Shortest path Algorithm

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Shortest path Algorithm

• Operational Research
• By
• TMJA Cooray

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• An important problem in OR is to find the
  shortest path in a network from a given point to
  any other point in the network.

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• Definition A directed edge in a network N is
  often called an arc. We write uv to denote an
  arc directed from u to v. The set of arcs of N is
  denoted by A(N).
• We shall consider a directed and acyclic ( no
  directed cycle) network N, here.
• Every acyclic directed network contains a
  source , say x.
• An acyclic ordering of the vertices of a directed
  network N is an ordering of V(N) such that vertex
  x comes before vertex y whenever A(N) contains an
  arc directed from x to y.
• This will imply that x is the only source in N,
  since every other vertex must have at least one
  arc directed to into it.

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• ResultLet N be an acyclic directed network. Then
  we can give the vertices of N an acyclic
  ordering.
• Proof Since N is acyclic, N contains a source,
  say x. Give x the label v1 and delete x and all
  the arcs incident with x, from N to form a new
  network N1N-x.
• The network N1 is also acyclic and hence contains
  a source. Say y, give y the label v2 and delete y
  and the incident edges to form the next network.
  N2N1-y.
• Iterate this procedure until all vertices are
  labeled. Then we get v1,v2,vn an acyclic
  ordering of the vertices of N.

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• The vertex a is the only source in N, give a the
  label v1 and delete ab,ai and ah to give the
  network N1.
• The vertex i is the only source in N1.give i the
  label v2 and delete ib, ic,ig and ih to give N2.
• The vertices b and h are the two sources of N2.
  Choose one arbitrarily say b, label as v3.delete
  b and edges bc to form N3.
• N3 contains two sources c and h. select c and
  label as v4. Delete c and cd, cf, and cg to give
  network N4.
• N4 contains the unique source h and give it the
  label v5. delete h and hg. Iterating the
  procedure gives successively gv6,dv7,fv8 and
  ev9 .The acyclic ordering is complete. ?

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• Acyclic network with the vertices labeled in an
  acyclic order.

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4 7
4 3 2 5
9 3 5

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Definition

• Denote the weight on the arc/edge uv
  A(N), by w(uv), for all uv A(N),.
• The length of a directed path P ,to be the sum of
  the weights on the arc of P and denote it by
  len(P).
• The lsp(x,y) denote the length of a shortest
  directed path from a vertex x to a vertex y.

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Algorithm to find a shortest path from a source
to every other vertex in an acyclic directed
network (v1,v2,..vn vertices in N))

• 1.(Initialization step) Put the vertex v1 into T,
  so that V(T)v1,give v1 the label lsp(v1,v1)0
• 2.(Recursive step) Suppose V(T)v1,v2,..vk and
  that the vertex vj has the label lsp(v1,vj), for
  j1,2,k. Choose vi so that vivk1 A(N), and
  lsp(v1, vi)w(vivk1 ) is as small as possible.
• Add the vertex vk1. and the arc vivk1 to T
  give vk1 the label lsp(v1, vk1) lsp(v1,
  vi)w(vivk1 )
• 3.(stop condition) Stop when T contains all the
  vertices of N.

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• We apply this algorithm to find the length of
  the shortest directed path from the vertex a in
  the network N shown in slide 7
• to every other vertex in V(N).
• (The first step is to give V(N) an acyclic
  ordering. As shown in slide 7.)
• Put v1 into T, give v1 label lsp(v1,v1)0
• Consider v2. The only arc into v2 is the arc v1v2
  with weight 7. we therefore add v2 and the arc
  v1v2 to T, and give v2 the label
  lsp(v1,v2)lsp(v1,v1)w(v1v2)07

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• Consider v3. there are two arcs into v3.
• The arc v1v3 lsp(v1,v3)lsp(v1,v1)w(v1v3)088
• Arc v2v3 lsp(v1,v3)lsp(v1,v2)w(v2v3)7613
• The minimum is 8, hence we add v3 and the arc
  v1v3 to T, and give v3 the label 8.
• Consider v4 two arcs v2v4 and v3v4into v4 give
  respectively
• lsp(v1,v4)lsp(v1,v2)w(v2v4)7411
• lsp(v1,v4)lsp(v1,v3)w(v3v4)8614
• Minimum is 8. Give v4 the label 11.add v1v4 to T

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• Continuing the same procedure,
• Consider v5 label v5 as 9 and add v1v5 to T.
• Consider v6 label v6 as 12 and arc v2v6 to T.
• Consider v7 label as 15 and v6v7 to T.
• Consider v8 label as 15 and v4v8 to T .
• Consider v9 label as 19 and v8v9 to T
• Every vertex has been labeled with the length of
  the shortest path from v1.
• Stop the procedure.
• T is now a spanning tree

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• Spanning tree with the labeling
• Shortest path

(8)
(11)
v4
v3
v7
(15)
(7)
v9
v2
v1
(19)
(0)
v5
v8
(9)
v6
(15)
(12)
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Longest path algorithm.

• Definition
• llp(x,y) denotes the length of the longest
  directed path from a vertex x to a vertex y.

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Algorithm to find a longest path from a source
to every other vertex in an acyclic directed
network (v1,v2,..vn vertices in N))

• 1.(Initialization step) Put the vertex v1 into T,
  so that V(T)v1,give v1 the label lsp(v1,v1)0
• 2.(Recursive step) Suppose V(T)v1,v2,..vk and
  that the vertex vj has the label llp(v1,vj), for
  j1,2,k. Choose vi so that vivk1A(N),and
  llp(v1, vi)w(vivk1 ) is as large as possible.
• Add the vertex vk1 and the arcvivk1 to T
  give vk1 the label llp(v1, vk1) llp(v1,
  vi)w(vivk1 )
• 3.(stop condition) Stop when T contains all the
  vertices of N.

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• Spanning tree with the labeling
• ( longest path)

(13)
v4
v3
(19)
v7
(29)
(0)
(7)
v9
v2
(36)
v1
v5
(10)
v8
(26)
v6
(32)
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