Is There a Problem With No Asymptotically Optimal Algorithm?

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SUMMARY

The discussion centers on the existence of computational problems for which no asymptotically optimal algorithm exists. Specifically, it questions whether, for any algorithm x solving a problem P, there can be another algorithm y with a worst-case runtime T(y) that is strictly faster than T(x) in terms of little-o notation. The conversation highlights the complexity of defining "faster than linear" and the implications of logarithmic runtimes, suggesting that while algorithms exist with runtimes of O(ln(x)), the quest for a universally faster algorithm remains ambiguous.

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mXSCNT
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I was just wondering. Is there a computational problem P for which there is no asymptotically least-time solution?

In other words: let x be an algorithm which is a solution to P. Denote by T(x), the worst-case runtime of x as a function of its input size. Is it possible that for any algorithm x that solves P, there exists an algorithm y that solves P, where T(y) = o(T(x))?

o is little-o notation, so in other words that would mean T(y) = O(T(x)) and T(y) is not Θ(T(x)).
 
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Well, there are algorithms that take time O(ln(x)). Does that count? You just want something that is faster than linear, or do you want something that isn't asymptotic to ANYthing? What are you after? O(ln(ln(ln ... forever ... (x)))))...)? That wouldn't even make any sense. Take the log enough times and eventually you get a negative number, and you take it again, and poof.
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edit:
holy cow, I didn't notice what date this was written. The thing just showed me a bunch of related posts and I was perusing them.
 

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