Networks! Topics in Contemporary Mathematics! MA 103! Summer II, 2013!

Transcription

1 1! Networks! Topics in Contemporary Mathematics! MA 103! Summer II, 2013!

2 2! Content Trees Spanning Trees Kruskal s Algorithm The Shortest Network Connecting Three Points Shortest Networks for Four or More Points

3 Network Our defini*on of a network is going to be really simple essen*ally, a network is a graph that is connected. In this context the term is most commonly used when the graph models a real life network.

4 Network Typically, the vertices of a network (sometimes called nodes or terminals) are objects transmitting stations, computer servers, places, cell phones, people, and so on. The edges of a network (which in this context are often called links) indicate connections among the objects wires, cables, roads, Internet connections, social connections, and so on.

5 Network Some networks are centrally planned and carefully designed to meet certain goals and objectives. Often these types of networks are very expensive to build, and one of the primary considerations when designing such networks is minimizing their cost. This certainly applies to networks of roads, fiber-optic cable lines, rail lines, power lines, and so on.

6 Optimal Network The general theme of this lesson is the problem of finding op4mal networks connec4ng a set of points. Op2mal means shortest, cheapest, or fastest, depending on whether the cost variable is distance, money, or 2me. Thus, the design of an optimal network involves two basic goals: (1) to make sure that all the vertices (stations, places, people, etc.) end up connected to the network and (2) to minimize the total cost of the network.

7 Trees For obvious reasons, problems of this type are known as minimum network problems. The backbone of a minimum network is a special type of graph called a tree. We will starts with a discussion of the proper*es of trees.

8 Example 1 The Amazonian Cable Network The Amazonia Telephone Company is contracted to provide telephone, cable, and Internet service to the seven small mining towns shown.

9 Example 1 The Amazonian Cable Network These towns are located deep in the heart of the Amazon jungle, which makes the project particularly difficult and expensive. In this environment the most practical and environmentally friendly option is to create a network of underground fiberoptic cable lines connecting the towns. In addition, it makes sense to bury the underground cable lines along the already existing roads connecting the towns. (How would you maintain and repair the lines otherwise?)

10 Example 1 The Amazonian Cable Network Here is a weighted graph model describing the situa*on. The ver*ces of the graph represent the towns, the edges of the graph represent the exis*ng roads, and the weight of each edge represents the cost (in millions of dollars) of crea*ng a fiber op*c cable connec*on along that par*cular road.

11 Example 1 The Amazonian Cable Network The problem facing the engineers and planners at the Amazonia Telephone Company is to build a cable network that (1) utilizes the existing network of roads, (2) connects all the towns, and (3) has the least cost. The challenge, of course, is in meeting the last requirement.

12 Language of Graphs Reformulate the three requirements of Example 1 in the language of graphs: 1. The network must be a subgraph of the original graph (in other words, its edges must come from the original graph). 3. The network must span the original graph (in other words, it must include all the ver9ces of the original graph). 3. The network must be minimal (in other words, the total weight of the network should be as small as possible).

13 Minimal Network - Cannot Have Any Circuits The last requirement has an important corollary a minimal network cannot have any circuits. Why not? Imagine that the solid edges in the figure represent already exis*ng links in a minimal network. Why would you then build the link between X and Y and close the circuit? The edge XY would be a redundant link of the network.

14 A Few Formal Definitions A network is just another name for a connected graph. (This terminology is most commonly used when the graph models a real life situa*on.) When the network has weights associated to the edges, we call it a weighted network. A network with no circuits is called a tree.

15 A Few Formal Definitions A spanning tree of a network is a subgraph that connects all the ver*ces of the network and has no circuits. Among all spanning trees of a weighted network, one with least total weight is called a minimum spanning tree (MST) of the network.

16 Example 2 Networks, Trees, and Spanning Trees The six graphs on the following slides all have the same set of vertices (A through L). Let s imagine, for the purposes of illustration, that these vertices represent computer labs at a university, and that the edges are Ethernet connections between pairs of labs.

17 Example 2 Networks, Trees, and Spanning Trees In this figure, there is no network the graph is disconnected. Recall that a network is a connected graph.

18 Example 2 Networks, Trees, and Spanning Trees The graph is now connected and we have a network. However, in this network there are several circuits (for example, K, H, I, J, K) that create redundant connec*ons. In other words, this network is not a tree.

19 Example 2 Networks, Trees, and Spanning Trees In this figure, there is a par*al tree connec*ng some of, but not all, the labs. G and I are leu out, so once again, we have no network.

