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Assignment 3
COMP 250
1 Introduction
In this assignment, we extend the powerful idea of binary search trees from one dimension to
multiple dimensions. In particular, we will work with points in <
k where k can be any positive
integer. Such points might represent image data, audio data, or text. For example, machine learning
methods often map such common data to feature vectors in a k − D feature space, and then use
these features in various ways.1
In binary search, one is given a key and tries to find that key amongst many that are stored in a
data structure. We will examine a related but different problem for k − D data. Our problem will
be to take a given point, which we will call a query point rather than a “key”, and search for the
nearest point in the data set. In the field of Machine Learning, the nearest point is typically referred
to as the nearest neighbor.
One way to find the nearest point would be to iterate by brute force over all data points, compute the distance from the query point to each data point, and then return the data point that has
the minimum distance.2 However, this brute force search is inefficient, for the same reason that
linear search through a list is inefficient relative to binary search, since in the worst case it requires
examining each data point. One would instead like to have a “nearest point” method that is must
faster, by restricting the search to a small subset of the data points.
1.1 KD-Trees
One way to restrict the search for a query point is to use a tree data structure called a kd-tree. This
is a binary tree which is similar to a binary search tree, but there are important differences. One
difference between a kd-tree and a binary search tree concerns the data itself. In a binary search
tree, the data points (keys) are ordered as if on a line, or in Java terms they are Comparable. In a
kd-tree, there is no such overall ordering. Instead, we only have an ordering within each of the k
dimensions, and this ordering can differ between dimensions. For example, consider k = 2, and let
points be denoted (x1, x2). We can order two points by their x1 value and we can order two points
by their x2 value, but we don’t want to order (x1, x2) and (x
0
1
, x0
2
) in general. For example, in the
case x1 < x0
1 but x2 > x0
2
, we do not want to impose an ordering of these two points.
Another important difference is how binary search trees and kd-trees are typically used. With
a binary search tree, one typically searches for an exact match for the search key. With a kd-tree,
one typically searches for the nearest data point. We will use the Euclidean distance. Give a query
point xq ∈ Z
k we want to find a data point x[ ] that minimizes the squared distance,
sqrdist(xq[ ], x[ ]) ≡
X
i
(xq[i] − x[i])2
.
We used squared distance rather than distance because taking the square root buys us no extra
information about which point is closest.
Note that there may be multiple data points that are the same (squared) distance from a query
point xq. One could define the problem slightly differently by returning all points in this non-unique
case. Or, one could ask for the nearest n points, and indeed this is common. We will just keep it
simple and ask for a single nearest point. If multiple points lie at that same nearest distance, then
any of these points may be returned.
Kd-trees which are a particular type of binary tree. Each internal node implicitly defines a
k−dimensional region in <
k
, which we refer to as a hyperrectangle, and which will be defined
1https://en.wikipedia.org/wiki/Feature_(machine_learning)
2Note that the nearest point may be non-unique.
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below. An internal node has two (non-null) children which represent two spatially disjoint (nonintersecting) hyperrectangles that are contained within the parent hyperrectangle. We will refer to
the two children as the low child and high child. 3
Data points are stored at the leaf nodes, namely each leaf node has one Datum object. Internal
nodes do not store references to Datum objects. Rather, for each internal node, we associate a set
of data points, namely the Datum objects stored in the descendant leaf nodes.
Each internal node’s hyperrectangle is defined by data points that are associated with each node,
as follows. For each of the k dimensions, the range of the hyperrectangle in that dimension is the
difference of the maximum and minimum values of the data points. An example will be given later
to illustrate. The range of a node in each dimension will be used when constructing the tree, as
discussed below.
1.1.1 Example
Let’s consider a kd-tree node that is associated with ten 2D points (k = 2). The points are shown
below. The horizontal axis is dimension x0 and the vertical axis is dimension is x1, which correspond to array representation x[0] and x[1]. The red rectangle indicates the minimum and maximum
values of the data points in the two dimensions. We will discuss the significance of these values
below.
Figure 1: 10 data points in a 2D space. The red rectangle is the hyperrectangle defined by the low
and high values in the two dimensions.
