---
id: 5a23c84252665b21eecc7ecb
title: K-D ツリー
challengeType: 1
forumTopicId: 302295
dashedName: k-d-tree
---

# --description--

k-d ツリー (*k*-次元ツリーの略称) は、k 次元空間の点を整理するためのスペース分割データ構造です。 k-d ツリーは、多次元探索鍵に関わる探索 (例えば、範囲探索や最近傍探索) など、いくつかの応用にとって有用なデータ構造です。 k-d ツリーはバイナリ空間分割ツリーの特殊ケースです。 ただし、k-d ツリーは高次元空間での効率的な最近傍探索には適していません。 原則として、次元が *k* の場合、データ内の点の数 *N* は、*N* ≫ 2<sup><i>k</i></sup> であるべきです。 そうでなければ、k-d ツリーが高次元データで使用される場合、ツリーの点のほとんどを評価することになり、効率はしらみつぶし探索と大差なく、代わりに近似最近傍探索など他の方法が使用されることになります。

# --instructions--

k-d ツリーを使って最近傍探索を実行する関数を記述してください。 関数は次の 2 つのパラメータをとります。k 次元ポイントの配列と、関数が返す最近傍を求めるための単一の k 次元ポイントです。 k 次元ポイントは、k 要素の配列として与えられます。

# --hints--

`kdNN` は関数とします。

```js
assert(typeof kdNN == 'function');
```

`kdNN([[[2, 3], [5, 4], [9, 6], [4, 7], [8, 1], [7, 2]], [9, 2])` は配列を返す必要があります。

```js
assert(
  Array.isArray(
    kdNN(
      [
        [2, 3],
        [5, 4],
        [9, 6],
        [4, 7],
        [8, 1],
        [7, 2]
      ],
      [9, 2]
    )
  )
);
```

`kdNN([[[2, 3], [5, 4], [9, 6], [4, 7], [8, 1], [7, 2]], [9, 2])` は `[ 8, 1 ]` を返す必要があります。

```js
assert.deepEqual(
  kdNN(
    [
      [2, 3],
      [5, 4],
      [9, 6],
      [4, 7],
      [8, 1],
      [7, 2]
    ],
    [9, 2]
  ),
  [8, 1]
);
```

`kdNN([[[2, 3], [5, 4], [9, 6], [4, 7], [8, 1], [7, 2]], [7, 1])` は `[ 8, 1 ]` を返す必要があります。

```js
assert.deepEqual(
  kdNN(
    [
      [2, 3],
      [5, 4],
      [9, 6],
      [4, 7],
      [8, 1],
      [7, 2]
    ],
    [7, 1]
  ),
  [8, 1]
);
```

`kdNN([[[2, 3], [5, 4], [9, 6], [4, 7], [8, 1], [7, 2]], [3, 2])` は `[ 2, 3 ]` を返す必要があります。

```js
assert.deepEqual(
  kdNN(
    [
      [2, 3],
      [5, 4],
      [9, 6],
      [4, 7],
      [8, 1],
      [7, 2]
    ],
    [3, 2]
  ),
  [2, 3]
);
```

`kdNN([[2, 3, 1], [9, 4, 5], [4, 6, 7], [1, 2, 5], [7, 8, 9], [3, 6, 1]], [1, 2, 3])` は `[ 1, 2, 5 ]` を返す必要があります。

```js
assert.deepEqual(
  kdNN(
    [
      [2, 3, 1],
      [9, 4, 5],
      [4, 6, 7],
      [1, 2, 5],
      [7, 8, 9],
      [3, 6, 1]
    ],
    [1, 2, 3]
  ),
  [1, 2, 5]
);
```

`kdNN([[2, 3, 1], [9, 4, 5], [4, 6, 7], [1, 2, 5], [7, 8, 9], [3, 6, 1]], [4, 5, 6])` は `[ 4, 6, 7 ]` を返す必要があります。

```js
assert.deepEqual(
  kdNN(
    [
      [2, 3, 1],
      [9, 4, 5],
      [4, 6, 7],
      [1, 2, 5],
      [7, 8, 9],
      [3, 6, 1]
    ],
    [4, 5, 6]
  ),
  [4, 6, 7]
);
```

