代写CMPSC 465 Data Structures & Algorithms Spring 2024 Worksheet 2代做Prolog

CMPSC 465

Data Structures & Algorithms

Spring 2024

Worksheet 2

1. Growth Rate. Sort the following expressions from slowest to fastest growth rate. (You may assume all logarithms have base 2.)

2. Find run time. How long does the recursive multiplication algorithm (given below) take to multiply a non-negative n-bit number by a non-negative m-bit number? Justify your answer.

Algorithm 1 multiply(x,y)

Input: An n-bit integer x and an m-bit integer y, where x, y ≥ 0

Output: Their product x · y

if y = 0 then

   return 0

end if

z = multiply(x,⌊2/y⌋)

if y = even then

   return 2z

else

   return x+2z

end if

3. Computing Factorials. Consider the problem of computing N! = 1×2×··· ×N.

1. If N is a b-bit number, how many bits long is N! (Θ notation suffices)?

2. Consider the simple algorithm to compute N! that does the multiplication in sequence, 1×2×3×...×N. Analyze its running time.

4. Fibonacci growth. The Fibonacci numbers F0, F1, F2 ... are defined by the recurrence F0 = 0, F1 = 1, and Fn = Fn−1 + Fn−2. In this problem, we will confirm that this sequence grows exponentially fast and obtain some bounds on its growth.

(a) Use induction to prove that Fn ≥ 2 0.5n for n ≥ 6.

(b) Find a constant c > 0 such that Fn ≥ 2cn for all n ≥ 3. Show that your answer is correct.

(c) Find the maximum constant c > 0 for which Fn = Ω(2cn).