Algorithms and Data Structures

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N.Wirth. Algorithms and Data Structures. Oberon version 60 a[j] := x END END END END ShellSort Analysis of Shellsort. The analysis of this algorithm poses some very difficult mathematical problems, many of which have not yet been solved. In particular, it is not known which choice of increments yields the best results. One surprising fact, however, is that they should not be multiples of each other. This will avoid the phenomenon evident from the example given above in which each sorting pass combines two chains that before had no interaction whatsoever. It is indeed desirable that interaction between various chains takes place as often as possible, and the following theorem holds: If a k-sorted sequence is i-sorted, then it remains k-sorted. Knuth [2.8] indicates evidence that a reasonable choice of increments is the sequence (written in reverse order) 1, 4, 13, 40, 121, ... where h k-1 = 3h k +1, h T = 1, and T = k × ⎣log 3 (n)⎦ - 1. He also recommends the sequence 1, 3, 7, 15, 31, ... where h k-1 = 2h k +1, h T = 1, and T = k × ⎣log 2 (n)⎦ - 1. For the latter choice, mathematical analysis yields an effort proportional to n 1.2 required for sorting n items with the Shellsort algorithm. Although this is a significant improvement over n 2 , we will not expound further on this method, since even better algorithms are known. 2.3.2 Tree Sort The method of sorting by straight selection is based on the repeated selection of the least key among n items, then among the remaining n-1 items, etc. Clearly, finding the least key among n items requires n-1 comparisons, finding it among n-1 items needs n-2 comparisons, etc., and the sum of the first n-1 integers is (n 2 -n)/2. So how can this selection sort possibly be improved? It can be improved only by retaining from each scan more information than just the identification of the single least item. For instance, with n/2 comparisons it is possible to determine the smaller key of each pair of items, with another n/4 comparisons the smaller of each pair of such smaller keys can be selected, and so on. With only n-1 comparisons, we can construct a selection tree as shown in Fig. 2.3. and identify the root as the desired least key [2.2]. 06 12 06 44 12 18 06 44 55 12 42 94 18 06 67 Fig. 2.3. Repeated selection among two keys The second step now consists of descending down along the path marked by the least key and eliminating it by successively replacing it by either an empty hole at the bottom, or by the item at the alternative branch at intermediate nodes (see Figs. 2.4 and 2.5). Again, the item emerging at the root of the tree has the (now second) smallest key and can be eliminated. After n such selection steps, the tree is empty (i.e., full of holes), and the sorting process is terminated. It should be noted that each of the n selection steps requires only log(n) comparisons. Therefore, the total selection process requires only on the order of n*log(n) elementary operations in addition to the n steps required by the construction of the tree. This is a very significant improvement over the straight methods requiring n 2 steps, and even over
N.Wirth. Algorithms and Data Structures. Oberon version 61 Shellsort that requires n 1.2 steps. Naturally, the task of bookkeeping has become more elaborate, and therefore the complexity of individual steps is greater in the tree sort method; after all, in order to retain the increased amount of information gained from the initial pass, some sort of tree structure has to be created. Our next task is to find methods of organizing this information efficiently. 12 44 12 18 44 55 12 42 94 18 67 Fig. 2.4. Selecting the least key 12 12 18 44 12 18 67 44 55 12 42 94 18 67 Fig. 2.5. Refilling the holes Of course, it would seem particularly desirable to eliminate the need for the holes that in the end populate the entire tree and are the source of many unnecessary comparisons. Moreover, a way should be found to represent the tree of n items in n units of storage, instead of in 2n - 1 units as shown above. These goals are indeed achieved by a method called Heapsort by its inventor J. Williams [2-14]; it is plain that this method represents a drastic improvement over more conventional tree sorting approaches. A heap is defined as a sequence of keys h L , h L+1 , ... , h R (L ≥ 0) such that h i < h 2i+1 and h i < h 2i+2 for i = L ... R/2-1. If a binary tree is represented as an array as shown in Fig. 2.6, then it follows that the sort trees in Figs. 2.7 and 2.8 are heaps, and in particular that the element h 0 of a heap is its least element: h 0 = min(h 0 , h 1 , ... , h n-1 ) h 0 h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 h 10 h 11 h 12 h 13 h 14 Fig. 2.6. Array viewed as a binary tree h 0 42 06 55 94 18 12
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