`kdNN([[2, 3, 1], [9, 4, 5], [4, 6, 7], [1, 2, 5], [7, 8, 9], [3, 6, 1]], [8, 8, 8])` は `[ 7, 8, 9 ]` を返す必要があります。

```js
assert.deepEqual(
  kdNN(
    [
      [2, 3, 1],
      [9, 4, 5],
      [4, 6, 7],
      [1, 2, 5],
      [7, 8, 9],
      [3, 6, 1]
    ],
    [8, 8, 8]
  ),
  [7, 8, 9]
);
```

# --seed--

## --seed-contents--

```js
function kdNN(fpoints, fpoint) {

}
```

# --solutions--

```js
function kdNN(fpoints, fpoint) {
  function Node(obj, dimension, parent) {
    this.obj = obj;
    this.left = null;
    this.right = null;
    this.parent = parent;
    this.dimension = dimension;
  }

  function kdTree(points, metric, dimensions) {
    var self = this;

    function buildTree(points, depth, parent) {
      var dim = depth % dimensions.length,
        median,
        node;

      if (points.length === 0) {
        return null;
      }
      if (points.length === 1) {
        return new Node(points[0], dim, parent);
      }

      points.sort(function(a, b) {
        return a[dimensions[dim]] - b[dimensions[dim]];
      });

      median = Math.floor(points.length / 2);
      node = new Node(points[median], dim, parent);
      node.left = buildTree(points.slice(0, median), depth + 1, node);
      node.right = buildTree(points.slice(median + 1), depth + 1, node);

      return node;
    }

    this.root = buildTree(points, 0, null);

    this.insert = function(point) {
      function innerSearch(node, parent) {
        if (node === null) {
          return parent;
        }

        var dimension = dimensions[node.dimension];
        if (point[dimension] < node.obj[dimension]) {
          return innerSearch(node.left, node);
        } else {
          return innerSearch(node.right, node);
        }
      }

      var insertPosition = innerSearch(this.root, null),
        newNode,
        dimension;

      if (insertPosition === null) {
        this.root = new Node(point, 0, null);
        return;
      }

      newNode = new Node(
        point,
        (insertPosition.dimension + 1) % dimensions.length,
        insertPosition
      );
      dimension = dimensions[insertPosition.dimension];

      if (point[dimension] < insertPosition.obj[dimension]) {
        insertPosition.left = newNode;
      } else {
        insertPosition.right = newNode;
      }
    };

    this.nearest = function(point, maxNodes, maxDistance) {
      var i, result, bestNodes;

      bestNodes = new BinaryHeap(function(e) {
        return -e[1];
      });

      function nearestSearch(node) {
        var bestChild,
          dimension = dimensions[node.dimension],
          ownDistance = metric(point, node.obj),
          linearPoint = {},
          linearDistance,
          otherChild,
          i;

        function saveNode(node, distance) {
          bestNodes.push([node, distance]);
          if (bestNodes.size() > maxNodes) {
            bestNodes.pop();
          }
        }

        for (i = 0; i < dimensions.length; i += 1) {
          if (i === node.dimension) {
            linearPoint[dimensions[i]] = point[dimensions[i]];
          } else {
            linearPoint[dimensions[i]] = node.obj[dimensions[i]];
          }
        }

        linearDistance = metric(linearPoint, node.obj);

        if (node.right === null && node.left === null) {
          if (
            bestNodes.size() < maxNodes ||
            ownDistance < bestNodes.peek()[1]
          ) {
            saveNode(node, ownDistance);
          }
          return;
        }

        if (node.right === null) {
          bestChild = node.left;
        } else if (node.left === null) {
          bestChild = node.right;
        } else {
          if (point[dimension] < node.obj[dimension]) {
            bestChild = node.left;
          } else {
            bestChild = node.right;
          }
        }

        nearestSearch(bestChild);

        if (bestNodes.size() < maxNodes || ownDistance < bestNodes.peek()[1]) {
          saveNode(node, ownDistance);
        }

        if (
          bestNodes.size() < maxNodes ||
          Math.abs(linearDistance) < bestNodes.peek()[1]
        ) {
          if (bestChild === node.left) {
            otherChild = node.right;
          } else {
