470 lines
12 KiB
Markdown
470 lines
12 KiB
Markdown
---
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id: 5a23c84252665b21eecc7ecb
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title: K-D ツリー
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challengeType: 5
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forumTopicId: 302295
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dashedName: k-d-tree
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---
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# --description--
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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。
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# --instructions--
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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。
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# --hints--
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`kdNN` は関数とします。
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```js
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assert(typeof kdNN == 'function');
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```
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`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。
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```js
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assert(
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Array.isArray(
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kdNN(
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[
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[2, 3],
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[5, 4],
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[9, 6],
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[4, 7],
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[8, 1],
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[7, 2]
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],
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[9, 2]
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)
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)
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);
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```
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`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。
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```js
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assert.deepEqual(
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kdNN(
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[
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[2, 3],
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[5, 4],
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[9, 6],
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[4, 7],
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[8, 1],
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[7, 2]
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],
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[9, 2]
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),
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[8, 1]
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);
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```
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`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。
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```js
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assert.deepEqual(
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kdNN(
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[
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[2, 3],
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[5, 4],
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[9, 6],
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[4, 7],
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[8, 1],
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[7, 2]
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],
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[7, 1]
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),
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[8, 1]
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);
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```
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`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。
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```js
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assert.deepEqual(
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kdNN(
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[
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[2, 3],
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[5, 4],
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[9, 6],
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[4, 7],
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[8, 1],
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[7, 2]
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],
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[3, 2]
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),
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[2, 3]
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);
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```
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`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。
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```js
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assert.deepEqual(
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kdNN(
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[
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[2, 3, 1],
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[9, 4, 5],
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[4, 6, 7],
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[1, 2, 5],
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[7, 8, 9],
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[3, 6, 1]
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],
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[1, 2, 3]
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),
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[1, 2, 5]
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);
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```
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`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。
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```js
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assert.deepEqual(
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kdNN(
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[
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[2, 3, 1],
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[9, 4, 5],
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[4, 6, 7],
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[1, 2, 5],
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[7, 8, 9],
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[3, 6, 1]
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],
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[4, 5, 6]
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),
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[4, 6, 7]
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);
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```
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`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。
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```js
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assert.deepEqual(
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kdNN(
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[
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[2, 3, 1],
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[9, 4, 5],
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[4, 6, 7],
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[1, 2, 5],
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[7, 8, 9],
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[3, 6, 1]
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],
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[8, 8, 8]
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),
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[7, 8, 9]
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);
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```
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# --seed--
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## --seed-contents--
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```js
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function kdNN(fpoints, fpoint) {
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}
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```
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# --solutions--
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```js
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function kdNN(fpoints, fpoint) {
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function Node(obj, dimension, parent) {
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this.obj = obj;
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this.left = null;
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this.right = null;
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this.parent = parent;
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this.dimension = dimension;
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}
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function kdTree(points, metric, dimensions) {
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var self = this;
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function buildTree(points, depth, parent) {
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var dim = depth % dimensions.length,
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median,
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node;
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if (points.length === 0) {
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return null;
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}
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if (points.length === 1) {
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return new Node(points[0], dim, parent);
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}
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points.sort(function(a, b) {
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return a[dimensions[dim]] - b[dimensions[dim]];
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});
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median = Math.floor(points.length / 2);
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node = new Node(points[median], dim, parent);
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node.left = buildTree(points.slice(0, median), depth + 1, node);
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node.right = buildTree(points.slice(median + 1), depth + 1, node);
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return node;
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}
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this.root = buildTree(points, 0, null);
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this.insert = function(point) {
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function innerSearch(node, parent) {
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if (node === null) {
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return parent;
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}
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var dimension = dimensions[node.dimension];
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if (point[dimension] < node.obj[dimension]) {
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return innerSearch(node.left, node);
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} else {
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return innerSearch(node.right, node);
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}
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}
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var insertPosition = innerSearch(this.root, null),
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newNode,
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dimension;
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if (insertPosition === null) {
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this.root = new Node(point, 0, null);
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return;
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}
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newNode = new Node(
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point,
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(insertPosition.dimension + 1) % dimensions.length,
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insertPosition
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);
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dimension = dimensions[insertPosition.dimension];
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if (point[dimension] < insertPosition.obj[dimension]) {
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insertPosition.left = newNode;
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} else {
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insertPosition.right = newNode;
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}
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};
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this.nearest = function(point, maxNodes, maxDistance) {
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var i, result, bestNodes;
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bestNodes = new BinaryHeap(function(e) {
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return -e[1];
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});
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function nearestSearch(node) {
