---
id: 5951a53863c8a34f02bf1bdc
title: 最近点対の問題
challengeType: 1
forumTopicId: 302232
dashedName: closest-pair-problem
---

# --description--

2次元の任意の点の集合の中から最も近い2点を見つける関数を提供します。

簡単な解決策は、(*brute-force algorithm*と呼ばれる)$O(n^2)$ アルゴリズムを使用することです。疑似コード(インデックス使用)は単純に以下のようになります。

<pre><strong>bruteForceClosestPair</strong> of P(1), P(2), ... P(N)
<strong>if</strong> N &#x3C; 2 <strong>then</strong>
  <strong>return</strong> ∞
<strong>else</strong>
  minDistance ← |P(1) - P(2)|
  minPoints ← { P(1), P(2) }
  <strong>foreach</strong> i ∈ [1, N-1]
    <strong>foreach</strong> j ∈ [i+1, N]
      <strong>if</strong> |P(i) - P(j)| &#x3C; minDistance <strong>then</strong>
        minDistance ← |P(i) - P(j)|
        minPoints ← { P(i), P(j) }
      <strong>endif</strong>
    <strong>endfor</strong>
  <strong>endfor</strong>
  <strong>return</strong> minDistance, minPoints
<strong>endif</strong>
</pre>

より好ましいアルゴリズムは、$O(n\log n)$の再帰的な分割統治法に基づいており、疑似コードは以下のようになります。

<pre><strong>closestPair</strong> of (xP, yP)
  where xP is P(1) .. P(N) sorted by x coordinate, and
  yP is P(1) .. P(N) sorted by y coordinate (ascending order)
<strong>if</strong> N ≤ 3 <strong>then</strong>
  <strong>return</strong> closest points of xP using brute-force algorithm
<strong>else</strong>
  xL ← points of xP from 1 to ⌈N/2⌉
  xR ← points of xP from ⌈N/2⌉+1 to N
  xm ← xP(⌈N/2⌉)<sub>x</sub>
  yL ← { p ∈ yP : p<sub>x</sub> ≤ xm }
  yR ← { p ∈ yP : p<sub>x</sub> > xm }
  (dL, pairL) ← closestPair of (xL, yL)
  (dR, pairR) ← closestPair of (xR, yR)
  (dmin, pairMin) ← (dR, pairR)
  <strong>if</strong> dL &#x3C; dR <strong>then</strong>
    (dmin, pairMin) ← (dL, pairL)
  <strong>endif</strong>
  yS ← { p ∈ yP : |xm - p<sub>x</sub>| &#x3C; dmin }
  nS ← number of points in yS
  (closest, closestPair) ← (dmin, pairMin)
  <strong>for</strong> i <strong>from</strong> 1 <strong>to</strong> nS - 1
    k ← i + 1
    <strong>while</strong> k ≤ nS <strong>and</strong> yS(k)<sub>y</sub> - yS(i)<sub>y</sub> &#x3C; dmin
      <strong>if</strong> |yS(k) - yS(i)| &#x3C; closest <strong>then</strong>
        (closest, closestPair) ← (|yS(k) - yS(i)|, {yS(k), yS(i)})
      <strong>endif</strong>
      k ← k + 1
    <strong>endwhile</strong>
  <strong>endfor</strong>
  <strong>return</strong> closest, closestPair
<strong>endif</strong>
</pre>

入力には、引数を `x` と `y` メンバーが数字に設定されている `Point` オブジェクトの配列であると想定します。 `distance` と `pair` の key:value ペア (最も近い2点のペア) を含むオブジェクトを返します。

例えば、入力配列 `points` を持つ `getClosestPair` の場合

```js
const points = [
  new Point(1, 2),
  new Point(3, 3),
  new Point(2, 2)
];
```

次のものを返します。

```js
{
  distance: 1,
  pair: [
    {
      x: 1,
      y: 2
    },
    {
      x: 2,
      y: 2
    }
  ]
}
```

**注記: ** `pair` 配列を `x` の値で昇順に並べ替えます。


# --hints--

`getClosestPair` という関数です。

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

`getClosestPair(points1).distance` は `0.0894096443343775` になります。

```js
assert.equal(getClosestPair(points1).distance, answer1.distance);
```

