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Thè initial claim, So, is true before the loop begins 2 If Si-1 is true before iteration i begins, then one can show that S will be true after iteration i is ovet 3. 1 (for the correctness of arrayMax), but let us nevertheless níethod more example here. In particular, let us consider applying the. 12, which searches for n element x in an array A. To show arrayFind to be correct, we use a loop invariant argument. That is, we inductively define statements, S, for i = 0, 1,... , n, that lead to the correctness of arrayFind.

Chapter 1. Algorithm Analysis 20 Ordering Functions by Their Growth Rates problem. are available an a1orithm A, Suppose two algorithms solving the same which has a running timé of 9(n), and an algorithm B, which has a running time of 9(n2). Which one is better7 The little-oh notation says that n is o(n2), which implies that algorithm A is asymptotically better than algorithm B, although. for a given (small) value of n, it is possible for algorithm B to have lower running time than algorithm A. the above tables, the benefits of algorithm A over algorithm B will become clear.

22. 22. 22. 22: A collection of loop methods. 7. l5 Show that if f(n) is O(g(n)) is O(g(n) +h(n)). lS is O(g(n)) and d(n) is O(h(n)), then the summationf(n) +d(n) = O(f(n) + g(n)). if and only if g(n) is Q(f(n)). Show that if p(n) is a polynomial in n, then logp(n) is O(logn). l9 Show that (n + l)S is O(nS). 20 Show that 2n+1 is O(2n). 21 Show that n is o(nlogn). 23 Show that 0)( n). n310gn is Q(n3). 24 Show that if(n)l is O(f(n)) always greater than 1. 25 Justify the fact that if d(n) is O(f(n)) d(n)e(n) is O(f(n)g(n)).

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