Create manchar_algorithm.md

Author: the-ethan-hunt

Competitive Coding/Strings/String Search/Manachar_algorithm/manchar_algorithm.md

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+# Manachar's Algorithm
+
+Manacher's algorithm is a very handy algorithm with a short implementation that
+can make many programming tasks, such as finding the number of palindromic substrings
+or finding the longest palindromic substring, very easy and efficient. The running
+time of Manacher's algorithm is *O(N)* where *N*  is the length of the input string.
+
+## Implementation
+
+Let:
+
+1. s be a string of N characters
+
+2. s2 be a derived string of s, comprising N * 2 + 1 elements, with each element
+corresponding to one of the following: the N characters in s, the N-1 boundaries
+among characters, and the boundaries before and after the first and last character
+respectively
+
+3. A boundary in s2 is equal to any other boundary in s2 with respect to element
+matching in palindromic length determination
+
+4. p be an array of palindromic span for each element in s2, from center to either
+outermost element, where each boundary is counted towards the length of a palindrome
+(e.g. a palindrome that is three elements long has a palindromic span of 1)
+
+5. c be the position of the center of the palindrome currently known to include a
+boundary closest to the right end of s2 (i.e., the length of the palindrome = p[c]*2+1)
+
+6. r be the position of the right-most boundary of this palindrome (i.e., r = c + p[c])
+
+7. i be the position of an element (i.e., a character or boundary) in s2 whose palindromic
+span is being determined, with i always to the right of c
+
+8. i2 be the mirrored position of i around c (e.g., {i, i2} = {6, 4}, {7, 3}, {8, 2},… when c = 5 (i.e., i2 = c * 2 - i)