Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 68
and it will use an implicit stack for this purpose.) The remedy lies in stacking the sort request for the longer
partition and in continuing directly with the further partitioning of the smaller section. In this case, the size of
the stack M can be limited to log(n).
The change necessary is localized in the section setting up new requests. It now reads
IF j - L < R - i THEN
IF i < R THEN (*stack request for sorting right partition*)
INC(s); low[s] := i; high[s] := R
END;
R := j (*continue sorting left partition*)
ELSE
IF L < j THEN (*stack request for sorting left parition*)
INC(s); low[s] := L; high[s] := j
END;
L := i (*continue sorting right partition*)
END
2.3.4 Finding the Median
The median of n items is defined as that item which is less than (or equal to) half of the n items and
which is larger than (or equal to) the other half of the n items. For example, the median of
16 12 99 95 18 87 10
is 18. The problem of finding the median is customarily connected with that of sorting, because the obvious
method of determining the median is to sort the n items and then to pick the item in the middle. But
partitioning yields a potentially much faster way of finding the median. The method to be displayed easily
generalizes to the problem of finding the k-th smallest of n items. Finding the median represents the special
case k = n/2.
The algorithm invented by C.A.R. Hoare [2-4] functions as follows. First, the partitioning operation of
Quicksort is applied with L = 0 and R = n-1 and with a k selected as splitting value x. The resulting index
values i and j are such that
1. a h < x for all h < i
2. a h > x for all h > j
3. i > j
There are three possible cases that may arise:
1. The splitting value x was too small; as a result, the limit between the two partitions is below the desired
value k. The partitioning process has to be repeated upon the elements a i ... a R (see Fig. 2.9).
≤
≥
L
j i
k
R
Fig. 2.9. Bound x too small
2. The chosen bound x was too large. The splitting operation has to be repeated on the partition a L ... a j
(see Fig. 2.10).