`getClosestPair(points1).pair` は `[ { x: 7.46489, y: 4.6268 }, { x: 7.46911, y: 4.71611 } ]` になります。

```js
assert.deepEqual(
  JSON.parse(JSON.stringify(getClosestPair(points1))).pair,
  answer1.pair
);
```

`getClosestPair(points2).distance` は `65.06919393998976` になります。

```js
assert.equal(getClosestPair(points2).distance, answer2.distance);
```

`getClosestPair(points2).pair` は `[ { x: 37134, y: 1963 }, { x: 37181, y: 2008 } ]` になります。

```js
assert.deepEqual(
  JSON.parse(JSON.stringify(getClosestPair(points2))).pair,
  answer2.pair
);
```

`getClosestPair(points3).distance` は `6754.625082119658` になります。

```js
assert.equal(getClosestPair(points3).distance, answer3.distance);
```

`getClosestPair(points3).pair` は `[ { x: 46817, y: 64975 }, { x: 48953, y: 58567 } ]` になります。

```js
assert.deepEqual(
  JSON.parse(JSON.stringify(getClosestPair(points3))).pair,
  answer3.pair
);
```

# --seed--

## --after-user-code--

```js
const points1 = [
    new Point(0.748501, 4.09624),
    new Point(3.00302, 5.26164),
    new Point(3.61878,  9.52232),
    new Point(7.46911,  4.71611),
    new Point(5.7819,   2.69367),
    new Point(2.34709,  8.74782),
    new Point(2.87169,  5.97774),
    new Point(6.33101,  0.463131),
    new Point(7.46489,  4.6268),
    new Point(1.45428,  0.087596)
];

const answer1 = {
  distance: 0.0894096443343775,
  pair: [
    {
      x: 7.46489,
      y: 4.6268
    },
    {
      x: 7.46911,
      y: 4.71611
    }
  ]
};

const points2 = [
  new Point(37100, 13118),
  new Point(37134, 1963),
  new Point(37181, 2008),
  new Point(37276, 21611),
  new Point(37307, 9320)
];

const answer2 = {
  distance: 65.06919393998976,
  pair: [
    {
      x: 37134,
      y: 1963
    },
    {
      x: 37181,
      y: 2008
    }
  ]
};

const points3 = [
  new Point(16910, 54699),
  new Point(14773, 61107),
  new Point(95547, 45344),
  new Point(95951, 17573),
  new Point(5824, 41072),
  new Point(8769, 52562),
  new Point(21182, 41881),
  new Point(53226, 45749),
  new Point(68180, 887),
  new Point(29322, 44017),
  new Point(46817, 64975),
  new Point(10501, 483),
  new Point(57094, 60703),
  new Point(23318, 35472),
  new Point(72452, 88070),
  new Point(67775, 28659),
  new Point(19450, 20518),
  new Point(17314, 26927),
  new Point(98088, 11164),
  new Point(25050, 56835),
  new Point(8364, 6892),
  new Point(37868, 18382),
  new Point(23723, 7701),
  new Point(55767, 11569),
  new Point(70721, 66707),
  new Point(31863, 9837),
  new Point(49358, 30795),
  new Point(13041, 39744),
  new Point(59635, 26523),
  new Point(25859, 1292),
  new Point(1551, 53890),
  new Point(70316, 94479),
  new Point(48549, 86338),
  new Point(46413, 92747),
  new Point(27186, 50426),
  new Point(27591, 22655),
  new Point(10905, 46153),
  new Point(40408, 84202),
  new Point(52821, 73520),
  new Point(84865, 77388),
  new Point(99819, 32527),
  new Point(34404, 75657),
  new Point(78457, 96615),
  new Point(42140, 5564),
  new Point(62175, 92342),
  new Point(54958, 67112),
  new Point(4092, 19709),
  new Point(99415, 60298),
  new Point(51090, 52158),
  new Point(48953, 58567)
];

const answer3 = {
  distance: 6754.625082119658,
  pair: [
    {
      x: 46817,
      y: 64975
    },
    {
      x: 48953,
      y: 58567
    }
  ]
}
```

