10.m way search tree

NITRKL

1. Data Structure and Algorithm (CS 102) Ashok K Turuk 1

2. 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. A node may be represented as A , (K 1 , A 1 ), (K 2 , A 2 ) …. (K m-1 , A m-1 ) where K i 1<= i <= m-1 are the keys and A i , 0<=i<=m-1 are the pointers to the subtree of T 2

3. m -Way Search Tree [2] If the node has k child nodes where k<=m , then the node can have only ( k-1 ) keys, K 1 , K 2 , …… K k-1 contained in the node such that K i < K i+1 and each of the keys partitions all the keys in the subtrees into k subsets [3] For a node A , (K 1 , A 1 ), (K 2 , A 2 ) , …. (K m-1 , A m-1 ) all key values in the subtree pointed to by A i are less than the key K i+1 , 0<=i<=m-2 and all key values in the subtree pointed to by A m-1 are greater than K m-1 3

4. m -Way Search Tree [4] Each of the subtree A i , 0<=i<=m-1 are also m-way search tree 4

5. 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 5

6. Searching in an m -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 6

7. Insertion in an m -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 7

8. Insertion in an m -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 8

9. Deletion in an m -Way Search Tree Let K be the key to be deleted from the m-way search tree. K A i A j K : Key A i , A j : Pointers to subtree 9

10. Deletion in an m -Way Search Tree [1] If (A i = A j = NULL) then delete K [2] If (A i  NULL, A j = NULL ) then choose the largest of the key elements K’ in the child node pointed to by A i and replace K by K’ . [3] If (A i = NULL, A j  NULL ) then choose the smallest of the key element K” from the subtree pointed to by A j , delete K” and replace K by K” . 10

11. Deletion in an m -Way Search Tree [4] If (A i  NULL, A j  NULL ) then choose the largest of the key elements K’ in the subtree pointed to by A i or the smallest of the key element K” from the subtree pointed to by A j to replace K . 11

12. 5 -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 Delete 151 12

13. 5 -Way Search Tree 18 44 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 13

14. 5 -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 Delete 262 14

15. 5 -Way Search Tree 18 44 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 15

16. 5 -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 Delete 12 16

17. 5 -Way Search Tree 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 Delete 12 17

18. B Trees B tree is 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 18

19. B Trees [iii] the number of keys in each internal node is one less than the number of child nodes and these keys partition the keys in the subtrees of nodes in a manner similar to that of m-way search trees [iv] all leaf nodes are on the same level 19

20. B Tree of order 5 48 31 45 56 64 85 87 88 100 112 X X X X X 49 51 52 X X X X 20 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

21. Searching a B Tree 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 21

22. Insertion in a B-Tree A key is inserted according to the following procedure [1] If the leaf node in which the key is to be inserted is not full, then the insertion is done in the node. A node is said to be full if it contains a maximum of (m-1) keys given the order of the B-tree to be m 22

23. Insertion in a B-Tree [2] If the node were to be full then insert the key in order into the existing set of keys in the node. Split the node at its median into two nodes at the same level, pushing the median element up by one level. Accommodate the median element in the parent node if it is not full. Otherwise repeat the same procedure and this may call for rearrangement of the keys in the root node or the formation of new root itself. 23

24. 5 -Way Search Tree 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

25. 5 -Way Search Tree 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

26. 5 -Way Search Tree 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

27. 5 -Way Search Tree 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

28. 5 -Way Search Tree 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

29. 5 -Way Search Tree 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

30. 5 -Way Search Tree 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

31. 5 -Way Search Tree 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

32. 5 -Way Search Tree 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

33. 5 -Way Search Tree 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

34. Deletion in a B-Tree It is desirable that a key in leaf node be removed. When a key in an internal node to be deleted, then we promote a successor or a predecessor of the key to be deleted ,to occupy the position of the deleted key and such a key is bound to occur in a leaf node. 34

35. Deletion in a B-Tree Removing a key from leaf node : If the node contain more than the minimum number of elements, then the key can be easily removed. If the leaf node contain just the minimum number of elements, then look for an element either from the left sibling node or right sibling node to fill the vacancy. 35

36. Deletion in a B-Tree If the left sibling has more than minimum number of keys, pull the largest key up into the parent node and move down the intervening entry from the parent node to the leaf node where key is deleted. Otherwise, pull the smallest key of the right sibling node to the parent node and move down the intervening parent element to the leaf node. 36

37. Deletion in a B-Tree If both the sibling node has minimum number of entries, then create a new leaf node out of the two leaf nodes and the intervening element of the parent node, ensuring the total number does not exceed the maximum limit for a node. If while borrowing the intervening element from the parent node, it leaves the number of keys in the parent node to be below the minimum number, then we propagate the process upwards ultimately resulting in a reduction of the height of B-tree 37

38. B-tree of Order 5 38 110 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

39. B-tree of Order 5 39 110 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

40. B-tree of Order 5 40 110 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

41. B-tree of Order 5 41 110 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

42. B-tree of Order 5 42 110 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

43. B-tree of Order 5 43 110 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

44. B-tree of Order 5 44 110 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

45. Heap Suppose H is a complete binary tree with n elements H is called a heap or maxheap if each node N of H has the following property Value at N is greater than or equal to the value at each of the children of N . 45

46. Heap 46 97 88 95 66 55 66 35 18 40 30 26 48 24 55 95 62 77 48 25 38 97 88 95 66 55 95 48 66 35 48 55 62 77 25 38 18 40 30 26 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

47. Inserting into a Heap Suppose H is a heap with N elements Suppose an ITEM of information is given. Insertion of ITEM into heap H is given as follows: [1] First adjoin ITEM at the end of H so that H is still a complete tree, but necessarily a heap [2] Let ITEM rise to its appropriate place in H so that H is finally a heap 47

48. Heap 48 97 88 95 66 55 66 35 18 40 30 26 48 24 55 95 62 77 48 25 38 97 88 95 66 55 95 48 66 35 48 55 62 77 25 38 18 40 30 26 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Insert 70 70

49. Heap 49 97 88 95 66 55 66 35 18 40 30 26 48 24 55 95 62 77 48 25 38 97 88 95 66 55 95 48 66 35 48 55 62 77 25 38 18 40 30 26 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Insert 70 70

50. Heap 50 97 88 95 66 55 66 35 18 40 30 26 70 24 55 95 62 77 48 25 38 97 88 95 66 55 95 48 66 35 48 55 62 77 25 38 18 40 30 26 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Insert 70 48

51. Heap 51 97 88 95 66 70 66 35 18 40 30 26 55 24 55 95 62 77 48 25 38 97 88 95 66 55 95 48 66 35 48 55 62 77 25 38 18 40 30 26 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Insert 70 48

52. Build a Heap Build a heap from the following list 44, 30, 50, 22, 60, 55, 77, 55 52

53. 53 44 44, 30, 50, 22, 60, 55, 77, 55 44 30 44 30 50 50 30 44 50 30 44 22 Complete the Rest Insertion 77 55 60 50 30 22 44 55

54. Deleting the Root of a Heap Suppose H is a heap with N elements Suppose we want to delete the root R of H Deletion of root is accomplished as follows [1] Assign the root R to some variable ITEM [2] Replace the deleted node R by the last node L of H so that H is still a complete tree but necessarily a heap [3] Reheap . Let L sink to its appropriate place in H so that H is finally a heap. 54

55. 55 95 8 5 7 55 33 15 30 6 5 20 15 22 22 8 5 7 55 33 15 30 6 5 20 15 85 22 7 55 33 15 30 6 5 20 15 85 55 7 22 33 15 30 6 5 20 15