Analysis of Time complexity in Binary Search Tree

Transcription

1 Computing For Nation Development, February 2 26, 2010 Bharati Vidyapeeth s Institute of Computer Applications and Management, New Delhi Analysis of Time complexity in Binary Search Tree Pawan Jindal 1, Amit Kumar 2 and Shishir Kumar 3 1,2,3 Deptt. of Computer Science & Engineering, Jaypee Institute of Engineering & Technology, Guna, M.P. India, ID: 1 pawan.jindal@jiet.ac.in, 2 amit.kumar@jiet.ac.in, 3 shishir.kumar@jiet.ac.in ABSTRACT Data structure is essential for designing of software s. As most of operating systems uses tree or tree like data structure to store the data in it. Trees are used in text processing, searching algorithms, sorting algorithms, compiler designing etc. Trees are also used to represent phrase structure of sentences, which is crucial to language processing programs. In BST the key value of a particular node is greater than the key value of its left child node but less than or equal to its right child node value. The operations of data storage and organization can be done efficiently with Binary Search Tree. In this paper, the nonlinear data structure BST is being studied with the different operations like searching, insertion, deletion, updating etc. along with the derivation of time complexities of these operations in the worst case, best case and average case and explained with the help of diagram. Some advantages of BST over binary tree in the terms of time complexities is being studied. So we came to the conclusion that trees are the most important data structure to store, retrieve, update data etc. and they are being used for creating databases which is being used world wide and millions of the people are getting benefit from it. KEYWORDS Binary search tree, Data structure, time complexity, worst case, average case, best case. 1. INTRODUCTION As data structure[1,2,,6,8,9] are being used to store the data, to retrieve the data, to update the data and for some more useful operations. BST are much better than binary tree in terms of time complexity as well as space complexity for operations like insertion, searching, updation, deletion etc. Data structure is heart of computer science because it is being used for designing of software. The keys in a binary search tree must satisfy binary search-tree property. According to BST[16,17,18,21,22,26] property If x be a node in a binary search tree. If y is a node in the left sub tree of x, then key[y]< key[x]. If y is a node in the right sub tree of x, then key[x] key[y]. If every node in BST have only left child (except leaf node) i.e. any node in BST does not have right child then BST is termed as left skewed BST as shown in figure 1. If every node in BST have only right child (except leaf node) i.e. any node in BST does not have left child then BST is termed as right skewed BST as shown in figure 3 Figure 1. Left skewed BST with height 4 As height of BST is directly proportional to the time complexity of an operation. If the five nodes as in figure 1 is represented as in figure 2. As height of BST in figure 2 <height of BST in figure 1. So time complexity[3,4,7,8,10] of BST in figure 2 will be lesser than that of BST in figure 1 for any operation although the total number of nodes and the key values in the nodes are exactly same in both BST. Figure 2. BST with the exactly same number of nodes and with the same key value as in Figure 1 but with height Figure 3. Right skewed BST with height 4. 39

2 Figure 4. BST with the exactly same number of nodes and with the same key value as in Figure 3 but with height INSERTION Generally algorithm[11,12,14,1] may be defined as the finite set of instructions which are being executed in a systematic way gives the solution to a particular problem. Every operation like inserting a particular node, searcing a particular node, deleting a particular node in BST has a particular algorithm which when executed in the systematic way executes the particular operation. As root node is most important node in BST because all operations like searching, inserting, deleting etc. start from the root node of BST. If the values,40,60,3,78 have to be inserted in to initially empty BST in the systematic way then the resultant BST which is formed is shown below 3 40 Figure. BST with height As asymptotic notation[1,2,3,,6] like Θ-notation(Theta notation), О notation(big O notation), Ω-notation(big omega notation), o-notation(little-oh notation), ω-notation(little ω notation) are being used to calculate the time complexities of different operations like searching, insertion, deletion, updation etc. The time complexity of a particular operation is further divided in to the following parts:- worst case time complexity, average case time complexity, best case time complexity. If the initially BST is left skewed BST or right skewed BST and an operation of insertion of a particular element is done which increases the height of initial BST with one then it is the case of insertion of node with the maximum time complexity. If the initial BST is as shown below So the total number of comparison which are being done =4. As the total number of nodes in initial BST is 4 so worst case time complexity after insertion = О(n). If the node which we want to insert become the root node then that case will be of the best case time complexity So best case time complexity for insertion = О(1). Average case[13,24,28,29] time complexity for insertion a node in BST=O(lgn) 78 Figure 6. BST with height 3. Then after inserting a node with value 88 the resultant tree is shown below Figure 7. BST with height SEARCHING The worst case time complexity for searching a particular node in BST= О(n). If the BST is left skewed BST or right skewed BST and we want to search that element which is leaf node of the tree then that case will be of the worst case tome complexity. As BST shown by figure 8 is right skewed BST and we want to search 99 then total number of comparison for searching will be= which is equal to the total number of nodes in BST So the time complexity will be О(n). If the node which we want to search is the root node then that case will be of the best case time complexity So best case time complexity for searching = О(1). Average case time complexity for searching a node in BST=O(lgn) Figure 8. BST with height

