Thompson construction
Convert regex to NFA compositionally. Union, concatenation, Kleene star each produce standard NFA fragment.
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NFA simulation
Track current set of NFA states. On each input char, compute next set. Linear time via bitset / hash set.
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Backtracking
Try each option; recurse on failure. Simple, supports backreferences. Exponential worst case (ReDoS attacks).
Hybrid: RE2
Google's RE2: Thompson-style NFA/DFA. No backreferences. Linear guaranteed. Handles adversarial input safely.