cs324fig8.cdr
Wednesday, October 22, 2014 8:12:41 AM
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Transitive Closure
Composite Default screen
➪ Transitive relation
Recall, digraphs are
useful to model
dependence such
as dependent tasks.
They can be used
to model relations
in general.
➪ Modeling relations
➪ Adjacency matrix
Quiz
Define formally the
transitive closure of
digraph.
Quiz
Give an example
for a relation
that is not transitive.
➪ Transitive closure of digraph
a
b
c
d
KAU • CPCS-324
© 2014 Dr. Muhammad Al-Hashimi
1
cs324fig8.cdr
Wednesday, October 22, 2014 8:12:41 AM
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Transitive Closure of Digraph
Applications
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An important
reachability question
in al these case is:
given two elements,
are they related?
➪ It’s about reachability
➪ Cells in spreadsheet
➪ Lineage (family) trees
➪ OOP: class hierarchy analysis
➪ Connecting flights
KAU • CPCS-324
© 2014 Dr. Muhammad Al-Hashimi
2
cs324fig8.cdr
Wednesday, October 22, 2014 8:12:42 AM
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Transitive Closure of Digraph
Computation
Composite Default screen
➪ DFS-based TC
Notice how the code
template relates to
pictorial representation of graph.
d
...
0
0
1
2
3
Is there a starting
point which answers
the question for paths
of length 1?
a
c
a
b
c
d
...
Notice how
“phantom” nodes
simulate positions
in the adjacency
matrix.
b
2
b
a
a
a
3
4
c
d
d
b
➪ Warshall’s algorithm: idea
Quiz
What does the number in () mean? How
does it relate to path
length?
KAU • CPCS-324
Can it be used
to generate
answer for path
of length 2?
© 2014 Dr. Muhammad Al-Hashimi
3
Warshall’s Algorithm
cs324fig8.cdr
Wednesday, October 22, 2014 8:12:42 AM
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Principle
Composite Default screen
Rules
Look at element
k in my row and
column.
a
1
0
(next)
(previous)
b
c
1
d
Main insight
1
2
3
4
a
b
c
d
0
0
0
1
1
0
0
0
0
0
0
1
0
1
0
0
b
d
0
0
0
1
1
0
0
1
0
0
0
1
0
1
0
0
Exercise
Verify that there
is a path from d
to b that passes
through a. (Answer next slide)
KAU • CPCS-324
© 2014 Dr. Muhammad Al-Hashimi
4
cs324fig8.cdr
Wednesday, October 22, 2014 8:12:42 AM
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Warshall’s Algorithm
Example
Composite Default screen
a
b
c
d
Exercise
Verify the 1 circled
green. Check b (vertex 2) in its row and
column in R(1).
a
b
c
d
KAU • CPCS-324
© 2014 Dr. Muhammad Al-Hashimi
5
cs324fig8.cdr
Wednesday, October 22, 2014 8:12:42 AM
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Warshall’s Algorithm
Composite Default screen
can we add an edge to find a path between i,j of length k+1?
inherit previous
path of length k
➪ Performance
© 2014 Dr. Muhammad Al-Hashimi
6
check vertex k
in my row (i)
Exercise
Compare to
performance of
DFS-based TC.
check vertex k in
my column (j)
KAU • CPCS-324
While We Are At It ...
cs324fig8.cdr
Wednesday, October 22, 2014 8:12:42 AM
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Composite Default screen
➪ Distance matrix
Weight can be interpreted as distance or
cost, something to
be minimized.
a
2
7
3 6
c
b
1
d
Quiz
What does infinity
means?
KAU • CPCS-324
© 2014 Dr. Muhammad Al-Hashimi
7
Shortest Distance
cs324fig8.cdr
Wednesday, October 22, 2014 8:12:42 AM
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Look at vertex
k=1 (vertex a)
in my row and
column, if sum
smaller replace.
Composite Default screen
a
2
2
3
4
b
7
3 6
c
1
1
d
2
Exercise
Verify in the graph the
length of a path from
b to c that passes
through a.
1
3
5
2
3
4
Exercise
Verify the 7
in green.
(Answer
next slide).
KAU • CPCS-324
© 2014 Dr. Muhammad Al-Hashimi
8
Shortest Paths
cs324fig8.cdr
Wednesday, October 22, 2014 8:12:42 AM
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Composite Default screen
Exercise
What’s the path? Verify
its length.
KAU • CPCS-324
© 2014 Dr. Muhammad Al-Hashimi
9
All-Pairs Shortest-Paths Problem
cs324fig8.cdr
Wednesday, October 22, 2014 8:12:42 AM
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Floyd’s Algorithm
➪ Performance
KAU • CPCS-324
© 2014 Dr. Muhammad Al-Hashimi
10
Dynamic “Programming”
cs324fig8.cdr
Wednesday, October 22, 2014 8:12:42 AM
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Composite Default screen
Quiz
State the principle as
required by dynamic
programming.
Quiz
What’s the simpler
subproblem solved
by each stage?
➪ Principle of optimality
➪ Review Warshall-Floyd idea
➪ Originally a math method
➪ In algorithms
Solve simpler
subproblems in
each stage, not
necessarily smaller
in terms of input
size.
© 2014 Dr. Muhammad Al-Hashimi
➪ Solve a problem in stages
➪ Space efficiency concern
11
KAU • CPCS-324
cs324fig8.cdr
Wednesday, October 22, 2014 8:12:42 AM
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Graph Implementation
TC Package
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Checkpoint 8 expectations
KAU • CPCS-324
© 2014 Dr. Muhammad Al-Hashimi
12
Class Exercise
cs324fig8.cdr
Wednesday, October 22, 2014 8:12:42 AM
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Do it by hand!
a
a
b
2
7
3 6
c
c
d
For all pairs, is there a path?
b
1
d
For all pairs, what’s the shortest distance?
What’s the cheapest way to connect the vertices?
© 2014 Dr. Muhammad Al-Hashimi
13
KAU • CPCS-324
Check Your Answers
cs324fig8.cdr
Wednesday, October 22, 2014 8:12:43 AM
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Composite Default screen
a
a
b
2
7
3 6
c
c
d
b
1
d
KAU • CPCS-324
© 2014 Dr. Muhammad Al-Hashimi
14