20 Example 2 Networks, Trees, and Spanning Trees This figure shows a tree that spans (i.e., reaches) all the ver*ces. We now have a network connec*ng all the labs and without any redundant connec9ons. Here, the tree is the network.

21 Example 2 Networks, Trees, and Spanning Trees This figure shows (in red) the same tree as in the previous slide but now highlighted inside of a larger network. In this case we describe the red tree as a spanning tree of the larger network.

22 Example 2 Networks, Trees, and Spanning Trees This figure shows a different spanning tree for the same network as in the previous slide.

23 Properties of Trees As graphs go, trees occupy an important niche between disconnected graphs and over connected graphs. A tree is special by virtue of the fact that it is barely connected. This means several things:

24 Properties of Trees For any two ver9ces X and Y of a tree, there is one and only one path joining X to Y. (If there were two different paths joining X and Y, then these two paths would form a circuit, as shown.)

25 Properties of Trees Every edge of a tree is a bridge. (Suppose that some edge AB is not a bridge. Then without AB the graph is s*ll connected, so there must be an alterna*ve path from A to B. This would imply that the edge AB is part of a circuit as illustrated.)

26 Properties of Trees Among all networks with N ver*ces, a tree is the one with the fewest number of edges.

27 Example 3 Connect the Dots (and Then Stop) Imagine the following connect-the-dots game: Start with eight isolated vertices. The object of the game is to create a network connecting the vertices by adding edges, one at a time. You are free to create any network you want. In this game, bridges are good and circuits are bad. (Imagine, for example, that for each bridge in your network you get a $10 reward, but for each circuit in your network you pay a $10 penalty.)

28 Example 3 Connect the Dots (and Then Stop) We will let M denote the number of edges you have added at any point in time. In the early stages of the game the graph is disconnected.

29 Example 3 Connect the Dots (and Then Stop) For M = 7, the graph becomes connected. Each of these networks is a tree, and thus each of the seven edges is a bridge. Stop here and you will come out $70 richer.

30 Example 3 Connect the Dots (and Then Stop) Interestingly, this is as good as it will get. When M = 8, the graph will have a circuit it just can t be avoided. In addition, none of the edges in that circuit can be bridges of the graph. As a consequence, the larger the circuit that we create, the fewer the bridges left in the graph. Some examples are shown on the next slide.

31 Example 3 Connect the Dots (and Then Stop) A circuit, 5 bridges A circuit, 2 bridges A circuit, 1 bridge

32 Example 3 Connect the Dots (and Then Stop) As the value of M increases, the number of circuits goes up (very quickly) and the number of bridges goes down.

33 3 Key Properties of 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.

34 3 Key Properties of 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.

35 3 Key Properties of 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.

36 Disconnected Graph Notice that a disconnected graph (i.e., not a network) can have N vertices and N 1 edges, as shown.

37 Reformulate Property 3 to... PROPERTY 4 If a network has N vertices and M edges, then M N 1. [For convenience, we will refer to the difference 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.)

38 Spanning Trees In the case of a network with positive redundancy, there are many trees within the network that connect its vertices these are the spanning trees of the network.

39 Example 4 Counting Spanning Trees The network in the figure has N = 8 ver*ces and M = 8 edges. The redundancy of the network is R = M (N 1) = 1, so to find a spanning tree we will have to discard one edge.

40 Example 4 Counting 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 if we exclude any one of the three edges, then we will have a spanning tree.

41 Example 4 Counting Spanning Trees Thus, the network has three different spanning trees.

42 Example 4 Counting Spanning Trees The network in the Figure has M = 9 edges and N = 8 ver*ces. The redundancy of the network is R = 2, so to find a spanning tree we will have to discard two edges. Edges AB and AH are bridges of the network, so they will have to be part of any spanning tree.

43 Example 4 Counting Spanning Trees The other seven edges are split into two separate circuits (B, C, G, B of length 3 and C, D, E, F, C of length 4). A spanning tree can be found by bus*ng each of the two circuits. This means excluding any one of the three edges of circuit B, C, G, B and any one of the four edges of circuit C, D, E, F, C.

44 Example 4 Counting Spanning Trees For example, if we exclude BC and CD, we get the spanning tree shown.

45 Example 4 Counting Spanning Trees We could also exclude BC and DE and get the spanning tree shown.

46 Example 4 Counting Spanning Trees Given that there are 3 4 = 12 different ways to choose an edge from the circuit of length 3 and an edge from the circuit of length 4, there are 12 spanning trees. We have already shown two, here s one more.

