---
id: 5951a53863c8a34f02bf1bdc
title: Problem des dichtesten Punktpaares
challengeType: 1
forumTopicId: 302232
dashedName: closest-pair-problem
---
# --description--
Provide a function to find the closest two points among a set of given points in two dimensions.
Die einfache Lösung ist ein $O(n^2)$-Algorithmus (den wir *Brute-force-Algorithmus* nennen können); der Pseudocode (unter Verwendung von Indizes) könnte einfach sein:
bruteForceClosestPair of P(1), P(2), ... P(N)
if N < 2 then
return ∞
else
minDistance ← |P(1) - P(2)|
minPoints ← { P(1), P(2) }
foreach i ∈ [1, N-1]
foreach j ∈ [i+1, N]
if |P(i) - P(j)| < minDistance then
minDistance ← |P(i) - P(j)|
minPoints ← { P(i), P(j) }
endif
endfor
endfor
return minDistance, minPoints
endif
Ein besserer Algorithmus basiert auf dem rekursiven Divide-and-Conquer-Ansatz, der $O(n\log n)$ ist, ein Pseudocode sein könnte:
closestPair von (xP, yP)
wobei xP P(1) .. ist P(N) sortiert nach der x-Koordinate, und
yP ist P(1) .. P(N) sorted by y coordinate (ascending order)
if N ≤ 3 then
return closest points of xP using brute-force algorithm
else
xL ← points of xP from 1 to ⌈N/2⌉
xR ← points of xP from ⌈N/2⌉+1 to N
xm ← xP(⌈N/2⌉)x
yL ← { p ∈ yP : px ≤ xm }
yR ← { p ∈ yP : px > xm }
(dL, pairL) ← closestPair of (xL, yL)
(dR, pairR) ← closestPair of (xR, yR)
(dmin, pairMin) ← (dR, pairR)
if dL < dR then
(dmin, pairMin) ← (dL, pairL)
endif
yS ← { p ∈ yP : |xm - px| < dmin }
nS ← number of points in yS
(closest, closestPair) ← (dmin, pairMin)
for i from 1 to nS - 1
k ← i + 1
while k ≤ nS and yS(k)y - yS(i)y < dmin
if |yS(k) - yS(i)| < closest then
(closest, closestPair) ← (|yS(k) - yS(i)|, {yS(k), yS(i)})
endif
k ← k + 1
endwhile
endfor
return closest, closestPair
endif
Für die Eingabe wird erwartet, dass das Argument ein Array von `Point`-Objekten mit `x` und `y` Mitglieder auf Zahlen gesetzt. Gibt ein Objekt zurück, das die Schlüssel:Wert-Paare für `distance` und `pair` (das Paar der beiden nächstgelegenen Punkte) enthält.
Zum Beispiel `getClosestPair` mit dem Eingabefeld `points`:
```js
const points = [
new Point(1, 2),
new Point(3, 3),
new Point(2, 2)
];
```
Würde zurückgeben:
```js
{
distance: 1,
pair: [
{
x: 1,
y: 2
},
{
x: 2,
y: 2
}
]
}
```
**Note:** sortiere die `pair`-Anordnung nach ihren `x`-Werten in aufsteigender Reihenfolge.
# --hints--
`getClosestPair` sollte eine Funktion sein.
```js
assert(typeof getClosestPair === 'function');
```
`getClosestPair(points1).distance` sollte `0.0894096443343775` sein.
```js
assert.equal(getClosestPair(points1).distance, answer1.distance);
```
`getClosestPair(points1).pair` sollte `[ { x: 7.46489, y: 4.6268 }, { x: 7.46911, y: 4.71611 } ]` sein.
```js
assert.deepEqual(
JSON.parse(JSON.stringify(getClosestPair(points1))).pair,
answer1.pair
);
```
`getClosestPair(points2).distance` sollte `65.06919393998976` sein.
```js
assert.equal(getClosestPair(points2).distance, answer2.distance);
```
`getClosestPair(points2).pair` sollte `[ { x: 37134, y: 1963 }, { x: 37181, y: 2008 } ]` sein.
```js
assert.deepEqual(
JSON.parse(JSON.stringify(getClosestPair(points2))).pair,
answer2.pair
);
```
`getClosestPair(points3).distance` sollte `6754.625082119658` sein.
```js
assert.equal(getClosestPair(points3).distance, answer3.distance);
```
`getClosestPair(points3).pair` sollte `[ { x: 46817, y: 64975 }, { x: 48953, y: 58567 } ]` sein.
```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);
}
```