Algorithms and Data Structures

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N.Wirth. Algorithms and Data Structures. Oberon version 54 moves are C min = n - 1 M min = 2*(n - 1) C ave = (n 2 - n)/4 M ave = (n 2 + 3n - 4)/4 C max = (n 2 - 3n + 2)/2 M max = (n 2 - n)/2 The minimal numbers occur if the items are initially in order; the worst case occurs if the items are initially in reverse order. In this sense, sorting by insertion exhibits a truly natural behavior. It is plain that the given algorithm also describes a stable sorting process: it leaves the order of items with equal keys unchanged. The algorithm of straight insertion is easily improved by noting that the destination sequence a 0 ... a i-1 , in which the new item has to be inserted, is already ordered. Therefore, a faster method of determining the insertion point can be used. The obvious choice is a binary search that samples the destination sequence in the middle and continues bisecting until the insertion point is found. The modified sorting algorithm is called binary insertion. PROCEDURE BinaryInsertion; (* ADenS2_Sorts *) VAR i, j, m, L, R: INTEGER; x: Item; BEGIN FOR i := 1 TO n-1 DO x := a[i]; L := 0; R := i; WHILE L < R DO m := (L+R) DIV 2; IF a[m]
N.Wirth. Algorithms and Data Structures. Oberon version 55 improvement is by no means drastic: the important term M is still of the order n 2 . And, in fact, sorting the already sorted array takes more time than does straight insertion with sequential search. This example demonstrates that an "obvious improvement" often has much less drastic consequences than one is first inclined to estimate and that in some cases (that do occur) the "improvement" may actually turn out to be a deterioration. After all, sorting by insertion does not appear to be a very suitable method for digital computers: insertion of an item with the subsequent shifting of an entire row of items by a single position is uneconomical. One should expect better results from a method in which moves of items are only performed upon single items and over longer distances. This idea leads to sorting by selection. 2.2.2 Sorting by Straight Selection This method is based on the following principle: 1. Select the item with the least key. 2. Exchange it with the first item a 0 . 3. Then repeat these operations with the remaining n-1 items, then with n-2 items, until only one item — the largest — is left. This method is shown on the same eight keys as in Table 2.1. Initial keys 44 55 12 42 94 18 06 67 i=1 06 55 12 42 94 18 44 67 i=2 06 12 55 42 94 18 44 67 i=3 06 12 18 42 94 55 44 67 i=4 06 12 18 42 94 55 44 67 i=5 06 12 18 42 44 55 94 67 i=6 06 12 18 42 44 55 94 67 i=7 06 12 18 42 44 55 67 94 Table 2.2. A Sample Process of Straight Selection Sorting. The algorithm is formulated as follows: FOR i := 0 TO n-1 DO assign the index of the least item of a i ... a n-1 to k; exchange a i with a k END This method, called straight selection, is in some sense the opposite of straight insertion: Straight insertion considers in each step only the one next item of the source sequence and all items of the destination array to find the insertion point; straight selection considers all items of the source array to find the one with the least key and to be deposited as the one next item of the destination sequence. PROCEDURE StraightSelection; (* ADenS2_Sorts *) VAR i, j, k: INTEGER; x: Item; BEGIN FOR i := 0 TO n-2 DO k := i; x := a[i]; FOR j := i+1 TO n-1 DO IF a[j] < x THEN k := j; x := a[k] END END; a[k] := a[i]; a[i] := x END
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