Using definition of , O and to establish your claims:

I.a: Show that for any real constants a and b, where b > 0,

(n + a)^b = (n^b ).

I.b: Explain why the statement The running time of algorithm

A is at least O(n² ) is meaningless.

I.c: Justify your answers:

Is 2ⁿ⁺¹ = O(2ⁿ )? Is 2²ⁿ = O(2ⁿ)?

For each of the following a e problems, design (or give) an algorithm. Based on your obtained algorithm, give (i) a natural size metric for its inputs; (ii) its basic operation; (iii) whether the basic operation count can be different for inputs of the same size; and (iv) What is the efficiency class of this algorithm:

a. Computing n!

b. Euclids algorithm

c. Cutting a stick: A sticks n inches long needs to be cut into n 1-inch pieces. Outline an algorithm that performs this task with the minimum number of cuts if several pieces of the stick can be cut at the same time. Also give a formula for the minimum number of cuts.

d. Merge sort

e. Lighter or heavier? You have n > 2 identicallooking coins and a two-pan balance scale with no weights. One of the coins is a fake, but you do not know whether it is lighter or heavier than the genuine coins, which all weigh the same. Design a (1) algorithm to determine whether the fake coin is lighter or heavier than the others.