Name: ____________________________ Login: _______ (CS61B Spring 2005 Midterm 2, Shewchuk) 2
Problem 1. (8 points) Quickies.
a. (1 point) The worst-case time to find the second-smallest key in a 2-3-4 tree with n keys (and no
duplicate keys) is Θ (______________________), if the operation that finds it is implemented in an
intelligent way.
b. (1 point) Suppose Algorithm A runs in Ο (n log n) worst-case time, and Algorithm B runs in Ω(n2)
worst-case time. Is it always true that, given the same input, Algorithm A takes less time than Algorithm
B? If “yes,” explain why. If “no,” give a counterexample.
c. (1 point) Suppose the variable e references an Exception. Suppose the method that declared this
variable executes the line System.out.println(e);. Is this an acceptable thing to do in Java? If
so, what does Java do when it executes the line? (We aren't asking what string gets printed; instead, tell
us what Java does internally to print an exception.) If not, explain why it's not acceptable, and describe
the error we get.
d. (1 point) Draw a four-node tree, with numerical keys (one in each node), that could be either a valid
binary heap or a valid binary search tree.
e. (2 points) Draw the binary search tree that results if you start with an empty binary search tree and
insert the following keys in the following order: 0, 8, 2, 5, 3, 7, 1; then remove 8.
f. (2 points) Your friend tells you that his binary search tree implementation rebalances the tree after every
insert and delete operation, so that it always has the minimum possible height. This seems like a good
idea, because the height of the search tree never exceeds log2 n. But it's really a terrible idea, because
(unlike with a 2-3-4 tree) it cannot be done quickly. Give an argument why your friend's
implementation must sometimes take Ω(n) time per operation. (You get to pick the tree and the
sequence of insert/delete operations.)
g. (1 bonus point) What is the central tenet of Britney Theory?
Conversion to PDF by Arpad Kovacs using OpenOffice.org 2.0.4, 2007
