Draw the 7 element array showing the number of comparisons needed to find each element

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

For each problem show all work. Homework must be submitted in a Word document

1. A binary search of a sorted array with 3 unique elements where the search is always successful can be viewed as an array with each element having a probability of being selected = 1/3 and the number of comparisons a binary search requires to find the element is shown inside the array.

Then the average number of comparisons would be A(3) = (1/3) [2 + 1 + 2] = 5/3

Find the A(7) for an array with 7 elements for the same binary search.

(a) Draw the 7-element array showing the number of comparisons needed to find each element.

(b) Determine A(7)

(c) Using what you learned from parts (a) and (b), find the summation expression for A(n). To simplify your work, assume that n = 2k – 1. Use k in your summation. Leave your answer in closed form in terms of n.

(d) Simplify your a(n) for large values of n.

2. Consider the algorithm below

Precondition: n is a non-negative integer

function f ( n )

{ temp = 0

if (n != 0)

{ for (i = 1; i <= 3; i++) temp = temp + n * f(n-1) return temp }

else return 1;

Solve for the closed form by repeated substitution

3. The recurrence relation is given as an = 2an-1 + 3an-2 Use the method linear homogeneous characteristic roots to solve for the closed form, with given initial conditions a0 = 2 and a1 = 4

(a) Find the general solution

(b) Find the Specific solution

(c) Use your closed form result to find a

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