Tutorial questions
Algorithms
2019 Semester 1Tutorial 2
The University of Sydney
School of Computer Science
Problem 1
a) Show that log(n!) is O(nlog n).
b) (Harder) Show that log(n!) is Ω(nlog n).
Solution:
a) n! = n.(n−1). . . . .2.1≤n.n. . . . .n =nn. Taking logs of both sides gives the required
answer.
b) n!≥n
2. . . . . n
2(n
2times) = n
2n/2. Taking logs of both sides gives log(n!) ≤n
2log n
2=
n
2(log(n)−log(2)) ∈Ω(nlog n).
Problem 2
Which operations are standard for the following data structures? How much time does each operation
require?
1. Queue
2. Stack
3. Binary search tree
4. Balanced binary search tree (AVL)
Solution:
1: enqueue, dequeue, first, size O(1)
2: push, pop, top, size O(1)
3: insert, delete, find O(n)
4: insert, delete, find O(log n)
Problem 3
(a) Draw the queue Qand the output when the following sequence of operations are performed:
enqueue(5)
enqueue(2)
first()
enqueue(3)
dequeue()
first()
size()
(b) Draw the stack Sand the output when the following sequence of operations are performed:
push(5)
push(2)
top()
push(3)
1
