I know that measuring asymptotic complexity can be based on any resources you have, whether it's time, memory usage, number of comparisons, etc. But when it comes to sorting something, I realize we normally associate the asymptotic notation with the basic operations like number of swaps/steps or number of comparisons.

More specifically, I find that all of those *Big-O*s in sorting algorithms are based on the number of comparisons, and not the number of swaps (for example, in selection sort). So we assume here that comparisons are more expensive than the actual steps it takes to sort itself? Why is comparison so popular?

In bubble sort, if we also include the number of "bubbling" you have in the worst case scenario, isn't the upper bound going to increase by another n^{2} approximately? That increase is a lot. So why do we ignore it here?

Correct me if I'm wrong. Thanks.

Answer:

The reason that comparisons are such an important factor, is because you could always store things by pointer, worst comes to worst, and make swaps the same for any types of keys. These simple system tricks will not work for comparisons. Comparing strings by lexicographic order is inherently more expensive than comparing integers.

As for your point regarding the *n ^{2}* of bubble-sort: it does not change the order of growth at all.

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