---
id: 5951a53863c8a34f02bf1bdc
title: Problema della coppia più vicina
challengeType: 5
forumTopicId: 302232
dashedName: closest-pair-problem
---
# --description--
Crea una funzione che trova i due punti più vicini tra quelli in un set di punti in due dimensioni.
La soluzione diretta è un algoritmo $O(n^2)$ (che possiamo chiamare *algoritmo a forza bruta* (brute-force)); lo pseudo-codice (usando indici) potrebbe essere semplicemente:
bruteForceClosestPair di 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
Un algoritmo migliore è basato sull'approccio ricorsivo dividi e conquista, che è $O(n\log n)$, uno pseudocodice potrebbe essere:
closestPair 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)
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
Per l'input, aspettati che l'argomento sia un array di oggetti `Point` con i membri `x` e `y` impostati come numeri. Restituisci un oggetto contenete le coppie chiave-valore per `distance` e `pair` (la coppia dei due punti più vicini).
Per esempio `getClosestPair` con input array `points`:
```js
const points = [
new Point(1, 2),
new Point(3, 3),
new Point(2, 2)
];
```
Restituirebbe:
```js
{
distance: 1,
pair: [
{
x: 1,
y: 2
},
{
x: 2,
y: 2
}
]
}
```
**Nota:** Ordina l'array `pair` secondo i valori `x` in ordine crescente.
# --hints--
`getClosestPair` dovrebbe essere una funzione.
```js
assert(typeof getClosestPair === 'function');
```
`getClosestPair(points1).distance` dovrebbe essere `0.0894096443343775`.
```js
assert.equal(getClosestPair(points1).distance, answer1.distance);
```
`getClosestPair(points1).pair` dovrebbe essere `[ { x: 7.46489, y: 4.6268 }, { x: 7.46911, y: 4.71611 } ]`.
```js
assert.deepEqual(
JSON.parse(JSON.stringify(getClosestPair(points1))).pair,
answer1.pair
);
```
`getClosestPair(points2).distance` dovrebbe essere `65.06919393998976`.
```js
assert.equal(getClosestPair(points2).distance, answer2.distance);
```
`getClosestPair(points2).pair` dovrebbe essere `[ { x: 37134, y: 1963 }, { x: 37181, y: 2008 } ]`.
```js
assert.deepEqual(
JSON.parse(JSON.stringify(getClosestPair(points2))).pair,
answer2.pair
);
```
`getClosestPair(points3).distance` dovrebbe essere `6754.625082119658`.
```js
assert.equal(getClosestPair(points3).distance, answer3.distance);
```
`getClosestPair(points3).pair` dovrebbe essere `[ { x: 46817, y: 64975 }, { x: 48953, y: 58567 } ]`.
```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);
}
```