Dynamic k-d Tree

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2012-09-08

08:13

前回のあらすじ：バランスが壊れたらぶっ壊して作りなおすと amortized O(log n)．

Scapegoat Treeでは，単純な二分探索木に破壊と再生テクニックを導入して計算量を低下させていたが，破壊と再生が本当に役立つのは

木だが，回転操作は難しい
一方，ゼロからバランスしたものを作るのはわりと楽

という構造が相手のときだと思う．このようなデータ構造の代表にk-d木があるので，今回はk-d木を破壊と再生で平衡化する手法を説明する．なお，これは前回文献として紹介したOvermars(1983)に載っている内容なので，歴史的にはScapegoat Tree(Andersson(1989), Galperin-Rivest(1993)) よりも古い．

k-d木は深さごとに軸が変わる二分探索木のことで，空間からの点の探索などに使われる．この上で「二分探索木の回転」みたいな操作をやろうとすると，うんざりできる程度にはややこしいことに気づく．一方で，ゼロから完全平衡したk-d木を作るのは難しくなくて，軸に応じた中央値をとって左右を再帰的に構成するだけでいける．計算量は中央値を O(n) でとって f(n) = 2 f(n/2) + O(n) の f(n) = O(n log n)．以下にその実装例を示す（クエリなし．つくるだけ）．

// 29 lines
const int D = 2;
struct kdTree {
  struct Node {
    int x, y;
    Node *ch[2];
    Node(int x, int y) : x(x), y(y) { ch[0] = ch[1] = 0; }
  } *r;
  struct Compare {
    int d;
    Compare(int d) : d(d) { }
    bool operator()(Node *i, Node *j) {
      return d == 0 ? (i->x < j->x) : (i->y < j->y);
    }
  };
  Node *build(Node **b, Node **e, int d) {
    if (e - b <= 0) return 0;
    Node **m = b + (e - b) / 2;
    nth_element(b, m, e, Compare(d)); 
    (*m)->ch[0] = build(b, m, (d+1)%D);
    (*m)->ch[1] = build(m+1, e, (d+1)%D);
    return *m;
  }
  kdTree(int p[][2], int n) {
    Node *node[n];
    for (int i = 0; i < n; ++i) 
      node[i] = new Node(p[i][0], p[i][1]);
    r = build(node, node+n, 0);
  }
};

さて，ここからがポイント．まずは平衡化を後回しにして要素の追加と削除を入れていく．

追加は普通の二分探索木と同じでよい．要素の追加は葉にしか発生しないので，特に難しいことはない．削除は問題で，中間ノードが消えると以降のk-d条件が意味不明になってしまう．そこで削除は「削除したマーク」をつけて以降無視する，という実装をとる．これは破壊と再生テクニックのときに使うテクニックでもあった．

// 53 lines = 29 + 24 
const int D = 2;
struct kdTree {
  struct Node {
    int x, y;
    bool removed;
    Node *ch[2];
    Node(int x, int y) : x(x), y(y), removed(false) { ch[0] = ch[1] = 0; }
  } *r;
//
// ...omit...
//
  Node *insert(Node *t, Node *p, int d) {
    if (!t) return p;
    int b = !Compare(d)(p, t); 
    t->ch[b] = insert(t->ch[b], p, (d+1)%D);
    return t;
  }
  void insert(int x, int y) { r = insert(r, new Node(x,y), 0); }

  Node *find(Node *t, Node *p, int d) {
    if (!t) return 0;
    Node *f = 0;
    if (!t->removed && t->x == p->x && t->y == p->y) f = t;
    if (!f && !Compare(d)(p,t)) f = find(t->ch[1], p, (d+1)%D);
    if (!f && !Compare(d)(t,p)) f = find(t->ch[0], p, (d+1)%D);
    return f;
  }
  Node *find(int x, int y) { Node n(x,y); return find(r, &n, 0); }

  void remove(int x, int y) { 
    Node *f = find(x, y);
    if (!f) return;
    f->removed = true;
  }
};

