I was going through this tutorial on the analysis of the Selection sort algorithm (personal.denison.edu/~kretchmar/272/SelectionSortAnalysis.pdf). I have been spending quite some time on understanding algorithmic analysis and have not been completely successful.
If you look at the PDF, there are certain ‘times’ associated with c3,c4 an c5. I have no idea why the author added the summation symbol, why he chose the top and bottom indexes and why he chose the ‘(i+1)’ after the first summation. I understand that the summation notation is a compact way to express the sum of a set of numbers… but I can’t seem to complete the puzzle.
Thanks
1) He’s summing the values from the outer loop (line 1). That’s why he uses i (the index variable from the outer loop).
2) It’s traditional to write Σ expressions with the start value less than the end value. However, in this case the number of times the inner loop will execute for some i is n – i. When i is 1 (the start of the loop), the inner loop will execute n – 1 times, and when i gets up to n – 1 at the end of the outer loop, the inner loop will execute 1 time (n – (n – 1)). So you might want to write that as sum(i = n-1 -> 1), but as I said the tradition is to write it from smallest to largest.
3) The loop executes i times, but the loop test executes i + 1 times: i times successfully and once unsuccessfully, terminating the loop. Hence the value summed inside the loop body is i, but the value summed for the loop termination test itself (i.e.. the for statement) is i + 1.