Quadtree Nearest Neighbour Algorithm

Question

I have implemented a quadtree structure for n points as well as a method for returning an array of points within a given rectangle. I can't seem to find an algorithm to efficiently find the point that is closest to another given point. Am I missing something obvious? I assume a recursive solution is the correct approach?

Am working in Objective C but pseudo code would be fine. Additionally I am actually storing lat, long data and the distance between points is along a great circle.

EDIT: This is my tree insert and subdivide code

- (BOOL)insert:(id<PASQuadTreeDataPoint>)dataPoint {

    BOOL pointAdded = false;

    // If the point lies within the region
    if(CGRectContainsPoint(self.region, dataPoint.point)) {

        // If there are less than 4 points then add this point
        if(self.dataPoints.count < kMaxPointsPerNode) {
            [self.dataPoints addObject:dataPoint];
            pointAdded = true;
        }
        else {

            // Subdivide into 4 quadrants if not already subdivided
            if(northEast == nil) [self subdivide];

            // Attempt to add the point to one of the 4 subdivided quadrants
            if([northEast insert:dataPoint]) return true;
            if([southEast insert:dataPoint]) return true;
            if([southWest insert:dataPoint]) return true;
            if([northWest insert:dataPoint]) return true;
        }
    }

    return pointAdded;
}

- (void)subdivide {

    // Compute the half width and the origin
    CGFloat width = self.region.size.width * 0.5f;
    CGFloat height = self.region.size.height * 0.5f;
    CGFloat x = self.region.origin.x;
    CGFloat y = self.region.origin.y;

    // Create a new child quadtree with the region divided into quarters
    self.northEast = [PASQuadTree quadTreeWithRegion:CGRectMake(x + width, y, width, height)];
    self.southEast = [PASQuadTree quadTreeWithRegion:CGRectMake(x + width, y + height, width, height)];
    self.southWest = [PASQuadTree quadTreeWithRegion:CGRectMake(x, y + height, width, height)];
    self.northWest = [PASQuadTree quadTreeWithRegion:CGRectMake(x, y, width, height)];
}

EDIT: Have written this code to find the smallest node (leaf) that would contain the point:

-(PASQuadTree *)nodeThatWouldContainPoint:(CGPoint)point {

    PASQuadTree *node = nil;

    // If the point is within the region
    if (CGRectContainsPoint(self.region, point)) {

        // Set the node to this node
        node = self;

        // If the node has children
        if (self.northEast != nil) {

            // Recursively check each child to see if it would contain the point
            PASQuadTree *childNode = [self.northEast nodeThatWouldContainPoint:point];
            if (!childNode) childNode = [self.southEast nodeThatWouldContainPoint:point];
            if (!childNode) childNode = [self.southWest nodeThatWouldContainPoint:point];
            if (!childNode) childNode = [self.northWest nodeThatWouldContainPoint:point];
            if (childNode) node = childNode;
        }
    }

    return node;
}

Closer but no cigar!

Answer 1

Find the smallest square with your search point at the center and exactly one other point inside that rectangle (you need to do logn number of searches).

Let x be the distance to the other point.

Then find all the points within a square whose side is 2x and centered around your first point. For each point within this square, calculate the distance from search point and find the closest.

UPDATE: How to find one square centered around search point that contains exactly one other point?

Find a random point. Let the distance to that random point be x. Find all points within square of size x centered around search point. If there are non zero number of points within this square, then select a point at random and repeat. If there are no points, increase search square size to (2-0.5)*x (in next step (2-0.25)*x and so on.