Excursions in Modern Mathematics, 7e: 7.4 - 2Copyright © 2010 Pearson Education, Inc. 7 The Mathematics of Networks 7.1Trees 7.2Spanning Trees 7.3 Kruskal’s

Excursions in Modern Mathematics, 7e: 7.4 - 2Copyright © 2010 Pearson Education, Inc.

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


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 Network


The Amazonia Telephone Company has relocated to Australia and now calls itself the 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.

Example 7.8 The Outback Cable Network


The three towns are connected by three unpaved straight roads.


What is the cheapest underground fiber-optic cable network connecting the three towns?


The Australian outback is flat scrub with few major roads 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 (along the 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 is as short as possible. We shall call such a network the shortest network (SN).



The MST can always be found using Kruskal’s algorithm, and it gives us a ceiling


on the length of the shortest network. In this example the MST consists of two (any two) of the three sides of the equilateral triangle, and its length is 1000 miles.


The T-network is clearly shorter. The length of the height AJ of the triangle can be


computed using properties of 30-60-90 triangles and is approximately 433 miles. It follows that the length of this network is approximately 933 miles.


The Y-network shown is shorter than the T-network. There is a “Y”-junction at S, with three equal 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 come together at the junction point S, forming equal 120º angles.


It is not hard to convince oneself that the Y-network is the shortest network connecting the three towns. An informal argument goes like this: Since the original layout of the towns 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.



Our discussion of shortest networks will rely heavily on a careful analysis of the junction points of the network we create. In some

Shortest Network

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.


In other cases, the junction points are points that were not in the original network–they are new points introduced in the process of creating the network.

Shortest Network


We call such points interior junction points of the network. In addition, the junction point S has the special property that it makes for

Shortest Network

a “perfect Y” junction–three branches coming together at equal 120º angles. We will call such a junction point a Steiner point of the network.


■ A junction point of a network is a point where two or more branches of the network come together.

■ A native junction point is a junction point that is also one of the original vertices of the graph.

JUNCTION POINTS


■ A junction point that is not a native junction point is called an interior junction point of the network.

■ An interior junction point formed by three branches coming together at equal 120º angles is called a Steiner point of the network.

JUNCTION POINTS


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

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 again to find the shortest network connecting the three terminals. Given what we learned in Example 7.8, a reasonable guess is that the shortest network is going to require an interior junction point, somewhere inside the Japan- Guam-Hawaii triangle. But where?



The junction point S is located in such a way that three branches of the network coming out of S form equal 120º angles–in other words,


S is a Steiner point of the network.


The length of the shortest network is 5180 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-line lengths because of the uneven nature of the ocean floor. The actual length of submarine cable used in TPC-3 is about 5690 miles.



From Examples 7.8 and 7.9 we are tempted to conclude that the key to finding the shortest network connecting three points (cities) A, B, and C is to find a Steiner point 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 120º angles 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!

Kruskal’s Algorithm


Off and on, there is talk of building a high-speed rail connection between Los Angeles

Example 7.10 A High-Speed Rail Network

and Las Vegas. To make it more interesting, let’s add a third city–Salt Lake City–to the mix.


The straight-line distances between the three cities are shown in Fig. 7-21(a). The


mathematical question again is, What is the shortest network connecting these three cities?


The first observation is a simple but important general property of triangles illustrated: For any triangle ABC and interior point S, angle ASC must be bigger than angle ABC.


It follows that for angle ASC to measure 120º, the measure of angle ABC must be less than 120º.


Unfortunately (or fortunately–depends on how you look at it), the angle ABC in this example measures about 155º; therefore, there is no Steiner junction point inside the triangle.


Without a Steiner junction point, how do we find the shortest network?


The answer turns out to be surprisingly simple: In this situation the shortest network consists of the two shortest sides of the triangle, as shown.


Please notice that this shortest network happens to be the minimum spanning tree as well!


■ If one of the angles of the triangle is 120º or more, the shortest network linking the three vertices consists of the two shortest sides of the triangle.

THE SHORTEST NETWORK CONNECTING THREE POINTS

In this situation, the shortest network coincides with the minimum spanning tree.


■ If all three angles of the triangle are less than 120º, the shortest network is obtained by finding a Steiner point S

THE SHORTEST NETWORK CONNECTING THREE POINTS

inside the triangle and joining S to each of the vertices.


Finding the shortest network connecting three points is a problem with a long and an interesting history going back some 400 years. In the early 1600s Italian Evangelista Torricelli came up with a remarkably simple and elegant method for locating 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 understand why it works is a few facts from high school geometry. Here it goes:

Torricelli’s Construction


Suppose A, B, and C form a triangle such that all three angles of the triangle are less than 120º.

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 on opposite sides of BC.



Step 2

Circumscribe a circle around equilateral triangle BCX.



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!


Documents

Excursions in Modern Mathematics, 7e: 7.4 - 2Copyright © 2010 Pearson Education, Inc. 7 The Mathematics of Networks 7.1Trees 7.2Spanning Trees 7.3 Kruskal’s