# algtech - Algorithm Analysis Techniques To review the...

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Algorithm Analysis Techniques To review the following: Order analysis Summation techniques Recursion Recurrences A running example: Given an array of integers (positive, zero, negative), find a sequence of contiguous locations that result in the largest sum, among all possible such sequences.

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Order Notations: Suppose f ( n ) and g ( n ) are two functions defined on integers n 0 and f ( n ) > 0, g ( n ) > 0. Definition of Big-O : We write f ( n ) = O( g ( n )), or, simply f = O( g ), if there exist positive constants c and k (i.e. independent of n ), such that f ( n ) c g ( n ) for all n k . Example: If f ( n ) = 3 n + 12, g ( n ) = n , then f = O( g ) because 3 n + 12 4 n when n 12. (Thus, c = 4 and k = 12.) Definition of Little-o: We write f ( n ) = o( g ( n )), or, simply f = o( g ), if Example: 3 n + 12 = o( n 2 ) because Definition of Theta Θ : We write f ( n ) = Θ ( g ( n )), or, simply f = Θ ( g ), if f = O( g ) and g = O( f ). Example: 3 n + 12 = Θ ( n ). . 0 lim ) ( ) ( = n g n f n . 0 ) 12 3 ( lim 12 3 lim 2 2 = + = + n n n n n n
Theorems (facts, rules) about order notations: (1) If and if c 0, then f = Θ ( g ). (2) If then f = O( g ). (That is, little-o implies big-O.) (3) If then g = o( f ). (4) If f 1 = O( g 1 ) and f 2 = O( g 2 ), then f 1 + f 2 = O( g 1 + g 2 ) and f 1 f 2 = O( g 1 g 2 ). (5) f 1 + f 2 = Θ (max( f 1 , f 2 )). (For example, (2 n + 12) + (lg n ) = Θ (2 n + 12) = Θ ( n ). Note: When doesn’t exist (e.g., because the value of the ratio oscillates as n grows), we cannot conclude the relative magnitude between f and g . , ) ( ) ( lim c n g n f n = , 0 ) ( ) ( lim = n g n f n , ) ( ) ( lim = n g n f n ) ( ) ( lim n g n f n

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Typical growth rates in increasing order of magnitudes: Function Name c constant lg n logarithmic n linear n lg n n lg n (between linear and quadratic?) n 2 quadratic n 3 cubic 2 n exponential n ! factorial 2 n n ! super-factorial (?)
Summation rules and formulas: (1) (2) where k is independent of the summation index i and independent of n . (3)

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## algtech - Algorithm Analysis Techniques To review the...

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