Add Z-function algorithm implementation (#2067)

* Add Z-function algorithm implementation

* Spelling correction

* Reference url correction

* Add additional function as an example of z-function usage, change docstrings for functions

* Fix flake8 errors

* Update z_function.py

Co-authored-by: Christian Clauss <cclauss@me.com>

strings/z_function.py

@@ -0,0 +1,89 @@
+"""
+https://cp-algorithms.com/string/z-function.html
+
+Z-function or Z algorithm
+
+Efficient algorithm for pattern occurrence in a string
+
+Time Complexity: O(n) - where n is the length of the string
+
+"""
+
+
+def z_function(input_str: str) -> list:
+    """
+    For the given string this function computes value for each index,
+    which represents the maximal length substring starting from the index
+    and is the same as the prefix of the same size
+
+    e.x.  for string 'abab' for second index value would be 2
+
+    For the value of the first element the algorithm always returns 0
+
+    >>> z_function("abracadabra")
+    [0, 0, 0, 1, 0, 1, 0, 4, 0, 0, 1]
+    >>> z_function("aaaa")
+    [0, 3, 2, 1]
+    >>> z_function("zxxzxxz")
+    [0, 0, 0, 4, 0, 0, 1]
+    """
+    z_result = [0] * len(input_str)
+
+    # initialize interval's left pointer and right pointer
+    left_pointer, right_pointer = 0, 0
+
+    for i in range(1, len(input_str)):
+        # case when current index is inside the interval
+        if i <= right_pointer:
+            min_edge = min(right_pointer - i + 1, z_result[i - left_pointer])
+            z_result[i] = min_edge
+
+        while go_next(i, z_result, input_str):
+            z_result[i] += 1
+
+        # if new index's result gives us more right interval,
+        # we've to update left_pointer and right_pointer
+        if i + z_result[i] - 1 > right_pointer:
+            left_pointer, right_pointer = i, i + z_result[i] - 1
+
+    return z_result
+
+
+def go_next(i, z_result, s):
+    """
+    Check if we have to move forward to the next characters or not
+    """
+    return i + z_result[i] < len(s) and s[z_result[i]] == s[i + z_result[i]]
+
+
+def find_pattern(pattern: str, input_str: str) -> int:
+    """
+    Example of using z-function for pattern occurrence
+    Given function returns the number of times 'pattern'
+    appears in 'input_str' as a substring
+
+    >>> find_pattern("abr", "abracadabra")
+    2
+    >>> find_pattern("a", "aaaa")
+    4
+    >>> find_pattern("xz", "zxxzxxz")
+    2
+    """
+    answer = 0
+    # concatenate 'pattern' and 'input_str' and call z_function
+    # with concatenated string
+    z_result = z_function(pattern + input_str)
+
+    for val in z_result:
+        # if value is greater then length of the pattern string
+        # that means this index is starting position of substring
+        # which is equal to pattern string
+        if val >= len(pattern):
+            answer += 1
+
+    return answer
+
+
+if __name__ == "__main__":
+    import doctest
+    doctest.testmod()