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What are some interesting open or solved-but-hard questions around problems having linear-time solutions? Ala riffle shuffles.

I'm especially curious about problems which people believe to be linear-time but for which either no linear-time algorithm is known or for which finding one proved difficult.

As a technical matter, I'm perhaps most interested in weak-linear-time on a single-tape nondeterministic machine, but other linear-time models are still interesting, ala deterministic or multi-tape models. I'm aware that strong-linear-time on a single tape nondeterministic machine is regular, hence linear-time on a single-tape deterministic machine is regular too.

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    $\begingroup$ Minimum Spanning Tree algorithm. From Wikipedia: ... Whether the problem can be solved deterministically in linear time by a comparison-based algorithm remains an open question .... See also this question for a couple of "hard to implement" linear time algorithms. $\endgroup$ – Marzio De Biasi Mar 22 '13 at 15:01
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    $\begingroup$ What is "weak" (or, for that matter, "strong") linear time? $\endgroup$ – Joshua Grochow Mar 22 '13 at 15:51
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    $\begingroup$ @MarzioDeBiasi: Your comment should be an answer! $\endgroup$ – Jeffε Mar 22 '13 at 17:57
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    $\begingroup$ So, are you interested in algorithms (described in a random-access model of computation) or complexity results (framed in terms of Turing machines)? $\endgroup$ – Jeffε Mar 22 '13 at 18:44
  • $\begingroup$ Yeah, Marzio's comment answer is plenty to keep me chugging along here, thank you! $\endgroup$ – Jeff Burdges Mar 23 '13 at 18:02

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