2. Consider the following recursive algorithm Algorithm T(n) //Input: A non-nega

Question

2. Consider the following recursive algorithm Algorithm T(n) //Input: A non-negative integer n if n = 0 return 1 elseif n= 1 return 5 else return 8T(n-1)-16T(n-2)+9 Set up a recurrence relation for this function's output values T(n). You need to give the initial conditions. (15pts) a. b. Solve the recurrence relation to determine what this algorithm computes, i.e., give the exact mathematical form for T(n). (10pts) 3. Sort the list 19, 13, 2, 7,9, 1, 5 in an ascending order by Insertion Sort. You should

Explanation / Answer

A recursive algorithm is an algorithm which calls itself with "smaller (or simpler)" input values, and which obtains the result for the current input by applying simple operations to the returned value for the smaller (or simpler) input. More generally if a problem can be solved utilizing solutions to smaller versions of the same problem, and the smaller versions reduce to easily solvable cases, then one can use a recursive algorithm to solve that problem. For example, the elements of a recursively defined set, or the value of a recursively defined function can be obtained by a recursive algorithm.

If a set or a function is defined recursively, then a recursive algorithm to compute its members or values mirrors the definition. Initial steps of the recursive algorithm correspond to the basis clause of the recursive definition and they identify the basis elements. They are then followed by steps corresponding to the inductive clause, which reduce the computation for an element of one generation to that of elements of the immediately preceding generation.

Algorithm 1: Even(positive integer k)
Input: k , a positive integer
Output: k-th even natural number (the first even being 0)

Algorithm:
if k = 1, then return 0;
else return Even(k-1) + 2 .

Here the computation of Even(k) is reduced to that of Even for a smaller input value, that is Even(k-1). Even(k) eventually becomes Even(1) which is 0 by the first line. For example, to compute Even(3), Algorithm Even(k) is called with k = 2. In the computation of Even(2), Algorithm Even(k) is called with k = 1. Since Even(1) = 0, 0 is returned for the computation of Even(2), and Even(2) = Even(1) + 2 = 2 is obtained. This value 2 for Even(2) is now returned to the computation of Even(3), and Even(3) = Even(2) + 2 = 4 is obtained.
As can be seen by comparing this algorithm with the recursive definition of the set of nonnegative even numbers, the first line of the algorithm corresponds to the basis clause of the definition, and the second line corresponds to the inductive clause.

By way of comparison, let us see how the same problem can be solved by an iterative algorithm.

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