Unformatted text preview: T .
If G is the union of disjoint directed cycles of length at most T . Then
Z = ∅. On the other hand, the case where Z will contain the highest
number of elements is when G is the union of disjoint directed cycles of
length T + 1. In this case two elements of each cycle will have to be in Z
in order for the remaining elements of the cycle to be within distance T
from an element in Z . Since the number of cycles of length T + 1 is less
or equal to TN , we get Z  = 2 · TN . Thus, for a general G,
+1
+1
Z  ≤ 2 · N
2N
≤
,
T +1
T so Z  is O(N/T ).
(c) Note: Given an element y ∈ Y , the algorithm 3.7 ﬁnds K ∈ Y (recall that
P = C = K = Y ) such that g (K ) = eK (x) = y , that is, considering the
fact that G is the union of disjoint directed cycles, K is such that (K, y )
is an edge in G.
Thus, if y is a member of a cycle of length 1 (that is (y, y ) is an edge),
then the algorithm won’t enter the while loop and K = y .
If g (y ) = y ; We know that, by construction of Z , either y is a member of a
loop with length at most T , or there exists zj = y such that the distance
between y and zj is at most T .
• Suppose y is a member of a cycle with length at mos...
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This document was uploaded on 02/24/2014.
 Spring '12

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