Algorithms and Data Structures

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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 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).
N.Wirth. Algorithms and Data Structures. Oberon version 69 ≤ ≥ L k Fig. 2.10. Bound x too large j i R 3. j < k < i: the element a k splits the array into two partitions in the specified proportions and therefore is the desired quantile (see Fig. 2.11). ≤ ≥ L Fig. 2.11. Correct bound j k i R The splitting process has to be repeated until case 3 arises. This iteration is expressed by the following piece of program: L := 0; R := n; WHILE L < R-1 DO x := a[k]; partition (a[L] ... a[R-1]); IF j < k THEN L := i END; IF k < i THEN R := j END END For a formal proof of the correctness of this algorithm, the reader is referred to the original article by Hoare. The entire procedure Find is readily derived from this. PROCEDURE Find (k: INTEGER); (* ADenS2_Sorts *) (*reorder a such that a[k] is k-th largest*) VAR L, R, i, j: INTEGER; w, x: Item; BEGIN L := 0; R := n-1; WHILE L < R-1 DO x := a[k]; i := L; j := R; REPEAT WHILE a[i] < x DO i := i+1 END; WHILE x < a[j] DO j := j-1 END; IF i j; IF j < k THEN L := i END; IF k < i THEN R := j END END END Find If we assume that on the average each split halves the size of the partition in which the desired quantile lies, then the number of necessary comparisons is n + n/2 + n/4 + ... + 1 ≈ 2n
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