then theanswer is a minimum spanning tree, such as the one shown. The lengthof the MST is 1500 miles.

6. Example 7.11 Four-City Networks On the other hand, if interior junction points are allowed, somewhat shorter networks are possible. One obvious improvement is the network shown,with an X-junction located at O, the center of the square. The length of this networkis approximately 1414 miles.

7. Example 7.11 Four-City Networks We can shorten the network even more if we place not one but two interiorjunction points inside the square. It’s not immediately obvious that this is a goodmove, but once we recognize that two junction points might be better than one,then it’s not hard to see that the best option is to make the two interior junctionpoints Steiner points. There are two different networks possible with two Steinerpoints inside the square, and they are shown on the next slide.

8. Example 7.11 Four-City Networks These twonetworks are essentially equal (one is a rotated version of the other) and clearlyhave the same length–approximately 1366 miles. It is impossible to shorten these any further– the two networks are the shortest networksconnecting the four cities.

9. Example 7.12 Four-City Networks: Part 2 Let’s repeat what we did in Example 7.11, but this time imagine that the fourcities are located at the vertices of a rectangle, as shown.

10. Example 7.12 Four-City Networks: Part 2 By now,we have some experience on our side, so we can cut to the chase. We know thatthe MST is 1000 miles long. That’s the easy part.

11. Example 7.12 Four-City Networks: Part 2 For the shortest network, an obviouscandidate would be a network with two interior Steiner junction points. There aretwo such networks shown.

12. Example 7.12 Four-City Networks: Part 2 The length of the network on the left is approximately 993 miles,while the length of the network on the right is approximately 920 miles–apretty significant difference.

13. Example 7.12 Four-City Networks: Part 2 It is obvious that the network on the left cannot be the shortest network. If thereis any justice, then the network on the right fits the pattern and ought to be the shortest. Infact, it is!

14. Example 7.12 Four-City Networks: Part 3 Let’s look at four cities once more. This time, imagine that the cities are located atthe vertices of a skinny trapezoid, as shown. The minimum spanning tree is also shown and it is 600 miles long.

15. Example 7.12 Four-City Networks: Part 3 What about the shortest network? We are fairly certain that we should be looking for a network with acouple of interior Steiner junction points. After a little trial and error, however,we realize that such a layout is impossible! The trapezoid is too skinny, or, to putit in a more formal way, the angles at A and B are greater than 120º. Sinceno Steiner points can be placed inside the trapezoid, the shortest network, whateverit is, will have to be one without Steiner junction points.

16. Example 7.12 Four-City Networks: Part 3 If not Steiner junction points, then how aboutother kinds of interior junction points? How about X-junctions, orT-junctions, or Y-junctions where the angles are not all 120º? As reasonable as this sounds, the only possibleinterior junction points in a shortest network are Steiner points.

17. THE INTERIOR JUNCTION RULE FOR SHORTEST NETWORKS In the shortest network the interior junction points are all Steiner points.

18. Shortest Networks The interior junction rule is an important and a powerful piece of informationin building shortest networks, and we will come back to it soon. Meanwhile, whatdoes it tell us about the situation of Example 7.13? It tells us that the shortest network cannot have any interior junction points. Steiner junction points are impossible because of the geometry; other types of junction points do not work because ofthe interior junction rule.

19. Shortest Networks But we also know that the shortest network without interior junction points is the minimum spanning tree! Conclusion: For the four citiesof Example 7.13, the shortest network is the minimum spanning tree!

20. Example 7.14 Four-City Networks: Part 4 For the last time, let’s look at four cities A,B,C, and D. This time, the cities sit asshown. The MST is shown and its length is 1000 miles.

21. Example 7.14 Four-City Networks: Part 4 In searching for the shortest network, the MST is agood starting point–it might even be the shortest network. Ifthe shortest network is not the MST, then it will have interior junction points,which by the interior junction rule will have to be Steiner points. Because of thelayout of these cities, it is geometrically impossible to build a network with twointerior Steiner points. On the other hand, there are three possible networks witha single interior Steiner point:

22. Example 7.14 Four-City Networks: Part 4 We have now narrowed the list of contenders for the shortest network title tofour: the MST or one of the networks shown. All we have to do now to figure which one is the SN is to compute their lengths. The length of theMST is 1000 miles (we knew that!).

23. Example 7.14 Four-City Networks: Part 4 This last network is the shortest in our list and thus the shortest network connecting, the four cities. 1325.04 miles 1325.04 miles 981.86 miles

24. Shortest Networks What happens when the number of cities grows–five, six, one hundred?How do we find the shortest network? Here, mathematicians face a situationanalogous to the one discussed in Chapter 6–no optimal and efficient algorithmis currently known. At this point, the best we can do is to take advantage of thefollowing rule, which we will informally call the shortest network rule.

25. THE SHORTEST NETWORK RULE The shortest network connecting a set of points is either ■a minimum spanning tree (no interior junction points) or ■a Steiner tree. [A Steiner tree is a network with no circuits (i.e., a tree) suchthat all interior junction points are Steiner points.]

26. Shortest Network Algorithm 1. Make a list of all possible Steiner trees connecting the points, and find theshortest network in that list. 2. Using Kruskal’s algorithm, find the minimum spanning tree connecting the points. 3. Compare the trees found in Steps 1 and 2. The shorter of these two is the desiredshortest network.

27. But it is Impractical • With as few as 10 points, we might have to compute over a million possible Steiner trees; with 20 cities, the number of possibleSteiner trees is in the billions. • Optimal(it guarantees the shortest net-work), but inefficient(as thenumber of points grows, theamount of effort needed to findthe solution grows too fast forthe algorithm to be practical).

28. Approximate Solutions • Settle for approximate solutions (in other words,if we are willing to accept ashort network–not necessarilythe shortest). • Sophisticated approximate algorithmsfor finding short networks can tackleproblems involving hundredsof points and efficiently produce short networks withlengths that are no more than1% off the shortest network.

29. Approximate Solutions • Even an old workhorselike Kruskal’s algorithm canbe used as a reasonably goodapproximate algorithm forfinding short networks. • For anyset of points–no matter howmany and how they are laidout–the MST is never thatmuch longer than the shortestnetwork: 13.4% longer in theworst case scenario, but usually3% or less.