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We analysis mentioned algorithms with assumption $n\to \infty$ that $n$ is input size. So for a small input or constant number of input, measuring the running time of those algorithms are not correct. For example maybe for a given input with small size, after analysis running time you get an equal complexity but in general some of those algorithm have huge ...


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The running time is measured independently of a particular input, and it is always written in asymptotic form (big-O notation). You can't know the exact time an algorithm will run, since there are a lot of unpredictable factors that contribute to it.


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You have misunderstood the analysis of randomized quicksort. The idea is that we can view pivot selection in randomized quicksort in the following way. We first select a permutation $\pi$ on the elements on the array. When choosing a pivot among some subarray $B$, we choose the element of $B$ which appears first in $\pi$. Suppose that $i < j$. We are ...


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If we had a way to canonicalize graphs efficiently, then we could canonicalize the graph before feeding it to the ML. Unfortunately, no polynomial-time algorithm for graph canonization is known. One naive algorithm for graph canonization runs in $O(n!)$ time. I don't believe the claim is literally correct that graph canonization requires $O(n!)$ time, but ...


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Obviously your code will be lower bounded by $\Omega \left(n \choose k \right)$ since you can't skip any combinations, then you wouldn't be generating them all. With this in mind we can do a little better analysis on the bound. Let's take a look at the recursion tree for this. Let's use a small example like 4 choose 2. Here is how your algorithm would work: ...


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