## Kd-Trees

Write a data type to represent a set of points in the unit square (all points have x- and y-coordinates between 0 and 1) using 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 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 an ordered symbol table
public boolean equals(Object that)              // does this point equal that?
public void draw()                              // draw to standard draw
public String toString()                        // string representation
}
```
Write an immutable data type RectHV.java that implements the following API.
```public class RectHV {
public RectHV(double xmin, double ymin,         // construct the rectangle [xmin, xmax] x [ymin, 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 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 distanceTo(Point p)               // Euclidean distance from point p to the closest point in rectangle
public double distance2To(Point p)              // square of Euclidean distance from point p to closest point in rectangle
public boolean equals(Object that)              // does this rectangle equal that?
public void draw()                              // draw to standard draw
public String toString()                        // string representation
}
```
Thoroughly test your data types before proceeding.

Brute-force implementation. Write a mutable data type PointSET.java that represents a set of points in the unit square. Implement the following API by using a red-black BST (using either SET or java.util.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 (if it is not already in the set)
public boolean contains(Point p)                // does the set contain the point p?
public void draw()                              // draw all of the points to standard draw
public Iterable<Point> range(RectHV rect)       // all points in the set that are inside the rectangle
public Point nearest(Point p)                   // a nearest neighbor in the set to p; null if set is empty
}
```
Your implementation should support insert() and contains() in time proportional to the logarithm of the number of points in the set; it should support nearest() and range() in time proportional to the number of points in the set.

2d-tree implementation. Write a mutable data type KdTree.java that uses a 2d-tree to implements the same API as PointSET. 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.

• Search and insert. The algorithms for search and insert are similar to those for BSTs, 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 BST 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 a 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 the 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. As a memory model, use the section of the textbook on memory analysis.

• 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 this 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 algs4.jar). Finally, submit a readme.txt file and answer the questions.

Midterm evaluation.  Please fill out the following anonymous questionnaire to provide us with feedback to help us improve this course.

This assignment was developed by Kevin Wayne.