Write a data type to represent a set of points in the unit square (all points have

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

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

Use the immutable data type RectHV.java (not part ofpublic 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? public void draw() // draw to standard draw public String toString() // string representation }

Do not modify these data types.public class RectHV { public RectHV(double xmin, double ymin, // construct the rectangle [xmin, xmax] x [ymin, ymax] double xmax, double ymax) // throw a java.lang.IllegalArgumentException 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(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? public void draw() // draw to standard draw public String toString() // string representation }

**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` from `algs4.jar` or `java.util.TreeSet`).

Your implementation should supportpublic 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(Point2D p) // add the point p to the set (if it is not already in the set) public boolean contains(Point2D p) // does the set contain the point p? public void draw() // draw all of the points to standard draw public Iterable<Point2D> range(RectHV rect) // all points in the set that are inside the rectangle public Point2D nearest(Point2D p) // a nearest neighbor in the set to p; null if set is empty }

**2d-tree implementation.**
Write a mutable data type `KdTree.java` that uses a 2d-tree to
implement the same API (but replace `PointSET` with `KdTree`).
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.

*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). That is, a node is searched only if it might contain a point that is closer than the best one found so far. The effectiveness of the pruning rule depends on quickly finding a nearby point. To do this, organize your recursive method so that when there are two possible subtrees to go down, you choose first*the subtree that is on the same side of the splitting line as the query point*; the closest point found while exploring the first subtree may enable pruning of the second subtree.

**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 a file (specified as a command-line argument) 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 a file (specified as a command-line argument) 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 of running time and memory usage.**
Analyze the effectiveness of your approach to this problem by giving estimates of
its time and space requirements.

- 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*, using the memory-cost model from lecture and Section 1.4 of the textbook. Count all memory that is used by your 2d-tree, including memory for the nodes, points, and rectangles. For the purposes of this assignment, assume that each`Point2D`object uses 32 bytes. - Give the expected running time in seconds (using tilde notation)
to build a 2d-tree on
*N*uniformly random points in the unit square. (Do not count the time to read in or generate the points.) - 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 uniformly 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 only `PointSET.java` and
and `KdTree.java`.
We will supply `Point2D.java`, `RectHV.java`, `stdlib.jar`, and `algs4.jar`.
You may not call any library functions other than
those in `java.lang`, `java.util`, `stdlib.jar`, and `algs4.jar`.
Finally, submit a
readme.txt file and answer the questions.

This assignment was developed by Kevin Wayne.