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

threads of b. Describe and analyze an efficient algorithm for Bob to match up all
of his nuts and bolts.
C-11.26
Show how to use a deterministic O(n)-time selection algorithm to sort a
sequence of n elements in O(n log n) worst-case time.
C-11.27
Given an unsorted sequence S of n comparable elements, and an integer k, give
an O(nlogk) expected-time algorithm for finding the O(k) elements that have
rank …n/k…, 2…n/k…, 3 …n/k…, and so on.
C-11.28
Let S be a sequence of n insert and removeMin operations, where all the keys
involved are integers in the range [0,n − 1]. Describe an algorithm running in
O(nlog* n) for determining the answer to each removeMin.
C-11.29
Space aliens have given us a program, alienSplit, that can take a sequence S of n
integers and partition S in O(n) time into sequences s 1 , S 2 , …, Sk of size at most
…n/k… each, such that the elements in Si are less than or equal to every
element in Si +1 , for i = 1,2,…, k − 1, for a fixed number, k < n. Show how to
use alienSplit to sort S in O(nlogn/logk) time.
C-11.30
Karen has a new way to do path compression in a tree-based union/find partition
data structure starting at a node v. She puts all the nodes that are on the path
from v to the root in a set S. Then she scans through S and sets the parent pointer
of each node in S to its parent's parent pointer (recall that the parent pointer of
the root points to itself). If this pass changed the value of any node's parent
pointer, then she repeats this process, and goes on repeating this process until
she makes a scan through S that does not change any node's parent value. Show
that Karen's algorithm is correct and analyze its running time for a path of
length h.
C-11.31
This problem deals with modification of the quick-select algorithm to make it
deterministic yet still run in O(n) time on an n-element sequence. The idea is to
modify the way we choose the pivot so that it is chosen deterministically, not
randomly, as follows:
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