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algorithmtimeanalysisrecurrence

Algorithm analysis recurrence comparision

Let algorithms X: T(n) = 7T(n/2)+n^2, and Y: T’(n) = aT’(n/4)+n^2

I need to find the largest value for a, such that algorithm Y is asymptotically faster than algorithm X.

Solution

  • Using masters theorem :-

        X : T(n) = O(n^log2(7))

So to get an asymptotically faster algorithm using masters theorem   

    2 <= log4(a) < log2(7)

 Finding max value such that the condition is true :-

    log4(a) < log2(7)     
    log2(a)/log2(4) < log2(7)   
    log2(a)/2 < log2(7)
    log2(a) < 2*log2(7)
    a < 7^2   =  a < 49

 As a is no of subproblems it needs to be integer therefore :-

 a = 48 is max value that a can take so that Y is asymptotically faster than X