Offen we need to ask Google Maps, what’s the nearest restaurant/hotel/whatever nearby? Then Google Maps will take your GPS information (latitude, longitude), and do a search on the map to find the nearest location. This is a multidimensional nearest neighbor search problem, in which case k-d tree can be useful. K-d tree is a binary tree of k-dimensional data, and is interesting that it splits the left and right children by different dimensions at different depth of the tree. Similar to other binary trees, searches take O(log n) time on average.

Here’s Java implementation for nearest location search using kd-tree:

import javax.annotation.Nonnull ; import javax.annotation.Nullable ; import java.util.ArrayList ; import java.util.Collections ; import java.util.Comparator ; import java.util.List ; import static java . lang . Math . cos ; import static java . lang . Math . sin ; import static java . lang . Math . toRadians ; class LocationKDTree { private static final int K = 3 ; // 3-d tree private final Node tree ; public LocationKDTree ( @Nonnull final List < Location > locations ) { final List < Node > nodes = new ArrayList <>( locations . size ()); for ( final Location location : locations ) { nodes . add ( new Node ( location )); } tree = buildTree ( nodes , 0 ); } @Nullable public Location findNearest ( final double latitude , final double longitude ) { final Node node = findNearest ( tree , new Node ( latitude , longitude ), 0 ); return node == null ? null : node . location ; } private static Node findNearest ( final Node current , final Node target , final int depth ) { final int axis = depth % K ; final int direction = getComparator ( axis ). compare ( target , current ); final Node next = ( direction < 0 ) ? current . left : current . right ; final Node other = ( direction < 0 ) ? current . right : current . left ; Node best = ( next == null ) ? current : findNearest ( next , target , depth + 1 ); if ( current . euclideanDistance ( target ) < best . euclideanDistance ( target )) { best = current ; } if ( other != null ) { if ( current . verticalDistance ( target , axis ) < best . euclideanDistance ( target )) { final Node possibleBest = findNearest ( other , target , depth + 1 ); if ( possibleBest . euclideanDistance ( target ) < best . euclideanDistance ( target )) { best = possibleBest ; } } } return best ; } @Nullable private static Node buildTree ( final List < Node > items , final int depth ) { if ( items . isEmpty ()) { return null ; } Collections . sort ( items , getComparator ( depth % K )); final int index = items . size () / 2 ; final Node root = items . get ( index ); root . left = buildTree ( items . subList ( 0 , index ), depth + 1 ); root . right = buildTree ( items . subList ( index + 1 , items . size ()), depth + 1 ); return root ; } private static class Node { Node left ; Node right ; Location location ; final double [] point = new double [ K ]; Node ( final double latitude , final double longitude ) { point [ 0 ] = ( double ) ( cos ( toRadians ( latitude )) * cos ( toRadians ( longitude ))); point [ 1 ] = ( double ) ( cos ( toRadians ( latitude )) * sin ( toRadians ( longitude ))); point [ 2 ] = ( double ) ( sin ( toRadians ( latitude ))); } Node ( final Location location ) { this ( location . latitude , location . longitude ); this . location = location ; } double euclideanDistance ( final Node that ) { final double x = this . point [ 0 ] - that . point [ 0 ]; final double y = this . point [ 1 ] - that . point [ 1 ]; final double z = this . point [ 2 ] - that . point [ 2 ]; return x * x + y * y + z * z ; } double verticalDistance ( final Node that , final int axis ) { final double d = this . point [ axis ] - that . point [ axis ]; return d * d ; } } private static Comparator < Node > getComparator ( final int i ) { return NodeComparator . values ()[ i ]; } private static enum NodeComparator implements Comparator < Node > { x { @Override public int compare ( final Node a , final Node b ) { return Double . compare ( a . point [ 0 ], b . point [ 0 ]); } }, y { @Override public int compare ( final Node a , final Node b ) { return Double . compare ( a . point [ 1 ], b . point [ 1 ]); } }, z { @Override public int compare ( final Node a , final Node b ) { return Double . compare ( a . point [ 2 ], b . point [ 2 ]); } } } } class Location { public double latitude ; public double longitude ; public String name ; }

The code should be quite straightforward to use: First you need to initialized a `LocationKDTree`

object with a list of locations, and then you can find the nearest location of given latitude/longitude by `findNearest`

method. One thing worth noting is that the tree is built as a 3-d tree, because you know, the earth is round! The trick is to convert the (latitude, longitude) pair to a (x, y, z) coordinate by:

x = cos ( toRadians ( latitude )) * cos ( toRadians ( longitude )) y = cos ( toRadians ( latitude )) * sin ( toRadians ( longitude )) z = sin ( toRadians ( latitude ))

`toRadians`

is a Java function that converts an angel from degree measure to radian measure.