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 and computer animation to speeding up neural networks and data mining.

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
Do not modify these data types.

Brute-force implementation. Write a mutable data type that uses a red–black BST to represent a symbol table whose keys are two-dimensional points, by implementing the following API:

public class PointST<Value> {

    // construct an empty symbol table of points 
    public PointST()

    // is the symbol table empty? 
    public boolean isEmpty()

    // number of points
    public int size()

    // associate the value val with point p
    public void put(Point2D p, Value val)

    // value associated with point p 
    public Value get(Point2D p)

    // does the symbol table contain point p? 
    public boolean contains(Point2D p)

    // all points in the symbol table 
    public Iterable<Point2D> points()

    // all points that are inside the rectangle (or on the boundary) 
    public Iterable<Point2D> range(RectHV rect)

    // a nearest neighbor of point p; null if the symbol table is empty 
    public Point2D nearest(Point2D p)

    // unit testing (required)
    public static void main(String[] args)


Implementation requirements.  You must use either RedBlackBST or java.util.TreeMap; do not implement your own red–black BST.

Corner cases.  Throw an IllegalArgumentException if any argument is null.

Unit testing.  Your main() method must call each public constructor and method directly and help verify that they work as prescribed (e.g., by printing results to standard output).

Performance requirements.  In the worst case, your implementation must support size() in constant time; put(), get() and contains() in \(\Theta(\log n\)) time; and points(), nearest(), and range() in \(\Theta(n)\) time, where n is 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 entire plane [(−∞, −∞), (+∞, +∞ )]; 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. Analyze the effectiveness of your approach to this problem by estimating how many many nearest-neighbor searches per second that each of your two implementations can perform for input1M.txt (1 million points), where the query points are uniformly random points in the unit square. Count only the time for the nearest-neighbor searches, not the time to read and insert the points.

Challenge for the bored.  Add the following method to

public Iterable<Point2D> nearest(Point2D p, int k)
This method returns the k points that are closest to the query point (in any order); return all n points in the data structure if nk. It must 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.


file points 10 24
readme.txt 6

Reminder: You can lose up to 4 points for poor style and up to 4 points for inadequate unit testing.

This assignment was developed by Kevin Wayne, with boid simulation by Josh Hug.
Copyright © 2004.