Question

Got a question on this step count problem. I think I have the basic idea but struggling to get the math straight on paper. First the pseudo code:

for i = 1 to n
    x = x + 1
    j = n
    while (j>1)
        j = j/2

and my interpretation:

n/2 + 1

n/2

n/2

n/2 <– top part of summation for line 4 (the while loop)

Σ logn+2

i=1 <– bottom part of summation for line 4 (the while loop)

n/2 <– top part of summation for line 5 (body of the while loop)

Σ logn+1

i=1 <– bottom part of summation for line 5 (body of the while loop)

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So the first three lines in step count aren’t bad to see, but then the two summations inside the loop are a bit tricky to me. So for the first one logn + 2 I can split into two steps:

n/2

Σ 2

i=1

and

n/2

Σ logn

i=1

so I’m used to dealing with straight summation of n, not n/2. But I’ll give it a quick shot.

2*((n^2+n)/2) or rather just drop the fraction for this one so it’s straight: n^2+n.

Then for the logn piece:

logn*((n^2+n)/2) = ? Now I’m not really sure here. Not sure this even simplifies down at all. If anyone has any suggestions on this I’d appreciate it, but I think what I got there is it for this half. So I combine the two together to get my final answer for the first summation line 4 of the pseudo code:

logn * (n^2+n)/2 + n^2 + n

And for the second summation I think it would then be:

logn * n^2 + n

(can borrow a bit from what I had already completed above. Just dropped the part where I had 2 * (n^2+n)/2 and instead I’ll simply have two sets of the half which combine to make one n^2+n if that makes any sense I think?

Please let me know if I’ve totally lost it. Thank you!!

Answer 1

I think you’re analysis is a bit off. First, notice that what happens in the body of the for loop is independent of the value of i or x. So the for loop executes n times and the same number of steps happen in the for loop body each time. The body of the while loop executes floor(log₂(n)) times. That’s about all the analysis that’s needed. The result is n*floor(log₂(n)). (If by “step” you mean statement executions, you need to modify this a bit to account for the number of statements in each loop body.)