Sorting Algorithm that minimizes the maximum number of comparisons in which individual items are involved

Question

I'm interested in finding a comparison sorting algorithm that minimizes the number of times each single element is compared with others during the run of the algorithm.

For a randomly sorted list, I'm interested in two distributions: the number of comparisons that are needed to sort a list (this is the traditional criterion) and the number of comparisons in which each single element of the list is involved.

Among the algorithms that have a good performance in terms of the number of comparisons, say achieving O(n log(n)) on average, I would like to find out the one for which, on average, the number of times a single element is compared with others is minimized.

I supposed that the theoretical minimum is O(log(n)) which is obtained by dividing the above figure on the total number of comparisons by n.

I'm also interested in the case where data are likely to be already ordered to some extent.

Is perhaps a simulation the best way to go about finding an answer?

(My previous question has been put on hold - This is now a very clear question, if you can't understand it then please explain why)

Answer 1

Yes you definitely should do simulations.
There you will implicitely set the size and pre-ordering constraints in a way that may allow more specific statements than the general question you rose.

There can , however, not be a clear answer to such question in general.

Big-O deals with asymptotic behaviour while your question seem to target smaller problem sizes. So Big-O could hint on the best candidates for sufficiently large input sets to a sort run. (But, eg if you are interested in size<=5 the results may be completely different!) For getting proper estimate on comparison operations you would need to analyze each individual algorithm.

At the end, the result (for a given algorithm) will necesarily be specific to the dataset being sorted.

Also, on avarage is not well defined in your context. I'd assume you intend to refer to the number of comparisions on the participating objects for a given sort and not avarage over a (sufficiently large) set of sort runs.

Even within a single algorithm the distribution of comparisions an individual object is taking place in may show a large standard deviation in one case and be (nearly) equally distributed in another case.

As complexity of a sorting algorithm is determined by the total number of comparisons (and position changes thereof), I do not assume there will be much from therotical analysis contributing to an answer.

Maybe you can add some background on what would make an answer to your question "interesting" in a practical sense?