Question 1 (27 points):
In lib280-asn5 you are provided with a fully functional 2-3 tree class called TwoThreeTree280. It
implements the KeyedBasicDict280 interface and therefore supports the operations we saw in class.
It does not, however, implement KeyedDict280 which adds additional operations including all of the
methods in KeyedLinearIterator280 which, in turn, includes all of the public operations on a cursor.
Note that KeyedDict280 is the same interface that is implemented by KeyedChainedHashTable280 so
you should be somewhat familiar with it from the previous assignment.
The task for this question is to extend the TwoThreeTree280 to a class called
IterableTwoThreeTree280 which allows linear iteration over the key-element pairs stored in the twothree tree in ascending keyorder. We will achieve this by adding additional references to leaf nodes
so that the leaf nodes form a bi-linked list. Note that adding this feature to a 2-3 tree results in exactly
a B+ tree of order 3 (see textbook Section 14.1). We aren’t going to call it a B+ tree class though,
because we will be specifically a B+ tree of order 3, and higher-order B+ trees will not be supported.
Figure 1 in the Appendix shows the differences between a 2-3 tree and a B+ tree of order 3 containing
the same elements. The algorithms for insertion and deletion are the same in both kinds of tree, except
2 | P a g e
that in the case of the B+ tree, references to/from the predecessor and successor leaf nodes in keyorder have to be adjusted to maintain the bi-linked list of leaf nodes.
The full class hierarchy of IterableTwoThreeTree280 is shown in Figure 2 of the Appendix. The
hierarchy of tree node classes is shown in Figure 3 of the Appendix.
To implement the IterableTwoThreeTree280, the following tasks must be carried out:
1. Extend LeafTwoThreeNode280 so that it has extra references to its predecessor and successor leaf
nodes. This has been done for you in the class LinkedLeafTwoThreeNode280.
2. Override the createNewLeafNode protected method so that it
LinkedLeafTwoThreeNode280 objects. This has already been done.
3. (10 points) Override the insert and delete methods of TwoThreeTree280 with modified versions that
correctly maintain the additional predecessor and successor references in the LinkedLeafTwoThreeNode280.
Each leaf node should always point to the the leaf node immediately to the left of it (the predecessor) and to
the right of it (the successor) even if they are not siblings. Of course, the leaf node with the smallest key has
no predecessor and the leaf node with the largest key has no successor.
In IterableTwoThreeTree280, the insert and delete methods from TwoThreeTree280 already have been
copied, and TODO comments have been inserted indicating where you need to add additional code to
maintain the additional leaf node references. The comments also provide a few hints. You should not
have to modify any of the existing code for insert or delete, just add new code to deal with the linking
and unlinking of leaf nodes from their successors and predecessors.
4. (12 points) Implement the additional methods required by KeyedDict280 (and, by extension,
KeyedLinearIterator280). Some of these have been done for you, others have not. TODO comments in
IterableTwoThreeTree280 indicate which methods you need to implement and maybe even a hint or two.
5. (5 points) In the main() function, write a regression test to test the methods required by KeyedDict280
(and, by extension, KeyedLinearIterator280). You to not need to explicitly test the insertion and deletion
methods since testing of the methods from KeyedLinearIterator280 will reveal any problems with the
new leaf node linkages, but you will need to insert and delete items to create test cases.
You must test all of the methods listed in the interfaces that are coloured blue in Figure 2 of the
Appendix. Warning: there is a small but non-zero probability that there are bugs in the methods in the
blue-coloured classes for which implementations were provided, so treat them as if you implemented
them yourself. A local class called Loot has been defined in the main method for you use as the data
items to insert into the tree for testing. This class implements the type of item depicted in Figure 1 in
the Appendix consisting of the name of a magic item from a fantasy game, and its value in gold
pieces. The item keys are the item names (Strings). The data item is an integer and the key is a
Hint: The toStringByLevel() method prints not only the 2-3 tree’s structure, but also displays current
linear ordering of the nodes that results from following the successor links in the leaf nodes, beginning
with the leftmost leaf node. This may be helpful for the debugging of step 2.
Question 2 (45 points):
In Question 2 you will be implementing a k-D tree. We begin with introducing some algorithms that you
will need. Then we will present what you must do.
