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![m-Way Search Tree
(vi) Each of the subtreeAi , 0 ≤ i ≤ m-1 are also m-way search tree
m-Way SearchTree [ m=4]
Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal](https://image.slidesharecdn.com/8m-waytreebtree-250427101503-23f8db98/75/8_m-way-tree_Btree-data-structures-and-algorithms-5-2048.jpg)
![m-Way Search Tree [ m=5]
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Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal](https://image.slidesharecdn.com/8m-waytreebtree-250427101503-23f8db98/75/8_m-way-tree_Btree-data-structures-and-algorithms-6-2048.jpg)
![m-Way Search Tree
(vi) Each of the subtreeAi , 0 ≤ i ≤ m-1 are also m-way search tree
m-Way SearchTree [ m=4]
Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal](https://crownmelresort.com/image.slidesharecdn.com/8m-waytreebtree-250427101503-23f8db98/75/8_m-way-tree_Btree-data-structures-and-algorithms-5-2048.jpg)
![m-Way Search Tree [ m=5]
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Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal](https://crownmelresort.com/image.slidesharecdn.com/8m-waytreebtree-250427101503-23f8db98/75/8_m-way-tree_Btree-data-structures-and-algorithms-6-2048.jpg)
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8_m-way tree_Btree data structures and algorithms
- 1. Advanced Data Structures (CS365) m-waySearchTree Dr. Sourav Kanti Addya email: souravkaddya@nitk.edu.in Department of Computer Science and Engineering National Institute of Technology Karnataka, Surathkal
- 2. Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 3. m-Way Search Tree An m-way search tree T may be an empty tree. If T is non-empty, it satisfies the following properties: (i) For some integer m known as the order of the tree, each node has at most m child nodes. (ii) A node may be represented as A0 , (K1,A1), (K2,A2) …. (Km-1 ,Am-1 ) where Ki , 1 ≤ i ≤ m-1 are the keys and Ai, 0 ≤ i ≤ m-1 are the pointers to the subtree of T (iii) If the node has c child nodes where c ≤ m, then the node can have only (c-1) keys, K1 , K2 , …… Kc-1 (iv) The keys in a node are ordered, i.e., K1<K2< …… <Kc-1 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 4. m-Way Search Tree (v)For a node A0 , (K1 ,A1), (K2 ,A2) , …. (Km-1 ,Am-1 ), if Si is the subtree pointed byAi, 0 ≤ i ≤ m-1then Key(S0)<K1 Key(Sm-1)>Km-1 Ki < Key(Si) < Ki+1 , 1 ≤ i ≤ m-2 A0 A1 A2 A3 < < > Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 5. m-Way Search Tree (vi)Each of the subtreeAi , 0 ≤ i ≤ m-1 are also m-way search tree m-Way SearchTree [ m=4] Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 6. m-Way Search Tree[ m=5] 18 44 76 198 X X 7 12 X X 80 92 141 262 8 10 148 151 172 186 X X X X X X X X X X X 77 X X 272 286 350 X X X X Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 7. Searching in anm-Way Search Tree 18 44 76 198 X X 7 12 X X 80 92 141 262 8 10 148 151 172 186 X X X X X X X X X X X 77 X X 272 286 350 X X X X Look for 77 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 8. Insertion in anm-Way Search Tree 18 44 76 198 X X 7 12 X X 80 92 141 262 8 10 148 151 172 186 X X X X X X X X X X X 77 X X 272 286 350 X X X X Insert 6 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 9. Insertion in anm-Way Search Tree 18 44 76 198 X X 7 12 X X 80 92 141 262 8 10 148 151 172 186 X X X X X X X X X X X 77 X X 272 286 350 X X X X Insert 6 6 X Insert 146 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 10. Insertion in anm-Way Search Tree 18 44 76 198 X X 7 12 X X 80 92 141 262 8 10 148 151 172 186 X X X X X X X X X X 77 X X 272 286 350 X X X X Insert 6 6 X Insert 146 146 X X Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 11. Deletion in anm-Way Search Tree Let K be the key to be deleted from the m-way search tree. K Ai Aj K : Key Ai ,Aj : Pointers to subtree Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 12. Deletion in anm-Way Search Tree 1) If (Ai = Aj = NULL) then delete K 2) If (Ai NULL, Aj = NULL ) then choose the largest of the key elements K’ in the subtree pointed to by Ai and replace K by K’. 3) If (Ai = NULL, Aj NULL ) then choose the smallest of the key element K” from the subtree pointed to by Aj , delete K” and replace K by K”. 4) If (Ai NULL, Aj NULL ) then choose the largest of the key elements K’ in the subtree pointed to by Ai or the smallest of the key element K” from the subtree pointed to by Aj to replace K. Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 13. 