The challenge
Given a number of points on a plane, your task is to find two points with the smallest distance between them in linearithmic O(n log n) time.
Example
1 2 3 4 5 6 7 8 9
1
2 . A
3 . D
4 . F
5 . C
6
7 . E
8 . B
9 . G
For the plane above, the input will be:
[
[2,2], // A
[2,8], // B
[5,5], // C
[6,3], // D
[6,7], // E
[7,4], // F
[7,9] // G
]
=> closest pair is: [[6,3],[7,4]] or [[7,4],[6,3]]
(both answers are valid)
The two points that are closest to each other are D and F.
Expected answer should be an array with both points in any order.
Goal
The goal is to come up with a function that can find two closest points for any arbitrary array of points, in a linearithmic time.
Point class is preloaded for you as:
public class Point {
public double x, y;
public Point() {
x = y = 0.0;
}
public Point(double x, double y) {
this.x = x;
this.y = y;
}
@Override
public String toString() {
return String.format("(%f, %f)", x, y);
}
@Override
public int hashCode() {
return Double.hashCode(x) ^ Double.hashCode(y);
}
@Override
public boolean equals(Object obj) {
if (obj instanceof Point) {
Point other = (Point) obj;
return x == other.x && y == other.y;
} else {
return false;
}
}
}
More information on wikipedia.
The solution in Java code
Option 1:
import java.util.*;
import java.lang.*;
public class Solution {
public static List<Point> closestPair(List<Point> points) {
final List<Point> pair = new ArrayList<Point>();
final List<Point> arr = new ArrayList<>(points);
arr.sort((a, b) -> Double.valueOf(a.x).compareTo(Double.valueOf(b.x)));
final int n = points.size();
double l = Double.valueOf(Integer.MAX_VALUE);
double tolerance = Math.sqrt(l);
int a = 0, b = 0;
for (int i = 0; i + 1 < n; i++) {
for (int j = i + 1; j < n; j++) {
if (arr.get(j).x >= arr.get(i).x + tolerance) break;
final double ls = Math.pow(arr.get(i).x - arr.get(j).x, 2) + Math.pow(arr.get(i).y - arr.get(j).y, 2);
if (ls < l) {
l = ls;
tolerance = Math.sqrt(l);
a = i;
b = j;
}
}
}
pair.add(arr.get(a));
pair.add(arr.get(b));
return pair;
}
}
Option 2:
import java.util.Arrays;
import java.util.*;
import java.util.stream.Collectors;
public class Solution {
public static List<Point> closestPair(List<Point> points) {
return divideAndConquer(points.stream().sorted((o1, o2) -> Double.compare(o1.x,o2.x))
.collect(Collectors.toList()).toArray(new Point[points.size()]),points.size());
}
private static List<Point> divideAndConquer(Point[] pointSet, int size){
if(size<=3){
double minDist=Double.MAX_VALUE;
Point aMin = null;
Point bMin = null;
for(int i=0; i<size-1;i++){
Point a = pointSet[i];
for(int j=i+1; j<size; j++){
Point b = pointSet[j];
double dist = Math.abs(a.x-b.x)+Math.abs(a.y-b.y);
if(dist<minDist ){
minDist=dist;
aMin = a;
bMin = b;
}
}
}
return Arrays.asList(aMin,bMin);
}
int mid = size/2;
Point midPoint = pointSet[mid];
List<Point> lMinPoints = divideAndConquer(Arrays.copyOfRange(pointSet, 0, mid), mid);
List<Point> rMinPoints = divideAndConquer(Arrays.copyOfRange(pointSet, mid, size), size-mid);
double mDistL = calculateDist(lMinPoints.get(0),lMinPoints.get(1));
double mDistR = calculateDist(rMinPoints.get(0),rMinPoints.get(1));
Point aMin = null;
Point bMin = null;
double minDist;
if(mDistL<mDistR){
minDist = mDistL;
aMin = lMinPoints.get(0);
bMin = lMinPoints.get(1);
} else{
minDist = mDistR;
aMin = rMinPoints.get(0);
bMin = rMinPoints.get(1);
}
double streamMinDist = minDist;
Point[] midSetSorted = Arrays.stream(pointSet).filter(point -> Math.abs(midPoint.x-point.x)<streamMinDist)
.sorted((o1, o2) -> Double.compare(o1.y,o2.y)).toArray(Point[]::new);
if(midSetSorted.length==0){
return Arrays.asList(aMin,bMin);
}
for(int i=0; i<midSetSorted.length-1; i++){
Point point = midSetSorted[i];
for(int j=i+1; j<midSetSorted.length && Math.abs(midSetSorted[j].y - point.y) < minDist; j++){
if(calculateDist(midSetSorted[i],midSetSorted[j])<minDist){
minDist = calculateDist(midSetSorted[i],midSetSorted[j]);
aMin = midSetSorted[j];
bMin = midSetSorted[i];
}
}
}
return Arrays.asList(aMin,bMin);
}
private static double calculateDist(Point a, Point b){
return Math.abs(a.x-b.x)+Math.abs(a.y-b.y);
}
}
Test cases to validate our solution
import java.util.Arrays;
import java.util.Comparator;
import java.util.List;
import org.junit.Assert;
import org.junit.FixMethodOrder;
import org.junit.Test;
import org.junit.runners.MethodSorters;
@FixMethodOrder(MethodSorters.NAME_ASCENDING)
public class SolutionTest {
@Test
public void test01_Example() {
List<Point> points = Arrays.asList(
new Point(2, 2), //A
new Point(2, 8), //B
new Point(5, 5), //C
new Point(6, 3), //D
new Point(6, 7), //E
new Point(7, 4), //F
new Point(7, 9) //G
);
List<Point> result = Solution.closestPair(points);
List<Point> expected = Arrays.asList(new Point(6, 3), new Point(7, 4));
verify(expected, result);
}
@Test
public void test02_TwoPoints() {
List<Point> points = Arrays.asList(
new Point(2, 2),
new Point(6, 3)
);
List<Point> result = Solution.closestPair(points);
List<Point> expected = Arrays.asList(new Point(6, 3), new Point(2, 2));
verify(expected, result);
}
@Test
public void test03_DuplicatedPoint() {
List<Point> points = Arrays.asList(
new Point(2, 2), //A
new Point(2, 8), //B
new Point(5, 5), //C
new Point(5, 5), //C
new Point(6, 3), //D
new Point(6, 7), //E
new Point(7, 4), //F
new Point(7, 9) //G
);
List<Point> result = Solution.closestPair(points);
List<Point> expected = Arrays.asList(new Point(5, 5), new Point(5,5));
verify(expected, result);
}
private void verify(List<Point> expected, List<Point> actual) {
Comparator<Point> comparer = Comparator.<Point>comparingDouble(p -> p.x);
Assert.assertNotNull("Returned array cannot be null.", actual);
Assert.assertEquals("Expected exactly two points.", 2, actual.size());
Assert.assertFalse("Returned points must not be null.", actual.get(0) == null || actual.get(1) == null);
expected.sort(comparer);
actual.sort(comparer);
boolean eq = expected.get(0).x == actual.get(0).x && expected.get(0).y == actual.get(0).y
&& expected.get(1).x == actual.get(1).x && expected.get(1).y == actual.get(1).y;
Assert.assertTrue(String.format("Expected: %s, Actual: %s", expected.toString(), actual.toString()), eq);
}
}