3 | P a g e
Helper Algorithms for Implementing k-dimensional Trees
As we saw in class, in order to build a k-D tree we need to be able to find the median of a set of
elements efficiently. The “j-th smallest element” algorithm will do this for us. If we have an array of n
elements, then finding the n/2-smallest element is the same as finding the median.
Below is a version of the j-th smallest element algorithm that operates on a subarray of an array
specified by offsets le f t and right (inclusive). It places at offset j (le f t j right) the element that belongs
at offset j if the subarray were sorted. Moreover, all of the elements in the subarray smaller than that
belonging at offset j are placed between offsets le f t and j 1 and all of the elements in the subarray
larger than that element are placed between offsets j + 1 and right (but there is no guarantee on the
ordering of any of these elements!). Thus, if we want to find the median element of a subarray bounded
by le f t and right, we can call jSmallest(list, left, right, (left+right)/2)
The offset (le f t + right)/2 (integer division!) is always the element in the middle of the subarray between
offsets le f t and right because the average of two numbers is always equal to the number halfway in
Al go ri th m j Sm al le st ( list , left , right , j )
list – array of c o m p a r a b l e elements
left – offset of start of subarray for which we want the median element
right – offset of end of subarray for which we want the median element
j – we want to find the element that belongs at array index j
To find the median of the subarray between array indices ’ left ’ and ’ right ’ , pass in j = (
right + left )/2.
P r e c o n d i t i o n : left <= j <= right P r e c o n d i t i o n : all elements in ’ list ’ are unique ( things get messy o th er wi se !) P o s t c o n d i t i o n : the element x that belongs at index j if the subarray were sorted is in position j . Elements in the subarray smaller than x are to the left of offset j and the elements in the subarray larger than x are to the right of offset j . if ( right > left )
4 | P a g e
// P ar ti tio n the subarray using the last element , list [ right ] , as a pivot .
// The index of the pivot after p a r t i t i o n i n g is returned .
// This is exactly the same par ti ti on al go ri th m used by qu ic
p i v o t I n d e x := p ar tit io n ( list , left , right )
// If the p i v o t I n d e x is equal to j, then we found the j – th smallest
// element and it is in the right place ! Yay!
// If the position j is smaller than the pivot index , we know that
// the j – th smallest element must be between left , and pivotIndex -1 , so
// r e c u r s i v e l y look for the j – th smallest element in that subarray :
if j < p i v o t I n d e x jS ma ll es t ( list , left , pivotIndex -1 , j ) // Otherwise , the position j must be larger than the pivotIndex , // so the j - th smallest element must be between p i v o t I n d e x +1 and right . else if j > p i v o t I n d e x
jS ma ll es t ( list , p i v o t I n d e x +1 , right , j )
// Otherwise , the pivot ended up at list [j] , and the pivot *is* the
// j – th smallest element and we ’re done .
Notice that there is nothing returned by jSmallest, rather, it is the postcondition that is important. The
postcondition is simply that the element of the subarray specified by left and right that belongs at index j if
the subarray were sorted is placed at index j and that elements between le f t and j = 1 are smaller than the
j-th smallest element and the elements between j + 1 and right are larger than the j-th smallest element.
There are no guarantees on ordering of the elements within these parts of the subarray except that they
are smaller and larger than the the element at index j, respectively. This means that if you invoke this
algorithm with j = (right + le f t)/2 then you will end up with the median element in the median position of the
subarray, all smaller elements to its left (though unordered) and all larger elements to its right (though
unordered), which is just what you need to implement the tree-building algorithm! NOTE: for this algorithm
to work on arrays of NDPoint280 objects you will need an additional parameter d that specifies which
dimension (coordinate) of the points is to be used to compare points. An advantage of making this algorithm
operate on subarrays is that you can use it to build the k-d tree without using any additional storage — your
input is just one array of NDPoint280 objects and you can do all the work without any additional arrays —
just work with the correct subarrays.
You may have noticed that jSmallest uses the partition algorithm partition the elements of the subarray
using a pivot. The pseudocode for the partition algorithm used by the jSmallest algorithm is given
below. Note that in your implementation, you will, again, need to add a parameter d to denote which
dimension of the n-dimensional points should be used for comparison of NDPoint280 objects.
5 | P a g e
// P ar ti tio n a subarray using its last element as a pivot .
