检查java中splay树操作的复杂性

Question

I have implemented splay tree (insert, search, delete operation) in Java. Now I want to check if the complexity of the algorithm is O(logn) or not. Is there any way to check this by varying the input values (number of nodes) and checking the run time in seconds? Say, by putting input values like 1000, 100000 and checking the run time or is there any other way?

Answer 1

Strictly speaking, you cannot find the time complexity of the algorithm by running it for some values of n . Let's assume that you've run it for values n_1, n_2, ..., n_k . If the algorithm makes n^2 operations for any n <= max(n_1, ..., n_k) and exactly 10^100 operations for any larger value of n , it has a constant time complexity, even though it would look like a quadratic one from the points you have.

However, you can assess the number of operations it takes to complete on an input of a size n (I wouldn't even call it time complexity here, as the latter has a strict formal definition) by running on some values of n and looking at ratios T(n1) / T(n2) and n1 / n2 . But even in case of a "real" algorithm (in sense that it is not a pathological case described in the first paragraph), you should be careful with the "structure" of the input (for example, a quick sort algorithm that takes the first element as pivot runs in O(n log n) on average for a random input, so it would look like an O(n log n) if you generate random arrays of different sizes. However, it runs in O(n^2) time for a reversed sorted array).

To sum it up, if you need to figure out if it's fast enough from a practical point of view and you have an idea how a typical input to your algorithm looks like, you can try generating inputs of different sizes and see how the execution time grows.

However, if you need a bound on the runtime in a mathematical sense, you need to prove some properties and bounds of your algorithm mathematically.

In your case, I would say that testing on random inputs can be a reasonable idea (because there is a mathematical proof that the time complexity of one operation is O(log n) for a splay tree), so you just need to check that the tree you have implemented is indeed a correct splay tree. One note: I'd recommend to try different patterns of queries (like inserting elements in sorted/reverse order and so on) as even unbalanced trees can work pretty fast as long as the input is "random".