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Unformatted text preview: Asymptotic Notation In order to express algorithm performance, we need a notation that ignores constant factors; and focuses on the behavior as the size of the input increases. Defn: Let f : N R . O ( f ( n )) is the set of all functions g : N R such that for some natural number n and some strictly positive real number c , g ( n ) cf ( n ) whenever n n . 1 Examples 2 n 2 O ( n 2 ) . 3 n 2 + 17 n + 350 O ( n 2 ) . n 2 negationslash O ( n ) . O ( n 2 + 5 n ) O (2 n 2 + 4) . O (2 n 2 + 4) O ( n 2 + 5 n ) . O (2 n 2 + 4) = O ( n 2 + 5 n ) . O ( n ) O ( n 2 ) . O ( n ) negationslash = O ( n 2 ) . 2 Precondition: A [0 ..n 1] is an array of Number s, n is a Nat . Postcondition: Returns the maximum subsequence sum of A . MaxSumBU ( A [0 ..n 1] ) m 0; msuf // Invariant: m is the maximum subsequence sum // of A [0 ..i 1] , // msuf is the maximum suffix sum for A [0 ..i 1] for i to n 1 msuf Max (0 , msuf + A [ i ]) m Max ( m, msuf ) return m 3 Theorem 3.8 Suppose f 1 ( n ) O ( g 1 ( n )) and f 2 ( n ) O ( g 2 ( n )) . Then 1. f 1 ( n...
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 Spring '09
 Howell

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