Excursions in Modern Mathematics Sixth Edition

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Excursions in Modern MathematicsSixth Edition

• Peter Tannenbaum

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Chapter 7The Mathematics of Networks

• The Cost of Being Connected

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The Mathematics of NetworksOutline/learning
Objectives

• To identify and use a graph to model minimum
  network problems.
• To classify which graphs are trees.
• To implement Kruskals algorithm to find a
  minimal spanning tree.

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The Mathematics of NetworksOutline/learning
Objectives

• To understand Torricellis construction for
  finding a Steiner point.
• To recognize when the shortest network connecting
  three points uses a Steiner point.
• To understand basic properties of the shortest
  network connecting a set of (more than three)
  points.

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The Mathematics of Networks

• 7.1 Trees

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The Mathematics of Networks

• Network
• Another name for a connected graph.
• Tree
• A network with no circuits.
• Spanning Tree
• A subgraph that connects all the vertices of the
  network and has no circuits.
• Minimum Spanning Tree (MST)
• Among all spanning trees of a weighted network,
  one with the least total weight.

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The Mathematics of Networks Tree or Not?
The graphs in (a) and (b) are disconnected, so
they are not even networks, let alone trees.
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The Mathematics of Networks Tree or Not?
The graphs in (c) and (d) are networks that have
circuits so neither of the is a tree.
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The Mathematics of Networks Tree or Not?
The graphs in (e) and (f) are networks with no
circuits, so they are indeed trees.
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The Mathematics of Networks Tree or Not?
The structure of a family tree (g) and the
structure formed by the bonds of some molecules
(h) are also trees.
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The Mathematics of Networks Summary of Key
Properties

• 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.

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The Mathematics of Networks Summary of Key
Properties

• 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.

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The Mathematics of Networks Summary of Key
Properties

• 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.

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The Mathematics of Networks
Notice that a disconnected graph (not a network)
can have N vertices and N 1 edges.
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The Mathematics of Networks

• 7.2 Spanning Trees

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The Mathematics of Networks Summary of Key
Properties

• Property 4
• If a network has N vertices and M edges, then M ?
  N 1. 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.

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The Mathematics of Networks Counting
Spanning Trees

The network in (a) has N 8 vertices and M 8
edges. The redundancy of the network is R 1,
so to find a spanning tree we will have to
discard one edge.
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The Mathematics of Networks 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
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The Mathematics of Networks Counting
Spanning Trees

if we exclude any of the three edges we will have
a spanning tree. Thus, the network has three
different spanning trees (b), (c), and (d).
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The Mathematics of Networks

• 7.3 Kruskals Algorithm

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The Mathematics of Networks

• There are several well-known algorithms for
  finding minimum spanning trees. In this section
  we will discuss one of the the nicest of these,
  callled Kruskals algorithm.

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The Mathematics of Networks

What is the minimum spanning tree (MST) of the
network shown in (b)?
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The Mathematics of Networks

We will use Kruskals algorithm to find the MST
of the network. Step 1. Among all the possible
links, we choose the cheapest one, in this case
GF (at a cost of 42 million). This link is
going to be a part of the MST, and we mark it in
red as shown in (a).
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The Mathematics of Networks

Kruskals algorithm 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
cheapest link available is AD at 49 million.
Again, we choose it for the MST and mark it in
red.
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The Mathematics of Networks

Kruskals algorithm Step 4. For the next
cheapest link there is a tie between AB and DG,
both at 51 million. But we can rule out AB it
would create a circuit in the MST, and we cant
have that!) The link DG, on the other hand, is
just fine, so we mark in red and make ti part of
the MST.
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The Mathematics of Networks

Kruskals algorithm 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.
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The Mathematics of Networks

Kruskals algorithm Step 6 (cont.). 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.
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The Mathematics of Networks

Kruskals algorithm Step Wait a second we
are finished! We can tell we are done six links
is exactly what is needed for an MST on seven
vertices (N 1). Figure (c) shows the MST in
red. The total cost of the network is 299
million.
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The Mathematics of Networks

• 7.4 The Shortest Network Connecting Three Points

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The Mathematics of Networks

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

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The Mathematics of Networks

• Here, cheapest means shortest, so the name of
  the game to design a network that is as short as
  possible. We shall call such a network the
  shortest network (SN).

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The Mathematics of Networks

• The search for the shortest network often starts
  with a look at the minimum spanning tree. The
  MST can always be found using Kruskals algorithm
  and it gives us a ceiling on the length of the
  shortest network.

