By Dr Antonio Gulli

This publication provides a set of Dynamic programming difficulties, their resolution, and the C++ code relating to them.

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**Sample text**

Boolean Parenthesization – given a Boolean expression of and, or, xor, true, false, find the number of ways to parenthesize and evaluate to true 32. Games of Coins – a set of coins are aligned and two players can pick one coin from each side in turn. Maximize the value of the coins picked by the first player 33. Maximum value contiguous subsequence – given an array of real numbers, find a contiguous subsequence with max sum 34. Balanced partition – given an array of integers between 0 and, divide the integers into two sets such that the difference of their sums is minimized 35.

29. Given a Binary Search Tree, find the size of the largest independent set of nodes Solution A set of nodes in a tree is independent if there are no edges directly connecting them in a father-children relation. A dynamic programming solution is based on the consideration that a node can be either part of the largest independent set or not. In the former case the size of the largest set will be the sum of the largest set of grandchildren plus 1 because we skip the grandchildren’s father. In the latter case the size of the largest set is the sum of the largest set of children because by construction we do not add any new node to the set.

For each position we define: and Code Left as exercise Complexity Complexity is. 20. Box Stacking – Given a set of 3d boxes, compute the largest stack of boxes. A box can be stacked only on top of another box with larger base. Solution This problem is a variation of the longest increasing sequence. Each box has three dimensions which can be rotated to form the base: either (), or, or. If we suppose having infinite amount of each box types, then we can solve this problem by computing the LIS for the remaining dimensions of the box.