Notes on Suffix Array and Manacher Algorithm

Recently I am working on some data structure questions, and I kind of dive into the “Longest Palindromic Substring” problem, master(do I?) the Suffix Array and Manacher algorithm. I found that they are impressively amazing and have something in common.

What is Suffix Array?

A suffix array is a sorted array of all suffixes of a string. For example, “abracadabra”:

$$\\newcommand\\T{\\Rule{0pt}{1em}{.3em}} \\begin{array}{|c|c|c|} \\hline \\text{i} & \\text{sa[i]} & \\text{S[sa[i]...]} \\T \\\\\\hline \\text{0} & \\text{11} & \\text{(empty string)} \\\\\\hline \\text{1} & \\text{10} & \\text{a} \\\\\\hline \\text{2} & \\text{7} & \\text{abra} \\\\\\hline \\text{3} & \\text{0} & \\text{abracadabra} \\\\\\hline \\text{4} & \\text{3} & \\text{acadabra} \\\\\\hline \\text{5} & \\text{5} & \\text{adabra} \\\\\\hline \\text{6} & \\text{8} & \\text{bra} \\\\\\hline \\text{7} & \\text{1} & \\text{bracadabra} \\\\\\hline \\text{8} & \\text{4} & \\text{cadabra} \\\\\\hline \\text{9} & \\text{6} & \\text{dabra} \\\\\\hline \\text{10} & \\text{9} & \\text{ra} \\\\\\hline \\text{11} & \\text{2} & \\text{racadabra} \\\\\\hline \\end{array}$$

How to compute suffix array efficiently?

Manber and Myers invented a algorithm with runtime complexity $O(n log^{2} n)$. The basic idea is doubling. Start with the character at each location, we first compute the orders of substrings with length 2 at each location, then use the results to compute the orders of substrings with length 4. The assessed prefix length doubles in each iteration of the algorithm until a prefix is unique (or length >= n) and provides the rank of the associated suffix.

What is Manacher Algorithm?