The purpose of this assignment is to create a symbol table data type whose keys are two-dimensional points. Well use a 2d-tree to support efficient range search (find all the points contained in a query rectangle) and k-nearest neighbor search (find k points that are closest 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.

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 for points in the plane. Here is the subset of its API that you may use:

method description

Use the immutable data type RectHV for axis-aligned rectangles. Here is the subset of its API that you may use:

method description

Symbol Table API Here is a Java interface PointST<Value> specifying the API for a symbol table data type whose keys are two-dimensional points represented as Point2D objects:

method description

boolean isEmpty() is the symbol table empty?

int size() number of points in the symbol table

void put(Point2D p, Value val) associate the value val with point p Value get(Point2D p) value associated with point p boolean contains(Point2D p) does the symbol table contain the point p?

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

Iterable<Point2D> range(RectHV rect) all points in the symbol table that are inside the rectangle rect

Point2D nearest(Point2D p) a nearest neighbor to point p; null if the symbol table is empty Iterable<Point2D> nearest(Point2D p, int k) k points that are closest to point p

Problem 1. (Brute-force Implementation) Write a mutable data type BrutePointST that implements the above API using a red-black BST (RedBlackBST).

Corner cases. Throw a java.lang.NullPointerException if any argument is null.

Performance requirements. Your implementation should support put(), get() and contains() in time proportional to the logarithm of the number of points in the set in the worst case; it should support points(), range(), and nearest() in time proportional to the number of points in the symbol table.

Problem 2. (2d-tree Implementation) Write a mutable data type KdTreePointST that uses a 2d-tree to implement the above symbol table API. 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.
• Level-order traversal. The points() method should return the points in level-order: first the root, then all children of the root (from left/bottom to right/top), then all grandchildren of the root (from left to right), and so forth. The level-order traversal of the 2d-tree above is (0.7, 0.2), (0.5, 0.4), (0.9, 0.6), (0.2, 0.3), (0.4, 0.7).

The prime advantage of a 2d-tree over a BST is that it supports efficient implementation of range search, nearest neighbor, and k-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.

• 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). That is, you should search a subtree 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, you should search a node 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.
• k-nearest neighbor search. Use the technique from kd-tree nearest neighbor search described above.

Corner cases. Throw a java.lang.NullPointerException if any argument is null.

Acknowledgements This project is an adaptation of the Kd-Trees assignment developed at Princeton University by Kevin Wayne, with boid simulation by Josh Hug.