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In the Handbook of Exact String Matching algorithms, it describes the transition set for a basic string matching automaton as:

for

  • q in Q (q is a prefix of x)
  • a in $\Sigma$
  • (q, a, qa) is in E if and only if qa is also a prefix of x
  • otherwise (q, a, p) is in E such that p is the longest suffix of qa which is a prefix of x.

I'm having difficulty understanding the last statement. Why would there be a state transition to a suffix?

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Assume that you build an automaton for finding occurrences of $bububa$ in the input string. When the automaton reads the input string $bubububa$, then it after reading $bububu$, it must not just return to state $\epsilon$, as otherwise an occurrence of $bububa$ is not detected. Also, jumping back to state $bu$ is would not do the trick, as then again, the $bububa$ starting in letter 3 of the word would be missed. Rather, after seeing a $u$ in state $bubub$, the automaton must transition to $bubu$ in order to no miss the possibility that only the first two letters are to be ignored.

The last line in the definition of $E$ describes how the back transitions in the automaton are found: in order not to miss any occurrence of $bububa$, the state in the automaton jumps back as little as possible.

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