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T(n) = b'T() + n²logn for some constants b, r > 0. (a) Use the Master theorem and come up with a polynomial upper bound for T(n). (b) What value of d did you use in the previous part? Can that be improved to give a tighter upper bound for T(n)? (c) What is the tightest upper bound that Master theorem allows you to provide for T(n)?

T(n) = b'T() + n²logn for some constants b, r > 0. (a) Use the Master theorem and come up with a polynomial upper bound for T(n). (b) What value of d did you use in the previous part? Can that be improved to give a tighter upper bound for T(n)? (c) What is the tightest upper bound that Master theorem allows you to provide for T(n)?

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3. Do not get intimidated by the terms on the recurrence. Write out the next two levels for the recurrence to figure out what level i looks like and then add up the general term across the range of i. (Refer to the lecture on Master theorem and the derivation to set up the summation and then execute the summation)

3. (Recurrence resolution) Consider the following recurrence : T(n) = b'T() + n² logn for some constants b, r > 0. (a) Use the Master theorem and come up with a polynomial upper bound for T(n). (b) What value of d did you use in the previous part? Can that be improved to give a tighter upper bound for T(n)? (c) What is the tightest upper bound that Master theorem allows you to provide for T(n)? (d) Provide a lower bound for T(n). (hint: this part should take you lesser time than sneezing) (e) Now solve the above recurrence precisely using a recurrence tree as follows: Draw the recurrence tree. What is the cost at level i? (assume that the top level is labelled level 1) • How many levels does this recurrence tree have? • What is the cost at level i? • Add up all the costs across all the levels to get a bound for T(n).
Transcribed Image Text:3. (Recurrence resolution) Consider the following recurrence : T(n) = b'T() + n² logn for some constants b, r > 0. (a) Use the Master theorem and come up with a polynomial upper bound for T(n). (b) What value of d did you use in the previous part? Can that be improved to give a tighter upper bound for T(n)? (c) What is the tightest upper bound that Master theorem allows you to provide for T(n)? (d) Provide a lower bound for T(n). (hint: this part should take you lesser time than sneezing) (e) Now solve the above recurrence precisely using a recurrence tree as follows: Draw the recurrence tree. What is the cost at level i? (assume that the top level is labelled level 1) • How many levels does this recurrence tree have? • What is the cost at level i? • Add up all the costs across all the levels to get a bound for T(n).
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