Question

Algorithms help As a new event in the Wizard-Lympics Tournament, students must compete in pairs in the following game. An even number n of magical items are laid out in a row, with the i-th item having a value of v(i) in Knuts for each i = 1,2,...,n. The players alternate turns, and on each turn, the player may choose either the first item or the last item remaining in the row, then remove it from the row and add it to their pile. The player with the highest-valued pile at the end wins. (The value of a player’s pile at the end is simply the sum of the value of the items in the pile.) We will use dynamic programming to help us analyze this game.

1. For a given vector v of values, let F(i,j) denote the maximum value that the first player can definitely achieve using the values v[i],...,v[j]—this means a tight, exact upper bound, an amount which the first player cannot beat, but that the first player can in fact achieve if he/she plays optimally. Write down a recurrence relation for F(i,j) and prove that it is correct. Be sure to include a base case! You can assume that this problem has optimal substructure.

2. Based on your recurrence from part (a), describe a dynamic programming table and the order in which it should be filled in.

3. Write pseudo-code for the dynamic programming solution you described in part (b) and analyze its run-time.

Answer 1