Lazy DFAs (2010) are a clever optimization to mitigate the O(2^m) blowup of DFA construction, by only constructing the states that you actually visit. lazy DFAs reduce the theoretical automata construction time to either O(2^m) or O(n), whichever is lower. you could argue that it’s theoretically no longer linear time, since you could have a regex that creates a new state for every character in the input, but in practice you will keep revisiting the same states. for all intents and purposes it behaves more like O(n) with some initial wind-up time. the main downside of lazy DFAs is that they are more complex to implement, and you have to ship a compiler as part of your regex algorithm. i want to highlight Rust regex and RE2 as excellent implementations of this approach, which you can also see in the benchmarks.
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Nothing in the argument required the arc to be a semicircle. If the arc has length xxx times the circumference (where x≤1/2x \leq 1/2x≤1/2), each anchor's event has probability xN−1x^{N-1}xN−1 instead of (1/2)N−1(1/2)^{N-1}(1/2)N−1. Mutual exclusivity still holds because the gap is at least 1−x≥1/21 - x \geq 1/21−x≥1/2 of the circumference, too large for any other anchor's arc to bridge: