More Graph Algorithms

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More Graph Algorithms

• 15 April 2003

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Applications of Graphs

• Graph theory is used in dealing with problems
  which have a fairly natural graph/network
  structure, for example
• road networks nodes are towns or road junctions,
  arcs are the roads
• communication networks telephone systems
• computer systems
• We are reading this document via the Web. This is
  a graph, with each document (file) being a node
  and each hypertext link an arc.

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Königsberg Bridge Problem

• Graph theory has a relatively long history in
  classical mathematics. In 1736 Euler solved the
  problem of whether, given the map on the next
  slide of the city of Königsberg in Germany,
  someone could make a complete tour, crossing over
  all 7 bridges over the river Pregel, and return
  to their starting point without crossing any
  bridge more than once.
• The town of Königsberg was in northern Germany
  and is now part of Russia.

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The Map
5
Eulerian Paths

• A path that solves the problem is called an
  Eulerian path. Eulerian paths have interesting
  mathematical and computational properties.
• Euler reasoned that for such a journey to be
  possible that each land mass should have an even
  number of bridges connected to it, or if the
  journey would begin at one land mass and end at
  another, then exactly those two land masses could
  have an odd number of connecting bridges while
  all other land masses must have an even number of
  connecting bridges.

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A Path Problem

• http//www.planemath.com/activities/flightpath/fli
  ghtpath1.html

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Shortest Path Problem

• Given a directed graph in which each edge has a
  nonnegative weight or cost, find a path of least
  total weight from a given vertex, called the
  source, to every other vertex in the graph.

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The Main Idea (Dijkstras Algorithm)

• We keep a set S of vertices whose closest
  distances to the source, vertex 0, are known and
  add one vertex to S at each stage.
• We maintain a table of distances D that gives,
  for each vertex v, the distance from 0 to v along
  a path all of whose vertices are in S, except
  possibly the last one.
• To determine what vertex to add to S at each
  step, we apply the greedy criterion of choosing
  the vertex v with the smallest distance recorded
  in D, such that v is not already in S.

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Greedy Criteria

• Choose the vertex v with the smallest distance
  recorded in the table D, such that v is not
  already in S.

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Why Dijkstras Algorithm Works

• Prove that the distance recorded in D really is
  the length of the shortest path from source to v.
  
• Suppose that there were a shorter path from
  source to v, such as shown on the next slide.
  This path first leaves S to go to some vertex x,
  then goes on to v (possibly even reentering S
  along the way). But if this path is shorter than
  the marked path to v, then its initial segment
  from source to x is also shorter, so that the
  greedy criterion would have chosen x rather than
  v as the next vertex to add to S, since we would
  have had distancesource to x lt distancesource
  to v.

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Proof
v
Marked path
Hypothetical shortest path
x
S
0
Source
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Why Dijkstras Algorithm Works-2

• When we add v to S, we think of v as now marked
  and also mark the shortest path from source to v.
  
• Next, we update the entries of D by checking, for
  each vertex w not in S, whether a path through v
  and then directly to w is shorter than the
  previously recorded distance to w.

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Dijkstras Algorithm
14
Dijkstras Algorithm
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Dijkstras Algorithm
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Dijkstras Algorithm Demo

• http//www-b2.is.tokushima-u.ac.jp/ikeda/suuri/di
  jkstra/Dijkstra.shtml
• For graphs with n vertices and a arcs, the time
  required by Dijkstra's algorithm is O(n2). It
  will be reduced to O(a log n) if a heap is used
  to keep v ? V-Si d(source, v) lt infinity.

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Spanning Trees

• Consider the following problem.
• In the diagram shown below we have four wells in
  an offshore oilfield (nodes 1 to 4 below) and an
  on-shore terminal (node 5 below). The four wells
  in this field must be connected together via a
  pipeline network to the on-shore terminal.

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5 is Shore, 1, 2, 3, 4 are Wells
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Two Spanning Trees
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Minimal Spanning Tree

• This particular problem is a specific example of
  a more general problem - namely given a graph
  (such as that shown on previous slide) which
  edges would we use so that
• the total cost of the edges used is a minimum
  and
• all the points are connected together.
• This problem is called the minimal spanning tree
  (MST) problem.

