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Apr 16, 2025
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Change text in Aho Corasick #1451

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2 changes: 1 addition & 1 deletion src/string/aho_corasick.md
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Expand Up @@ -209,7 +209,7 @@ Assume that at the moment we stand in a vertex $v$ and consider a character $c$.
1. $go[v][c] = -1$. In this case, we may assign $go[v][c] = go[u][c]$, which is already known by the induction hypothesis;
2. $go[v][c] = w \neq -1$. In this case, we may assign $link[w] = go[u][c]$.

In this way, we spend $O(1)$ time per each pair of a vertex and a character, making the running time $O(mk)$. The major overhead here is that we copy a lot of transitions from $u$ in the first case, while the transitions of the second case form the trie and sum up to $m$ over all vertices. To avoid the copying of $go[u][c]$, we may use a persistent array data structure, using which we initially copy $go[u]$ into $go[v]$ and then only update values for characters in which the transition would differ. This leads to the $O(m \log k)$ algorithm.
In this way, we spend $O(1)$ time per each pair of a vertex and a character, making the running time $O(mk)$. The major overhead here is that we copy a lot of transitions from $u$ in the first case, while the transitions of the second case form the trie and sum up to $m$ over all vertices. To avoid the copying of $go[u][c]$, we may use a persistent array data structure, using which we initially copy $go[u]$ into $go[v]$ and then only update values for characters in which the transition would differ. This leads to the $O(n \log k)$ algorithm.

## Applications

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