3 Divide and conquer algorithm
You are given a rooted tree T with n nodes. The tree may not be binary. You want to find the
depth of T. Here, we define the depth of a node v in the tree to be the maximum number of nodes
along a path from v to a leaf under v (going downwards). The depth of T is the depth of the root
of T. Now give a divide and conquer algorithm for finding the depth of a tree. You need to analyze
its running time.
4 Recurrence
Solve the following recurrence. Show the steps.
1. $T(n) = \begin{cases} 12T(n/4) + n^{1.5}, & \text{if } x \ge 4\\ 1, & \text{otherwise} \end{cases}$
2. $T(n) = \begin{cases} T(\sqrt{n}) + \log n, & \text{if } x \ge 4\\ 1, & \text{otherwise} \end{cases}$
3. $T(n) = T(n/2) + \Theta(1)$, which is the running time of the binary search.
