10版 - 龙江大地锻造冰雪经济新引擎

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Each point \(p \in \mathbb{H}^n\) has tangent vectors \(\frac{\partial}{\partial x^i} in T_p M\) (which we write as the partial derivatives) at \(p\) given local coordinates (i.e. a basis \(\text{span}\{x^1,\dots,x^n\} = T_p M\)). The collection \(\bigl\{\frac{\partial}{\partial x^1}\big|_p,\dots,\frac{\partial}{\partial x^n}\big|_p\bigr\}\) forms a basis of \(T_pM\).

both of these approaches use NFAs under the hood, which means O(m * n) matching. our approach is fundamentally different: we encode lookaround information directly in the automaton via derivatives, which gives us O(n) matching with a small constant. the trade-off is that we restrict lookarounds to a normalized form (?<=R1)R2(?=R3) where R1/R2/R3 themselves don’t contain lookarounds. the oracle-based approaches support more general nesting, but pay for it in the matching loop. one open question i have is how they handle memory for the oracle table - if you read a gigabyte of text, do you keep a gigabyte-sized table in memory for each lookaround in the pattern?

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