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CSE 2320 Section 060/061 Spring 1999

Homework 4

Due: April 8, 1999, in class (April 9, 1999, 5:00pm for -10%)

1.
Show the Red-Black trees that result after each successive insertion of the keys 23, 85, 26, 24, 6, 30, 51, 94, 69, 73 (in this order) into an initially empty Red-Black tree.

2.
Show the Red-Black trees that result after each successive deletion of the keys 10, 8, 7, 4, 2, 3, 11, 12, 5, 1, 9, 6 (in this order) from the following Red-Black tree.


\begin{figure}
\centerline{\psfig{figure=figures/h42.ps}}
\end{figure}

3.
Show the B-trees that result after each successive insertion of the 22 keys A - V (in alphabetical order) into an initially empty B-tree with minimum degree t = 3.

4.
Show the B-trees that result after each successive deletion of the keys Q, B, P, G, L, J, N, O, F, H, Y, K, I, R, E (in this order) from the following B-tree with minimum degree t = 2. When you have a choice between a left or right sibling, always choose the left sibling. Always try Case 2a (predecessor) before Case 2b (successor).


\begin{figure}
\centerline{\psfig{figure=figures/h44.ps}}
\end{figure}

5.
Consider the problem of constructing an optimal binary search tree (in terms of search time) for a set of integer keys, given the frequencies with which each key will be accessed. Assume there are n keys with values 1 to n, and key i has frequency fi such that $0 \leq f_i \leq
1$ and $\sum_{i=1}^{n} f_i = 1$. The goal is to minimize the value of $\sum_{i=1}^{n} f_i d_i $ where di is the depth of key i in the binary search tree.

(a)
Let c(i,j) be the optimal cost of a binary search tree containing the keys $i \ldots j$. Assuming the optimal choice for the root of this tree is key k, give an expression for c(i,j) in terms of c(i,k-1) and c(k+1,j).

(b)
Show that this problem exhibits optimal substructure.

(c)
Define a recursive solution for computing c(i,j) and write pseudocode for a divide-and-conquer algorithm implementing your solution.

(d)
Give a recurrence T(n) for the running time of your recursive solution in part c, where n=j-i+1, and show that $T(n) = \Omega(2^n)$.

(e)
Show that the recursive solution has overlapping subproblems and compute the number of unique subproblems in terms of n.

(f)
Write pseudocode for a O(n3) bottom-up, dynamic programming solution to the optimal binary search tree problem. Justify the O(n3)running time of your solution. Note that your solution must keep track of the root keys for each subtree and must include a procedure for actually constructing the tree.


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