Consider the following recurrence: A(n) = == = {4 A(n/2 if n = 1 4A(n/2)+n otherwise...

 

Transcribed Image Text:

Consider the following recurrence: A(n) = == = {4 A(n/2 if n = 1 4A(n/2)+n otherwise We want to find an exact closed form of this equation by using the tree method. (a) (2 points) Draw your recurrence tree (as shown in the class; see lecture 18, slide 19). (b) (1 point) What is the size of the input at level i (as in class, call the root level O)? (c) (2 points) what is the number of nodes at level i? Note: let i = 0 indicate the level corresponding to the root node. So, when i = 0, your expression should be equal to 1. (d) (3 points) What is the total work at the i-th recursive level? Be sure to show your work for full credit. (e) (1 point) What is the last level of the tree? (f) (2 points) What is the work done in the base case? (g) (3 points) Combine your answers from previous parts to get an expression for the total work. Simplify the expression to a closed form. Be sure to show your work for full credit. (h) (1 point) From the closed form you computed above, give a tight-O bound. No need to prove it. (Hint: You may need to simplify the closed form.) Consider the following recurrence: A(n) = == = {4 A(n/2 if n = 1 4A(n/2)+n otherwise We want to find an exact closed form of this equation by using the tree method. (a) (2 points) Draw your recurrence tree (as shown in the class; see lecture 18, slide 19). (b) (1 point) What is the size of the input at level i (as in class, call the root level O)? (c) (2 points) what is the number of nodes at level i? Note: let i = 0 indicate the level corresponding to the root node. So, when i = 0, your expression should be equal to 1. (d) (3 points) What is the total work at the i-th recursive level? Be sure to show your work for full credit. (e) (1 point) What is the last level of the tree? (f) (2 points) What is the work done in the base case? (g) (3 points) Combine your answers from previous parts to get an expression for the total work. Simplify the expression to a closed form. Be sure to show your work for full credit. (h) (1 point) From the closed form you computed above, give a tight-O bound. No need to prove it. (Hint: You may need to simplify the closed form.)

Answer rating: 100% (QA)

0 AC A2 A12 A212 A24 5 L n 2 A12 An4 An4 A14 5 24 2 At the ith recursive level the... View the full answer