Produce an equation that asymptotically describes the following algorithms runtime: define algorithm_1(input): x = 2 y = 2 x = ((y + z) * 80)/4 print x, y, z define algorithm_2(input): x = 1 y = 5 loop from x to size(input) * 4: y = y +5 print x, y define algorithm_3(input): x = user_input() y = user_input() z = 0 loop i = 0 to size(input): if x

1. First code :: 

-> In this we have 3 operations and all the operation take constant time.

We are considering (input = n). 

Lets take time taken for :: 

• x = 2 , y = 2  -> a    (Constant time)
• x = ((y + z) * 80) /4   -> b  (Constant time)
• print x ,y, z   -> c  (Constant time)

Here all are constant operations. So,

T(n) = a + b + c , which can also be written as ::

T(n) = k (Constant)

2. Second code :: 

-> In this code we have a loop and so the operation will take linear time. 

We are considering (input = n). 

Lets take time taken for :: 

• x = 1 , y = 5  -> a   (Constant time)
• loop, (y = y + 5) ,  (print x, y) -> 4*n  (Linear time)

T(n) = a + 4*n       or,

T(n) = a + b*n     ( a, b = constant) 

3. Third code :: 

-> In this code we have a loop and so the operation will take linear time. 

We are considering (input = n). 

Lets take time taken for :: 

• x and y input -> e (Constant time)
• z = 0 -> f (constant time)
• loop -> g* n  (linear time)
• if x < y   -> h* n  (linear time)
• Other operations -> k  (linear time)

T(n) = (e + f) + (g+h) *n + k     

- We assume :: (e + f) = a , (g+h) = b

T(n) = a + bn + k    ( a, b, n are constants)