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..) 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

Operations Research : Applications and Algorithms
Operations Research : Applications and Algorithms
4th Edition
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Wayne L. Winston
Chapter11: Nonlinear Programming
Section: Chapter Questions
Problem 14RP
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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):
Transcribed Image Text: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?
Transcribed Image Text: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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