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By Nikos Karacapilidis

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2 Complexity Theory 32 Therefore, no direct recursive calls are needed. As a consequence, the performance increases drastically. The divide-and-conquer algorithm for the Fibonachi-numbers reads as follows, the array f [Iis used t o store the results: algorithm fib-dynamic(n) begin i f n < 3 then return 1; f[l]:= 1; f [a] := 1; for i := 3 , 4 , . . , n do f [i] := f [i- 1 1 f [i - 21 returnf [n]; end + Since the algorithm contains just one loop it runs in O(n) time. The last basic programming principle which is presented here is backtracking.

Since the tree is not empty and 24 is smaller than the object at the root (33), a recursive call with the left subtree occurs. Here, number 17 is stored at the root. Thus, the procedure is called with the right subtree of the first subtree. In the next step again the search continues in the right subtree where finally the object is found. The algorithm performs one descent into the tree. Hence, its time complexity is O(h) if h is the height of the search tree. If the search tree is complete, the height is h E O(logn), where n is the number of elements in the tree.

N/2 do comment divide set begin B 2. - A 2 ,. C z. - A z+n/2; end mergesort(n/2, {Bl, . . ,Bn12)); comment sort subsets mergesort(n/2, {GI,. . , CnI2)); x := 1;y := I ; comment largest elements of sequences for i := 1,.. ,n do comment merge sorted subsets if x 5 n/2 AND B, < C, then Ai := B,; x := x 1; else A 2. - cy ,. y := y I ; end '- + + The hierarchy of recursive calls of mergesort(4, {5,2,3,1)) is displayed in the upper part of Fig. 10. In the lower part the merging of the sorted subset is shown.

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