Triple nested loop where outer and inner loop are dependent on n

Question

I am trying to count the exact/total number of iterations these nested for-loops are executed:

procedure P (integer n);
 for (i: 1 to n)
  for (j: 1 to 10)
   for (k: n to n+5)
    x := x + 1;

I know that the first for loop runs (n-1) times, the second loop runs 10-1 (9) times, thus making the time complexity so far 9(n-1). I know that the the third loop will run n+5-n (5) times, but I am not completely sure because of the n involved. I believe with this calculation added, the total runtime will be 5(9(n-1)) which will translate to O(n). In general, I am wondering if this approach of multiplying the iterations of each loop is a good approach to solving worst case running time and big-Theta estimates. Thanks in advance.

score 0 · Accepted Answer · edited Apr 13 '17 at 12:48

In general, I am wondering if this approach of multiplying the iterations of each loop is a good approach to solving worst case running time and big-Theta estimates.

Basically, yes. But you don't really multiply: As explained here at length, you have nested sums. In your case,

$\qquad\displaystyle \sum_{i=1}^n \sum_{j=1}^{10} \sum_{k=n}^{n+5} 1$

(if we count the number of assignment operations) is easy to evaluate because the loops do not depend on each other. It looks like multiplication here, but if you get things like

  for i = 1 to n
      for j = 1 to i
          foo

that does not work anymore.

Note that worst-case considerations and asymptotics do not enter the picture here.

Triple nested loop where outer and inner loop are dependent on n

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