MATHEMATICAL THOUGHT
AND PRACTICE
Chapter 7: The Mathematics of Networks The Cost of Being Connected
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.”
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...
Network
Twitter Network flowingdata.com
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.
Optimal Network
www.juliesjournalonline.com Social Networking Collage
The Amazonian Cable Network
The telephone company must lay fiber-optic lines along the roads between towns.
Vertices represent the towns, edges represent existing roads, and
weights represent costs in millions of dollars to lay the cables along that road.
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).
Minimal Network – No Circuits
Circuits cannot be part of minimal networks.
The edge XY would be a redundant link of the
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).
Formal Definitions
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.
Networks, Trees, and Spanning Trees
No network–graph is
No network – partial tree
Networks, Trees, and Spanning Trees
Network, spanning tree, no
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.
Properties of Trees
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.)
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.
As M increases, the number of circuits goes up and bridges goes down.
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.
Spanning Trees
An American Haunting
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.
The network has three different 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.
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.
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).
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.
What is the optimal fiber-optic cable network connecting the seven towns shown?
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.
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.
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!
Kruskal’s Algorithm
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?
Shortest Network
The Outback Cable Network
The towns are connected by
unpaved straight roads. What is the cheapest
fiber-optic network connecting the three towns?
The MST gives us a ceiling of1000 miles.
The T-network is shorter,
933 miles, using 30-60-90 triangles.
The Y-network is shortest at 866 miles.
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).
Third Trans-Pacific Cable Network
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?
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.
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.
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.
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?
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.
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.
Four-City Networks
Imagine four cities (A, B, C, and D) that need to be connected. What does the optimal network look like?
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.
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.
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.
This time we change the dimensions of the rectangle, as shown.
We know that the MST is 1000 miles long.
For the shortest network, an obvious candidate would be a network with two interior Steiner
junction points. There are two such networks shown.
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!
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.
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.
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.
Four-City 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.
Shortest Networks
This time, our cities sit as shown.
The MST is shown and its length is 1000 miles.
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:
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
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.
Shortest Network Algorithm
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
Shortest Network Algorithm Impractical
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.
Homework
Online homework for Chapter 7 Paper homework for Chapter 7 Online quiz for Chapter 7
