The.Algorithm.Design.Manual.Springer-Verlag.1998

Exercises
used to select the pivot. Although Quicksort is on average, certain partition rules cause
Quicksort to take time if the array is already sorted.
(a) Suppose we always pick the pivot element to be the key from the last position of the subarray.
On a sorted array, does Quicksort now take , , or ?
(b) Suppose we always pick the pivot element to be the key from the middle position of the
subarray. On a sorted array, does Quicksort now take , , or ?
(c) Suppose we always pick the pivot element to be the key of the median element of the first
three keys of the subarray. (The median of three keys is the middle value, so the median of 5, 3, 8
is five.) On a sorted array, does Quicksort now take , , or ?
(d) Suppose we always pick the pivot element to be the key of the median element of the first,
last, and middle elements of the subarray. On a sorted array, does Quicksort now take ,
, or ?
14. (*) Given a set S of n integers and an integer T, give an algorithm to test whether k of
the integers in S sum up to T.
15. (**) Design a data structure that allows one to search, insert, and delete an integer X in O(1) time
(i.e. constant time, independent of the total number of integers stored). Assume that and
that there are m+n units of space available for the symbol table, where m is the maximum number
of integers that can be in the table at any one time. (Hint: use two arrays A[1..n] and B[1..m].) You
are not allowed to initialize either A or B, as that would take O(m) or O(n) operations. This means
the arrays are full of random garbage to begin with, so you must be very careful.
16. (*) Let P be a simple, but not necessarily convex, polygon and q an arbitrary point not necessarily
in P. Design an efficient algorithm to find a line segment originating from q that intersects the
maximum number of edges of P. In other words, if standing at point q, in what direction should
you aim a gun so the bullet will go through the largest number of walls. A bullet through a vertex
of P gets credit for only one wall. An algorithm is possible.
17. (**) The onion of a set of n points is the series of convex polygons that result from finding the
convex hull, striping it from the point set, and repeating until no more points are left. Give an
algorithm for determining the onion of a point set.
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