            otherChild = node.left;
          }
          if (otherChild !== null) {
            nearestSearch(otherChild);
          }
        }
      }

      if (maxDistance) {
        for (i = 0; i < maxNodes; i += 1) {
          bestNodes.push([null, maxDistance]);
        }
      }

      if (self.root) nearestSearch(self.root);

      result = [];

      for (i = 0; i < Math.min(maxNodes, bestNodes.content.length); i += 1) {
        if (bestNodes.content[i][0]) {
          result.push([bestNodes.content[i][0].obj, bestNodes.content[i][1]]);
        }
      }
      return result;
    };
  }

  function BinaryHeap(scoreFunction) {
    this.content = [];
    this.scoreFunction = scoreFunction;
  }

  BinaryHeap.prototype = {
    push: function(element) {
      // Add the new element to the end of the array.
      this.content.push(element);
      // Allow it to bubble up.
      this.bubbleUp(this.content.length - 1);
    },

    pop: function() {
      // Store the first element so we can return it later.
      var result = this.content[0];
      // Get the element at the end of the array.
      var end = this.content.pop();
      // If there are any elements left, put the end element at the
      // start, and let it sink down.
      if (this.content.length > 0) {
        this.content[0] = end;
        this.sinkDown(0);
      }
      return result;
    },

    peek: function() {
      return this.content[0];
    },

    size: function() {
      return this.content.length;
    },

    bubbleUp: function(n) {
      // Fetch the element that has to be moved.
      var element = this.content[n];
      // When at 0, an element can not go up any further.
      while (n > 0) {
        // Compute the parent element's index, and fetch it.
        var parentN = Math.floor((n + 1) / 2) - 1,
          parent = this.content[parentN];
        // Swap the elements if the parent is greater.
        if (this.scoreFunction(element) < this.scoreFunction(parent)) {
          this.content[parentN] = element;
          this.content[n] = parent;
          // Update 'n' to continue at the new position.
          n = parentN;
        }
        // Found a parent that is less, no need to move it further.
        else {
          break;
        }
      }
    },

    sinkDown: function(n) {
      // Look up the target element and its score.
      var length = this.content.length,
        element = this.content[n],
        elemScore = this.scoreFunction(element);

      while (true) {
        // Compute the indices of the child elements.
        var child2N = (n + 1) * 2,
          child1N = child2N - 1;
        // This is used to store the new position of the element,
        // if any.
        var swap = null;
        // If the first child exists (is inside the array)...
        if (child1N < length) {
          // Look it up and compute its score.
          var child1 = this.content[child1N],
            child1Score = this.scoreFunction(child1);
          // If the score is less than our element's, we need to swap.
          if (child1Score < elemScore) swap = child1N;
        }
        // Do the same checks for the other child.
        if (child2N < length) {
          var child2 = this.content[child2N],
            child2Score = this.scoreFunction(child2);
          if (child2Score < (swap == null ? elemScore : child1Score)) {
            swap = child2N;
          }
        }

        // If the element needs to be moved, swap it, and continue.
        if (swap != null) {
          this.content[n] = this.content[swap];
          this.content[swap] = element;
          n = swap;
        }
        // Otherwise, we are done.
        else {
          break;
        }
      }
    }
  };

  var dims = [];

  for (var i = 0; i < fpoint.length; i++) dims.push(i);

  var tree = new kdTree(
    fpoints,
    function(e1, e2) {
      var d = 0;
      var e3 = e1;
      if (!Array.isArray(e1)) {
        e3 = [];
        for (var key in e1) e3.push(e1[key]);

        e1 = e3;
      }
      e1.forEach(function(e, i) {
        var sqd = e1[i] - e2[i];
        d += sqd * sqd;
      });
      return d;
    },
    dims
  );

  return tree.nearest(fpoint, 1, 1000)[0][0];
}
```
