Sqrt Tree

Sqrt Tree 是一个可以在 的时间内预处理，并在 的时间内回答在某个区间上进行某个运算的询问，且能在 的时间内单点修改的数据结构。这个运算需要满足结合率，如 。

C++模板

SqrtTreeItem op(const SqrtTreeItem &a, const SqrtTreeItem &b);

int log2Up(int n) {
  int res = 0;
  while ((1 << res) < n) {
    res++;
  }
  return res;
}

class SqrtTree {
 private:
  int n, lg, indexSz;
  vector<SqrtTreeItem> v;
  vector<int> clz, layers, onLayer;
  vector<vector<SqrtTreeItem>> pref, suf, between;

  void buildBlock(int layer, int l, int r) {
    pref[layer][l] = v[l];
    for (int i = l + 1; i < r; i++) {
      pref[layer][i] = op(pref[layer][i - 1], v[i]);
    }
    suf[layer][r - 1] = v[r - 1];
    for (int i = r - 2; i >= l; i--) {
      suf[layer][i] = op(v[i], suf[layer][i + 1]);
    }
  }

  void buildBetween(int layer, int lBound, int rBound, int betweenOffs) {
    int bSzLog = (layers[layer] + 1) >> 1;
    int bCntLog = layers[layer] >> 1;
    int bSz = 1 << bSzLog;
    int bCnt = (rBound - lBound + bSz - 1) >> bSzLog;
    for (int i = 0; i < bCnt; i++) {
      SqrtTreeItem ans;
      for (int j = i; j < bCnt; j++) {
        SqrtTreeItem add = suf[layer][lBound + (j << bSzLog)];
        ans = (i == j) ? add : op(ans, add);
        between[layer - 1][betweenOffs + lBound + (i << bCntLog) + j] = ans;
      }
    }
  }

  void buildBetweenZero() {
    int bSzLog = (lg + 1) >> 1;
    for (int i = 0; i < indexSz; i++) {
      v[n + i] = suf[0][i << bSzLog];
    }
    build(1, n, n + indexSz, (1 << lg) - n);
  }

  void updateBetweenZero(int bid) {
    int bSzLog = (lg + 1) >> 1;
    v[n + bid] = suf[0][bid << bSzLog];
    update(1, n, n + indexSz, (1 << lg) - n, n + bid);
  }

  void build(int layer, int lBound, int rBound, int betweenOffs) {
    if (layer >= (int)layers.size()) {
      return;
    }
    int bSz = 1 << ((layers[layer] + 1) >> 1);
    for (int l = lBound; l < rBound; l += bSz) {
      int r = min(l + bSz, rBound);
      buildBlock(layer, l, r);
      build(layer + 1, l, r, betweenOffs);
    }
    if (layer == 0) {
      buildBetweenZero();
    } else {
      buildBetween(layer, lBound, rBound, betweenOffs);
    }
  }

  void update(int layer, int lBound, int rBound, int betweenOffs, int x) {
    if (layer >= (int)layers.size()) {
      return;
    }
    int bSzLog = (layers[layer] + 1) >> 1;
    int bSz = 1 << bSzLog;
    int blockIdx = (x - lBound) >> bSzLog;
    int l = lBound + (blockIdx << bSzLog);
    int r = min(l + bSz, rBound);
    buildBlock(layer, l, r);
    if (layer == 0) {
      updateBetweenZero(blockIdx);
    } else {
      buildBetween(layer, lBound, rBound, betweenOffs);
    }
    update(layer + 1, l, r, betweenOffs, x);
  }

  SqrtTreeItem query(int l, int r, int betweenOffs, int base) {
    if (l == r) {
      return v[l];
    }
    if (l + 1 == r) {
      return op(v[l], v[r]);
    }
    int layer = onLayer[clz[(l - base) ^ (r - base)]];
    int bSzLog = (layers[layer] + 1) >> 1;
    int bCntLog = layers[layer] >> 1;
    int lBound = (((l - base) >> layers[layer]) << layers[layer]) + base;
    int lBlock = ((l - lBound) >> bSzLog) + 1;
    int rBlock = ((r - lBound) >> bSzLog) - 1;
    SqrtTreeItem ans = suf[layer][l];
    if (lBlock <= rBlock) {
      SqrtTreeItem add =
          (layer == 0) ? (query(n + lBlock, n + rBlock, (1 << lg) - n, n))
                       : (between[layer - 1][betweenOffs + lBound +
                                             (lBlock << bCntLog) + rBlock]);
      ans = op(ans, add);
    }
    ans = op(ans, pref[layer][r]);
    return ans;
  }

 public:
  SqrtTreeItem query(int l, int r) { return query(l, r, 0, 0); }

  void update(int x, const SqrtTreeItem &item) {
    v[x] = item;
    update(0, 0, n, 0, x);
  }

  SqrtTree(const vector<SqrtTreeItem> &a)
      : n((int)a.size()), lg(log2Up(n)), v(a), clz(1 << lg), onLayer(lg + 1) {
    clz[0] = 0;
    for (int i = 1; i < (int)clz.size(); i++) {
      clz[i] = clz[i >> 1] + 1;
    }
    int tlg = lg;
    while (tlg > 1) {
      onLayer[tlg] = (int)layers.size();
      layers.push_back(tlg);
      tlg = (tlg + 1) >> 1;
    }
    for (int i = lg - 1; i >= 0; i--) {
      onLayer[i] = max(onLayer[i], onLayer[i + 1]);
    }
    int betweenLayers = max(0, (int)layers.size() - 1);
    int bSzLog = (lg + 1) >> 1;
    int bSz = 1 << bSzLog;
    indexSz = (n + bSz - 1) >> bSzLog;
    v.resize(n + indexSz);
    pref.assign(layers.size(), vector<SqrtTreeItem>(n + indexSz));
    suf.assign(layers.size(), vector<SqrtTreeItem>(n + indexSz));
    between.assign(betweenLayers, vector<SqrtTreeItem>((1 << lg) + bSz));
    build(0, 0, n, 0);
  }
};

用法示例(查询某个区间的和，并修改某个点的值):

// 定义元素类型和操作
typedef int SqrtTreeItem;
SqrtTreeItem op(const SqrtTreeItem &a, const SqrtTreeItem &b) {
    return a + b; // 求和示例
}

int main() {
    vector<SqrtTreeItem> data = {1, 2, 3, 4};
    SqrtTree tree(data);
    
    // 查询区间[0, 3]的和
    cout << tree.query(0, 3) << endl; // 输出10
    
    // 更新位置2的元素为5
    tree.update(2, 5);
    
    // 查询区间[1, 3]的和
    cout << tree.query(1, 3) << endl; // 输出11 (2+5+4)
    
    return 0;
}