SOLVED CMPT 280– Intermediate Data Structures and Algorithms Assignment 5

  Your Tasks Question 1 (33 points): In lib280-asn6 you are provided with a fully functional 2-3 tree class called TwoThreeTree280. Recall that 2-3 trees are keyed dictionaries. As such, the TwoThreeTree280 class implements the KeyedBasicDict280 interface. This interface adds the methods obtain(k), delete(k) and has(k), and set(x) (replace the item whose key matches they key of x with the item x). Presently, TwoThreeTree280 does not 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 keyed data items stored in the two-three 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 17.1). We aren’t going to call it a B+ tree class though, because we are implementing specifically a B+ tree of order 3, and higher-order B+ trees will not be supported. Our IterableTwoThreeTree280 class will be exactly a B+ tree of order 3. Figure 1 in the Appendix shows the differences between a 2-3 tree (without iteration) and a B+ tree of order 3 containing the same elements, with the linking of the leaf nodes to support iteration. The algorithms for insertion and deletion are the same in both kinds of tree, except that in the case of the B+ tree, references to/from the predecessor and successor leaf nodes in key-order 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 ?? of the Appendix. To implement the IterableTwoThreeTree280, the following tasks must be carried out: 1. Make an extension of LeafTwoThreeNode280 that adds references to its predecessor and successor leaf nodes. This has already been done for you in the class LinkedLeafTwoThreeNode280. 2. Override the TwoThreeTree280::createNewLeafNode() method by adding a new protected method in IterableTwoThreeTree280 that it returns a new LinkedLeafTwoThreeNode280 object instead of a TwoThreeNode280 object. This has already been done for you. 3. In IterableTwoThreeTree280, 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 in the tree. 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. Maintaining these links is very similar to inserting and removing nodes from the middle of a doubly-linked list. 4. 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. In this class, the Page 2 linear iterator interface allows positioning of the cursor along the leaf-level of the tree. The cursor can never be positioned at an internal node. 5. 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. This is because you will need to insert and delete items to create test cases for those methods in KeyedLinearIterator280 You must test all of the methods listed in the interfaces that are coloured blue in Figure 2 of the Appendix. Use instances of the local class called Loot (which has been defined in the main() method) 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). Hint: The toStringByLevel() method you’ve been given 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 (30 points): For this question 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. 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 left and right (inclusive). It places at offset j (where left ≤ 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 left 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! The only element guaranteed to be in its sorted position is the one that belongs at offset j. Thus, if we want to find the median element of a subarray of the array list bounded by offsets left and right, we can call jSmallest(list, left, right, (left+right)/2) The offset (left + right)/2 (integer division!) is always the element in the middle of the subarray between offsets left and right because the average of two numbers is always equal to the number halfway in between them. The j-smallest algorithm is presented in its entirety on the next page. Page 3 ✞ ☎ Algorithm jSmallest ( list , left , right , j ) list - array of comparable 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. Precondition : left <= j <= right Precondition : all elements in ’list ’ are unique ( things get messy otherwise !) Postcondition : the element x that belongs at offset j , if the subarray were sorted , is at offset 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 ) // Partition the subarray using the last element , list [ right ] , as a pivot . // The index of the pivot after partitioning is returned . // This is exactly the same partition algorithm used by quicksort . pivotIndex := partition ( list , left , right ) // If the pivotIndex 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 // recursively look for the j-th smallest element in that subarray : if j < pivotIndex jSmallest ( list , left , pivotIndex -1 , j ) // Otherwise , the position j must be larger than the pivotIndex , // so the j-th smallest element must be between pivotIndex +1 and right . else if j > pivotIndex jSmallest ( list , pivotIndex +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 + left)/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. Page 4 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. ✞ ☎ // Partition a subarray using its last element as a pivot . Algorithm partition ( list , left , right ) list - array of comparable elements left - lower limit on subarray to be partitioned right - upper limit on subarray to be partitioned Precondition : all elements in ’list ’ are unique ( things get messy otherwise !) Postcondition : 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 guarantee about the ordering of the elements before and after the pivot . returns the offset at which the pivot element ended up pivot = list [ right ] swapOffset = left for i = left to right -1 if ( list [ i ] <= pivot ) swap list [ i ] and list [ swapOffset ] swapOffset = swapOffset + 1 swap list [ right ] and list [ swapOffset ] return