Lecture No. 10
3.2.1 Sieve Technique
The reason for introducing this algorithm is that it illustrates a very important special case
of divide-and-conquer, which I call the sieve technique. We think of divide-and-conquer
as breaking the problem into a small number of smaller subproblems, which are then
solved recursively. The sieve technique is a special case, where the number of
subproblems is just 1.
The sieve technique works in phases as follows. It applies to problems where we are
interested in finding a single item from a larger set of n items. We do not know which
item is of interest, however after doing some amount of analysis of the data, taking say
Θ(nk) time, for some constant k, we find that we do not know what the desired the item
is, but we can identify a large enough number of elements that cannot be the desired
value, and can be eliminated from further consideration. In particular “large enough”
means that the number of items is at least some fixed constant fraction of n (e.g. n/2,
n/3). Then we solve the problem recursively on whatever items remain. Each of the
resulting recursive solutions then do the same thing, eliminating a constant fraction of the
remaining set.
3.2.2 Applying the Sieve to Selection
To see more concretely how the sieve technique works, let us apply it to the selection
problem. We will begin with the given array A[1..n]. We will pick an item from the
array, called the pivot element which we will denote by x. We will talk about how an item
is chosen as the pivot later; for now just think of it as a random element of A.
We then partition A into three parts.
1. A[q] contains the pivot element x,
2. A[1..q - 1] will contain all the elements that are less than x and
3. A[q + 1..n] will contains all elements that are greater than x.
Within each sub array, the items may appear in any order. The following figure shows a
partitioned array:
Figure 3.4: A[p..r] Partitioned about the pivot x
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