1.1.2 Constructing a KD-tree
We will not give a formal definition of kd-tree. Instead we give an algorithm for constructing
one, given list of data points is <
k
. The algorithm starts by constructing a root node and proceeds
recursively. When constructing a node, we pass in an array of the data points associated with that
node. If the array contains just one data point, then the node is a leaf node. Otherwise, the node is
an internal node (with one exception which we return to below). The set of data points associated
with an internal node is then partitioned into two non-empty sets, which are used to define the
two children of that node. If the internal node has two data points associated with it, then the two
children nodes will be leaf nodes. If an internal node has three data points associated with it, then
one child node will be internal with two points associated with it and the other child node will be
a leaf. There is one special case to consider, namely that all the data points at a node are equal
3You can think of them as ‘left’ and ‘right’, respectively, if you like but keep in mind that we are in a k dimensional
space. In a 3D space, you might have left/right, up/down, back/front. We will use the same words for each dimension,
namely low/high.
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(duplicates). In this case, the node is considered a leaf node, and the duplicate data points are
removed.
Because the kd-tree will be used to find nearest point to targets, we would like the data points at
each node to be a compact as possible in <
k
. Put another way, we would like the points associated
with each child to be as close to each other as possible, and the points associated with the two
children to be as far apart from each other as possible. This will allow us to sometimes only search
the data points associated with one of the children, which speeds up the search. To try to achieve
this compactness property when partitioning the set of data points at a node, we compute which of
the k dimensions has the largest range of values, where range is defined by the maximum minus the
minimum value within a given dimension. We then partition or split the data points into two sets
based only on their coordinate values in this dimension. For this assignment, we require that you
follow this rule, namely you must choose the splitting dimension d ∈ {1, . . . , k} according to this
maximum range criterion. This guarantees, in particular, that the range of values in this splitting
dimension is greater than zero. (If there are two dimensions d and d
0
that both have the greatest
range of values, then either may be chosen.)
Once the splitting dimension d of a node is chosen, the algorithm chooses a splitting value.
Data points associated with that node whose value x[d] in coordinate d is less than or equal to the
splitting value go in the low child’s subtree, and the remaining data points go into the high child’s
subtree.
How to choose the splitting value? We would like the tree to be as balanced as possible, and
so we try to choose a splitting value that partitions the data points at a node such that there are a
roughly equal number of data points associated with the two children. (This is similar to the role
of the pivot in quicksort.) It may not possible to choose a rule that is fast and always achieves this
goal, however. For example, choosing splitting value to be the median value works well if the data
point values in dimension d are distinct, but if there are repeats then the median method might not
work well. (Also, choosing the median quickly is not obvious.) We will leave it up to you how to
choose this splitting value. We suggest a simple choice for the splitting value to be the average of
the min and max values in the splitting dimension. This choice at least guarantees that we partition
the data points at a node into two sets (assuming the points are different).
1.1.3 Example (continued)
In our example, the data has a larger range in the x0 dimension than in the x1 dimension and so we
split in the x0 dimension.
Figure 2: For this figure and subsequent figures, the splitting plane is defined by the median of
the x0 values of the 10 data points. By convention, points on the splitting plane belong to the low
half. Although the median split is very common, for your implementation, we strongly suggest that
you do not attempt to split using the median, but instead split using some other value such as the
average of the low and high values in the splitting dimension. This will be easier to do.
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The figure below shows the result of splitting recursively as described above. On the left, the
original bounding rectangle is partitioned into a nested set of rectangles. This is done by choosing a
set of splitting planes (dashed black lines) for each rectangle. The splitting planes are labelled with
letters a to h. On the right is shown the KD-tree structure that is defined by the splitting planes and
the corresponding nesting of rectangles. For example, the large bounding rectangle is partitioned
into two rectangles by the splitting plane a, which corresponds to the root node of the tree.
Consider an internal node d and its subtree, and the corresponding splitting plane labelled d in
the figure on the left. This subtree defined by d is the low (left) child of node b and thus the points
associated with this subtree lie on the low side of splitting plane b, namely these are the points
{p4, p3, p5}. (Recall that points that lie on a splitting plane are included on the low side, not the
high side.) The splitting plane d splits this set into two sets {p4, p3} and {p5}. The set {p4, p3} is
in turn split by plane e, and corresponding node e’s children are two leaves containing these two
points. The point {p5} becomes a leaf which is the high child of node d. We encourage to examine
other internal nodes and ask which regions they correspond to and what are the data points within
these regions.
