Theorem 1 (Master Theorem).Let a 1 and b> 1 be constants, let f(n) be a function f: N→ R+, and let T(n) be defined onthe nonnegative integers by the recurrenceT(n) = aT(n/b) + f(n).Then T(n) has the following asymptotic bounds:1. if f(n) = O(nloga-) for some constant e > 0, then T(n) = (nossa),2. if f(n) € (noga) then T(n) = O(noga log n), and3. if f(n) = N(nloa+E) for some e > 0 and if af (n/b) ≤ df (n) for some constant d < 1 andall sufficiently large n, then T(n) = (f(n)).Use the Master Theorem above to solve the following recurrences when possible (notethat not every blank needs to be filled in in every problem): 1.) T(n) = T(n/3) + cf(n) =Which is growing faster: f(n) or nWhich case of the Master Theorem does that potentially put us in?log, a =versus n'nlog, a?If you're potentially in case one or three, is it possible to find an epsilon which makeseither f(n) = O(noga-) (if you're in case one) or f(n) = N(nloa+e) (if you're incase three) true? Choose one or show an inequality.If you're potentially in case three and there is an e, try to find a constant d < 1 suchthat af (n/b) ≤ df (n) for large enough n's.What can you conclude?

T(n)=T(n/3)+c f(n) = c, nlogba = n0=c T(n) = θ(logn) Explanation:Please refer to solution…

..) T(n) = T(n/3) + c f(n) = versus n log, a = Which is growing faster: f(n) or nos, a? Which case of the Master Theorem does that potentially put us in? If you're potentially in case one or three is it possible to find an ensilon which n

..) T(n) = T(n/3) + c f(n) = versus n log, a = Which is growing faster: f(n) or nos, a? Which case of the Master Theorem does that potentially put us in? If you're potentially in case one or three is it possible to find an ensilon which n

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

Theorem 1 (Master Theorem).
Let a 1 and b> 1 be constants, let f(n) be a function f: N→ R+, and let T(n) be defined on
the nonnegative integers by the recurrence
T(n) = aT(n/b) + f(n).
Then T(n) has the following asymptotic bounds:
1. if f(n) = O(nloga-) for some constant e > 0, then T(n) = (nossa),
2. if f(n) € (noga) then T(n) = O(noga log n), and
3. if f(n) = N(nloa+E) for some e > 0 and if af (n/b) ≤ df (n) for some constant d < 1 and
all sufficiently large n, then T(n) = (f(n)).
Use the Master Theorem above to solve the following recurrences when possible (note
that not every blank needs to be filled in in every problem):

1.) T(n) = T(n/3) + c
f(n) =
Which is growing faster: f(n) or n
Which case of the Master Theorem does that potentially put us in?
log, a =
versus n'
nlog, a?
If you're potentially in case one or three, is it possible to find an epsilon which makes
either f(n) = O(noga-) (if you're in case one) or f(n) = N(nloa+e) (if you're in
case three) true? Choose one or show an inequality.
If you're potentially in case three and there is an e, try to find a constant d < 1 such
that af (n/b) ≤ df (n) for large enough n's.
What can you conclude?

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