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Formal Definition

• A minimal spanning tree of a connected network is
  a spanning tree such that the sum of the weights
  of its edges is as small as possible.

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Oil Rigs

• For example, in the given graph, one possible
  structure connecting all the points together
  would consist of the edges 1-2, 2-3, 3-4, 4-5 and
  5-1, but there are other structures where the
  total cost of the edges used is smaller than in
  this structure (e.g. drop any edge in the cycle
  1-2-3-4-5-1 formed above).

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Kruskals Algorithm

• Points to note
• for an n-node graph the solution involves (n-1)
  edges
• the solution can be easily arrived at by using an
  algorithm due to Kruskal (developed in the mid
  1950s)
• The final structure is called the minimal
  spanning tree (MST) of the graph.

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Kruskal Algorithm

• The algorithm is
• For a graph with n nodes keep adding the shortest
  (least cost) edge while avoiding the creation of
  cycles until (n-1) edges have been added
• Note that the Kruskal algorithm only applies to
  graphs in which all the edges are undirected.

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Edge Table
u v w
1 5 2
1 2 3
1 4 3
4 5 4
2 5 5
2 3 6
1 3 7
2 4 8
3 4 10
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Example

• For the graph shown earlier, applying the Kruskal
  algorithm and starting with the shortest (least
  cost) edge, we have
• Edge Cost Decision
• 1-5 2 add to tree
• 1-2 3 add to tree
• 1-4 3 add to tree
• 4-5 4 reject as forms cycle 1-5-4-1
• 2-5 5 reject as forms cycle 1-5-2-1
• 2-3 6 add to tree
• Stop since 4 edges have been added and these are
  all we need for the MST of a graph with 5 nodes.

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MST

• The MST of the oil rig graph consists of the
  edges 1-5, 1-2, 1-4 and 2-3 (total cost 14)

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Kruskal Demo

• http//www-b2.is.tokushima-u.ac.jp/ikeda/suuri/kr
  uskal/Kruskal.shtml

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Prims Algorithm Demo

• http//www-b2.is.tokushima-u.ac.jp/ikeda/suuri/di
  jkstra/Prim.html

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Limitations

• The principal limitation of the pipeline network
  shown is (in practical terms) that we have
  ignored the fact that some pipelines carry more
  oil than others and this means that they must
  therefore be of larger diameter and hence more
  expensive (per unit length).
• We have the situation that the cost of a pipeline
  depends upon the amount of oil it must carrybut
  we do not know how much oil any particular
  pipeline is carrying until we have decided the
  entire pipeline network.

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Extensions

• Iterative application of MST calculations
  provides one way of solving such practical
  pipeline design problems. Typically we first set
  a cost for each pipeline, then
• solve the MST problem
• calculate how much oil is carried in each
  pipeline
• adjust pipeline costs accordingly and reiterate

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Extensions

• Other difficulties include
• mixing oil of different grades
• gases given off
• use of compressors
• geographic/political problems in getting the
  pipeline to go where we want
• network very vulnerable to a failure in an edge
  or at a node
• time periods the flow from each well will vary
  over the years.

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Bottom Line

• A pioneering application of MST models to
  pipeline network design occurred in the early
  1960s in the design of a gas pipeline network in
  the Gulf of Mexico.
• Savings resulted of the order of tens of millions
  of dollars. This illustrates how nice and neat
  theoretical problems (like the MST) can be used
  as building blocks to solve complex practical
  problems.

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Network Design

• The oil rig problem can be considered as a
  network design problem. Such problems typically
  involve a number of distinct points (nodes) and
  connections between pairs of nodes (edges) and
  the problem is to design a suitable network to
  connect together a number of the nodes at minimum
  cost and which satisfies a number of criteria.
  For example we may want a minimum cost network
  which still functions after a failure in any one
  of the edges.
• Such problems are increasingly importantin
  particular in the telecommunications industry. In
  the simplest case the nodes are given, each
  possible edge has a cost associated with its use
  and the problem is which edges should be used so
  as to minimize total cost while ensuring that the
  network is connected. This problem is exactly the
  MST problem dealt with here.