Trie

A Trie maintains a set $S$ of strings in ${\Sigma }^{\ast }$ by maintaining a deterministic finite automaton that accepts and only accepts strings in $S$. Specifically, let $T$ be a set of all prefixes of strings in $S$, $\delta$ be a function in $(T\cup \{q_{\perp }\})\times \Sigma \to T\cup \{q_{\perp }\}$ satisfying

$$\delta (s,\sigma )=\{\begin{array}{ll}s\sigma ,& s\sigma \in T\\ q_{\perp },& s\sigma \notin T\end{array}$$

Then it is easy to prove that $M=(T\cup \{q_{\perp }\},\Sigma ,\delta ,\epsilon ,S)$ is a deterministic finite automaton that accepts and only accepts strings in $S$. This costs a space of $\mathcal{O}(|\Sigma |{\sum }_{s\in S}|s|)$.

Add

Add updates $S$ to $S\cup \{s\}$ in $\mathcal{O}(|s|)$ time and $\mathcal{O}(1)$ space.

Updating the values in $\delta$ that is changed by $s$ yields an algorithm that solves the problem in $\mathcal{O}(|s|)$ time and $\mathcal{O}(1)$ space.

int o = 1;
for (char c : s) {
	if (!next[o][c]) {
		next[o][c] = next.size();
		next.emplace_back();
		f.push_back(false);
	}
	o = next[o][c];
}
f[o] = true;

Find

Find checks if $s\in S$ in $\mathcal{O}(|s|)$ time and $\mathcal{O}(1)$ space.

Running $s$ on $M$ yields an algorithm that solves the problem in $\mathcal{O}(|s|)$ time and $\mathcal{O}(1)$ space.

int o = 1;
for (char c : s) {
	o = next[o][c];
}
return f[o];