5-Way Search Tree 1844 76 198 X X 7 12 X X 80 92 141 262 8 10 148 151 172 186 X X X X X X X X X X X 77 X X 272 286 350 X X X X Delete 151 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 14. 5-Way Search Tree 1844 76 198 X X 7 12 X X 80 92 141 262 8 10 148 172 186 X X X X X X X X X X 77 X X 272 286 350 X X X X Delete 151 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 15. 5-Way Search Tree 1844 76 198 X X 7 12 X X 80 92 141 262 8 10 148 151 172 186 X X X X X X X X X X X 77 X X 272 286 350 X X X X Delete 12 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 16. 5-Way Search Tree 1844 76 198 X X 7 10 X X 80 92 141 262 8 148 151 172 186 X X X X X X X X X X 77 X X 272 286 350 X X X X Delete 12 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 17. 5-Way Search Tree 1844 76 198 X X 7 12 X X 80 92 141 262 8 10 148 151 172 186 X X X X X X X X X X X 77 X X 272 286 350 X X X X Delete 18 --Delete 12 --Replace 18 by 12 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 18. 5-Way Search Tree Delete18 --Delete 12 --Replace 18 by 12 18 44 76 198 X X 7 10 X X 80 92 141 262 8 148 151 172 186 X X X X X X X X X X 77 X X 272 286 350 X X X X Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 19. 5-Way Search Tree Delete18 --Delete 12 --Replace 18 by 12 12 44 76 198 X X 7 10 X X 80 92 141 262 8 148 151 172 186 X X X X X X X X X X 77 X X 272 286 350 X X X X Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 20. 5-Way Search Tree 1844 76 198 X X 7 12 X X 80 92 141 262 8 10 148 151 172 186 X X X X X X X X X X X 77 X X 272 286 350 X X X X Delete 262 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 21. 5-Way Search Tree 1844 76 198 X X 7 12 X X 80 92 141 272 8 10 148 151 172 186 X X X X X X X X X X X 77 X X 286 350 X X X Delete 262 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 22. 5-Way Search Tree 1844 76 198 X X 7 12 X X 80 92 141 262 8 10 148 151 172 186 X X X X X X X X X X X 77 X X 272 286 350 X X X X Delete 198 --delete186 --replace 198 by 186 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 23. 5-Way Search Tree 1844 76 198 X X 7 12 X X 80 92 141 262 8 10 148 151 172 X X X X X X X X X X 77 X X 272 286 350 X X X X Delete 198 --delete186 --replace 198 by 186 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 24. 5-Way Search Tree 1844 76 186 X X 7 12 X X 80 92 141 262 8 10 148 151 172 X X X X X X X X X X 77 X X 272 286 350 X X X X Delete 198 --delete186 --replace 198 by 186 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 25. 5-Way Search Tree 1844 76 198 X X 7 12 X X 80 92 141 262 8 10 148 151 172 186 X X X X X X X X X X X 77 X X 272 286 350 X X X X Delete 198 --delete 262 --replace 198 by 262 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 26. 5-Way Search Tree 1844 76 198 X X 7 12 X X 80 92 141 272 8 10 148 151 172 186 X X X X X X X X X X X 77 X X 286 350 X X X Delete 198 --delete 262 --replace 198 by 262 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 27. 5-Way Search Tree 1844 76 262 X X 7 12 X X 80 92 141 272 8 10 148 151 172 186 X X X X X X X X X X X 77 X X 286 350 X X X Delete 198 --delete 262 --replace 198 by 262 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 28. B Trees B treeis a balanced m-way search tree A B tree of order m, if non empty, is an m-way search tree in which i. the root has at least two child nodes and at most m child nodes ii. internal nodes except the root have at least m/2 child nodes and at most m child nodes iii. all leaf nodes are on the same level Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 29. B Tree oforder 5 48 31 45 56 64 85 87 88 100 112 X X X X X 49 51 52 X X X X 46 47 36 40 42 10 18 21 X X X X X X X X X X X X X X 58 62 67 75 X X X Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 30. Searching a BTree Searching for a key in a B-tree is similar to the one on an m-way search tree. The number of accesses depends on the height h of the B-tree Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 31. Insertion in aB-Tree 1. Attempt to insert the new key into a leaf 2. If this would result in that leaf becoming too big, split the leaf into two, promoting the middle key to the leaf’s parent 3. If this would result in the parent becoming too big, split the parent into two, promoting the middle key 4. This strategy might have to be repeated all the way to the top 5. If necessary, the root is split in two and the middle key is promoted to a new root, making the tree one level higher Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 32. Dr. Sourav KantiAddya|| Dept of CSE, NITK Surathkal