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The Mathematics of Networks

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

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The Mathematics of Networks

• It is not hard to find a network connecting the
  three towns shorter than the MST. The T- network
  (b) is clearly shorter. The length of the
  segment CJ is approximately 433 miles. The
  length of this network is 933 miles.

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The Mathematics of Networks

• We can do better. The Y- network shown in (c) is
  even shorter than the T- network. In this
  network there is a Y- junction at S, with three
  equal branches connecting S to each of A, B, and
  C.

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The Mathematics of Networks

• This network is approximately 866 miles long. A
  key feature is the way the three branches come
  together at the junction point S, forming equal
  120? angles.

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The Mathematics of Networks

• Before we move on, we need to discuss briefly the
  notion of a junction point on a network. ( A
  junction point in a network is any point where
  two or more segments of the network come together.

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The Mathematics of Networks

• The MST in (a) has a junction point at A, then
  network in (b), has a junction point at J, and
  the shortest network in (c) has a junction point
  at S.

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The Mathematics of Networks

• There are three important terms that we will use
  in connection with junction points
• In a network connecting a set of vertices, a
  junction point is said to be native junction
  point if it is located at one of the vertices.
• A nonnative junction point located somewhere
  other than at one of the original vertices is
  called an interior junction point of the network.
• An interior junction point consisting of three
  line segments coming together forming equal 120?
  angles

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The Mathematics of Networks

• There are three important terms that we will use
  in connection with junction points (continued)
• An interior junction point consisting of three
  line segments coming together forming equal 120?
  angles (a perfect Y- junction if you will) is
  called a Steiner point of the network.

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The Mathematics of Networks

• The Shortest Network Connecting Three Points
• If one of the angles of the triangle is 120? or
  more, the shortest network linking the three
  vertices consists of the two shortest sides of
  the triangle (a).

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The Mathematics of Networks

• The Shortest Network Connecting Three Points
• If all three angles of the triangle are less than
  120? , the shortest network is obtained by
  finding a Steiner point S inside the triangle and
  joining S to each of the vertices (b).

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The Mathematics of Networks

• Finding the Steiner Point Torricellis
  Construction
• Suppose A, B, and C form a triangle such that all
  three angles of the triangle are less than 120?
  (a).

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The Mathematics of Networks

• Finding the Steiner Point Torricellis
  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 (b).

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The Mathematics of Networks

• Finding the Steiner Point Torricellis
  Construction
• Step 2. Circumscribe a circle around equilateral
  triangle BCX (c).

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The Mathematics of Networks

• Finding the Steiner Point Torricellis
  Construction
• Step 3. Join X to A with a straight line (d).
  The point of intersection of the line segment XA
  with the circle is the Steiner point!

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The Mathematics of Networks

• 7.5 Shortest Networks for Four or More Points

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The Mathematics of Networks

• When it comes to finding shortest networks,
  things get really interesting when we have to
  connect four points.

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The Mathematics of Networks

• What does the optimal network connecting these
  four cities (A, B, C, and D) look like? Suppose
  the cities sit o the vertices of a square 500
  miles on each side as shown in (a).

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The Mathematics of Networks

• If we dont want to create any interior junction
  points in the network, then the answer is a
  minimum spanning tree, such as in (b). The
  length of the MST is 1500 miles.

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The Mathematics of Networks

• If interior junction points are allowed, somewhat
  shorter networks are possible. An improvement is
  the network (d) with and X-junction located at O,
  the center of the square. The length is
  approximately 1414 miles.

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The Mathematics of Networks

• We can shorten the network even more if we place
  not one but two interior junction points inside
  the square. There are two different networks
  possible with two Steiner points inside the
  square as in (d) and (e).

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The Mathematics of 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.

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The Mathematics of Networks

• The only possible interior junction points in a
  shortest network are Steiner points. For
  convenience, we will call this the interior
  junction rule for shortest networks.

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The Mathematics of Networks

• The Shortest Network Rule
• a minimum spanning tree (no interior junction
  points) or
• A Steiner tree. A Steiner tree is a network with
  no circuits (a tree) such that all interior
  junction points are Steiner points.

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The Mathematics of Networks
Conclusion

• Discussed the problem of creating an optimal
  network.
• Found out that Kruskals algorithm is a simple
  algorithm for finding MSTs.
• By allowing interior junction points, solve the
  shortest network connecting the points.