By Manoochehr Azmoodeh (auth.)
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Extra resources for Abstract Data Types and Algorithms
While C do S; and repeatS until C; For both these loop constructs we should find the number of times the loop will be executed and multiply this by the complexities of C and S. For instance, the loop of b_search function is executed log2 n times. Thus, its time complexity is log2 n (0(1)) = O(log2 n). This simple analysis may not work for all loops. In these cases other properties of the program should be used to infer the time complexity. As an example, let us look at the following hypothetical program fragment: size i := 1i while i m; I: begin < n do i:=i+l; statement,; .
As an example, let us consider the operation of adding an extra number to the end of a list of numbers stored in the memory. Let us denote this operation as add_to_ end(n) where n should be added to the end of the list. To be able to do this, we aim to write a procedure and we should consider the following details: • Where are the elements of this list stored in the main memory? • Are the elements stored in consecutive locations of memory or not? • If not, how are the elements linked together? Abstract Data Types and Program Design 27 • How is the parameter of this procedure (the new number) to be passed to the procedure?
Based on these we can derive T(n) ! = 1+1 n=O 1 + 3 + 2 + 1 + 1 + 3 + T(O) n =1 1 + 3 + 2 + 1 + 1 + 3 + T(n/2) n > 1 The Complexity of Algorithms 17 The assumption is made that C is not in the array (and thus the complexity is the worst case). For n = 0, one test (U < L) and one assignment are needed. For n = 1, one test (U < L), two arithmetic and an assignment (index := (L+U) div 2), one array access and an assignment (element := A[index]), one test (element= C) and finally a recursive call are needed, the latter requiring one subtraction, one assignment and a control transfer to the procedure.