COS 226 Programming Assignment

Kd-Trees

Remark: this assignment is brand new. So please email Kevin with any typos, ambiguities, or clarifications.

Write a data type using a 2d-tree to represent a set of points in the unit square (all points have x- and y-coordinates between 0 and 1) to support efficient range search (find all of the points contained in a query rectangle) and nearest neighbor search (find the closest point to a query point). 2d-trees have numerous applications from classifying astronomical objects to computer animation to speeding up neural networks to mining data to image retrieval.

Geometric primitives. To get started, implement two geometric primitives: one for points in the plane and one for axis-aligned rectangles in the plane.

Write an immutable data type Point.java that implements the following API.

public class Point implements Comparable<Point> {
   public Point(double x, double y)                // construct the point (x, y)
   public double x()                               // x-coordinate
   public double y()                               // y-coordinate
   public double distanceTo(Point that)            // Euclidean distance between two points
   public double distance2To(Point that)           // square of Euclidean distance between two points
   public int compareTo(Point that)                // for use in a symbol table
   public boolean equals(Object z)                 // does this point equal z?
   public void draw()                              // draw to standard draw
   public String toString()                        // string representation
}

public class RectHV {
   public RectHV(double xmin, double ymin,         // construct the rectangle [xmin, ymin] x [xmax, ymax]
                 double xmax, double ymax)         // throw an exception if (xmin > xmax) or (ymin > ymax)
   public double xmin()                            // minimum x-coordinate of rectangle
   public double ymin()                            // minimum y-coordinate of rectangle
   public double xmax()                            // maximum x-coordinate of rectangle
   public double ymax()                            // maximum y-coordinate of rectangle
   public boolean contains(Point p)                // does this rectangle contain p?
   public boolean intersects(RectHV that)          // do the two rectangles intersect?
   public double distanceTo(Point p)               // Euclidean distance from p to closest point in rectangle
   public double distance2To(Point p)              // square of Euclidean distance from p to closest point in rectangle
   public boolean equals(Object z)                 // does this rectangle equal z?
   public void draw()                              // draw to standard draw
   public String toString()                        // string representation
}

Brute-force implementation. Write a data type PointSET.java that represents a set of points in the unit square. Implement the following API by using a red-black tree (using either SET or TreeSet).

public class PointSET {
   public PointSET()                               // construct an empty set of points
   public boolean isEmpty()                        // is the set empty?
   public int size()                               // number of points in the set
   public void insert(Point p)                     // add the point p to the set
   public boolean contains(Point p)                // does the set contain p?
   public void draw()                              // draw all of the points to standard draw
   public Point nearest(Point p)                   // nearest neighbor in the set to p (null if set is empty)
   public Iterable<Point> range(RectHV rect)       // points in the set that are in the rectangle
}

2d-tree implementation. Write a data type KdTree.java that implements the same API as PointSET, but using a 2d-tree instead of a red-black tree. A 2d-tree is a generalization of a binary search tree to two-dimensional keys. The idea is to build a binary search tree with points in the nodes, using the x- and y-coordinates of the points as keys in strictly alternating sequence.

• Search and insert. The algorithms for search and insert are similar to those for binary search trees, but at the root we use the x-coordinate (if the point to be inserted has a smaller x-coordinate than the point at the root, go left; otherwise go right); then at the next level, we use the y-coordinate (if the point to be inserted has a smaller y-coordinate than the point in the node, go left; otherwise go right); then at the next level the x-coordinate and so forth.

insert (0.7, 0.2)	insert (0.5, 0.4)	insert (0.2, 0.3)	insert (0.4, 0.7)	insert (0.9, 0.6)

• Draw. A 2d-tree divides the unit square in a simple way: all the points to the left of the root go in the left subtree; all those to the right go in the right subtree; and so forth, recursively. Your draw() method should draw all of the points to standard draw in black and the subdivisions in red (for vertical splits) and blue (for horizontal splits). This method need not be efficient—it is primarily for debugging.

The prime advantage of a 2d-tree over a binary search tree is that it supports efficient implementation of range search and nearest neighbor search. Each node corresponds to an axis-aligned rectangle in the unit square, which encloses all of the points in its subtree. The root corresponds to the unit square; the left and right children of the root corresponds to the two rectangles split by the x-coordinate of the point at the root; and so forth.

• Range search. To find all points contained in a given query rectangle, start at the root and recursively search for points in both subtrees using the following pruning rule: if the query rectangle does not intersect the rectangle corresponding to a node, there is no need to explore that node (or its subtrees). A subtree is searched only if it might contain a point contained in the query rectangle.

• Nearest neighbor search. To find the closest point to a given query point, start at the root and recursively search in both subtrees using the following pruning rule: if the closest point discovered so far is closer than the distance between the query point and the rectangle corresponding to a node, there is no need to explore that node (or its subtrees). A subtree is searched only if it might contain a point that is closer than the best one found so far. The effectiveness of the pruning strategy depends on quickly finding a nearby point. To do this, organize your recursive method so that it begins by moving down the 2d-tree by considering the same sequence of nodes that would be considered if you were inserting the query point.

Clients.  You may use the following interactive client programs to test and debug your code.

• KdTreeVisualizer.java computes and draws the 2d-tree that results from the sequence of points clicked by the user in the standard drawing window.

• RangeSearchVisualizer.java reads a sequence of points from standard input and inserts those points into a 2d-tree. Then, it performs range searches on the axis-aligned rectangles dragged by the user in the standard drawing window.

• NearestNeighborVisualizer.java reads a sequence of points from standard input and inserts those points into a 2d-tree. Then, it performs nearest neighbor queries on point corresponding to the location of the mouse in the standard drawing window.

Analysis.  Analyze your approach to this problem giving estimates of its time and space requirements by answering the relevant questions in the readme.txt file. In particular:

• Give the total memory usage in bytes (using tilde notation) of your 2d-tree data structure as a function of the number of points N. Count all memory that is used by your 2d-tree, including memory for the nodes, points, and rectangles.

• Give the expected running time in seconds (using tilde notation) to build a 2d-tree on N random points in the unit square. (Do not count the time to read in the points from standard input.)

• How many nearest neighbor calculations can your 2d-tree implementation perform per second for input100K.txt (100,000 points) and input1M.txt (1 million points), where the query points are random points in the unit square? (Do not count the time to read in the points or to build the 2d-tree.) Repeat the question but with the brute-force implementation.

Submission.  Submit Point.java, RectHV.java, PointSET.java, and KDTree.java and any other files needed by your program (excluding those in stdlib.jar and adt.jar). Finally, submit a readme.txt file and answer the questions.