CMPSCI 311: Theory of Algorithms

David Mix Barrington

Fall, 2003

Discussion Notes #2

from Wednesday 17 Sept 2003

Here we solve some recurrence relations of various types. In each case find the function of n that satisfies the recurrence and the initial conditions, and prove by induction that your answer is correct. In the case of the last three, you need only find a solution that is correct when n is a power of two. Please answer the questions during the discussion period.

• Question 1: A(n) = A(n-1) + 3n² -3n +1; A(0) = 0

• Question 2: B(n) = 2B(n-1) - B(n-2); B(0) = 0, B(1) = 1

• Question 3: C(n) = 3C(n-1) - 2C(n-2); C(0)=0, C(1)=1

• Question 4: D(n) = 2D(n/2) + 3; D(1) = 1 (solve for n=2^k)

• Question 5: E(n) = 2E(n/2) + n; E(1) = 0 (solve for n=2^k)

• Question 6: F(n) = 2F(n/2) + n²; F(1) = 2 (solve for n=2^k)

Last modified 17 September 2003