mwbockma notes.pdf - CIS 477 9/11 Recurrence Relations the Master Method Notes by Maxwell Bockmann Review of recurrence relations ▪ Recurrence

mwbockma notes.pdf - CIS 477 9/11 Recurrence Relations the...

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CIS 477 9/11: Recurrence Relations & the Master Method Notes by Maxwell Bockmann Review of recurrence relations : Recurrence relations are equations that recursively define a function in terms of earlier values. The simplest method to solve one is to guess the solution, then prove it by induction. In today’s lecture, we learned about solving recurrences using the Master Method . The formula for recurrences is as follows: T(n) = aT * (n/b) + O(n d ) This is defined where a > 0, b > 1, d 0, and a, b, d are all constants. T(n) is the runtime of the algorithm. “a” is the number of recursive calls. “b” is the size of the recursive calls. “d” is the “other work” of the algorithm. There are three cases for this method: Case 1: If the constant d > log b a, then T(N) = O(n d ) Case 2: If the constant d = log b a, then T(N) = O(n d log n) Case 3: If the constant d < log b a, then T(N) = O(n log b a ) For examples on how to bound a recurrence, see examples/practice section. Proof sketch of the Master Method :
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We take several aspects of this tree generated by a recurrence to build the master method:
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  • Fall '15
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