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Author: trekhleb

src/algorithms/string/z-algorithm/README.md

@@ -1,18 +1,18 @@
-# Z-algorithm
+# Z Algorithm
 
 The Z-algorithm finds occurrences of a "word" `W` 
-within a main "text string" `T` in linear time.
+within a main "text string" `T` in linear time `O(|W| + |T|)`.
 
 Given a string `S` of length `n`, the algorithm produces 
-an array, `Z` where `Z[i]` represents the ongest substring 
+an array, `Z` where `Z[i]` represents the longest substring 
 starting from `S[i]` which is also a prefix of `S`. Finding
 `Z` for the string obtained by concatenating the word, `W` 
 with a nonce character, say `$` followed by the text, `T`,
 helps with pattern matching, for if there is some index `i`
 such that `Z[i]` equals the pattern length, then the pattern
 must be present at that point.
 
-While the `Z` array can be computed with two nested loops, the
+While the `Z` array can be computed with two nested loops in `O(|W| * |T|)` time, the
 following strategy shows how to obtain it in linear time, based 
 on the idea that as we iterate over the letters in the string 
 (index `i` from `1` to `n - 1`), we maintain an interval `[L, R]`
@@ -21,7 +21,42 @@ and `S[L...R]` is a prefix that is also a substring (if no such
 interval exists, just let `L = R =  - 1`). For `i = 1`, we can 
 simply compute `L` and `R` by comparing `S[0...]` to `S[1...]`.
 
+**Example of Z array**
+
+```
+Index            0   1   2   3   4   5   6   7   8   9  10  11 
+Text             a   a   b   c   a   a   b   x   a   a   a   z
+Z values         X   1   0   0   3   1   0   0   2   2   1   0 
+```
+
+Other examples
+
+```
+str =  a a a a a a
+Z[] =  x 5 4 3 2 1
+```
+
+```
+str =  a a b a a c d
+Z[] =  x 1 0 2 1 0 0
+```
+
+```
+str =  a b a b a b a b
+Z[] =  x 0 6 0 4 0 2 0
+```
+
+**Example of Z box**
+
+![z-box](https://ivanyu.me/wp-content/uploads/2014/09/zalg1.png)
+
 ## Complexity
 
 - **Time:** `O(|W| + |T|)`
-- **Space:** `O(|W|)`
+- **Space:** `O(|W|)`
+
+## References
+
+- [GeeksForGeeks](https://www.geeksforgeeks.org/z-algorithm-linear-time-pattern-searching-algorithm/)
+- [YouTube](https://www.youtube.com/watch?v=CpZh4eF8QBw&t=0s&list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8&index=70)
+- [Z Algorithm by Ivan Yurchenko](https://ivanyu.me/blog/2013/10/15/z-algorithm/)