- 33. • Suppose westart with an empty B-tree and keys arrive in the following order:1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45 • We want to construct a B-tree of order 5 • The first four items go into the root: • To put the fifth item in the root would over-fill it • Therefore, when 25 arrives, pick the middle key to make a new root Constructing a B-tree 12 8 1 2 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 34. Constructing a B-tree Add25 to the tree 1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45 12 8 1 2 25 Exceeds Order. Promote middle and split. Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 35. Constructing a B-tree(contd.) 6, 14, 28 get added to the leaf nodes: 1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45 12 8 1 2 25 12 8 1 2 25 6 1 2 28 14 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 36. Constructing a B-tree(contd.) Adding 17 to the right leaf node would over-fill it, so we take the middle key, promote it (to the root) and split the leaf 1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45 1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45 12 8 2 25 6 1 2 28 14 28 17 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 37. Constructing a B-tree(contd.) 7, 52, 16, 48 get added to the leaf nodes 1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45 12 8 25 6 1 2 28 14 17 7 52 16 48 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 38. Constructing a B-tree(contd.) Adding 68 causes us to split the right most leaf, promoting 48 to the root 1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45 8 17 7 6 2 1 16 14 12 52 48 28 25 68 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 39. Constructing a B-tree(contd.) Adding 3 causes us to split the left most leaf 1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45 48 17 8 7 6 2 1 16 14 12 25 28 52 68 3 7 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 40. Constructing a B-tree(contd.) 1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45 Add 26, 29, 53, 55 then go into the leaves 48 17 8 3 1 2 6 7 52 68 25 28 16 14 12 26 29 53 55 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 41. Constructing a B-tree(contd.) Add 45 increases the trees level 1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45 48 17 8 3 29 28 26 25 68 55 53 52 16 14 12 6 7 1 2 45 Exceeds Order. Promote middle and split. Exceeds Order. Promote middle and split. Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 42. Exercise in Insertinga B-Tree Insert the following keys to a 5-way B-tree: 3, 7, 9, 23, 45, 1, 5, 14, 25, 24, 13, 11, 8, 19, 4, 31, 35, 56 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 43. Answer to Exercise Dr.Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 44. Delete from aB-tree 1. If the key is already in a leaf node, and removing it doesn’t cause that leaf node to have too few keys, then simply remove the key to be deleted. 2. If the key is not in a leaf then it is guaranteed (by the nature of a B-tree) that its predecessor or successor will be in a leaf -- in this case can we delete the key and promote the predecessor or successor key to the non-leaf deleted key’s position. Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 45. Removal from aB-tree (2) • If (1) or (2) lead to a leaf node containing less than the minimum number of keys then we have to look at the siblings immediately adjacent to the leaf in question: – 3: if one of them has more than the min’ number of keys then we can promote one of its keys to the parent and take the parent key into our lacking leaf – 4: if neither of them has more than the min’ number of keys then the lacking leaf and one of its neighbours can be combined with their shared parent (the opposite of promoting a key) and the new leaf will have the correct number of keys; if this step leave the parent with too few keys then we repeat the process up to the root itself, if required Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 46. Type #1: Simpleleaf deletion 12 29 52 2 7 9 15 22 56 69 72 31 43 Delete 2: Since there are enough keys in the node, just delete it Assuming a 5-way B-Tree, as before... Note when printed: this slide is animated Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 47. Type #2: Simplenon-leaf deletion 12 29 52 7 9 15 22 56 69 72 31 43 Delete 52 Borrow the predecessor or (in this case) successor 56 Note when printed: this slide is animated Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 48. Type #3: Enoughsiblings 12 29 7 9 15 22 69 56 31 43 Delete 22 Demote root key and promote leaf key Note when printed: this slide