CSCI 256 Design and analysis of algorithms Assignment 1 Due Friday, 2/9/2001

Numbered problems are from Cormen, Leiserson, and Rivest. You should do all of the problems, but only turn in those from the second section..

Practice problems:

• Prove xⁿ - yⁿ is divisible by x - y for all x /= y, n >= 0. Hint: Add and subtract xy^n-1 to the expression and factor.

• Prove that the regions formed by n circles in the plane can be colored with two colors such that any neighboring regions are colored differently. Note: Neighbors are those that share a boundary, not just a single point of intersection.

• Prove that any integer postage greater than 7 cents can be formed used only 3 and 5 cent stamps and that this can be done using at most two 5 cent stamps. Hint: You need 3 base cases!

Problems to be turned in:

• Problem 1.3-3 from CLR. You need only prove this for n that are exact powers of 2 (i.e., for n of the form 2^k for k >= 0).

• Problem 2.2-7 from CLR. Hint: Calculate phi squared and phi-hat squared and use them to simplify F_i = F_i-1 + F_i-2.

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