Kd-Trees

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Frequently Asked Questions

What should I do if a point has the same x-coordinate as the point in a node when inserting / searching in a 2d-tree? Go the right subtree as specified.

Can I assume that all x- or y-coordinates of points inserted into the KdTree will be between 0 and 1? Yes. You may also assume that the insert(), contains(), and nearest() methods in KdTree are passed points with x- and y-coordinates between 0 and 1. You may also assume that the range() method in KdTree is passed a rectangle that lies in the unit box.

What should I do if a point is inserted twice in the data structure? The data structure represents a set of points, so you should keep only one copy.

Is a point on the boundary of a rectangle considered inside it? Do two rectangle intersect if they have just one point in common? Yes and yes, consistent with the implementation of RectHV.java.

What does the notation [0.5, 0.75] × [0.25, 0.375] mean when specifying a rectangle? It is the Cartesian product of the x-interval [0.5, 0.75] and the y-interval [0.25, 0.375]: the rectangle that includes all points with both 0.5 ≤ x ≤ 0.75 and 0.25 ≤ y ≤ 0.375. Note that the arguments to the RectHV constructor are in the order xmin, ymin, xmax, and ymax but the toString() method uses the Cartesian product notation.

How should I scale the coordinate system when drawing? Don't, please keep the default range of 0 to 1.

How should I set the size and color of the points and rectangles when drawing? Use StdDraw.setPenColor(StdDraw.BLACK) and StdDraw.setPenRadius(0.01) before before drawing the points; use StdDraw.setPenColor(StdDraw.RED) or StdDraw.setPenColor(StdDraw.BLUE) and StdDraw.setPenRadius() before drawing the splitting lines.

What should range() return if there are no points in the range? It should return an Iterable<Point2D> object with zero points.

How much memory does a Point2D object use? For simplicity, assume that each Point2D object uses 32 bytes—in reality, it uses a bit more because of the Comparator instance variables.

How much memory does a RectHV object use? You can look at the code and analyze it.

I run out of memory when running some of the large sample files. What should I do? Be sure to ask Java for additional memory, e.g., java -Xmx1600m RangeSearchVisualizer input1M.txt.

Testing

Testing. A good way to test KdTree is to perform the same sequence of operations on both the PointSET and KdTree data types and identify any discrepancies. The sample clients RangeSearchVisualizer.java and NearestNeighborVisualizer.java take this approach.

Sample input files.   The zip file kdtree.zip contains many sample input files for testing.

• circleN.txt contains n points on the circumference of the circle centered on (0.5, 0.5) of radius 0.5.

Possible Progress Steps

1. Node data type. There are several reasonable ways to represent a node in a 2d-tree. One approach is to include the point, a link to the left/bottom subtree, a link to the right/top subtree, and an axis-aligned rectangle corresponding to the node.

   private static class Node {
      private Point2D p;      // the point
      private RectHV rect;    // the axis-aligned rectangle corresponding to this node
      private Node lb;        // the left/bottom subtree
      private Node rt;        // the right/top subtree
   }
   

   Unlike the Node class for BST, this Node class is static because it does not refer to a generic Key or Value type that depends on the object associated with the outer class. This saves the 8-byte inner class object overhead. (Making the Node class static in BST is also possible if you make the Node type itself generic as well). Also, since we don't need to implement the rank and select operations, there is no need to store the subtree size.

2. Writing KdTree. Start by writing isEmpty() and size(). These should be very easy. From there, write a simplified version of insert() which does everything except set up the RectHV for each node. Write the contains() method, and use this to test that insert() was implemented properly. Note that insert() and contains() are best implemented by using private helper methods analogous to those found on page 399 of the book or by looking at BST.java. We recommend using orientation as an argument to these helper methods.

   Now add the code to insert() which sets up the RectHV for each Node. Next, write draw(), and use this to test these rectangles. Finish up KdTree with the nearest and range methods. Finally, test your implementation using our interactive client programs as well as any other tests you'd like to conduct.

Optimizations

• Squared distances. Whenever you need to compare two Euclidean distances, it is often more efficient to compare the squares of the two distances to avoid the expensive operation of taking square roots. Everyone should implement this optimization because it is both easy to do and likely a bottleneck.

• Range search. Instead of checking whether the query rectangle intersects the rectangle corresponding to a node, it suffices to check only whether the query rectangle intersects the splitting line segment: if it does, then recursively search both subtrees; otherwise, recursively search the one subtree where points intersecting the query rectangle could be.

• Save memory. You are not required to explicitly store a RectHV in each 2d-tree node (though it is probably wise in your first version).