Learning algorithm: suffix array height of the strike

【definition】

[LCP] the full name of the longest common prefix, longest prefix between the two suffixes, we define the following

Meaning lcp (i, j) is the longest prefixes and suffixes i and j

 

Function [z] function z [i] represents the longest prefix and suffix i of the string

 

 

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 【Solve the problem】

Both were in solving this problem

I.e. seeking longest common prefix and suffix strings between suffixes and

 

But something is different,

Suffix array to finally determine that the longest common prefix lexicographic suffixes i-th and i + 1-th suffix

z functions to finally determine that the longest common prefix and suffix i of the string

 

Then find some other value by the longest common prefix

 

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Learning algorithm []

Suffix array []

  Suffix array can be obtained in the n-time lexicographically i-th and the i - 1 longest common prefix suffix

  And this function is usually named height 

  Meaning height [i] is, for lexicographically the i - longest common prefix and a suffix of the suffix of the i-th

  Is obtained has the following properties of: (s [i] denotes the i-th suffix)

  1. If i is smaller than j, LCP (i, j) = min {LCP (k - 1, k), i + 1 <= k <= j}

  LCP can take advantage of this demand with RMQ

  2. Define H [i] as follows: i-th suffix starting in front of him and suffix LCP lexicographically

  即： h [ i ] = height [ rank [ i ] ]

  Then there are, for i> 1 and rank [i]> 1  

   h [ i ]  >  h [ i - 1 ] - 1 ;

   

  Certify as follows:

  And j is the first i - 1 suffix beginning to embark on the front of the suffix ranking position

  Note: j is not the first i - 2

  In this case, the suffix j-th and the i - 1 th in the LCP suffix defined height [rank [i - 1]], i.e. h [i - 1]

  That is part of the right half of us to prove

  Then we discuss j + 1 and i (obtained by the i - 1 + 1) of the relationship:

  First, when j and i - 1 where letters are not equal, h [i - 1] is 0

  Clearly then h [i]> h [i - 1] - 1

  Second, when j and i - 1 is equal to the first letter of the case,  

  Clearly then, j, and i - LCP 1 is h [i - 1] - 1

  In the suffix ranking than i exam, and the suffix i LCP's longest, highest similarity is clearly in SA recently that away from him

  Immediate SA [rank [i - 1]] - 1 

  That is, h [i]> = h [i - 1] - 1

  QED

  

  So every time we find the longest prefix when are available from h [i - 1] begin retrieving

  Is analogous manacher

  code show as below:

void GetHeight() {
    int j, k = 0;
    for(int i = 1; i <= N; i++) {
        if(k) k--;
        int j = sa[rak[i] - 1];
        while(s[i + k] == s[j + k]) k++;
        Height[rak[i]] = k;
        printf("%d\n", k);
    }
}

 

 

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topic:

 

[SDOI 2008] Sandy card