Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 7 The Mathematics of Networks 7.1Trees 7.2Spanning Trees 7.3 Kruskal’s Algorithm 7.4The Shortest Network Connecting Three Points 7.5Shortest Networks for Four or More Points

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Minimum spanning trees represent the optimal way to connect a set of points based on one key assumption–that all the connections should be along the links of the network. But what if, in a manner of speaking, we don’t have to follow the road? What if we are free to create new vertices and links “outside” the original network? To clarify the distinction, let’s look at a new type of cable network problem. Shortest Networks for Four Points

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Imagine four cities (A, B, C, and D) that need to be connected by an underground fiber- optic cable network. Suppose that the cities sit on the vertices of a square 500 miles on Example 7.11Four-City Networks each side, as shown. What does the optimal network connecting these cities look like? It depends on the situation.

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. If we don’t want to create any interior junction points in the network (either because we don’t want to venture off the prescribed paths–as in the jungle scenario–or because the cost of Example 7.11Four-City Networks creating a new junction is too high), then the answer is a minimum spanning tree, such as the one shown. The length of the MST is 1500 miles.

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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 Example 7.11Four-City Networks at O, the center of the square. The length of this network is approximately 1414 miles.

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. We can shorten the network even more if we place not one but two interior junction points inside the square. It’s not immediately obvious that this is a good move, 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 junction points Steiner points. There are two different networks possible with two Steiner points inside the square, and they are shown on the next slide. Example 7.11Four-City Networks

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. These two networks are essentially equal (one is a rotated version of the other) and clearly have the same length–approximately 1366 miles. It is impossible to shorten these any further– Example 7.11Four-City Networks the two networks are the shortest networks connecting the four cities.

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Let’s repeat what we did in Example 7.11, but this time imagine that the four cities are located at the vertices of a rectangle, as shown. Example 7.12Four-City Networks: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. By now, we have some experience on our side, so we can cut to the chase. We know that the MST is 1000 miles long. That’s the easy part. Example 7.12Four-City Networks: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. For the shortest network, an obvious candidate would be a network with two interior Steiner junction points. There are two such networks shown. Example 7.12Four-City Networks: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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–a pretty significant difference. Example 7.12Four-City Networks: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. It is obvious that the network on the left cannot be the shortest network. If there is any justice, then the network on the right fits the pattern and ought to be the shortest. In fact, it is! Example 7.12Four-City Networks: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Let’s look at four cities once more. This time, imagine that the cities are located at the vertices of a skinny trapezoid, as shown. Example 7.12Four-City Networks: Part 3 The minimum spanning tree is also shown and it is 600 miles long.

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. What about the shortest network? We are fairly certain that we should be looking for a network with a couple 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 put it in a more formal way, the angles at A and B are greater than 120º. Since no Steiner points can be placed inside the trapezoid, the shortest network, whatever it is, will have to be one without Steiner junction points. Example 7.12Four-City Networks: Part 3

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. If not Steiner junction points, then how about other kinds of interior junction points? How about X-junctions, or T-junctions, or Example 7.12Four-City Networks: Part 3 Y-junctions where the angles are not all 120º? As reasonable as this sounds, the only possible interior junction points in a shortest network are Steiner points.

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. In the shortest network the interior junction points are all Steiner points. THE INTERIOR JUNCTION RULE FOR SHORTEST NETWORKS

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The interior junction rule is an important and a powerful piece of information in building shortest networks, and we will come back to it soon. Meanwhile, what does 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 of the interior junction rule. Shortest Networks

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. But we also know that the shortest network without interior junction points is the minimum spanning tree! Conclusion: For the four cities of Example 7.13, the shortest network is the minimum spanning tree! Shortest Networks

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. For the last time, let’s look at four cities A, B, C, and D. This time, the cities sit as shown. Example 7.14Four-City Networks: Part 4 The MST is shown and its length is 1000 miles.

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. In searching for the shortest network, the MST is a good starting point–it might even be the shortest network. If the 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 the layout of these cities, it is geometrically impossible to build a network with two interior Steiner points. On the other hand, there are three possible networks with a single interior Steiner point: Example 7.14Four-City Networks: Part 4

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. We have now narrowed the list of contenders for the shortest network title to four: 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 the MST is 1000 miles (we knew that!). Example 7.14Four-City Networks: Part 4

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. This last network is the shortest in our list and thus the shortest network connecting, the four cities. Example 7.14Four-City Networks: Part miles miles

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. What happens when the number of cities grows–five, six, one hundred? How do we find the shortest network? Here, mathematicians face a situation analogous to the one discussed in Chapter 6–no optimal and efficient algorithm is currently known. At this point, the best we can do is to take advantage of the following rule, which we will informally call the shortest network rule. Shortest Networks

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. ■ 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) such that all interior junction points are Steiner points.] THE SHORTEST NETWORK RULE The shortest network connecting a set of points is either

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 1.Make a list of all possible Steiner trees connecting the points, and find the shortest 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 desired shortest network. Shortest Network Algorithm

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. With as few as 10 points, we might have to compute over a million possible Steiner trees; with 20 cities, the number of possible Steiner trees is in the billions. Optimal (it guarantees the shortest net- work), but inefficient (as the number of points grows, the amount of effort needed to find the solution grows too fast for the algorithm to be practical). But it is Impractical

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Settle for approximate solutions (in other words, if we are willing to accept a short network–not necessarily the shortest). Sophisticated approximate algorithms for finding short networks can tackle problems involving hundreds of points and efficiently produce short networks with lengths that are no more than 1% off the shortest network. Approximate Solutions

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Even an old workhorse like Kruskal’s algorithm can be used as a reasonably good approximate algorithm for finding short networks. For any set of points–no matter how many and how they are laid out–the MST is never that much longer than the shortest network: 13.4% longer in the worst case scenario, but usually 3% or less. Approximate Solutions