Kakkot Tree- A Binary Search Tree with Caching

Transcription

1 Kakkot Tree- A Binary Search Tree with Caching Rajesh Ramachandran Associate Professor, Department of MCA Sree Narayana Gurukulam College of Engineering Kadayirippu post,ernakulam ryanrajesh@hotmail.com Abstract - Kakkot Tree is a Binary Searching Tree with the caching concept added. It is used to speed up the searching process that is carried on a Binary Search Tree. The time complexity of BST varies between O(n) and O(log n) depends upon the number of nodes that are present in left and right children. Kakkot Tree is a new data structure which is a variant of BST with some initial nodes which act like caching. Key Words: Searching, Binary Search Tree, Data Structure, Caching, Kakkot Tree. 1. Introduction. Binary Search Tree (BST) is a binary tree where all the root node values are greater than its left children and greater than its right children. The operations that can be performed on a BST include searching, insertion, deletion and other queries. The time complexity of searching depends on how the items are added or deleted in the BST. To some extent the searching can be speed up by balancing the BST for each addition and deletion operation. But this balancing process is required after the addition and deletion operation. Kakkot Tree is new data structure which is a variant of BST. Kakkot Tree is a Binary Search Tree with few nodes is added before the root of the BST. These added new nodes will act as Caching nodes. So when a query is executed to search for an item in the BST, first it searches the caching nodes and if it is evident in this list then the query succeeds and it return the value accordingly. Else it searches the remaining tree as usual in the BST. The size of caching nodes can be set according to the application. II Binary Search Tree A Binary Search Tree is a binary tree. It may be empty, if it is not empty then it satisfies the following properties. i) Each node has exactly one key and the keys in the tree are distinct ii) The keys(if any) in the left subtree are smaller than the key in the root iii) The keys(if any) in the right subtree are larger than the keys in the root iv) The left and right subtree are also binary search tree.[1][] The time required to search a BST varies between O(n) to O(log n), depending on the structure of the tree.[2] If the records are inserted in sorted order, the resulting tree contains all null left links, so that the tree search reduces to a sequential search yielding O(n) comparisons[2]. A Binary search tree that minimizes the expected number of comparisons for a given set of keys and probabilities is called optimum. The fastest known algorithm to produce an optimum binary search tree is O (n 2 ) in general case [2]. The most common operation performed on BST is searching for a key stored in the tree. Besides the search operation, Binary search tree can support such queries as MINIMUM, MAXIMUM, SUCCESSOR, and PREDECESSOR. These operations run in O (h) time where h are the height of the tree. []. Binary search tree is used in many searching application where data is constantly added or removed from the list. The advantage of using a binary-search tree over a linked list is that, if the tree is reasonably balanced, the searching, insertion and deletion all can be implemented in O(log n ) time complexity, where n is the number of stored items. For a linked list, although insert can be implemented to run in O(1) time, lookup and delete take O(N) time. III Caching Caching refers to the strategy of keeping a copy of the recently accessed data for future reference. This stored data can be accessed quickly compared to other data whenever a reference to the data occurs. This extremely fast access to data speeds the overall process and reduces the total execution time. The effectiveness of the cache mechanism is based on a property called locality of reference. Analysis of programs shows that most of their execution time is spent on routine in which many /12/$ IEEE

2 instructions are executed repeatedly. These instructions may constitute a simple loop, nested loops, or a few procedures that repeatedly call each other[]. IV.Kakkot Tree. Kakkot tree is variant of Binary Search Tree. A BST with few nodes which act as caching nodes.the following figure 1 show the diagrammatic representation of BST Figure 1 Binary Search Tree To create a Kakkot Tree we need to add some caching nodes to this tree. Figure 2 shows the Kakkot Tree with nine nodes and caching nodes. When an item being searched, first it search in the caching nodes and if the item resides in the caching nodes it returns the value, if not then it search remaining tree starting from the root as in the Binary search tree method. If the desired item is found in the tree then the value will be returned and the same data will be copied to the caching node for future reference. So next time onwards the recently accessed data will be available in the caching node. This will speed searching when the same data is searched immediately. The recently accessed data will always available in caching nodes and thus it can end the search within the caching nodes thus improves the time complexity. The number of caching nodes can be set at the time of creation of the Kakkot tree. If the size of the caching nodes increases the time complexity of searching also gets increased and if the number of caching node is n it becomes a linear search method and the time complexity will be O (n). So by carefully selecting the caching nodes size we can achieve better performance over BST. Figure 2 Kakkot Tree The operations that can be performed on Kakkot Tree data structure are a) Create() b) ISEMPTY() c) Search() d) Insert() e) Delete() The basic operations such as Search, Insert and Delete operation contains nested functions to check the presence of the item in the caching nodes to add item to the caching nodes and to remove an item from caching nodes. Abstract Data type of Kakkot Tree is explained in the following ADT. ADT Kakkot_Tree is Object: A finite set of nodes classified as caching nodes and normal nodes, either empty or consists of data. Normal nodes if not empty will contain root node,left subtree and right subtree.

