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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
 Derivative, Continuous function, Analytic function, Smooth function

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