Al go ri th m p ar ti tio n ( list , left , right )
list – array of c o m p a r a b l e elements left – lower limit on
subarray to be p a r t i t i o n e d
right – upper limit on subarray to be p a r t i t i o n e d
P r e c o n d i t i o n : all elements in ’ list ’ are unique ( things get messy o th er wi se !)
P o s t c o n d i t i o n : all elements smaller than the pivot appear in the leftmost
part of the subarray , then the pivot element , followed by the
elements larger than the pivot . There is no g ua ra nt ee
about the ordering of the elements before and after the pivot .
returns the offset at which the pivot
element ended up
pivot = points [ right ]
s w a p O f f s e t = left
for i = left to right -1 if ( points [ i ] <= pivot ) swap points [ i ] and points [ s w a p O f f s e t ] s w a p O f f s e t = s w a p O f f s e t + 1 swap points [ right ] and points [ s w a p O f f s e t ] return s w a p O f f s e t ; // return the offset where the pivot ended up Algoirthm for Building the Tree An algorithm for building a k-d tree from a set of k-dimensional points is given below. It is slightly more detailed than the version given in the lecture slides. It uses the jSmallest algorithm presented above. 6 | P a g e Al go ri th m kdtree ( pointArray , left , right , int depth ) p o i n t A r r a y - array of k - d i m e n s i o n a l points left - offset of start of subarray from which to build a kd - tree right - offset of end of subarray from which to build a kd - tree depth - the current depth in the pa rt ial ly built tree - note that the root of a tree has depth 0 and the $k$ d i m e n s i o n s of the points are numbered 0 through k -1. if p o i n t A r r a y is empty return null ; else // Select axis based on depth so that axis cycles through all // valid values . (k is the d i m e n s i o n a l i t y of the tree ) d = depth mod k ; m e d i a n O f f s e t = ( left + right )/2 // Put the median element in the correct position // This call assumes you have added the di me nsi on d pa ra me te r // to jS ma ll es t as d es cri be d earlier . jS ma ll es t ( pointArray , left , right , d , m e d i a n O f f s e t ) // Create node and c on str uc t subtrees node = a new id - tree node node . item = p o i n t A r r a y [ m e d i a n O f f s et] node . l ef tCh il d = kdtree ( pointArray , left , medianOffet -1 , depth +1); node . r i g h t C h i l d = kdtree ( p o i n t A r r a y m e d i a n O f f s e t +1 , right , depth +1); return node ; Implementing the k-D Tree – What You Must Do Implement a k-D tree. You must use the NDPoint280 class provided in the lib280.base package of lib280asn6 to represent your k-dimensional points. You must design and implement both a node class (KDNode280.java) and a tree class (KDTree280.java). Other than specific instructions given in this question, the design of these classes is up to you and you can use as much or as little of lib280 as you deem appropriate, and you may use whatever private/protected methods you deem necessary. A portion of the marks for this question will be awarded for the design/modularity/style of the implementation of your class. A portion of the marks for this question will be awarded for acceptable inline and javadoc commenting. Your ADT must support the following operations: Construct a new (balanced) k-D tree from a set of k-dimensional points (it must work for any k >
0). Perform a range search: given a pair of points (a1, a2, . . . ak) and (b1, b2, . . . , bk), ai <= bi for all i = 1 . . . k, return all of the points (c1, c2, . . . , ck) such that a1 c1 b1, a2 c2 b2, . . . , ak ck bk. In addition, you should write a test program that generates the correctness of your tree. The test program should consist of two parts: 7 | P a g e 1. Show that your class can correctly build a k-D tree from a set of points. For k=2, display the the kdimensional points that are given as input (use between 8 and 12 elements), followed by a graphical representation of the built tree (similar to the toStringByLevel() output in the trees we’ve done previously). Do this again for one other value of k, between 3 and 5 (your choice). 