Lecture 18 Directed graphs and binary relations

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Lecture 18 Directed graphs and binary relations

• July 23rd, 2003

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Directed Graph

• The representation of directed graph
• Adjacent matrix (5.1)
• Adjacent relation
• ni ? nj ? there is an arc in G from ni
  to nj
• Page 413, Example 1
• Practice 1

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Three equivalent set
Binary relations on n-element sets
Directed graphs with n nodes, no parallel arcs.
n X n Boolean matrix
?
?
Example two binary relations ? and s on a set
N, Let R and S be the boolean matrix for ? and s.
Then boolean matrix for ? U s is R V S
boolean matrix for ? s is R S
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Reachability

• Definition In a directed graph, node nj is
  reachable from node ni if there is a path from ni
  to nj
• A(2) AXA (Boolean matrix multiplication).
• Recall the boolean matrix operation?

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A theorem

• Theorem on boolean adjacency matrices and
  reachability -- If A is the Boolean adjacency
  matrix for a directed graph G with n nodes and no
  parallel arcs, then A(2)I,j1 if and only if
  there is a path of length m from node ni to nj
• Prove by induction

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Reachability matrix R

• If the number of nodes is n.
• R A V A(2) V A(3) V ..V A(n)
• then ni is reachable from nj if and only if
  entry i, j in R is positive.

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Three equivalent sets for reachability
(ni,nj) belongs to the transistive closure of ?
nj reachable from ni in G
Ri,j 1 where R A V A(2) V A(3) V ..V A(n)
?
?
Example 5 (Page 416)
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Warshalls Algorithm

• Warshalls algorithm computes a sequence of n1
  matrices M0, M1,M2,.,Mn. For each k, 0ltkltn,
  MkI,j1 if and only if there is a path in G
  from ni to nj whose interior nodes come only from
  the set of nodes n1,n2,. nk .

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Warshalls Algorithm (cont.)

• ALGORITHM WALSHALL
• Walshall(nXn Boolean matrix M)
• for k0 to n-1 do
• for i1 to n do
• for j1 to n do
• Mi,j Mi,jV(Mi,k1
  Mk1,j)
• end for
• end for
• end for
• end Warshall

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Warshalls Algorithm (cont.)

• M0A and MnR
• To compute Mk1 from Mk
• 1. consider column k1 in Mk1
• 2. Foe each row with a 0 entry in this column,
  copy that row to Mk1.
• 3. Foe each row with a 1 entry in this column, or
  that row with row k1 and write the resulting
  row in Mk1.
• Application of Warshalls Algorithm find out R
  in O(n3) time.

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Assignment

• Exercise 6.1 , Due on Next Monday.
• 1, 4, 5, 17,24