3 Functions: For all x Є Kakkot Tree, n,item Є Element, k Є Key KakkotTree Create(n) ::= Creates an empty Kakkot Tree with n caching nodes. Boolean IsEmpty(x) ::=if(x==empty Kakkot Tree) return True else return False. AddCache(x,item,k) ::= add item to the caching nodes of x, if caching nodes are full then replace an element and insert item with the key k. RemoveCache(x,k) ::= remove the item with key k from x from the caching nodes. IsCache(x,k) ::= if the item with key k is in caching nodes of x return the item else return false. Element Search(x,k) ::= if(iscache(k) then return item with key k Else (return NULL if no such element else return item with key k and AddCache(x,item,k)) Element Delete(x,k) ::= delete and return item with key k and RemoveCache(x,k) (if any) Void Insert(x,item,k) ::= Insert the element item with the key k into x and AddCache(x,item,k). End. Now we will see how the operations that are carried out in the Kakkot Tree. A. Creation: The create function will create an empty KakkotTree with the specified number of caching nodes.as mentioned above the if number of caching nodes increases it becomes a linear searching and the time complexity becomes O(n) where n is the number of items in the tree. B.Insertion: When a new item is to be inserted, like BST it search from the root of the tree and finds its position either in the left subtree or right subtree of root node based on the key of the item. When the position is identified then that item is inserted in its place by updating the parent child link information. At the same time the item is also inserted to the caching nodes. If there is no space in the caching nodes then one of the items will be removed from the caching nodes to accommodate the new element. C.Deletion: If an item is to be removed from the tree, search the item in the tree. If it is present in the tree remove the item by changing necessary parent child link and delete the node from the tree. Here we have to check the caching node also if the item is evident in the tree. If the item is in caching nodes remove the same from the caching node also. D.Searching: In Kakkot Tree the searching first take place in the caching nodes. If the search element with specified key is in the caching nodes then it returns the item and the search ends there. This makes the time complexity of Kakkot tree better compared to the BST. Since the probability of recently accessed data is more, this works well in most of the applications. If the search element is not in the caching nodes then it search the normal nodes in the Kakkot Tree as in the BST. If the element is in the normal nodes then it returns the data and the same will be copied to the caching nodes, this makes easy searching for the future references of same data element. If the search element is not in the normal nodes also then it returns false which indicates the item is not in the Kakkot Tree. E.Caching Replacement Sometimes caching replacement will be necessary to provide a space for the new elements which are added to the caching nodes. That is when caching nodes are full and if a new element is to be added to this, then one of the elements has to be removed from the caching nodes to give space for the new one. A random replacement method could be implemented for this purpose. Now we will see how Kakkot Tree works using an example. The figure 2 shows the Kakkot Tree with four caching nodes and nine normal nodes. Insertion To insert an item to the existing Kakkot tree, it start with root node and as in the BST it identifies the position of and element will be inserted. The data element will also inserted in the caching nodes. Here since the caching nodes are full to insert the data element we need to remove one item from the caching nodes. This selection can be made random so that replacement can be made fast. Figure show the Kakkot Tree after inserting the data element.

4 Figure Kakkot Tree after insertion After inserting the item the kakkot Tree will be as in the figure. One of the caching nodes data will be removed according to a random replacement selection method. Hence any of the data elements in the caching node can be replaced by the item. 1 Figure Kakkot Tree after deletion Deletion 1 If the item 1 is deleted the Kakkot Tree will be as in the figure. Since the item 1 is not in the caching nodes there is no change in the caching node. if the data element 1 is deleted then the Kakkot Tree will be as 1 Figure Kakkot Tree after insertion

5 1 Searching Figure Kakkot Tree If we want to search an item 2 then it search in the caching nodes. Since 2 is not in the caching nodes it searches the normal nodes, and there also 2 is not available so it returns false, means that the search item is not in the Kakkot Tree. If the desired element is, then the searching in the caching nodes itself the search ends and the corresponding data related to the key will be returned and no searching will take place in the normal nodes. if the searching is for the data element, as usual it first search in the caching nodes. Since is not in the caching nodes it then searches the normal nodes. Now it finds in the normal nodes so it returns the data element corresponding to the key value and also it adds the same to the caching nodes. Now the Kakkot Tree will be as in the figure. Figure Kakkot Tree V.Conclusion Kakkot Tree could be used as a binary search tree and the speed of searching an item is improved when compare to BST. In Kakkot Tree when an item is not in the caching nodes but in the normal nodes then it has to be copied to caching node. Similarly during insertion of an item to the tree, the item has to be added to both caching nodes and normal nodes. This additional operation is required in the Kakkot Tree. But due to property of locality of reference, it definitely speed up the searching and thus minimize the time required to find an item in the list. References [1]. Ellis Horowitz, Sartaj Sahni, Susan Anderson-Free, Fundamentals of Data Structure in C, 2 nd Ed.,University press(india) Pvt.Ltd.,,2009. [2]. Yedidyah Langsam, Moshe J Augenstein, Aaron M Tanenbaum, Data Structure using C and C++, 2 nd Ed., Prentice Hall India Pvt. Ltd., 200. []. Ellis Horowitz, Sartaj Sahni, Sanguthevar Rajasekaran, Fundamentals of Computer Algorithms, 2 nd Ed., University Press,2009. []. Udit Agarwal, Algorithms Design and Analysis, rd Ed., Dhanpat Rai & Co., 2011.

6 []. Ravi Jain, Member, Yi-Bing Lin, Charles Lo,Seshadri Mohan, A Caching Strategy to Reduce Network Impacts of PCS IEEE Journal on Selected Areas in Communications, Volume: 12, Issue:, 199, Page(s): []. Xin Min, Jixian Zhang, Lei Luo, Caching Strategy On Mobile Rich Media Engine, Proceeding of th IEEE International Conference on Computer and Information Technology (CIT 20), 20, Page(s): 21-2 []. Richard F Gilberg,Behrouz A Forouzan, Data Structures: A Pseudo Code approach with C, 2 nd Ed., Cengage Learning IndiaPvt.Ltd., []. Carl Hamacher,Zvonko Vranesic, Safwat Zaky, Computer Organization, th Ed.,Mc.Graw Hill, IEEE International Conference on Computational Intelligence and Computing Research