COS 226

Programming Assignment


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 represents a symbol table whose keys are two-dimensional points, by implementing the following API:

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

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

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

Corner cases.  Throw a java.lang.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 time proportional log n; and points(), nearest(), and range() in time proportional to n, 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 input100K.txt (100,000 points) and 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.

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