47 Example 5 Counting Spanning Trees This network has M = 9 edges and N = 8 ver*ces. Here the circuits B, C, G, B and C, D, E, G, C share a common edge CG. Determining which pairs of edges can be excluded in this case is a bit more complicated.

48 Example 6 Minimum Spanning Trees This example builds on the ideas introduced in Example 2. The network shown here is the same network as Example 2 now with weights added to the edges.

49 Example 6 Minimum Spanning Trees Vertices represent computer labs, and edges are potential Ethernet connections. The weights represent the cost in K s (1K = $1000) of installing the Ethernet connections.

50 Example 6 Minimum Spanning Trees Using the terminology we introduced earlier, the weighted network has a redundancy of R = 3, (the network has M = 14 ver4ces and N = 12 edges).

51 Example 6 Minimum Spanning Trees The network has many possible spanning trees, and our job is to find one with least weight, that is, a minimum spanning tree (MST) for the network. We know that we can find a spanning tree by excluding certain edges of the network (those that close circuits) from the spanning tree and that there are many different ways in which this can be done. Given that the edges now have weights, a reasonable strategy for sor*ng through the op*ons would be to always try to exclude from the network the most expensive edges.

52 Example 6 Minimum Spanning Trees To illustrate the point, let s look at circuit B, C, D, E, B on the leu side of the network. It makes sense to exclude CD (which costs $95,000 to build).

53 Example 6 Likewise, on the right side of the network we have a configura*on of two circuits K, H, I, K and K, J, I, K that share the common edge KI. Minimum Spanning Trees

54 Example 6 We exclude the two most expensive edges from these two circuits (KJ and KI). Minimum Spanning Trees

55 Example 6 At this point, all the circuits of the original network are busted, and we end up with the red spanning tree shown. Minimum Spanning Trees

56 Minimum Spanning Trees Is the red spanning tree obtained in Example 6 the MST of the original network? We would like to think it is, but how can we be sure, especially azer the lessons learned in Hamilton circuit? And even if it is, what assurances do we have that our simple strategy will work in more complicated graphs? These are the ques*ons we will answer next.

57 Kruskal s Algorithm Kruskal s algorithm is almost iden4cal to the cheapestlink algorithm: We build the minimum spanning tree one edge at a *me, choosing at each step the cheapest available edge. The only restric9on to our choice of edges is that we should never choose an edge that creates a circuit. (Having three or more edges coming out of a vertex, however, is now OK.) What is truly remarkable about Kruskal s algorithm is that unlike the cheapest link algorithm it always gives an op9mal solu9on.

58 Example 7 The Amazonian Cable Network: Part 2 What is the op*mal fiber op*c cable network connec*ng the seven towns shown?

59 Example 7 The Amazonian Cable Network: Part 2 The weighted graph shows the costs (in millions of dollars) of laying the cable lines along each of the poten*al links of the network.

60 Example 7 The Amazonian Cable Network: Part 2 The answer, as we now know, is to find the MST of the graph. We will use Kruskal s algorithm to do it.

61 Example 7 The Amazonian Cable Network: Part 2 Step 1. Among all the possible links, we choose the cheapest one, in this par*cular case GF (at a cost of $42 million). This link is going to be part of the MST, and we mark it in red (or any other color) as shown

62 Example 7 The Amazonian Cable Network: Part 2 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 next cheapest link available is AD at $49 million. Again, we choose it for the MST and mark it in red.

63 Example 7 The Amazonian Cable Network: Part 2 Step 4. For the next cheapest link there is a *e 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 it in red and make it part of the MST.

64 Example 7 The Amazonian Cable Network: Part 2 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. 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.

65 Example 7 The Amazonian Cable Network: Part 2 Step... Wait a second we are finished! Even without looking at a picture, we can tell we are done six links is exactly what is needed for an MST on seven ver2ces. The figure shows the MST in red. The total cost of the network is $299 million.

66 KRUSKAL S ALGORITHM Step 1 Pick the cheapest link available. (In case of a tie, pick one at random.) Mark it (say in red). Step 2. Pick the next cheapest link available and mark it. Steps 3, 4,..., N 1. Continue picking and marking the cheapest unmarked link available that does not create a circuit.

67 Kruskal s Algorithm As algorithms go, Kruskal s algorithm is as good as it gets. First, it is easy to implement. Second, it is an efficient algorithm. As we increase the number of ver*ces and edges of the network, the amount of work grows more or less propor*onally (roughly speaking, if finding the MST of a 30 city network takes you, say, 30 minutes, finding the MST of a 60 city network might take you 60 minutes).

