Programming Assignment 5: Kd-Trees

Most students find that this assignment is more time consuming than previous assignments. Start early!

Create a symbol table data type whose keys are two-dimensional points. Use a 2d-tree to support efficient range search (find all of the points contained in a query rectangle) and nearest neighbor search (find a closest point to a query point). 2d-trees have numerous applications, ranging from classifying astronomical objects to computer animation to speeding up neural networks to mining data to image retrieval.

Range search and k-nearest neighbor

Geometric primitives. To get started, use the following geometric primitives for points and axis-aligned rectangles in the plane.

Geometric primitives

Use the immutable data type Point2D (part of algs4.jar) for points in the plane. Here is the subset of its API that you may use:

public class Point2D implements Comparable<Point2D> {
   public Point2D(double x, double y)              // construct the point (x, y)
   public  double x()                              // x-coordinate 
   public  double y()                              // y-coordinate 
   public  double distanceSquaredTo(Point2D that)  // square of Euclidean distance between two points 
   public     int compareTo(Point2D that)          // for use in an ordered symbol table 
   public boolean equals(Object that)              // does this point equal that object? 
   public  String toString()                       // string representation 
Use the immutable data type RectHV (part of algs4.jar) for axis-aligned rectanges. Here is the subset of its API that you may use:
public class RectHV {
   public    RectHV(double xmin, double ymin,      // construct the rectangle [xmin, xmax] x [ymin, ymax] 
                    double xmax, double 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(Point2D p)              // does this rectangle contain the point p (either inside or on boundary)? 
   public boolean intersects(RectHV that)          // does this rectangle intersect that rectangle (at one or more points)? 
   public  double distanceSquaredTo(Point2D p)     // square of Euclidean distance from point p to closest point in rectangle 
   public boolean equals(Object that)              // does this rectangle equal that object? 
   public  String toString()                       // string representation 
Do not modify these data types.

Brute-force implementation. Write a mutable data type that is symbol table with Point2D. Implement the following API by using a red-black BST (using either RedBlackBST from algs4.jar or java.util.TreeMap); do not implement your own red-black BST.

public class PointST<Value> {
   public         PointST()                                // construct an empty symbol table of points 
   public           boolean isEmpty()                      // is the symbol table empty? 
   public               int size()                         // number of points 
   public              void put(Point2D p, Value val)      // associate the value val with point p
   public             Value get(Point2D p)                 // value associated with point p 
   public           boolean contains(Point2D p)            // does the symbol table contain point p? 
   public Iterable<Point2D> points()                       // all points in the symbol table 
   public Iterable<Point2D> range(RectHV rect)             // all points that are inside the rectangle 
   public           Point2D nearest(Point2D p)             // a nearest neighbor to point p; null if the symbol table is empty 
   public static void main(String[] args)                  // unit testing (required) 
Corner cases.  Throw a java.lang.NullPointerException if any argument is null. Performance requirements.  Your implementation should support size() in constant time; it should support put(), get() and contains() in time proportional to the logarithm of the number of points in the set in the worst case; and it should support points(), nearest(), and range() in time proportional to the number of points in the symbol table.

2d-tree implementation. Write a mutable data type that uses a 2d-tree to implement the same API (but renaming PointST to KdTreeST). A 2d-tree is a generalization of a BST to two-dimensional keys. The idea is to build a BST with points in the nodes, using the x- and y-coordinates of the points as keys in strictly alternating sequence, starting with the x-coordinates.

Insert (0.7, 0.2)

insert (0.7, 0.2)
Insert (0.5, 0.4)

insert (0.5, 0.4)
Insert (0.2, 0.3)

insert (0.2, 0.3)
Insert (0.4, 0.7)

insert (0.4, 0.7)
Insert (0.9, 0.6)

insert (0.9, 0.6)
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)

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

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

Analysis of running time and memory usage. Analyze the effectiveness of your approach to this problem by giving estimates of its time and space requirements.

Challenge for the bored.  Add the following method to

public Iterable<Point2D> nearest(Point2D p, int k)
This method should return the k points that are closest to the query point (in any order); return all N points in the data structure if Nk. It should do this in an efficient manner, i.e. using the technique from kd-tree nearest neighbor search, not from brute force. Once you've completed this class, you'll be able to run (which depends upon both and Behold their flocking majesty.

Submission.  Submit only and We will supply algs4.jar. Your may not call library functions except those in those in java.lang, java.util, and algs4.jar. Finally, submit a readme.txt file and answer the questions.

This assignment was developed by Kevin Wayne, with boid simulation by Josh Hug.