Show that if , where and are real numbers and an , then data-custom-editor="chemistry" is . Big-O, big-Theta, and big-Omega notation can be extended to functions in more than one variable. For example, the statement is means that there exist constants C, , and such that whenever and .
It is given that , where and are real numbers then we have to prove that is .
Thus x> max(1,)
Hence, it can be said that f(x) is with constants b=max(1,) and C=
Assume b=min (1,)
Thus x> min (1,)
Hence, it can be said that f(x) is with constants b=min (1,) and C=
As we know that is and is .
Hence, by applying the definition of Big-Theta Notation, f (x) is .
a) Describe in detail (and in English) the steps of an algorithm that finds the maximum and minimum of a sequence of elements by examining pairs of successive elements, keeping track of a temporary maximum and a temporary minimum. If n is odd, both the temporary maximum and temporary minimum should initially equal the first term, and if n is even, the temporary minimum and temporary maximum should be found by comparing the initial two elements. The temporary maximum and temporary minimum should be updated by comparing them with the maximum and minimum of the pair of elements being examined.
b) Express the algorithm described in part (a) in pseudocode.
c) How many comparisons of elements of the sequence are carried out by this algorithm? (Do not count comparisons used to determine whether the end of the sequence has been reached.) How does this compare to the number of comparisons used by the algorithm in Exercise 5?
a.) Explain the concept of a greedy algorithm.
b.) Prove the example of a greedy algorithm that produces an optimal solution and explain why it produces an optimal solution.
c.) Provide an example of a greedy algorithm that does not always produce an optimal solution and explain why it fails to do so.
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