“Partial match” table (aka “failure function”) in KMP (on wikipedia)

问题

I'm reading the KMP algorithm on wikipedia. There is one line of code in the "Description of pseudocode for the table-building algorithm" section that confuses me: let cnd ← T[cnd]

It has a comment: (second case: it doesn't, but we can fall back), I know we can fall back, but why T[cnd], is there a reason? Because it really confuses me.

Here is the complete pseudocode fot the table-building algorithm:

algorithm kmp_table:
    input:
        an array of characters, W (the word to be analyzed)
        an array of integers, T (the table to be filled)
    output:
        nothing (but during operation, it populates the table)

    define variables:
        an integer, pos ← 2 (the current position we are computing in T)
        an integer, cnd ← 0 (the zero-based index in W of the next 
character of the current candidate substring)

    (the first few values are fixed but different from what the algorithm 
might suggest)
    let T[0] ← -1, T[1] ← 0

    while pos < length(W) do
        (first case: the substring continues)
        if W[pos - 1] = W[cnd] then
            let cnd ← cnd + 1, T[pos] ← cnd, pos ← pos + 1

        (second case: it doesn't, but we can fall back)
        else if cnd > 0 then
            let cnd ← T[cnd]

        (third case: we have run out of candidates.  Note cnd = 0)
        else
            let T[pos] ← 0, pos ← pos + 1

回答1:

You can fall back to T[cnd] because it contains the length of the previous longest proper prefix of the pattern W which is also the proper suffix of W[0...cnd]. So if the current character at W[pos-1] matches the character at W[T[cnd]], you may extend the length of longest proper prefix of W[0...pos-1] (which is the first case).

I guess it's kind of like dynamic programming where you rely on previously computed values.

This explanation might help you.

来源：https://stackoverflow.com/questions/18911735/partial-match-table-aka-failure-function-in-kmp-on-wikipedia