Solution_hw1

Solution_hw1 - Solutions for Homework 1 problems 1 = 2 =...

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Unformatted text preview: Solutions for Homework 1 problems 1. = 2 ) = ( log )( = = 3 (log ) = 2( =2 Claim: Pf: = ) . (n) = ( log )( ∈( lim ∈( ∈( ) : note, → () () ) =2 (( )( ) ) = 2( )= → )( ) = ( )= 2 = lim = () () )( = (2 = lim → =0 ): Since ): lim ( ) ( → = = lim )( → ( ) = lim → ( ( )( ) ) ( ) = lim → ) → =0 ( ( ) ) ∈( lim → ( ): lim . → () () = lim ( . ) = lim → ( . ) = ) =0 → () () ∈( lim → ( ): lim . ) = lim → ( ) . = lim → . ( ) = =0 2. (a) True PF: (consider the positive part of the functions) f(n) + g(n) ≤ 2 max {f(n), g(n)} => f(n) + g(n) ∈ O(max{f(n),g(n)}) Max{f(n) + g(n)} ≤ f(n) + g(n) => max {f(n)+g(n)} ∈ O(f(n) + g(n)) (b) False, for instance, take f(n) = (c) False, for instance, take f(n) = 3n g(n) = 2n s(n) = n + 1 r(n) = n 3. S(n) = log( ) = k n 2t log( 2t k )= ( − )2 = +2 = 2 − 2−2 2= (2 − 1) − ( − 1)2 4. (a) ( )= 2 Let 2 + log (2) = 1 )+ =2 ℎ (2 ) = 2 (2 Define G (k) = T (2 ) → (1) = 1 → ( ) = 2 ( − 1) + … 2 ( − )= 2 2 − ( + 1) + 2 ( − ) … (2) = 2 + − −2 (1) + 2 2 ( )= 2 2 (b) The “+3” can be ignored ( ) = 16 4 + + 2− 2 =2 2 ( − )= ( )= 1 ≤4 Using the Master Theorem (Page 139 of the textbook) = 16 Now, ( ) ∈ ( ), ℎ =4 = log / log = 2 ( )= log ) ( ) ∈ ( ( ) log ) = ( 5. Sort the set S into an array L For each I = 1 … size (A) If BinarySearch(A, X-A[i]) then Return true Return false (1) ( ) ( ) ...
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This note was uploaded on 02/27/2010 for the course ECE 544 taught by Professor Ray during the Spring '10 term at Rutgers.

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