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Question:

I am trying to get the worst run-time complexity order on a couple of algorithms created. However I have run into a problem that I keep tending to select the wrong or wrong amount of fundamental operations for an algorithm.

To me it appears to be that the selection of the fundamental operation is more of an art than a science. After googling and reading my text boxes, I still have not found a good definition. So far I have defined it as "An operation that always occurs within an algorithms execution" such as a comparison or array manipulation.

But algorithms often have many comparisons that are always executed so which operation do you pick?

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Solution:1

I agree to some degree it's an art, so you should always clarify when writing documentation, etc.. But usually it's a "visit" to the underlying data structure. So like you said, for an array it's a comparison or a swap, for a hash map it may be a manual examination of a key, for a graph it's a visit to a vertex or edge, etc.

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Solution:2

Even practicing complexity theorists have disagreements about this sort of thing, so what follows may be a bit subjective: http://blog.computationalcomplexity.org/2009/05/shaving-logs-with-unit-cost.html

The purpose of big-O notation is to summarize the efficiency of an algorithm for the reader. In practical contexts, I am most concerned with how many clock cycles an algorithm takes, assuming that the big-O constant is neither extremely small or large (and ignoring the effects of the memory hierarchy); this is the "unit-cost" model alluded to in the linked post.

The reason to count comparisons for sorting algorithms is that the cost of a comparison depends on the type of the input data. You could say that a sorting algorithm takes O(c n log n) cycles where c is the expense of a comparison, but it's simpler in this case to count comparisons instead because the other work performed by the algorithm is O(n log n). There's a sorting algorithm that sorts the concatenation of n sorted arrays of length n in n^2 log n steps and n^2 comparisons; here, I would expect that the number of comparisons and the computational overhead be stated separately, because neither necessarily dominates the other.

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Solution:3

This only works when You have actually implemented the algorithm, but You could just use a profiler to see which operation is the bottleneck. That's a practical point of view. In theory, some assume that everything that is not the fundamental operation runs in zero time.

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Solution:4

The somewhat simple definition I have heard is:

The operation which is executed at least as many times as any other operation in the algorithm.

For example, in a sorting algorithm, these tend to be comparisons rather than assignments as you almost always have to visit and 'check' an element before you re-order it, but the check may not result in a re-ordering. So there will always be at-least as many comparisons as assignments.

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