By Manoochehr Azmoodeh (auth.)

ISBN-10: 0333512103

ISBN-13: 9780333512104

ISBN-10: 1349211516

ISBN-13: 9781349211517

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**Example text**

A recurrence equation is an equation which defines the time complexity of a recursive routine in terms of complexity of smaller versions of the same problem. 1. function b_search( C, L, U 1 integer) 1 integer; search A[LJ .. A[UJ to find C and return its index var index, element 1 integer; (~ begin if ( U < L> then b_search 1 = 0 ~) ( ~ the sequence is empty ~) (~ search unsuccessful ~) else begin UNITS [3 J [ 1] index 1= (l + U> div 2; (~ the middle of the array ~) [3] element 1= A[indexJ; [2] if element = C then [1] b_search 1= index (~ search successful ~) [1] else if element > C then [1] 1) b search I : b - search (C, L, index [3 + Hn/2) 1 - - else end; end (~ b- search - of b search I= b- search CC, index + 1, U> [3 + Hn/2)] ~) Now, T(n), the worst-case time complexity of this function can be defined in terms of the length of the sequence, U-L+ 1 = n.

To begin with let us introduce the recurrence relation R(n) as R(n) = The Complexity of Algorithms l 1 n=O 1 R(n-1) + R(n-2) n 19 = 1 n>l It is clear that F(n) > R(n) for all n. Now we should try to find a closed formula for R(n). 62)n for some k. 62n) This indicates that algorithm A requires at least an exponential amount of time. To calculate T(n): T(n) = 1 + T(n) n ~ t=O F(i) > 1 +x +x 2 +x 3 + ... 62n+l ). Again this suggests that the procedure seql is an exponential algorithm. However, the procedure seq2 can be 20 Abstract Data Types and Algorithms easily shown to be O(n).

Each procedure or function defines an operation by providing: its name; a description of its parameters; an optional value that it may return; and a specification of the relation between the parameters, the returned value and any other data values. When using such operations there is no need to refer to the body of the pro- Abstract Data Types and Program Design 29 cedure or function which contains detailed information about how it actually produces the result. Such an abstraction has many advantages.

### Abstract Data Types and Algorithms by Manoochehr Azmoodeh (auth.)

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