Chapter 3: Q4E (page 216)
Use the definition of “\(f(x)\) is \(O(g(x))\)” to show that \({2^x} + 17\) is \(O({3^x})\).
By the definition of “\(f(x)\)is \(O(g(x))\)”, \({2^x} + 17\)is\(O({3^x})\)with \(C = 2\) and \(k = 2\).
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Describe an algorithm that takes as input a list of integers in non decreasing order and produces the list of all values that occur more than once. (Recall that a list of integers is non decreasing if each integer in the list is at least as large as the previous integer in the list.)
How many comparisons does the insertion sort use to sort the list n, n – 1,…, 2, 1?
The binary insertion sort is a variation of the insertion sort that uses a binary search technique (see Exercise 44) rather than a linear search technique to insert the element in the correct place among the previously sorted elements.
List all the steps used by algorithm 1 to find the maximum of the list
1, 8, 12, 9, 11, 2, 14, 5, 10, 4
Specify the steps of an algorithm that locates an element in a list of increasing integers by successively splitting the list into four sublists of equal (or as close to equal as possible) size, and restricting the search to the appropriate piece. In a list of elements, the same element may appear several times. A mode of such a list is an element that occurs at least as often as each of the other elements; a list has more than one mode when more than one element appears the maximum number of times.
a.) State the definition of the fact that f(n)is , where and are functions from the set of positive integers to the set of real numbers.
b.) Use the definition of the fact that f(n)isdirectly to prove or disprove that is .
c.) Use the definition of the fact that f(n)isdirectly to prove or disprove thatis.
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