Use the master method to give tight asymptotic bounds

08 Jun Use the master method to give tight asymptotic bounds

Posted at 10:05h in 2General studies by Blogger

Question

1. Use the master method to give tight asymptotic bounds for the following recurrences.

a. T (n) = 2T (n/4) + 1.

b. T(n) = 2T (n/4) + √n.

c. T(n) = 2T (n/4) + n.

d. T(n) = 2T (n/4) + n2.

2. Professor Caesar wishes to develop a matrix-multiplication algorithm that is asymptotically faster than Strassen’s algorithm. His algorithm will use the divide and- conquer method, dividing each matrix into pieces of size n/4 X n/4, and the divide and combine steps together will take Θ(n2) time. He needs to determine how many sub problems his algorithm has to create in order to beat Strassen’s algorithm.

If his algorithm creates a sub problems, then the recurrence for the running time T(n) becomes T(n) = aT (n/4) + Θ(n2). What is the largest integer value of a for which Professor Caesar’s algorithm would be asymptotically faster than Strassen’s algorithm?

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