1 MATHEMATICAL THOUGHT AND PRACTICE Chapter 7: The Mathematics of Networks The Cost of Being Connected
4 Network 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.
5 Network Twitter Network flowingdata.com The vertices of a network (nodes, terminals) are objects transmitting stations, servers, places, cell phones, people... The edges of a network (links) indicate connections among objects wires, cables, roads, internet connections, social connections...
6 Optimal Network Social Networking Collage The design of an optimal network involves two basic goals: 1. To make sure that all the vertices connect to the network and 2. To minimize the total cost of the network.
7 The Amazonian Cable Network The telephone company must lay fiber-optic lines along the roads between towns.
8 The Amazonian Cable Network Vertices represent the towns, edges represent existing roads, and weights represent costs in millions of dollars to lay the cables along that road.
9 Language of Graphs 1. The network must be a subgraph of (edges come from) the original graph. 2. The network must span (include all vertices) the original graph. 3. The network must be minimal (total weight of network should be as small as possible).
10 Minimal Network No Circuits Circuits cannot be part of minimal networks. The edge XY would be a redundant link of the network.
11 Formal Definitions images.worldgallery.co.uk A network is a connected graph. A weighted network has weighted edges. A network with no circuits is called a tree. A spanning tree is a subgraph that connects all the vertices and has no circuits. The spanning tree with least total weight is called a minimum spanning tree (MST).
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13 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 that these vertices represent computer labs at a university, and that the edges are Ethernet connections between pairs of labs.
14 Networks, Trees, and Spanning Trees No network graph is disconnected. Network with circuits
15 Networks, Trees, and Spanning Trees No network partial tree Network, spanning tree, no redundancies
16 Properties of Trees A tree is special because it s barely connected. This means: 1. For any two vertices, there is one and only one path joining X to Y. 2. Every edge of a tree is a bridge. 3. Among all networks with N vertices, a tree has the fewest number of edges.
17 Connect the Dots (and Stop) Start with eight isolated vertices. Create a network connecting the vertices by adding edges, one at a time. Create any network you want. Bridges are good and circuits are bad. (Imagine each bridge gives you $10 reward, but each you pay $10.)
18 Connect the Dots (and 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.
19 Connect the Dots (and Stop) As M increases, the number of circuits goes up and bridges goes down.
20 Tree Properties
21 Tree Properties
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24 Spanning Trees An American Haunting 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.
25 Counting Spanning Trees Redundancy of the network is R = M (N 1)= 1. So to find a spanning tree we will have to discard one edge.
26 The network has three different spanning trees.
27 Counting Spanning Trees The network has M = 9 edges and N = 8 vertices. The redundancy of the network is R = 2, so to find a spanning tree we will have to discard two edges getting rid of circuits.
28 Counting Spanning Trees
29 Counting Spanning Trees This network has M = 9 edges and N = 8 vertices. Here the circuits share a common edge CG. Determining which pairs of edges can be excluded in this case is a bit more complicated.
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32 Find Minimum Spanning Trees (MST) Vertices represent computer labs, and edges are potential Ethernet connections. The weights are in thousands of dollars. The weighted network has a redundancy of R = 3 (M = 14 and N = 12).
33 Find Minimum Spanning Trees (MST) Is this spanning tree minimal? Can we be sure? And if so, what assurances do we have that this strategy will work in all graphs? These are the questions we will answer next.
34 Kruskal s Algorithm
35 The Amazonian Cable Network What is the optimal fiberoptic cable network connecting the seven towns shown?
36 The Amazonian Cable Network Find the MST. Step 1 Choose the cheapest link: GF ($42 mill.) Step 2 The next cheapest link is BD ($45 million) Step 3 The next cheapest link is AD at $49 million. Step 4 Next, AB and DG tie, but we rule out AB since it would create a circuit.
37 The Amazonian Cable Network Step 5 The next cheapest link is CD. Step 6 The next cheapest is BC, but that would create a circuit. The next is CF, but that creates another circuit. The next CE. Finished, with a cost of $299 million.
38 Kruskal s Algorithm personales.upv.es As algorithms go, Kruskal s algorithm is as good as it gets: easy and efficient. As we increase the number of vertices and edges, the work grows proportionally. Since Kruskal s algorithm is optimal, always finding the MST, we have solved our conundrum!
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42 Shortest Network MSTs represent the optimal way to connect existing vertices and edges of the network. But what if were free to create new vertices and links outside the original network?
