HW1 solns.pdf - Homework 1: Solutions ORIE 4520:...

ORIE 4520: Stochastics at Scale
Fall 2015
Homework 1: Solutions
Sid Banerjee ( [email protected] )
Problem 1: (Practice with Asymptotic Notation)
An essential requirement for understanding scaling behavior is comfort with asymptotic (or ‘big-O’)
notation. In this problem, you will prove some basic facts about such asymptotics.
Part (a)
Given any two functions f ( · ) and g ( · ), show that f ( n ) + g ( n ) = Θ(max { f ( n ) , g ( n ) } ).
Part (b)
An algorithm ALG consists of two tunable sub-algorithms ALG
A
and ALG
B
, which have to be
executed serially (i.e., one run of ALG involves first executing ALG
A
followed by ALG
B
). Moreover,
given any function f ( n ), we can tune the two algorithms such that one run of ALG
A
takes time
O ( f ( n )) and ALG
B
takes time O ( n/f ( n )). How should we choose f to minimize the overall runtime
of ALG (i.e., to ensure the runtime of ALG is O ( h ( n )) for the smallest-growing function h )?
How would your answer change if ALG
A
and ALG
B
could be executed in parallel, and we have
to wait for both to finish?
Part (c)
We are given a recursive algorithm which, given an input of size n , splits it into 2 problems of size
n/ 2, solves each recursively, and then combines the two parts in time O ( n ). Thus, if T ( n ) denotes
the runtime for the algorithm on an input of size n , then we have:
T ( n ) = 2 T ( n/ 2) + O ( n )
Prove that T ( n ) = O ( n log n ).
1

ORIE 4520: Stochastics at Scale
Fall 2015
Homework 1: Solutions
Sid Banerjee ( [email protected] )
Hint: Note that for a constant size input, the algorithm takes O (1) time. How many recursions
does it require to reduce a problem of size n to constant size subproblems? What is the total runtime
overhead at each recursive level?