68 Kruskal s Algorithm Last, but not least, Kruskal s algorithm is an op*mal algorithm it will always find a minimum spanning tree. Thus, we have reached a surprisingly sa*sfying end to the MST story: No mamer how complicated the network, we can find its minimum spanning tree by means of an algorithm that is easy to understand and implement, is efficient, and is also op*mal.

69 Shortest Network 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.!

70 Example 8 The Outback Cable Network The Outback Cable Network company has a contract to create an underground fiber-optic network connecting the towns of Alcie Springs, Booker Creek, and Camoorea three isolated.! The towns are equidistant, forming an equilateral triangle 500 miles on each side.!

71 Example 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?!

72 Example 8 The Outback Cable Network 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).

73 Example 8 The Outback Cable Network 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.!

74 Example 8 The Outback Cable Network The T-network is clearly shorter. The length of the height AJ of the triangle can be computed using properties of triangles and! is approximately 433 miles. It follows that the length of this network is approximately 933 miles.!

75 Example 8 The Outback Cable Network 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.!

76 Example 8 The Outback Cable Network 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.!

77 Shortest Network Our discussion of shortest networks will rely heavily on a careful analysis of the 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.!

78 Shortest Network 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.!

79 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 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.!

80 JUNCTION POINTS 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.

81 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.

82 Example 9 The Third Trans-Pacific Cable Network 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.!

83 Example 9 The Third Trans-Pacific Cable Network

84 Example 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 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?!

85 Example 9 The Third Trans-Pacific Cable Network 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.!

86 Example 9 The Third Trans-Pacific Cable 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.!

87 Kruskal s Algorithm From Examples 8 and 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!!

88 Example 10 A High-Speed Rail Network Off and on, there is talk of building a high-speed rail connection between Los Angeles and Las Vegas.! To make it more interesting, let s add a third city Salt Lake City to the mix.!

89 Example 10 A High-Speed Rail Network The straight-line distances between the three cities are shown here.! The mathematical question again is, What is the shortest network connecting these three cities?!

90 Example 10 A High-Speed Rail Network 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.!

91 Example 10 A High-Speed Rail Network 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?!

92 Example 10 A High-Speed Rail 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!

93 THE SHORTEST NETWORK CONNECTING THREE POINTS 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. In this situation, the shortest network coincides with the minimum spanning tree.

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

95 Torricelli s Construction 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:!

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

97 Step 1 TORRICELLI S CONSTRUCTION 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.

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

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

100 Shortest Networks for Four 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.!

101 Example 11 Four-City Networks 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 each side, as shown.! What does the optimal network connecting these cities look like? It depends on the situation.!

102 Example 11 Four-City Networks 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! 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.!

103 Example 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 network is approximately 1414 miles.!

104 Example 11 Four-City Networks 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.!

105 Example 11 Four-City Networks 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! the two networks are the shortest networks connecting the four cities.!

106 Example 12 Four-City Networks: Part 2 Let s repeat what we did in Example 11, but this time imagine that the four cities are located at the vertices of a rectangle, as shown.!

107 Example 12 Four-City Networks: Part 2 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.!

108 Example 12 Four-City Networks: Part 2 For the shortest network, an obvious candidate would be a network with two interior Steiner junction points. There are two such networks shown.!

109 Example 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 a pretty significant difference.!

110 Example 12 Four-City Networks: Part 2 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!!

111 Example 12 Four-City Networks: Part 3 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.! The minimum spanning tree is also shown and it is 600 miles long.!

112 Example 12 Four-City Networks: Part 3 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.!

113 Example 12 Four-City Networks: Part 3 If not Steiner junction points, then how about other kinds of interior junction points? How about X-junctions, or T-junctions, or! 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.!

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

115 Shortest Networks 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 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.!

116 Shortest Networks But we also know that the shortest network without interior junction points is the minimum spanning tree! Conclusion: For the four cities of Example 13, the shortest network is the minimum spanning tree!!

117 Example 14 Four-City Networks: Part 4 Let s look at four cities A, B, C, and D. This time, the cities sit as shown.! The MST is shown and its length is 1000 miles.!

118 Example 14 Four-City Networks: Part 4 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:!

119 Example 14 Four-City Networks: Part 4 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!).!

120 Example 14 Four-City Networks: Part miles miles This last network is the shortest in our list and thus the shortest network connecting, the four cities.! miles

121 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 situation where 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.!

122 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) such that all interior junction points are Steiner points.]

123 Shortest Network Algorithm 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.!

124 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 possible Steiner trees is in the billions.! Optimal (it guarantees the shortest network), 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).!

125 Approximate Solutions 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.!

126 Approximate Solutions 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.!