そして最後にこれを平衡化しよう．remove のときは remove されたノード数を覚えておいて，しきい値を超えたら Global Rebuilding するだけでよい．insert は Partial Rebuilding をしなければならないので，そのために各ノードにサイズ（自分以下の子の数）と，高さを追加情報として覚えさせる．そして insert が実行されたらに挿入パス上のノードにおいて log(サイズ) と高さを比較して必要なら Partial Rebuilding する．

全体のソースコードは次のようになる．計算量は若干の注意が必要で，Scapegoat Tree のときは Rebuilding が O(n) でできたので n 回にならすと O(1) になり，無視して O(log n) と見積もっていた．この場合は Rebuilding に O(n log n) かかるので，ならして O(log n) になるから，あわせて O(log n) となる．

// 78 lines = 53 + 25
const int D = 2;
struct kdTree {
  int removed;
  struct Node {
    int x, y;
    bool removed;
    Node *ch[2];
    int size, ht;
    Node(int x, int y) : x(x), y(y), removed(false) { ch[0] = ch[1] = 0; }
  } *r;
  int size(Node *t) { return t ? t->size : 0; }
  int ht(Node *t) { return t ? t->ht : 0; }
  Node *update(Node *t) {
    if (!t) return t;
    t->size = 1 + size(t->ch[0]) + size(t->ch[1]);
    t->ht = 1 + max(ht(t->ch[0]), ht(t->ch[1]));
    return t;
  }
  struct Compare {
    int d;
    Compare(int d) : d(d) { }
    bool operator()(Node *i, Node *j) {
      return d == 0 ? (i->x < j->x) : (i->y < j->y);
    }
  };
  Node *build(Node **b, Node **e, int d) {
    if (e - b <= 0) return 0;
    Node **m = b + (e - b) / 2;
    nth_element(b, m, e, Compare(d)); 
    (*m)->ch[0] = build(b, m, (d+1)%D);
    (*m)->ch[1] = build(m+1, e, (d+1)%D);
    return update(*m);
  }
  Node **flatten(Node *r, Node **buf) {
    if (!r) return buf;
    buf = flatten(r->ch[0], buf);
    if (!r->removed) *(buf++) = r;
    return flatten(r->ch[1], buf);
  }
  Node *rebuild(Node *r, int d) {
    Node *b[size(r)], **e = flatten(r, b);
    removed -= size(r) - (e - b);
    return build(b, e, d);
  }
  kdTree(int p[][2], int n) : removed(0) {
    Node *node[n];
    for (int i = 0; i < n; ++i) 
      node[i] = new Node(p[i][0], p[i][1]);
    r = build(node, node+n, 0);
  }
  Node *insert(Node *t, Node *p, int d) {
    if (!t) return update(p);
    int b = !Compare(d)(p, t); 
    t->ch[b] = insert(t->ch[b], p, (d+1)%D);
    t = update(t);
    if (3 * log(size(t)) < ht(t)) t = rebuild(t, d);
    return t;
  }
  void insert(int x, int y) { r = insert(r, new Node(x,y), 0); }

  Node *find(Node *t, Node *p, int d) {
    if (!t) return 0;
    Node *f = 0;
    if (!t->removed && t->x == p->x && t->y == p->y) f = t;
    if (!f && !Compare(d)(p,t)) f = find(t->ch[1], p, (d+1)%D);
    if (!f && !Compare(d)(t,p)) f = find(t->ch[0], p, (d+1)%D);
    return f;
  }
  Node *find(int x, int y) { Node n(x,y); return find(r, &n, 0); }

  void remove(int x, int y) { 
    Node *f = find(x, y);
    if (!f) return;
    f->removed = true;
    ++removed;
    if (removed*2 > r->size) r = rebuild(r, 0);
  }
};

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