Chapter 8
Step 3:
if
F
=
0, output "reflexive"; otherwise, output "not reflexive".
The algorithm requires at most
t
comparisons.
241
(b) We set up a matrix
A
=
[aij]
whose
(i,j)
entry is a 1 if and only if
(i,j)
En,
and otherwise O.
Step 1:
for
i
=
1 to
t
for
j
=
1 to
t
set
aij
=
o.
Step 2:
For
i =
1 to
t,
set the
(ai,
b
i )
entry of
A
equal to
l.
Step 3:
Set
F
=
0 and
i
=
l.
Step 4:
while
F
=
0 and
i
<
t,
for
j
= i
to
t
if
aij
#
aji
set
F
=
1;
replace
i
by
i
+
l.
end while
Step 5:
If
F
=
0, output "symmetric" and stop; otherwise output "not symmetric" and stop.
When
i =
1, the loop inside Step 4 requires
t
comparisons. When
i =
2, the loop requires
t
 1
comparisons, and so on. The number of comparisons is at most
t+ (t1)
+
...
+2+ 1
=
~t(t+
1),
so the algorithm is
O(t
2
).
(c) We set up a matrix
A
whose
(i,j)
entry is a 1 if and only if
(i,j)
E
n,
and otherwise O. The
relation fails to be antisymmetric if and only if some
aij
=
1
=
aji
with
i #
j,
and this occurs if
and only if
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 Summer '10
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 Graph Theory, Transitive relation, Symmetric relation, ij

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