GATE | GATE-CS-2015 (Mock Test) | Question 10
Select the correct asymptotic complexity of an algorithm with runtime T(n, n) where
T(x, c) = Θ(x) for c <= 2, T(c, y) = Θ(y) for c <= 2, and T(x, y) = Θ(x+y) + T(x/2, y/2)
Explanation: The recurrence is
T(x, y) = Θ(x+y) + T(x/2, y/2)
It can be written as below.
T(x, y) = Θ(x+y) + Θ((x+y)/2) + Θ((x+y)/4) + Θ((x+y)/8) ..... T(x, y) = Θ((x+y) + (x+y)/2 + (x+y)/4 + (x+y)/8 ..... )
The above expression forms a geometric series with ratio as 2 and starting element as (x+y)/2
T(x, y) is upper bounded by Θ(x+y) as sum of infinite series is 2(x+y). It is lower bounded by (x+y)
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