# Suppose f n is o g n prove that f n 100 n is o g n n

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Suppose F ( n ) is O ( G ( n )). Prove that F ( n ) + 100 n is O ( G ( n ) + n ). 46. Consider the following Java code fragment static void swap( int[] a, int i, int j) { int temp = a[i]; a[i] = a[j]; a[j] = temp; } static int foo( int j, int i) { int r = 1; for (int k = j; k < i; k++) r = (r * k) % 100 ; return r; } static void bar(int n) { int[] a = new int[n]; for (int i = 0; i < n; i++) a[ i ] = i+1; for (int i = 1; i < n; i++ ) swap( a, i, foo( 0, i) ); } Analyze the running time of bar in Big-Oh notation. 13

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47. In this problem you NOT allowed to use any of the theorems about Big-Oh stated in the lecture slides, the textbook, or the lab writeups. Your proof should rely only on the definition of Big-Oh. Prove that 2 n +1 is O (2 n ). 48. Consider the following code fragment for computing the nth Fibonacci number, F n . Let C ( n ) be the number of recursive calls made by this fragment when the input is n . Show that C ( n ) = C ( n - 1) + C ( n - 2) + 1. Prove using induction that C ( n ) > F n . public static long fib (int n) { if (n <= 1) return n; else return fib(n-1) + fib(n-2); } 49. Which of the following statements are true: (a) O ( n 3 + n 2 + n ) is the same as O ( n 2 ). (b) O (10 10 ) is different from O (1). (c) The running time of an algorithm depends on the underlying model of computation. (d) The worst-case running time of the following code fragment is O ( n 2 ): int x = 0; for (int i = 0; i < n; i++) if (false) for (int j = 0; j < i; j++) x++; 14
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