Project 5 (Kd-Trees) Solution

The purpose of this assignment is to create a symbol table data type whose keys are two-dimensional points. We’ll use a 2d-tree to support e cient range search ( nd all the points contained in a query rectangle) and k-nearest neighbor search ( nd 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 edu.princeton.cs.algs4.Point2D for points in the plane. Here is the subset of its API that you may use:




method
description




Point2D(double x, double y)
construct the point (x; y)
double x()
x-coordinate
double y()
y-coordinate
double distanceSquaredTo(Point2D that)
square of Euclidean distance between this point and that



Comparator<Point2D distanceToOrder() a comparator that compares two points by their distance to this point




boolean equals(Point2D that) does this point equal that?




String toString() a string representation of this point




Use the immutable data type edu.princeton.cs.algs4.RectHV for axis-aligned rectangles. Here is the subset of its API that you may use:





































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Project 5 (Kd-Trees)
Swami Iyer
















method
description














RectHV(double xmin, double ymin, double xmax, double ymax)
construct the rectangle [xmin; xmax] [ymin; ymax]




double xmin()
minimum x-coordinate of rectangle








double xmax()
maximum x-coordinate of rectangle








double ymin()
minimum y-coordinate of rectangle








double ymax()
maximum y-coordinate of rectangle








boolean contains(Point2D p)
does this rectangle contain the point p








(either inside or on boundary)?


















boolean intersects(RectHV that)
does this rectangle intersect








that rectangle (at one or more points)?










double distanceSquaredTo(Point2D p)
square of Euclidean distance from








point p to closest point in rectangle


















boolean equals(RectHV that)
does this rectangle equal that?








String toString()
a string representation of this rectangle






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 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 (edu.princeton.cs.algs4.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.







$ java B r u t e P o i n t S T 0.661633 0.287141 0.65 0.68 0.28 0.29 5 < data / input10K . txt




st . empty ()? false




st . size () = 10000




First 5 values :




3380




1585




8903




4168




5971




7265




st . contains ((0.661633 ,
0 . 2 8 7 1 4 1 ) ) ? true
st . range ([0.65 , 0.68]
x [0.28 , 0.29]):






(0.663908 , 0.28533
7)


(0.661633 , 0.28714
1)








(0.671793 , 0.28860
8)


st . nearest ((0.661633
,
0.287141)) = (0.663908 , 0.285337)




st . nearest ((0.661633
, 0.287141) , 5):
(0.663908 , 0.28533
7)








(0.658329 , 0.29003
9)


(0.671793 , 0.28860
8)






(0.65471 , 0.276885)


(0.668229 , 0.27648
2)













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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: rst 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 e cient 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 in nitely large square from [( ; ); (+1; +1)]; 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 nd 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 nd 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 e ectiveness of the pruning rule depends on quickly nding a nearby point. To do this, organize your recursive method so that when there are two possible subtrees to go down, you choose rst the subtree that is on the same side of the splitting line as the query point; the closest point found while exploring the rst 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.




java K d T r e e P o i n t S T 0.661633 0.287141 0.65 0.68 0.28 0.29 5 < data / input10K . txt




st . empty ()?
false
st . size () =
10000
First 5 values :



0




2




1




4




3







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Project 5 (Kd-Trees)
Swami Iyer
















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st . contains ((0.661633 ,
0 . 2 8 7 1 4 1 ) ) ? true


st . range ([0.65 , 0.68]
x [0.28 , 0.29]):










(0.671793 , 0.28860
8)




(0.663908 , 0.28533
7)












(0.661633 , 0.28714
1)




st . nearest ((0.661633
,
0.287141)) = (0.663908 , 0.285337)








st . nearest ((0.661633
, 0.287141) , 5):


(0.668229 , 0.27648
2)










(0.65471 , 0.276885)




(0.671793 , 0.28860
8)












(0.658329 , 0.29003
9)




(0.663908 , 0.28533
7)


















Data Under the data directory, we provide several sample input les for testing.




Visualization Clients In addition to the test clients provided in BrutePointST and KdTreePointST, you may use the following interactive client programs to test and debug your code:




RangeSearchVisualizer reads a sequence of points from a le (speci ed as a command-line argument) and inserts those points into BrutePointST and KdTreePointST based symbol tables brute and kdtree respectively. Then, it performs range searches on the axis-aligned rectangles dragged by the user in the standard drawing window, and displays the points obtained from brute in red and those obtained from kdtree in blue.







$ java R a n g e S e a r c h V i s u a l i z e r data / input100 . txt


















































































NearestNeighborVisualizer reads a sequence of points from a le (speci ed as a command-line argument) and inserts those points into BrutePointST and KdTreeSPointT based symbol tables brute and kdtree respectively. Then, it performs k-(speci ed as the second command-line argument) nearest neighbor queries on the point corresponding to the location of the mouse in the standard drawing window, and displays the neighbors obtained from brute in red and those obtained from kdtree in blue.







$ java N e a r e s t N e i g h b o r V i s u a l i z e r data / input100 . txt 5





















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BoidSimulator is an implementation of Craig Reynold’s Boids program1 to simulate the ocking behavior of birds, using a BrutePointST or KdTreePointST data type. The rst command-line argument speci es which data type to use (brute for BrutePointST or kdtree for KdTreePointST), the second argument speci es the number of boids, and the third argument speci es the number of friends each boid has. 2







$ java B o i d S i m u l a t o r brute 100 10


















































































$ java B o i d S i m u l a t o r kdtree 1000 10







 
See www.en.wikipedia.org/wiki/Boidsm.




 
Note that the program does not scale well with the number of boids when using BrutePointST, which is after all a brute-force implementation. However, the program does scale quite well when using KdTreePointST.
















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Files to Submit:




 
BrutePointST.java




 
KdTreePointST.java




 
report.txt













Before you submit:




Make sure your programs meet the input and output speci cations by running the following command on the terminal:







$ python r un _t e st s . py -v [ < problems ]




where the optional argument <problems lists the problems (Problem1, Problem2, etc.) you want to test; all the problems are tested if no argument is given.




Make sure your programs meet the style requirements by running the following command on the terminal:







$ c h e c k _ s t y l e < program




where <program is the .java le whose style you want to check.




Make sure your report isn’t too verbose, doesn’t contain lines that exceed 80 characters, and doesn’t contain spelling/grammatical mistakes
















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






















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