3 Analysis of Time Complexity in Binary Search Tree 4. DELETION The worst case time complexity for deletion = О(n) because In worst case the total number of comparison will be equal to the total number of nodes in BST If the node with the maximum value which we want to search is the root node then that case will be of the best case time complexity. So best case time complexity for searching the maximum element = О(1). Average case time complexity for finding the node with the maximum value in BST=O(lgn). The following table shows the minimum number of nodes as well as maximum number of node in BST with a specified height. Generally if h is the height of BST. Then minimum number of nodes in BST with height h= h+1. Also the maximum number of nodes in BST with height h is 2 h+1 1 which is also observed from the Table 1. Figure 9. Right skewed BST with height After deletion the node with key value 93 the resultant BST will be as shown below Figure 11. Left skewed BST with height FINDING THE MAXIMUM ELEMENT The worst case time complexity for finding the maximum element in BST = О(n). Figure 10. Right skewed BST with height If the node which we want to delete is the root node then that case will be of the best case time complexity So best case time complexity for deletion a particular node = О(1). Average case time complexity for deletion a node in BST=O(lgn) FINDING THE MINIMUM ELEMENT The worst case time complexity for finding the minimum element in BST = О(n) because In worst case the total number of comparison will be equal to the total number of nodes in BST. For searching the node with the key value 41. The total number of comparison will be 4 which is equal to the total number of nodes in BST. So worst case time complexity in BST = О(n). If the node with the minimum value which we want to search is the root node then that case will be of the best case time complexity. So best case time complexity for searching the minimum element = О(1). Average case time complexity for finding the node with the minimum value in BST=O(lgn) For finding 99 the maximum number of comparisons will be done. Figure 12. Right skewed BST with height 4. S.No. Height Minimum no. of nodes Maximum no. of nodes

4 Table 1 The root node of BST is assumed at level 0. The following table shows the minimum number of nodes as well as maximum number of node in BST at a particular level l. Generally if l is the level of BST. Then minimum number of nodes at a particular level l =1 Also the maximum number of nodes in BST at a particular level l= 2 l which is also observed from the table 2. The time complexities of BST can be further decreased by making the tree height balanced BST. If the balancing factor of BST at each and every node is either 0 or 1 or -1 then that BST will be Height balanced Binary Search Tree. If at a particular point, the balancing bactor of node comes out to be different than 0 or 1 or -1 Then restructuring of tree is being done by rotating a particular node (by use of left rotation or right rotation)to make height balanced binary search tree which is known as AVL Tree because Adelson, Velski and Landis gave the concept of height balanced binary search tree in S.No. Level No. Minimum no. of nodes at level l Maximum no.of nodes at level l Table 2 7. CONCLUSION In this paper the time complexities for different operations like searching, insertion, deletion, updating etc. is being studied in the case of BST along with the derivation of time complexities of these operations in the worst case, best case and average case. It has been found that BST are much better than binary tree due to decreased in the time complexity and space complexity in BST as compared to the binary tree. It has been analyzed that the time complexity of BST is maximum when BST is left skewed BST or right skewed BST because in this case the height of BST is approximately equal to the total number of nodes in BST and the time complexity of BST is minimum when BST is in the form of completely binary tree. The time complexities of BST is deceased by making BST height balanced tree by restructuring tree with the use of left rotation or right rotation is also explained briefly. 8. REFERENCES [1]. E. Horowitz and S. Sahni, "Fundamental of Computer Algorithms", Computer Science Press,1982. [2]. T. Cormen, C. Leiserson and R. Rivest, "Introduction to Algorithms", MIT Press, 1991 [3]. A.V. Aho, J.E. Hopcroft and J.D. Ullman, "The Design and Analysis of Computer Algorithms", Addison-Wesley, [4]. A.V. Aho, J.E. Hopcroft and J.D. Ullman, "Data Structures and Algorithms", Addison-Wesley, []. S. Baase, "Computer Algorithms: Introduction to Design and Analysis", Addison-Wesley [6]. L. Banachowski, A. Kreczmar and W. Rytter, "Analysis of Algorithms and Data Structures",Addison-Wesley, [7]. R. Baeza-Yates, "Teaching Algorithms", IV Iberoamerican Congress on Computer Science Education, Canela, Brazil, July 199. [8]. G. Tel, "Introduction to Distributed Algorithms"Cambridge Univ. Press 199. [9]. G- Brassard, and P. Bratley, "Algorithmics: Theory and Practice", Prentice-Hall, [10]. E.M. Reingold, J. Nievergelt and N. Deo, "Combinatorial Algorithms: Theory and Practice", Prentice-Hall, [11]. H. Will, "Algorithms and Complexity", Prentice-Hall, 1986 [12]. M. Crochemore and W. Rytter, "Text Algorithms", Oxford Press, [13]. J.S. Vitter and P. Flajolet", "Average-case analysis of Algorithms and Data Structures", Handbook of Theoretical Computer Science (volume A), Elsevier and MIT Press, 1990, [14]. T.J. Fountain, "Parallel Computing: Principles and Practice", Cambridge Univ. Press, 199. [1]. W. Frakes and R. Baeza-Yates, editors. "Information Retrieval: Data Structures and Algorithms", Prentice- Hall, Continued on Page No. 404

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