Problem description The Jim Thornton Coffee House chain is planning expansion into Calgary. It has selected many possible sites for new coffee houses. The possible sites are joined by roads that form a spanning tree. To eliminate competition with itself the company has determined that it should not choose two sites that are adjacent in this tree. From its market evaluation the company has also determined the expected profit per year for each site. Your job is to determine what sites Jim Thornton should choose for his coffee houses. a. Define this problem precisely by defining the required Input and the required Output. Use mathematical notation. b. Write a recursive equation for the function that maximizes the profit per year. Include a short argument that your equation is correct. c. Write an efficient algorithm that computes the maximum profit per year and also computes the sites that should be chosen. Your algorithm should run in time that is polynomial in the input size. d. What is the asymptotic running time of your algorithm? Defend your answer. Your algorithm and proofs should be precise and concise (as well of course as correct.) Elegance of your solution counts. 2
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