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Unformatted text preview: number of comparisons is six. Figure 1: Finding the median of ﬁve numbers 1 3. The recurrence is deﬁned as following: T ( n ) = ( O (1) if n < = 9 T ‡ 2 n 3 · + T ‡ n 3 · + cn otherwise where c is a positive constant. We solve this recursion by induction. Assume that T ( n ) ≥ dn lg n , when n ≤ k and d is a positive constant. As a base case, when n = 9, we take a hidden constant for O (1) as C such that C ≥ 9 d lg 9. Then, for n = k + 1, T ( k + 1) ≥ T ‡ 2( k + 1) 3 · + T ‡ ( k + 1) 3 · + c ( k + 1) ≥ d ‡ 2( k + 1) 3 · lg ‡ 2( k + 1) 3 · + d ‡ ( k + 1) 3 · lg ‡ ( k + 1) 3 · + c ( k + 1) = d ‡ 2( k + 1) 3 · (lg( k + 1)lg 3 2 ) + d ‡ ( k + 1) 3 · (lg( k + 1)lg 3) + c ( k + 1) = d ( k + 1)lg( k + 1)d ‡ 2( k + 1) 3 lg 3 2 + k + 1 3 lg 3 · + c ( k + 1) = d ( k + 1)lg( k + 1)d ( k + 1)(lg 32 3 ) + c ( k + 1) ≥ d ( k + 1)lg( k + 1) . Therefore, this recurrence belongs Ω( n lg n ) as long as d ≤ c log 32 / 3 . 2...
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 Spring '08
 Unkown
 Algorithms, LG, 3 k, Selection algorithm

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