9.9 KiB
id | title | challengeType | forumTopicId | dashedName |
---|---|---|---|---|
5951a53863c8a34f02bf1bdc | Closest-pair problem | 1 | 302232 | closest-pair-problem |
--description--
Provide a function to find the closest two points among a set of given points in two dimensions.
The straightforward solution is a O(n^2)
algorithm (which we can call brute-force algorithm); the pseudo-code (using indexes) could be simply:
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
A better algorithm is based on the recursive divide and conquer approach, which is O(n\log n)
a pseudo-code could be:
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
For the input, expect the argument to be an array of Point
objects with x
and y
members set to numbers. Return an object containing the key:value pairs for distance
and pair
(the pair of two closest points).
For example getClosestPair
with input array points
:
const points = [
new Point(1, 2),
new Point(3, 3),
new Point(2, 2)
];
Would return:
{
distance: 1,
pair: [
{
x: 1,
y: 2
},
{
x: 2,
y: 2
}
]
}
Note: Sort the pair
array by their x
values in incrementing order.
--hints--
getClosestPair
should be a function.
assert(typeof getClosestPair === 'function');
getClosestPair(points1).distance
should be 0.0894096443343775
.
assert.equal(getClosestPair(points1).distance, answer1.distance);
getClosestPair(points1).pair
should be [ { x: 7.46489, y: 4.6268 }, { x: 7.46911, y: 4.71611 } ]
.
assert.deepEqual(
JSON.parse(JSON.stringify(getClosestPair(points1))).pair,
answer1.pair
);
getClosestPair(points2).distance
should be 65.06919393998976
.
assert.equal(getClosestPair(points2).distance, answer2.distance);
getClosestPair(points2).pair
should be [ { x: 37134, y: 1963 }, { x: 37181, y: 2008 } ]
.
assert.deepEqual(
JSON.parse(JSON.stringify(getClosestPair(points2))).pair,
answer2.pair
);
getClosestPair(points3).distance
should be 6754.625082119658
.
assert.equal(getClosestPair(points3).distance, answer3.distance);
getClosestPair(points3).pair
should be [ { x: 46817, y: 64975 }, { x: 48953, y: 58567 } ]
.
assert.deepEqual(
JSON.parse(JSON.stringify(getClosestPair(points3))).pair,
answer3.pair
);
--seed--
--after-user-code--
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--
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--
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);
}