## Description

1. Define a family H of hash functions mapping keys from a universe U to the set

{0, 1, . . . , n − 1} to be -universal if for all pairs of distinct keys k, l ∈ U,

P r(h(k) = h(l)) ≤ ,

where the probability is over the random choice of the hash function h from the family

H. Show that an -universal family of hash functions must have

>

1

n

−

1

|U|

.

2. In lecture, we show a few examples of universal family of hash functions which satisfy

that for all pairs of distinct keys k, l ∈ U,

P r(h(k) = h(l)) = 1

n

,

where n is the size of the hash table and the probability is over the random choice of

the hash function h from the family H. Does there exist any example with |U| > n so

that for all pairs of distinct keys k, l ∈ U,

P r(h(k) = h(l)) <
1
n
?
(Hint: consider this problem together with the claim in Problem 1.)
3. Suppose we want to design a hash table containing at most 600 elements using linear
probing. We require that an unsuccessful search needs no more than 8.5 compares and
a successful search needs no more than 3 compares on average. Please determine a
proper hash table size.
4. A full node in a binary tree is a node with two children. Prove that the number of full
nodes plus one is equal to the number of leaves in a nonempty binary tree.
1
5. For the following tree, show the order in which the nodes are visited during the following
tree traversals (The nodes in the tree are from A to I):
(a) Pre-order depth-first traversal.
(b) Post-order depth-first traversal.
(c) In-order depth-first traversal.
(d) Level-order traversal.
A
/ \
/ \
/ \
/ \
B H
/ \ /
/ \ /
C E I
\ / \
D F G
6. In class, we showed a recursive way to realize in-order depth-first traversal of a binary
tree. In this problem, we ask you to design a nonrecursive algorithm that performs
in-order depth-first traversal. Assume the tree is stored using a linked structure with
node as
struct node {
int key;
node *left, *right;
}
You can either describe your algorithm in plain English or write pseudo-code. If you
choose to write pseudo-code, you should write in a way that can be easily understood.
Otherwise, you will get a zero for the problem. (Hint: consider using a stack as an
auxiliary data structure.)
7. Min Heap
(a) Suppose that we are inserting the keys 3, 1, 4, 1, 5, 9, 2, 6, 5, 4 one by one into
an initially empty min heap. Show the resulting min heap in the form of a tree.
(b) For the min heap you obtained for Problem (7a), show the resulting heap after
calling two dequeueMin operations.
2
8. Min heap initialization
Given a sequence of keys 3, 9, 7, 2, 5, 2, 8, 6, 1, 4, show the resulting min heap if
we initialize it with the efficient algorithm we talked in lecture that takes O(n) time
complexity on an array of n elements. Show the intermediate steps in the form of a
tree. Do not just write the final result.
3