big-theta

For the following code fragment, what is the order of growth in terms of N? int sum = 0; for (int i = 1; i <= N; i = i*2) for (int j = 1; j <= N; j = j*2) for (int k = 1; k <= i; k++) sum++; I have figured that there is lgN term, but I am stuck on evaluating this part : lgN(1 + 4 + 8 + 16 + ....). What will the last term of the sequence be? I need the last term to calculate the sum. You have a geometric progression in your outer loops, so there is a closed form for the sum of which you want to take the log: 1 + 2 + 4 + ... + 2^N = 2^(N+1) - 1 To be precise, your sum is 1 + ... + 2^(floor(ld(N)

I'm having trouble understanding how to make this into a formula. for (int i = 1; i <= N; i++) { for (int j = 1; j <= N; j += i) { I realize what happens, for every i++ you have 1 level of multiplication less of j. i = 1, you get j = 1, 2, 3, ..., 100 i = 2, you get j = 1, 3, 5, ..., 100 I'm not sure how to think this in terms of Big-theta. The total of j is N, N/2, N/3, N/4..., N/N (My conclusion) How would be best to try and think this as a function of N? So your question can be actually reduced to "What is the tight bound for the harmonic series 1/1 + 1/2 + 1/3 + ... + 1/N?" For which the

Started learning algorithms. I understand how to find theta-notation from a 'regular recurrence' like T(n) = Tf(n) + g(n) . But I am lost with this recurrence: problem 1-2e : T(n) = T(√n) + Θ(lg lg n) How do I choose the method to find theta? And what, uh, this recurrence is? I just do not quite understand notation-inside-a-recurrence thing. One trick that might useful would be to transform n into something else, like, say, 2 k . If we do this, you can rewrite the above as T(2 k ) = T(2 k/2 ) + Θ(log log 2 k ) = T(2 k ) = T(2 k/2 ) + Θ(log k) Now this looks like a recurrence that we might