Closest pair of points in linearithmic time in Java

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);
	}
}

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