Figure 3: The outer red rectangle is the region defined by the low and high bounds of the 10 data
points. The inner red rectangle is the region bounded by the data points below splitting node d ,
namely points p3, p4, p5. Notice that this region does not span the entire region to the left of the b
splitting plane.
1.2 Finding the nearest point
Given a kd-tree, one would like to to find the nearest data point to a given (input) query point. The
query point need not belong to the set of data points in the tree.
The search algorithm for the nearest point is recursive, starting at the root node. For any node
during the search, the query point’s coordinate value in the splitting dimension xq[d] is compared
to the node’s splitting value. You might think that you could restrict your search for the closest
data point to the query point xq by examining only the data points on the same side of the splitting
plane as xq. This approach would indeed be sufficient if we were trying to find an exact match of
the query in the tree. However, the query point may not have an exact match, and in that case the
nearest point to the query point might not always lie on the same side of the splitting plane of a
node, and so one might have to search through the points on both sides of the splitting plane.
If one were always to search through all points associated with a node, then the tree structure
would serve no purpose, and one would be better off just brute force searching the list of data points.
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The clever insight that makes kd-trees useful is that we can sometimes restrict our search only to
those data points on the same side of the splitting plane as the query point. The crucial condition
for restricting the search is as follows: if the distance from the query point to the closest point on
the same side of the splitting plane is less than the distance from the query point to the splitting
plane itself, then points on the opposite side of the splitting plane cannot be the nearest point and
hence do not need to be considered. If, however, the distance from the query point to the nearest
point on the same side is greater than the distance from the query point to the splitting plane, then
one does need to consider the possibility that a data point on the other side of the splitting plane
might be the closest point.
Note that if there are two data points that are the same distance from the target, then we might
ask how to break the tie to get a unique answer. It is possible to come with rules for doing so, but
we will not concern ourselves with this. Instead, in the case that are multiple closest points to some
given query point, if you return one of the points, then this is considered to be correct. This avoids
us having to specify a tie breaking policy.
1.2.1 Example (continued)
The figure below shows the first splitting plane, which is defined at the root node of the tree. It
also shows a query point xq in blue. Suppose we first search the data points on the same side of the
splitting plane as the query point. The nearest data point on the same side of the splitting plane is
found to be p1. A circle is drawn that is centered on the query point and passes through p1. Notice
that the distance from the query point to p1 is less than the distance from the query point to the
splitting plane. That is, the ball shown around the query point contains the nearest point but doesn’t
intersect the splitting plane. In this case, we can be sure that the nearest of all 10 data points is not
on the other side of the splitting plane, and so we don’t need to search the data points other side.
Figure 4: A query point xq (blue) and the nearest data point p1 on the same side as the splitting
plane.
Let’s now take an example with a different query point x
0
q
. Now the ball around the closest
point p1 to the query point that is on the same side of the splitting plane which has radius kx
0
q −p1k
intersects the splitting plane. In this case, we cannot be sure that p1 is indeed the closest point in
the data set, as there may be a point on the other side of the splitting plane that falls strictly within
the ball and hence is closer. We therefore also have to search the points on the other side of the
splitting plane too. (Indeed in this example, p2 is such a point.)
The description above should be sufficient for you to write a method for building a kd-tree for
a given set of data points, and for finding the closest data point to a given target.
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Figure 5: A query point x
0
q
(blue) and the nearest data point p1 on the same side as the splitting
plane, and the nearest point p2 on the opposite side of the splitting plane.
1.3 More info about kd-trees
Kd-trees were invented in the 1970’s. If you are interested, have look at some of the original
research papers that presented the method. (Click to access the paper.) As you can see from just
these two papers, there are many possible ways to define kd-trees.
• Bentley, J. L. (1975). ”Multidimensional binary search trees used for associative searching”.
Communications of the ACM.
• Friedman, J. H.; Bentley, J. L.; Finkel, R. A. (1977). ”An Algorithm for Finding Best
Matches in Logarithmic Expected Time”. ACM Transactions on Mathematical Software.
