Question

I’m working out the recurrence relation

T(n) = T(3/4 * n) + O(1)

It’s coming out to be O(log(n)), but I was told before hand that the solution is O(n). I can’t find where I’m going wrong – this looks just like the recurrence relation for binary search. Any suggestions?

Answer 1

Try substituting T(n) = c*n or T(n) = c * log n into the equation and solving. One of the two equations will be solvable.

You can also check your answer by evaluating the function for different values of n.

-- Define T in your preferred language
t n | n <= 1 = 1 | otherwise = t (3/4 * n) + 1

-- If it's O(log n), then T(n)/log(n) should be asymptotically constant
-- If it's O(n), then T(n)/n should be asymptotically constant
check1 n = t n / log n
check2 n = t n / n

print [check1 1e10, check1 1e11, check1 1e12, check1 1e13]
print [check2 1e10, check2 1e11, check2 1e12, check2 1e13]

One of these will converge to a small positive number, the other will go to zero or infinity.