Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

where bn is the time needed to partition a list for a given pivot and concatenate
the result sublists after the recursive calls return. Show, by induction, that T(n)
is O(nlogn).
C-11.8
Modify inPlaceQuickSort (Code Fragment 11.6) to handle the general case
efficiently when the input sequence, S, may have duplicate keys.
C-11.9
Describe a nonrecursive, in-place version of the quick-sort algorithm. The
algorithm should still be based on the same divide-and-conquer approach, but
use an explicit stack to process subproblems. Your algorithm should also
guarantee the stack depth is at most O(logn).
C-11.10
Show that randomized quick-sort runs in O(nlogn) time with probability at least
1 − 1/n, that is, with high probability, by answering the following:
a.
For ach input element x, define Ci,j(x) to be a 0/1 random variable that is 1
if and only if element x is in j + 1 subproblems that belong to size group i.
Argue why we need not define Ci,j for j > n.
b.
Let Xi,j be a 0/1 random variable that is 1 with probability 1/2j, independent
of any other events, and let L = …log 4/3 n…. Argue why
c.
.
Show t the expected value of
d.
is at
Show that the probability that
most 1/n 2 using the
Chernoff bound that states that if X is the sum of a finite number of
independent 0/1 random variables with expected value μ > 0, then Pr(X >
2……) < (4/e) −… , where e = 2.71828128....
e.
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