2. For the second of the two trees you displayed in part 1, perform at least three range searches. For each search, display the query range, execute the range search, and then display the list of points in the tree that were found to be in range. A sample test program output is given below. Implementation and Debugging Strategy In order to implement the tree-building algorithm kdtree you first need to implement jSmallest which, in turn requires partition. It is strongly suggested that you implement and thoroughly test partition before trying to implement jSmallest. In turn, throughly test jSmallest before you implement kdtree. If you don’t do this, I can tell you from experience that it will be a nightmare to debug. You need to be sure that each algorithm is correct before implementing the algorithms that depend on it, otherwise, if you run into a bug it will be very hard to determine in which method in the chain of dependent methods the bug is occurring. Grading Scheme Correctness: 35 points (for node and tree class implementations, and required console output) Design: 5 points Comments (inline and Javadoc): 5 points Sample Output Input 2 D points : (5.0 , 2.0) (9.0 , 10.0) (11.0 , 1.0) (4.0 , 3.0) (2.0 , 12.0) (3.0 , 7.0) (1.0 , 5.0) The 2 D tree built from these points is : 8 | P a g e 4: - 3: (9.0 , 10.0) 4: - 2: (5.0 , 2.0) 4: - 3: (11.0 , 1.0) 4: - 1: (4.0 , 3.0) 4: - 3: (2.0 , 12.0) 4: - 2: (3.0 , 7.0) 4: - 3: (1.0 , 5.0) 4: - Input 3 D points : (1.0 , 12.0 , 1.0) (18.0 , 1.0 , 2.0) (2.0 , 12.0 , 16.0) (7.0 , 3.0 , 3.0) (3.0 , 7.0 , 5.0) (16.0 , 4.0 , 4.0) (4.0 , 6.0 , 1.0) (5.0 , 5.0 , 17.0) 5: - 4: (5.0 , 5.0 , 17.0) 5: - 3: (16.0 , 4.0 , 4.0) 4: - 2: (7.0 , 3.0 , 3.0) 4: - 3: (18.0 , 1.0 , 2.0) 4: - 1: (4.0 , 6.0 , 1.0) 4: - 3: (1.0 , 12.0 , 1.0) 4: - 2: (2.0 , 12.0 , 16.0) 4: - 3: (3.0 , 7.0 , 5.0) 4: - Looking for points between (0.0 , 1.0 , 0.0) and (4.0 , 6.0 , 3.0). Found : (4.0 , 6.0 , 1.0) Looking for points between (0.0 , 1.0 , 0.0) and (8.0 , 7.0 , 4.0). Found : (7.0 , 3.0 , 3.0) 9 | P a g e (4.0 , 6.0 , 1.0) Looking for points between (0.0 , 1.0 , 0.0) and (17.0 , 9.0 , 10.0). Found : (16.0 , 4.0 , 4.0) (7.0 , 3.0 , 3.0) (3.0 , 7.0 , 5.0) (4.0 , 6.0 , 1.0) 3 Files Provided lib280-asn6: A copy of lib280 which includes: TheTwoThreeTree280 class and related node and position classes in the lib280.tree package for Question 1. Partially completed IterableTwoThreeTree280 class in the in lib280.tree package for Question 1. the NDPoint280 class in the lib280.base package for representing n-dimensional points for question 2; 4 What to Hand In IterableTwoThreeTree280.java: Your completed B+ Tree of order 3 for Question 1. KDNode280.java: The node class for your k-D tree from Question 2. KDTree280.java: Your k-D tree class for Question 2. A5q2.txt/doc/pdf: The console output from your test program for question 2, cut and paste from the Eclipse console window. Appendix 10 | P a g e Figure 1: Top: a 2-3 tree; Bottom: a B+ tree of order 3 containing the same elements. Here the keys are strings (describing magical items in a fantasy game world) and the data items are integers (representing the value, in gold pieces, of the object described by the key). Note that the trees are the same except for the extra linkages of the leaf nodes. 11 | P a g e Figure 2: Class hierarchy for IterableTwoThreeNode280. For methods, only type names of parameters are shown. 12 | P a g e Figure 3: UML Class Hierarchy for 2-3 Tree Nodes in lib280. Every method that might be needed for either an internal or a leaf node is defined in the common abstract ancestor class TwoThreeTree280 (note: because it is abstract, it cannot be instantiated). Subclasses InternalTwoThreeNode280 and LeafTwoThreeNode280 contain the data needed for the respective types of nodes, and definitions of each method appropriate to that type of node. Inherited methods that don’t make sense for a particular type of node (e.g. getData() on an internal node) are defined to throw exceptions. The actual type of a reference to a TwoThreeNode can be determined by calling isInternal which is defined by internal nodes to return true and is defined by leaf nodes to return false. The LinkedLeafTwoThreeNode280 extends the leaf node class to add predecessor and successor references to maintain the bi-linked list of leaf nodes in the B+ tree of order 3.