Prove asymptotic bounds for recursion relations, Mathematics

1. (‡) Prove asymptotic bounds for the following recursion relations. Tighter bounds will receive more marks. You may use the Master Theorem if it applies.

1. C(n) = 3C(n/2) + n

2. G(n) = G(n - 1) + 1/n

3. I(n) = I(n/2) + n/ lg(n)

2. Define a (p,q)-tree as a rooted tree where every internal node has between p and q (inclusive) children. Use the Master Theorem to give asymptotic bounds for the height of the tree. You can assume both p and q are constants with 2 ≤ p ≤ q.

3. (‡) Dominos

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A 2 × 10 rectangle filled with ten dominos, and a 2 × 2 × 10 box filled with ten slabs.

1. A domino is a 2×1 or 1×2 rectangle. How many different ways are there to completely fill a 2 × n rectangle with n dominos?

2. A slab is a three-dimensional box with dimensions 1 × 2 × 2, 2 × 1 × 2, or 2 × 2 × 1. How many different ways are there to fill a 2 × 2 × n box with n slabs? Set up a recurrence relation and give reasonable exponential upper and lower bounds.

Posted Date: 3/19/2013 5:13:49 AM | Location : United States







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