3.4.3 Suffix Array and LCP (DC3)

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/*

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 DC3/skew algorithm recursively constructs the suffix array of the suffixes starting at positions
not divisible by 3, uses that order to sort the remaining suffixes with radix sort, and merges the
two sorted groups in linear time.

- `SuffixArrayDC3(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) 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 <string>
#include <utility>
#include <vector>
using std::string;

class SuffixArrayDC3 {
  static bool leq(int a1, int a2, int b1, int b2) { return (a1 < b1) || (a1 == b1 && a2 <= b2); }

  static bool leq(int a1, int a2, int a3, int b1, int b2, int b3) {
    return (a1 < b1) || (a1 == b1 && leq(a2, a3, b2, b3));
  }

  template<class It>
  static void radix_pass(It a, It b, It r, int n, int K) {
    std::vector<int> cnt(K + 1);
    for (int i = 0; i < n; i++) {
      cnt[r[a[i]]]++;
    }
    for (int i = 1; i <= K; i++) {
      cnt[i] += cnt[i - 1];
    }
    for (int i = n - 1; i >= 0; i--) {
      b[--cnt[r[a[i]]]] = a[i];
    }
  }

  template<class It>
  static void suffix_array_dc3(It s, It sa, int n, int K) {
    int n0 = (n + 2) / 3, n1 = (n + 1) / 3, n2 = n / 3, n02 = n0 + n2;
    std::vector<int> s12(n02 + 3), sa12(n02 + 3), s0(n0), sa0(n0);
    s12[n02] = s12[n02 + 1] = s12[n02 + 2] = 0;
    sa12[n02] = sa12[n02 + 1] = sa12[n02 + 2] = 0;
    for (int i = 0, j = 0; i < n + n0 - n1; i++) {
      if (i % 3 != 0) {
        s12[j++] = i;
      }
    }
    radix_pass(s12.begin(), sa12.begin(), s + 2, n02, K);
    radix_pass(sa12.begin(), s12.begin(), s + 1, n02, K);
    radix_pass(s12.begin(), sa12.begin(), s, n02, K);
    int name = 0, c0 = -1, c1 = -1, c2 = -1;
    for (int i = 0; i < n02; i++) {
      if (s[sa12[i]] != c0 || s[sa12[i] + 1] != c1 || s[sa12[i] + 2] != c2) {
        name++;
        c0 = s[sa12[i]];
        c1 = s[sa12[i] + 1];
        c2 = s[sa12[i] + 2];
      }
      (sa12[i] % 3 == 1 ? s12[sa12[i] / 3] : s12[sa12[i] / 3 + n0]) = name;
    }
    if (name < n02) {
      suffix_array_dc3(s12.begin(), sa12.begin(), n02, name);
      for (int i = 0; i < n02; i++) {
        s12[sa12[i]] = i + 1;
      }
    } else {
      for (int i = 0; i < n02; i++) {
        sa12[s12[i] - 1] = i;
      }
    }
    for (int i = 0, j = 0; i < n02; i++) {
      if (sa12[i] < n0) {
        s0[j++] = 3 * sa12[i];
      }
    }
    radix_pass(s0.begin(), sa0.begin(), s, n0, K);
    for (int p = 0, t = n0 - n1, k = 0; k < n; k++) {
      int i = (sa12[t] < n0) ? 3 * sa12[t] + 1 : 3 * (sa12[t] - n0) + 2, j = sa0[p];
      if (sa12[t] < n0
              ? leq(s[i], s12[sa12[t] + n0], s[j], s12[j / 3])
              : leq(s[i], s[i + 1], s12[sa12[t] - n0 + 1], s[j], s[j + 1], s12[j / 3 + n0])) {
        sa[k] = i;
        if (++t == n02) {
          for (k++; p < n0; p++, k++) {
            sa[k] = sa0[p];
          }
        }
      } else {
        sa[k] = j;
        if (++p == n0) {
          for (k++; t < n02; t++, k++) {
            sa[k] = (sa12[t] < n0) ? 3 * sa12[t] + 1 : 3 * (sa12[t] - n0) + 2;
          }
        }
      }
    }
  }

  string s;
  std::vector<int> sa;

 public:
  explicit SuffixArrayDC3(const string &s) : s(s), sa(s.size() + 1) {
    int n = static_cast<int>(s.size());
    std::vector<int> scopy(n);
    for (int i = 0; i < n; i++) {
      scopy[i] = static_cast<unsigned char>(s[i]) + 1;
    }
    scopy.resize(n + 4);
    suffix_array_dc3(scopy.begin(), sa.begin(), n + 1, 256);
    sa.erase(sa.begin());
  }

  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 {};
    }
    std::vector<int> rank(n), lcp(n - 1);
    for (int i = 0; i < n; i++) {
      rank[sa[i]] = i;
    }
    for (int i = 0, k = 0; i < n; i++) {
      if (rank[i] < n - 1) {
        int j = sa[rank[i] + 1];
        while (std::max(i, j) + k < n && s[i + k] == s[j + k]) {
          k++;
        }
        lcp[rank[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() {
  SuffixArrayDC3 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;
}