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Suffix Array

Suffix array is a useful data structure which stores suffix of a string in alphabetical order. (https://en.wikipedia.org/wiki/Suffix_array)

For example:

raw = abaab
Suffix: abaab, baab, aab, ab, b
Sorted suffix: aab, ab, abaab, b, baab

Null character is considered less than any other character.

Suffix array of raw:

[2, 3, 0, 4, 1]

because suffix starting from position 2(aab) is the smallest, then comes suffix starting from position 3(ab), ..., and suffix starting from position 1 (baab) is the largest.

Doubling algorithm

This algorithm builds suffix array in O(nlogn) time complexity and O(CHAR_MAX + n) space complexity, based on sorting segments with length 1, 2, 4, 8, ..., until all segments are different, which is called "doubling algorithm".

(Paper: http://www.cas.mcmaster.ca/~bill/best/algorithms/07Taxonomy.pdf)

Example

Still take abaab as an example:

(Visualized version: https://web.stanford.edu/~mjkay/NewSuffixArrays.pdf)

Iteration 1

We sort all segments with length 1 in 'raw'(i.e. a b a a b), and relabel the original string.

x = 01001  // 'a' => 0, 'b' => 1. two different segments. Still have repeated segments, continue iteration.
sa = 02314  // because letter 'a' appears at 0, 2, 3, then letter 'b' at 1, 4

Here, x[i] stands for the order of raw[i].

Iteration 2 [Step = 1]

We sort all segments with length 2 in raw(i.e. 01 10 00 01 1$ in old_x, since each element in old_x represents exact one character in raw), and relabel it

x = 13012  // '00' => 0, 01 => 1, 1$ => 2, 10 => 3. 4 different segments. still have repeated segments, continue iteration.
sa = 20341  // 00 at position 2, 01 at position 0 and 3, 1$ at position 4, 10 at position 1

Here, x[i] stands for the order of raw[i...i+1]

Iteration 3 [Step = 2]

We sort all segments with length 4 in x(i.e. 10 31 02 1$ 2$ in old_x, since each element in old_x represents 2 characters in raw. If we want x[i] stands for raw[i...i+3], then we need to let the concatenation of old_x[i](which stands for raw[i...i+1]) and old_x[i+2] (which stands for raw[i+2...i+3]) become the key of length-4 segment)

/----------------new_x[0]-----------------\
/-----old_x[0]------\/-----old_x[2]---------\
.........................................................
|raw[0]  |   raw[1]  |   raw[2]  |   raw[3]  |   raw[4] |
.........................................................

x = 24013  // 02 => 0, 1$ => 1, 10 => 2, 2$ => 3, 31 => 4. All different, done!
sa = 23041  // 02 at position 2, 1$ at position 3, 10 at position 0, 2$ at position 4, 31 at position 1

Here, x[i] stands for the order of raw[i...i+3] of all length-4 substrings of raw.

How to sort key pairs

In each iteration, we need sort key pairs (x[i], x[i + step]) (0 <= i < n). The naive method is to use quicksort, thus we need to use O(nlogn) for each iteration, and O(nlognlogn) for the whole algorithm.

A better way is to look into the range of x[i]/raw[i]: the maximum is bounded by both the length of raw (n) [since our goal is to make each elements of x different] and the character set size CHAR_MAX [for raw[i]]. Therefore, we can use bucket sort.

For 2-keyword bucket sort, the basic idea is(just like radix sort):

  1. Sort elements based on the 2nd keyword
  2. Sort elements based on the 1st keyword (must be stable)

Stable bucket sort

How can we perform a stable bucket sort?

// Assume origin is [1, 2, 2, 0, 0]
for (int item : origin)
  bucket[item]++;
// bucket: [2, 1, 2]
for (size_t i=1; i<bucket_size; ++i)
  bucket[i] += bucket[i-1];
// bucket: [2, 3, 5]
for  (int i = origin.size() - 1; i >= 0; --i)
  result[ --bucket[ origin[i] ] ] = i
// result: [3, 4, 0, 1, 2]

Sort array by 2nd keyword, using existing array of previous iteration

So how can we get the 2nd keyword? Remember that the 2nd keyword for position i is x[i+step] (if i + step < N) or $ (if i + step >= N).

Take a look at the result of the 2nd iteration:

x   =   13012

and the key pairs to sort of the 3rd iteration:

  10 31 02 1$ 2$,

2nd keyword:

  0  1  2  $  $

But the sorted array by the 2nd keyword is already there! Sa of iteration 2:

sa  =   23041
new_sa == { N - step, N - step + 1, ..., N - 1, sa[origin1] - step, sa[origin2] - step, ... }

The smallest step elements must be the last step elements, because the second keyword of them is always $. Sort result of 2nd keyword in iteration 3:

new_sa= 3 4 0 1 2.

Explanation: Step = 2, thus 3 4 are the smallest elements. position 0, 1 are impossible for a second keyword: ignore them from sa, the remaining items in sa (2 3 4) is the order of 2nd keywords of position (0 1 2)