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93
(b)
Fill an array with
2
n
+1
elements, which forces a
f
nal growth to an
array of size
2
n
+1
. Now, do an arbitrarily long series of alternating
inserts and deletes. This will cause the array to repeatedly shrink and
grow, for bad (
Θ(
n
2
)
) performance.
(c)
If we shrink the array whenever the space use goes below 25%, we will
have the desired performance.
14.17
Each node can be visited only once. Thus, there is initially potential for

V

node visits. We can look at each edge only once (the edges out of a node are
visited when the node is visited). Thus, there is potential for

E

edge visits.
The initial call to DFS can expend a small part of that potential, or a large
part. But, the sum of all the calls to DFS must cost
Θ(

V

+

E

)
.
14.18
As with MovetoFront, the contribution of unsuccessful searches requiring
comparisons between keys A and B is independent of other keys. We have
an unsuccessful search for A if and only if we have had more requests for
B so far. Assuming that B is requested
R
B
times and A is requested
R
A
times with
R
B
>R
A
, we can only have unsuccessful searches twice the
number of times that A is requested. This happens at most for
R
A
requests
to B occurring before
R
A
requests to A. The remaining requests to B are
successful without encountering A. Thus, the Count heuristic can have cost
at most twice that of the optimal static ordering.
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View Full Document 15
Limits to Computation
15.1
This reduction provides an upper bound of
O(
n
log
n
)
for the problem of
maximum
f
nding (that is the time for the entire process), and a lower bound
of constant time for SORTING (since that is the time spent by Maximum
Finding in this process). Neither bound is particularly enlightening. There is
no true reduction from SORTING to Maximum Finding (in the sense that the
transformations do not dominate the cost) since SORTING is an intrinsicly
more dif
f
cult problem than Maximum Finding.
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This document was uploaded on 10/31/2011 for the course BCN 3431 at University of Florida.
 Fall '10
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