3.4.2 Suffix Array and LCP (Counting Sort)

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138

/*

Given a string $s$, a suffix array is the array of the smallest starting positions for the sorted
suffixes of $s$. That is, the $i$-th position of the suffix array stores the starting position of
the $i$-th lexicographically smallest suffix of $s$. For example, $s$ = `"cab"` has the suffixes
`"cab"`, `"ab"`, and `"b"`. When sorted, the indices of the suffixes are `"ab"`, `"b"`, and `"cab"`,
so the suffix array (assuming 0-based indices) is $[1, 2, 0]$.

For a string $s$ of length $n$, the longest common prefix (LCP) array of length $n - 1$ stores the
lengths of the longest common prefixes between all pairs of lexicographically adjacent suffixes in
$s$. For example, `"baa"` has the sorted suffixes `"a"`, `"aa"`, and `"baa"`, with an LCP array of
$[1, 0]$.

The original Manber-Myers doubling algorithm comparison-sorts the suffixes by their first $2^k$
characters for increasing $k$: each round orders suffixes by their pair of ranks from the previous
round, so a comparison costs O(1) and the full order emerges after O(log n) rounds. Replacing the
comparison sort of each round with counting-sort-style rank updates removes a logarithmic factor.

- `SuffixArrayCountingSort(s)` constructs a suffix array from string `s`. 
- `get_sa()` returns the constructed suffix array.
- `get_lcp()` returns the corresponding LCP array for the suffix array.
- `find(needle)` returns one position that `needle` occurs in `s` (not necessarily the first), or
  `std::string::npos` if it cannot be found. For a `needle` of length $m$, this implementation uses
  an O(m log n) binary search, but can be optimized to O(m + log n) by first computing the LCP-LR
  array using the LCP array.

Time Complexity:
- O(n log n) per call to the constructor, where $n$ is the length of `s`.
- O(1) per call to `get_sa()`.
- O(n) per call to `get_lcp()`, where $n$ is the length of `s`.
- O(m log n) per call to `find(needle)`, where $m$ is the length of `needle` and $n$ is the length
  of `s`.

Space Complexity:
- O(n) auxiliary for storage of the suffix and LCP arrays.
- O(n) auxiliary heap space for the constructor.
- O(1) auxiliary space for all other operations.

*/

#include <algorithm>
#include <numeric>
#include <string>
#include <utility>
#include <vector>
using std::string;

class SuffixArrayCountingSort {
  string s;
  std::vector<int> sa, rk;

 public:
  explicit SuffixArrayCountingSort(const string &s) : s(s), sa(s.size()), rk(s.size()) {
    int n = static_cast<int>(s.size());
    std::iota(sa.rbegin(), sa.rend(), 0);
    for (int i = 0; i < n; i++) {
      rk[i] = static_cast<unsigned char>(s[i]);
    }
    std::stable_sort(sa.begin(), sa.end(), [&](int i, int j) {
      return static_cast<unsigned char>(s[i]) < static_cast<unsigned char>(s[j]);
    });
    for (int gap = 1; gap < n; gap *= 2) {
      std::vector<int> prev_rk(rk), prev_sa(sa), cnt(n);
      std::iota(cnt.begin(), cnt.end(), 0);
      for (int i = 0; i < n; i++) {
        rk[sa[i]] = (i > 0 && prev_rk[sa[i - 1]] == prev_rk[sa[i]] && sa[i - 1] + gap < n &&
                     prev_rk[sa[i - 1] + gap / 2] == prev_rk[sa[i] + gap / 2])
                        ? rk[sa[i - 1]]
                        : i;
      }
      for (int i = 0; i < n; i++) {
        int s1 = prev_sa[i] - gap;
        if (s1 >= 0) {
          sa[cnt[rk[s1]]++] = s1;
        }
      }
    }
  }

  const std::vector<int> &get_sa() const { return sa; }

  std::vector<int> get_lcp() const {
    int n = static_cast<int>(s.size());
    if (n == 0) {
      return {};  // Avoid constructing a vector of size (size_t)(-1).
    }
    std::vector<int> lcp(n - 1);
    for (int i = 0, k = 0; i < n; i++) {
      if (rk[i] < n - 1) {
        int j = sa[rk[i] + 1];
        while (std::max(i, j) + k < n && s[i + k] == s[j + k]) {
          k++;
        }
        lcp[rk[i]] = k;
        if (k > 0) {
          k--;
        }
      }
    }
    return lcp;
  }

  size_t find(const string &needle) {
    if (needle.empty()) {
      return 0;
    }
    int lo = 0, hi = static_cast<int>(s.size()) - 1;
    while (lo <= hi) {
      int mid = lo + (hi - lo) / 2;
      int cmp = s.compare(sa[mid], needle.size(), needle);
      if (cmp < 0) {
        lo = mid + 1;
      } else if (cmp > 0) {
        hi = mid - 1;
      } else {
        return sa[mid];
      }
    }
    return string::npos;
  }
};

/*** Example Usage ***/

#include <cassert>
using namespace std;

int main() {
  SuffixArrayCountingSort sa("banana");
  vector<int> sarr = sa.get_sa(), lcp = sa.get_lcp();
  vector<int> sarr_expected{5, 3, 1, 0, 4, 2};
  vector<int> lcp_expected{1, 3, 0, 0, 2};
  assert(sarr == sarr_expected);
  assert(lcp == lcp_expected);
  assert(sa.find("ana") == 1);
  assert(sa.find("x") == string::npos);
  return 0;
}