There are ample resources about kd-trees on the web. For example, the KD-tree Wikipedia
page . While we encourage you to learn more if you are interested, be aware that these resources
will contain more information than you need to do this assignment, and so you would need to sift
through it and figure out what is important and what can be ignored.
This PDF should have all you need to do the assignment. If you require clarification, use the
discussion board. Also, share links to good resources and to resolve questions you might have.
(But please, before posting, use the search feature to check if you question has already been asked
and answered.)
2 Instructions
2.1 Starter code
In our implementation, the tree will store the data points at the leaves. We assume that the coordinate values of the data and query points are all int values, so the data and query points are all in
Z
k
rather than <
k
, where Z is a common symbol for the set of integers.
The starter code contains three classes:
• KDTree: This class has several fields including rootNode which is the root of the tree and
is of type KDNode which is an inner class. It also has an integer field k which is the number
of dimensions, i.e. Z
k
.
• Datum: This class holds the coordinate values x[ ] of a data point or query point.
• Tester: Some example tests to verify the correctness of your code.
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2.2 Your Task
Implement the following methods in the KDTree class:
• (50 points) KDNode constructor
The constructor KDNode is given an array of Datum objects, and it constructs a KDNode
object that is the root of a subtree whose leaves contain the given Datum objects.
The KDNode constructor should split the given data points such that, on average, about half
go into the low child and the other about half go into high child, and so the resulting tree is
approximately balanced. The height of the tree should be roughly log2
(N) where N is the
number of data points.
It is difficult to choose a splitting rule that guarantees an equal (half-half) split, and so we
will not require this. In particular, even if you were to choose the splitting value to be the
median value of the data points in the splitting dimension, a very unbalanced tree could still
result if many data points had that splitting value at many of the nodes. It is good enough
for you to let the split be the average of the min and max value in the splitting dimension.
If properly implemented, this will lead to roughly log2
(N) height of the test trees. We have
provided you with a helper method KDTree.height that returns the height of the tree, and
KDTree.meanDepth that computes mean depth of the leaves of a tree.
If there are duplicate data points in the input, then your constructor must put only one copy of
the duplicate points into the tree. It is up to you to find an efficient way to remove duplicates.
What you should not do is try to remove duplicates from the initial list of data points, since
this would take time O(N2K) – namely comparing all pairs of data points for equality on
each dimension – which is too slow for large data sets. We will test that you have correctly
and efficiently removed duplicates.
• (20 points) KDTreeIterator
The method kDTreeIterator returns an Iterator object that can be used to iterate
through all the data points. The order of points in your iterator must correspond to an inorder traversal of your kd-tree, namely it must visit all data points in the low subtree of any
node before it visits any of the data points in the high subtree.
We will test your iterator by examining the order of data points. (The Grader will have access
to the KDNode objects and various fields of your tree: we have given these fields package
visibility.)
The tester that we provide you does not provide a test of your iterator, however. We suggest
you create your own iterator test cases, and share these testers with each other. For example,
you could test a set of 1D data points and ensure data points are visited in their correct order.
It is fine if your iterator stores references to all the data points (Datum objects). There are
more space efficient ways to make iterators from trees, which represent only one path in the
tree at any time. But we do not require this space efficiency.
• (30 points) nearestPointInNode
The method nearestPointInNode takes a query point Datum and returns a data point
Datum that is as close to the query point as any other data point. There may be multiple
points that have the same nearest distance to the query point, and in this case your method
should return one of them.
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2.3 Submission
• Submit a single zipped file A3.zip that contains the KDTree.java file to the myCourses
assignment A3 folder. If you submit the other files in the directory, then they will be ignored.
Include your name and student ID number within the comment at the top of each file.
• Xiru (TA) will check for invalid submissions once before the solution deadline and once the
day after the solution deadline. He will notify students by email if their submission is invalid.
It is your responsibility to check your email.
• Invalid solutions will receive a grade of 0. You may resubmit a valid solution, but only up to
the two days late limit, and you will receive a late penalty. As with Assignments 1 and 2, the
grade you will receive is the maximum grade from your submissions.
• If you have any issues that you wish the TAs (graders) to be aware of, please include them in
the comment field in mycourses along with your submission. Otherwise leave the mycourses
comment field blank.
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