43 The Outback Cable Network The towns are connected by unpaved straight roads. What is the cheapest fiberoptic network connecting the three towns?
44 The Outback Cable Network The MST gives us a ceiling of1000 miles. The T-network is shorter, 933 miles, using triangles. The Y-network is shortest at 866 miles.
45 Shortest Network
46 Shortest Network
47 Shortest Network
48 Third Trans-Pacific Cable Network hawaiivacations.transamglobal.com In 1989, a consortium of telephone companies 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).
49 Third Trans-Pacific Cable Network Submarine cable costs $50 to $70 K per mile, so we need the shortest network. An interior junction point in the triangle? Where?
50 Third Trans-Pacific Cable Network The theoretical length of the shortest network is 5180 miles, but the uneven ocean floor adds as much as 10%, and the actual length is 5690 miles.
51 Kruskal s Algorithm From these examples, we might assume that the shortest network connecting three points joins at a Steiner point S inside the triangle. This is only true when we have a Steiner point inside the triangle.
52 A High-Speed Rail Network
53 A High-Speed Rail Network General property of triangles: For any triangle ABC and interior point S, angle ASC must be bigger than angle ABC. For ASC to be 120º, ABC must be less than 120º.
54 A High-Speed Rail Network Our angle is about 155º; so no Steiner junction point exists inside the triangle. Without a Steiner junction point, how do we find the shortest network?
55 A High-Speed Rail Network In this situation the shortest network consists of the two shortest sides of the triangle, which happens to be the minimum spanning tree.
56 Shortest Network
57 Shortest Network
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64 Torricelli s Construction In the 1600s Italian Evangelista Torricelli discovered a remarkably simple and elegant method for locating a Steiner point inside a triangle, using a straightedge and a compass.
67 Four-City Networks Imagine four cities (A, B, C, and D) that need to be connected. What does the optimal network look like?
68 Four-City Networks If we don t want to create any interior junction points in the network, then the answer is a minimum spanning tree, such as the one shown. The length of the MST is 1500 miles.
69 Four-City Networks If interior junction points are allowed, an X-junction located at O, the center of the square, would shorten the length to approximately 1414 miles.
70 Four-City Networks Even better is a network with two Steiner points: Two such networks exist (one is a rotated version of the other) having the same length about 1366 miles, which is the shortest possible.
71 Four-City Networks This time we change the dimensions of the rectangle, as shown.
72 Four-City Networks We know that the MST is 1000 miles long.
73 Four-City Networks For the shortest network, an obvious candidate would be a network with two interior Steiner junction points. There are two such networks shown.
74 Four-City Networks 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, the shortest possible network!
75 Four-City Networks This time, imagine that the cities are located at the vertices of a skinny trapezoid, as shown. The minimum spanning tree (in red) is 600 miles long.
76 Four-City Networks What about the shortest network? We should look for two interior Steiner junction points; however, since angles at A and B are greater than 120º, no Steiner points exist inside the trapezoid.
77 Four-City Networks If not Steiner points, how about X-, T-, or Y- junctions? In reality, the only possible interior junction points in a shortest network are Steiner points.
78 Shortest Networks Since we also know that the shortest network without interior junction points is the minimum spanning tree, then the MST must be the shortest network whenever no interior Steiner points exist.
79 Four-City Networks This time, our cities sit as shown. The MST is shown and its length is 1000 miles.
80 Four-City Networks The shortest network is either the MST or one with interior Steiner points. It s impossible to have two interior Steiner points, but there are three possible networks with a single interior Steiner point:
81 Four-City Networks miles miles This last network is the shortest in our list and thus the shortest network connecting, the four cities miles
82 Shortest Network
83 Shortest Network Algorithm simonnemichelle.files.wordpress.com 1. List all Steiner trees. 2. Using Kruskal s algorithm, find the minimum spanning tree. 3. Compare all trees. The shorter is the shortest network.
84 Shortest Network Algorithm Impractical farm1.static.flickr.com 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 but inefficient
85 Approximate Solutions Sophisticated approximate algorithms solve problems with hundreds of points and efficiently produce short networks less than 1% off the shortest network. Even simple Kruskal s algorithm can be used as a reasonably good approximate. For any set of points, the MST is never much longer than the shortest network: 13.4% longer at most, but usually 3% or less.
86 Homework Online homework for Chapter 7 Paper homework for Chapter 7 Online quiz for Chapter 7 Group projects meet with me this week