## --seed-contents--

```js
const Point = function(x, y) {
  this.x = x;
  this.y = y;
};
Point.prototype.getX = function() {
  return this.x;
};
Point.prototype.getY = function() {
  return this.y;
};

function getClosestPair(pointsArr) {

  return true;
}
```

# --solutions--

```js
const Point = function(x, y) {
  this.x = x;
  this.y = y;
};
Point.prototype.getX = function() {
  return this.x;
};
Point.prototype.getY = function() {
  return this.y;
};

const mergeSort = function mergeSort(points, comp) {
    if(points.length < 2) return points;

    var n = points.length,
        i = 0,
        j = 0,
        leftN = Math.floor(n / 2),
        rightN = leftN;

    var leftPart = mergeSort( points.slice(0, leftN), comp),
        rightPart = mergeSort( points.slice(rightN), comp );

    var sortedPart = [];

    while((i < leftPart.length) && (j < rightPart.length)) {
        if(comp(leftPart[i], rightPart[j]) < 0) {
            sortedPart.push(leftPart[i]);
            i += 1;
        }
        else {
            sortedPart.push(rightPart[j]);
            j += 1;
        }
    }
    while(i < leftPart.length) {
        sortedPart.push(leftPart[i]);
        i += 1;
    }
    while(j < rightPart.length) {
        sortedPart.push(rightPart[j]);
        j += 1;
    }
    return sortedPart;
};

const closestPair = function _closestPair(Px, Py) {
    if(Px.length < 2) return { distance: Infinity, pair: [ new Point(0, 0), new Point(0, 0) ] };
    if(Px.length < 3) {
        //find euclid distance
        var d = Math.sqrt( Math.pow(Math.abs(Px[1].x - Px[0].x), 2) + Math.pow(Math.abs(Px[1].y - Px[0].y), 2) );
        return {
            distance: d,
            pair: [ Px[0], Px[1] ]
        };
    }

    var n = Px.length,
        leftN = Math.floor(n / 2),
        rightN = leftN;

    var Xl = Px.slice(0, leftN),
        Xr = Px.slice(rightN),
        Xm = Xl[leftN - 1],
        Yl = [],
        Yr = [];
    //separate Py
    for(var i = 0; i < Py.length; i += 1) {
        if(Py[i].x <= Xm.x)
            Yl.push(Py[i]);
        else
            Yr.push(Py[i]);
    }

    var dLeft = _closestPair(Xl, Yl),
        dRight = _closestPair(Xr, Yr);

    var minDelta = dLeft.distance,
        closestPair = dLeft.pair;
    if(dLeft.distance > dRight.distance) {
        minDelta = dRight.distance;
        closestPair = dRight.pair;
    }

    //filter points around Xm within delta (minDelta)
    var closeY = [];
    for(i = 0; i < Py.length; i += 1) {
        if(Math.abs(Py[i].x - Xm.x) < minDelta) closeY.push(Py[i]);
    }
    //find min within delta. 8 steps max
    for(i = 0; i < closeY.length; i += 1) {
        for(var j = i + 1; j < Math.min( (i + 8), closeY.length ); j += 1) {
            var d = Math.sqrt( Math.pow(Math.abs(closeY[j].x - closeY[i].x), 2) + Math.pow(Math.abs(closeY[j].y - closeY[i].y), 2) );
            if(d < minDelta) {
                minDelta = d;
                closestPair = [ closeY[i], closeY[j] ]
            }
        }
    }

    return {
        distance: minDelta,
        pair: closestPair.sort((pointA, pointB) => pointA.x - pointB.x)
    };
};

function getClosestPair(points) {
  const sortX = function(a, b) { return (a.x < b.x) ? -1 : ((a.x > b.x) ? 1 : 0); }
  const sortY = function(a, b) { return (a.y < b.y) ? -1 : ((a.y > b.y) ? 1 : 0); }

  const Px = mergeSort(points, sortX);
  const Py = mergeSort(points, sortY);

  return closestPair(Px, Py);
}
```