is animated Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 49. Type #3: Enoughsiblings 12 29 7 9 15 31 69 56 43 Note when printed: this slide is animated Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 50. Type #4: Toofew keys in node and its siblings 12 29 56 7 9 15 22 69 72 31 43 Delete 72 Too few keys! Join back together Note when printed: this slide is animated Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 51. Type #4: Toofew keys in node and its siblings 12 29 7 9 15 22 69 56 31 43 Note when printed: this slide is animated Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 52. Exercise in Removalfrom a B-Tree Given 5-way B-tree created by these data (last exercise): 3, 7, 9, 23, 45, 1, 5, 14, 25, 24, 13, 11, 8, 19, 4, 31, 35, 56 Add these further keys: 2, 6,12 Delete these keys: 4, 5, 7, 3, 14 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 53. Answer to Exercise Dr.Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 54. 5-Way B Tree(insertion examples) 8 96 116 2 7 X X X 104 110 37 46 55 86 X X X X X X X X 137 145 X X X Insert 4, 5, 58, 6 in the order Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 55. 5-Way B Tree(insertion examples) 8 96 116 104 110 37 46 55 86 X X X X X X X X 137 145 X X X Search tree after inserting 4 2 4 7 X X X X Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 56. 5-Way B Tree(insertion examples) 8 96 116 104 110 37 46 55 86 X X X X X X X X 137 145 X X X Search tree after inserting 4, 5 2 4 5 7 X X X X X Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 57. 5-Way B Tree(insertion examples) 8 96 116 104 110 37 46 55 86 X X X X X X X X 137 145 X X X 2 4 5 7 X X X X X 37,46,55,58,86 Split the node at its median into two node, pushing the median element up by one level Insert 58 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 58. 5-Way B Tree(insertion examples) 104 110 X X X 137 145 X X X 2 4 5 7 X X X X X 37 46 X X X 58 86 X X X 8 96 116 Insert 55 in the root Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 59. 5-Way B Tree(insertion examples) 104 110 8 55 96 116 X X X 137 145 X X X Search tree after inserting 4, 5, 58 2 4 5 7 X X X X X 37 46 X X X 58 86 X X X Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 60. 5-Way B Tree(insertion examples) 104 110 8 55 96 116 X X X 137 145 X X X Insert 6 2 4 5 7 X X X X X 37 46 X X X 58 86 X X X 2,4,5,6,7 Split the node at its median into two node, pushing the median element up by one level Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 61. 5-Way B Tree(insertion examples) 104 110 8 55 96 116 X X X 137 145 X X X Insert 5 at the root 37 46 X X X 58 86 X X X 6 7 2 4 X X X X X X Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 62. 5-Way B Tree(insertion examples) 104 110 X X X 137 145 X X X Insert 5 at the root 37 46 X X X 58 86 X X X 6 7 2 4 X X X X X X 96 116 5 8 55 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 63. 5-Way B Tree(insertion examples) 104 110 X X X 137 145 X X X Insert 5 at the root 37 46 X X X 58 86 X X X 6 7 2 4 X X X X X X 96 116 5 8 55 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 64. B-tree of Order5 (deletion examples) 110 65 86 120 226 70 81 32 44 X X X 90 95 100 X X X X X X 115 118 200 221 X X X X X X X 300 440 550 601 X X X X X Delete 95, 226, 221, 70 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 65. B-tree of Order5 110 65 86 120 226 70 81 32 44 X X X 90 100 X X X X X X 115 118 200 221 X X X X X X 300 440 550 601 X X X X X B-tree after deleting 95 delete 226 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 66. B-tree of Order5 110 65 86 120 300 70 81 32 44 X X X 90 100 X X X X X X 115 118 200 221 X X X X X X 300 440 550 601 X X X X X Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 67. B-tree of Order5 110 65 86 120 300 70 81 32 44 X X X 90 100 X X X X X X 115 118 200 221 X X X X X X 440 550 601 X X X X B-tree after deleting 95, 226 Delete 221 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 68. B-tree of Order5 110 65 86 120 440 70 81 32 44 X X X 90 100 X X X X X X 115 118 200 300 X X X X X X 550 601 X X X B-tree after deleting 95, 226, 221 Delete 70 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 69. B-tree of Order5 110 65 86 120 440 65 81 32 44 X X X 90 100 X X X X X 115 118 200 300 X X X X X X 550 601 X X X Delete 65 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal
- 70. B-tree of Order5 110 86 120 440 65 81 32 44 X X X 90 100 X X X X X 115 118 200 300 X X X X X X 550 601 X X X B-tree after deleting 95, 226, 221, 70 Dr. Sourav KantiAddya || Dept of CSE, NITK Surathkal