swapOffset ; // return the offset where the pivot ended up ✝ ✆ Algorithm 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. ✞ ☎ Algorithm kdtree ( pointArray , left , right , int depth ) pointArray - array of k - dimensional 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 partially built tree - note that the root of a tree has depth 0 and the $k$ dimensions of the points are numbered 0 through k -1. if the specified subarray of pointArray is empty return null ; else // Select axis based on depth so that axis cycles through all // valid values . (k is the dimensionality of the tree ) d = depth mod k ; medianOffset = ( left + right )/2 // Put the median element in the correct position // This call assumes you have added the dimension d parameter Page 5 // to jSmallest as described earlier . jSmallest ( pointArray , left , right , d , medianOffset ) // Create node and construct subtrees node = a new kD - tree node node . item = pointArray [ medianOffset ] node . leftChild = kdtree ( pointArray , left , medianOffet -1 , depth +1); node . rightChild = kdtree ( pointArray medianOffset +1 , right , depth +1); return node ; ✝ ✆ Your Tasks Implementing the k-D Tree Implement a k-D tree. You must use the NDPoint280 class provided in the lib280.base package of lib280-asn6 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. You may use as much or as little of lib280 as you think is appropriate. You’ll be graded in the actual appropriateness of your choices. You should aim to make the class fit into lib280 and its hierarchy of data structures, but you should not force things by extending classes inappropriately. 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 k-D tree 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 . Note: your tree does not have to have operations that insert or remove individual NDPoints. In addition, you should write a test program that demonstrates the correctness of your tree. The test program should consist of two parts: 1. Show that your class can correctly build a k-D tree from a set of points. For k=2, display the set of k-dimensional points that you used 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 (shown in pseudocode, above) 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 Page 6 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. This is a fundamental principle that is crucial to designing complex software systems. Make sure each piece is correct before relying on it later. Keep in mind that the algorithms presented above as abstract pseudocode do not necessarily translate line-for-line into Java code. Likely much of the pseudocode you’ve seen up to this point *does* translate in a line-by-line fashion, but you need to get used to pseudocode that doesn’t, as this is the entire point of using pseudocode, such that it is language independent and omits implementation details while still conveying what the algorithm is supposed to do. 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 lib280 . tree built from these points is : 3: (9.0 , 10.0) 2: (5.0 , 2.0) 3: (11.0 , 1.0) 1: (4.0 , 3.0) 3: (2.0 , 12.0) 2: (3.0 , 7.0) 3: (1.0 , 5.0) Input 3 D points : (1.0 , 12.0 , 0.0) (18.0 , 1.0 , 2.0) (2.0 , 13.0 , 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) 4: (5.0 , 5.0 , 17.0) 3: (16.0 , 4.0 , 4.0) 4: - 2: (7.0 , 3.0 , 3.0) 3: (18.0 , 1.0 , 2.0) 1: (4.0 , 6.0 , 1.0) 3: (2.0 , 13.0 , 16.0) 2: (1.0 , 12.0 , 0.0) 3: (3.0 , 7.0 , 5.0) Page 7 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) (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) ✝ ✆ Page 8 2 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; 3 What to Hand In IterableTwoThreeTree280.java: Your completed tree for Question 1. KDNode280.java: The node class for your k-D tree from Question 1. KDTree280.java: Your k-D tree class for Question 1. a7q1.txt/doc/pdf: The console output from your test program for question 1, cut and paste from the IntelliJ console window. Page 9 Appendix Leather Armor Potion of Healing Vampiric Blade +1 Mace 2000 Blue Ioun Stone 20000 Leather Armor 10 Plate Armor 350 Potion of Healing 100 Vampiric Blade 12000 Plate Armor Blue Ioun Stone Leather Armor Potion of Healing Vampiric Blade Plate Armor Blue Ioun Stone +1 Mace 2000 Blue Ioun Stone 20000 Leather Armor 10 Plate Armor 350 Potion of Healing 100 Vampiric Blade 12000 Figure 1: Top: a 2-3 tree which does not support a linear iterator; 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 keyed data items contain the item name and an integer (representing the value, in gold pieces, of the object). Note that the trees are the same except for the extra linkages of the leaf nodes. Page 10 IterableTwoThreeTree280 #smallest: LinkedLeafTwoThreeNode280 #largest: LinkedLeafTwoThreeNode280 #cursor: LinkedLeafTwoThreeNode280 #prev: LinkedLeafTwoThreeNode280 K,I TwoThreeTree280 #rootNode: TwoThreeNode280 +height #createNewLeafNode #createNewInternalNode(TwoThreeNode, K, TwoThreeNode, K, TwoThreeNode) #find(K) #giveLeft(TwoThreeNode, TwoThreeNode) #giveRight(TwoThreeNode, TwoThreeNode) #stealLeft(TwoThreeNode, TwoThreeNode) #stealRight(TwoThreeNode, TwoThreeNode) +toString +toStringByLevel K,I ≪interface≫ Container280 clear isEmpty isFull ≪interface≫ KeyedBasicDict280 delete(K) has(K) insert(I) obtain(K) set(I) K,I ≪interface≫ KeyedDict280 deleteItem search(K) searchCeilingOf(K) setItem(I) K,I ≪interface≫ KeyedLinearIterator280 K,I ≪interface≫ CursorSaving280 currentPosition goPosition(CursorPosition280) ≪interface≫ KeyedCursor280 itemKey keyItemPair K,I ≪interface≫ LinearIterator280 after before goAfter goBefore goFirst goForth I ≪interface≫ Cursor280 item itemExists I Figure 2: Class hierarchy for IterableTwoThreeNode280. For methods, only type names of parameters are shown. Page 11