CS61B, Spring 1995
Midterm #2
Professor P. N. Hilfinger
Problem #1
Tell whether each of the following is true or false. In each case, give a brief explanation where possible; for
false entries, this should take the form of a counter-example. Assume that comparisons always take constant
time.
a. If all items in a heap are distinct, the second-smallest item in a heap has no children (assuming the heap is
ordered with the largest item at the top).
b. Suppose that an ordinary binary tree (not a search tree) obeys the heap property - that the value of any node
is at least as large as that of any of its children. Then inserting a new item into this tree while maintaining the
heap property requires time O(lg n), where n is the number of nodes in the tree.
c. Insertion sort always rquires Θ(n2) time to sort n elements.
d. Quicksort always requires Ω(nlg n) time to sort n elements.
e. Suppose that RB is a red-black tree and that M comparisons are required to discover that a key, x, is not
present in RB. Then if y also is not present in RB, it will require at least M/2 comparisons to discover this.
Problem #2
Assume that h(k, n) is a hash function that takes a key k and integer n and produces a number in the range
[0...n-1]. Assume that K1, K2, ... is an infinite sequence of keys, and that h distributes these keys "evenly."
More precisely, assume that for any M > 0, the number of keys k ∈ {K1, ... KM} such that h(k, n) = p is M/n ±
1, for each p ∈ [0...n - 1]. Answer the following, giving brief explanations for each answer.
a. Suppose that you are to input a value M and then insert K1, ..., KM into a hash table. How can you arrange
that each insertion runs in worst-case time O(1)?
b. Suppose that M is not explicitly input, and that instead you read keys K1, K2, ..., inserting them into a hash
table as you go, until you encounter some marker indicating the end of the data. Assume that you try to
manage the hash table so as to minimize the time required to do all the insertions. What is the (asymptotic)
worst-case time for inserting Km (that is, the mth item you insert) as a function of m? Why?
c. Under the same assumptions as (b), what is the total worst-case asymptotic time for doing all M insertions?
Why?
Problem #3
A threaded tree is essentially a tree whose nodes are additionally linked together into a list such that
following the list gives an inorder traversal of the tree. Consider the following representation.
class TTree {
public:
CS61B, Midterm #2, Spring 1995
CS61B, Spring 1995 Midterm #2 Professor P. N. Hilfinger 1
