A symbolic runtime analysis of the algorithm

1. Design a dynamic programming algorithm to calculate the average height of a binary search tree.
   1. The algorithm must solve the following problem:

                                    Input: a binary search tree T, with subtrees left[T] and right[T].

                                    Output: the average height (number of nodes on a simple path from root

   to leaf).

1. Provide an explanation of how your algorithm works
2. Formal pseudocode of the algorithm
3. A proof that the algorithm is correct
4. A symbolic runtime analysis of the algorithm

2. Extend the solution you developed for problem 1 to find the length of the longest path in any directed, acyclic graphs (DAGs).
   1. The algorithm must solve the following problem:

                                    Input: A Dag, G(V, E).

                                    Output: the length of the longest path in G.

2. Provide an explanation of how your algorithm works
3. Formal pseudocode of the algorithm
4. A proof that the algorithm is correct
5. A symbolic runtime analysis of the algorithm

3. Design an algorithm that solves the following decision problem:
   1. The algorithm must solve the following problem:

                                    Input: A Graph, G(V, E) and a vertex v ∈ V.

Output: YES if for every pair of vertices x and u, a path from x to u must pass through v; NO otherwise.

3. Provide an explanation of how your algorithm works
4. Formal pseudocode of the algorithm
5. A proof that the algorithm is correct
6. A symbolic runtime analysis of the algorithm