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var bestChild,
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dimension = dimensions[node.dimension],
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ownDistance = metric(point, node.obj),
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linearPoint = {},
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linearDistance,
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otherChild,
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i;
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function saveNode(node, distance) {
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bestNodes.push([node, distance]);
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if (bestNodes.size() > maxNodes) {
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bestNodes.pop();
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}
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}
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for (i = 0; i < dimensions.length; i += 1) {
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if (i === node.dimension) {
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linearPoint[dimensions[i]] = point[dimensions[i]];
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} else {
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linearPoint[dimensions[i]] = node.obj[dimensions[i]];
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}
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}
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linearDistance = metric(linearPoint, node.obj);
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if (node.right === null && node.left === null) {
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if (
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bestNodes.size() < maxNodes ||
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ownDistance < bestNodes.peek()[1]
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) {
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saveNode(node, ownDistance);
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}
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return;
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}
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if (node.right === null) {
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bestChild = node.left;
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} else if (node.left === null) {
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bestChild = node.right;
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} else {
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if (point[dimension] < node.obj[dimension]) {
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bestChild = node.left;
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} else {
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bestChild = node.right;
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}
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}
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nearestSearch(bestChild);
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if (bestNodes.size() < maxNodes || ownDistance < bestNodes.peek()[1]) {
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saveNode(node, ownDistance);
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}
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if (
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bestNodes.size() < maxNodes ||
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Math.abs(linearDistance) < bestNodes.peek()[1]
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) {
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if (bestChild === node.left) {
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otherChild = node.right;
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} else {
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otherChild = node.left;
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}
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if (otherChild !== null) {
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nearestSearch(otherChild);
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}
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}
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}
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if (maxDistance) {
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for (i = 0; i < maxNodes; i += 1) {
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bestNodes.push([null, maxDistance]);
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}
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}
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if (self.root) nearestSearch(self.root);
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result = [];
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for (i = 0; i < Math.min(maxNodes, bestNodes.content.length); i += 1) {
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if (bestNodes.content[i][0]) {
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result.push([bestNodes.content[i][0].obj, bestNodes.content[i][1]]);
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}
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}
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return result;
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};
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}
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function BinaryHeap(scoreFunction) {
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this.content = [];
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this.scoreFunction = scoreFunction;
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}
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BinaryHeap.prototype = {
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push: function(element) {
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// Add the new element to the end of the array.
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this.content.push(element);
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// Allow it to bubble up.
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this.bubbleUp(this.content.length - 1);
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},
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pop: function() {
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// Store the first element so we can return it later.
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var result = this.content[0];
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// Get the element at the end of the array.
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var end = this.content.pop();
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// If there are any elements left, put the end element at the
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// start, and let it sink down.
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if (this.content.length > 0) {
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this.content[0] = end;
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this.sinkDown(0);
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}
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return result;
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},
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peek: function() {
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return this.content[0];
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},
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size: function() {
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return this.content.length;
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},
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bubbleUp: function(n) {
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// Fetch the element that has to be moved.
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var element = this.content[n];
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// When at 0, an element can not go up any further.
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while (n > 0) {
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// Compute the parent element's index, and fetch it.
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var parentN = Math.floor((n + 1) / 2) - 1,
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parent = this.content[parentN];
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// Swap the elements if the parent is greater.
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if (this.scoreFunction(element) < this.scoreFunction(parent)) {
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this.content[parentN] = element;
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this.content[n] = parent;
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// Update 'n' to continue at the new position.
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n = parentN;
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}
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// Found a parent that is less, no need to move it further.
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else {
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break;
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}
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}
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},
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sinkDown: function(n) {
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// Look up the target element and its score.
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var length = this.content.length,
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element = this.content[n],
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elemScore = this.scoreFunction(element);
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while (true) {
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// Compute the indices of the child elements.
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var child2N = (n + 1) * 2,
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child1N = child2N - 1;
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// This is used to store the new position of the element,
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// if any.
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var swap = null;
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// If the first child exists (is inside the array)...
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if (child1N < length) {
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// Look it up and compute its score.
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var child1 = this.content[child1N],
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child1Score = this.scoreFunction(child1);
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// If the score is less than our element's, we need to swap.
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if (child1Score < elemScore) swap = child1N;
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}
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// Do the same checks for the other child.
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if (child2N < length) {
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var child2 = this.content[child2N],
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child2Score = this.scoreFunction(child2);
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if (child2Score < (swap == null ? elemScore : child1Score)) {
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swap = child2N;
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}
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}
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// If the element needs to be moved, swap it, and continue.
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if (swap != null) {
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this.content[n] = this.content[swap];
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this.content[swap] = element;
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n = swap;
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}
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// Otherwise, we are done.
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else {
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break;
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}
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}
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}
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};
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var dims = [];
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for (var i = 0; i < fpoint.length; i++) dims.push(i);
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var tree = new kdTree(
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fpoints,
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function(e1, e2) {
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var d = 0;
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var e3 = e1;
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if (!Array.isArray(e1)) {
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e3 = [];
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for (var key in e1) e3.push(e1[key]);
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e1 = e3;
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}
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e1.forEach(function(e, i) {
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var sqd = e1[i] - e2[i];
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d += sqd * sqd;
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});
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return d;
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},
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dims
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);
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return tree.nearest(fpoint, 1, 1000)[0][0];
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}
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```
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