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- 1. Algorithms 2 - 3 Tree 2 – 3 – 4 Tree Balanced Trees
- 2. 2-3 Trees each internal node has either 2 or 3 children all leaves are at the same level Features
- 3. 2-3 Trees with Ordered Nodes 2-node 3-node • leaf node can be either a 2-node or a 3-node
- 4. Why 2-3 tree ● Faster searching? ■ Actually, no. 2-3 tree is about as fast as an “equally balanced” binary tree, because you sometimes have to make 2 comparisons to get past a 3-node ● Easier to keep balanced? ■ Yes, definitely. ■ Insertion can split 3-nodes into 2-nodes, or promote 2-nodes to 3-nodes to keep tree approximately balanced!
- 5. Why is this better? ● Intuitively, you unbalance a binary tree when you add height to one path significantly more than other possible paths. ● With the 2-3 insert algorithm, you can only add height to the tree when you create a new root, and this adds one unit of height to all paths simultaneously. ● Hence, the average path length of the tree stays close to log N.
- 6. Example of 2-3 Tree
- 7. What did we gain? What is the time efficiency of searching for an item?
- 8. Gain: Ease of Keeping the Tree Balanced Binary Search Tree 2-3 Tree both trees after inserting items 39, 38, ... 32
- 9. Inserting Items Insert 39
- 10. Inserting Items Insert 38 insert in leaf divide leaf and move middle value up to parent result
- 11. Inserting Items Insert 37
- 12. Inserting Items Insert 36 insert in leaf divide leaf and move middle value up to parent overcrowded node
- 13. Inserting Items ... still inserting 36 divide overcrowded node, move middle value up to parent, attach children to smallest and largest result
- 14. Inserting Items After Insertion of 35, 34, 33
- 15. Inserting so far
- 16. Inserting so far
- 17. Inserting Items How do we insert 32?
- 18. Inserting Items creating a new root if necessary tree grows at the root
- 19. Inserting Items Final Result
- 20. 70 Deleting Items Delete 70 80
- 21. Deleting Items Deleting 70: swap 70 with inorder successor (80)
- 22. Deleting Items Deleting 70: ... get rid of 70
- 23. Deleting Items Result
- 24. Deleting Items Delete 100
- 25. Deleting Items Deleting 100
- 26. Deleting Items Result
- 27. Deleting Items Delete 80
- 28. Deleting Items Deleting 80 ...
- 29. Deleting Items Deleting 80 ...
- 30. Deleting Items Deleting 80 ...
- 31. Deleting Items Final Result comparison with binary search tree
- 32. Deletion Algorithm I 1. Locate node n, which contains item I 2. If node n is not a leaf swap I with inorder successor deletion always begins at a leaf 3. If leaf node n contains another item, just delete item I else try to redistribute nodes from siblings (see next slide) if not possible, merge node (see next slide) Deleting item I:
- 33. Deletion Algorithm II A sibling has 2 items: redistribute item between siblings and parent No sibling has 2 items: merge node move item from parent to sibling Redistribution Merging
- 34. Deletion Algorithm III Internal node n has no item left redistribute Redistribution not possible: merge node move item from parent to sibling adopt child of n If n's parent ends up without item, apply process recursively Redistribution Merging
- 35. Deletion Algorithm IV If merging process reaches the root and root is without item delete root
- 36. Operations of 2-3 Trees all operations have time complexity of log n
- 37. 2-3-4 Trees • similar to 2-3 trees • 4-nodes can have 3 items and 4 children 4-node
- 38. 2-3-4 Tree Example
- 39. 2-3-4 Tree: Insertion Insertion procedure: • similar to insertion in 2-3 trees • items are inserted at the leafs • since a 4-node cannot take another item, 4-nodes are split up during insertion process Strategy • on the way from the root down to the leaf: split up all 4-nodes "on the way" insertion can be done in one pass (remember: in 2-3 trees, a reverse pass might be necessary)
- 40. 2-3-4 Tree: Insertion Inserting 60, 30, 10, 20, 50, 40, 70, 80, 15, 90, 100
- 41. 2-3-4 Tree: Insertion Inserting 60, 30, 10, 20 ... ... 50, 40 ...
- 42. 2-3-4 Tree: Insertion Inserting 50, 40 ... ... 70, ...
- 43. 2-3-4 Tree: Insertion Inserting 70 ... ... 80, 15 ...
- 44. 2-3-4 Tree: Insertion Inserting 80, 15 ... ... 90 ...
- 45. 2-3-4 Tree: Insertion Inserting 90 ... ... 100 ...
- 46. 2-3-4 Tree: Insertion Inserting 100 ...
- 47. 2-3-4 Tree: Insertion Procedure Splitting 4-nodes during Insertion
- 48. 2-3-4 Tree: Insertion Procedure Splitting a 4-node whose parent is a 2-node during insertion
- 49. 2-3-4 Tree: Insertion Procedure Splitting a 4-node whose parent is a 3-node during insertion
- 50. 2-3-4 Tree: Deletion Deletion procedure: • similar to deletion in 2-3 trees • items are deleted at the leafs swap item of internal node with inorder successor • note: a 2-node leaf creates a problem Strategy (different strategies possible) • on the way from the root down to the leaf: turn 2-nodes (except root) into 3-nodes deletion can be done in one pass (remember: in 2-3 trees, a reverse pass might be necessary)
- 51. 2-3-4 Tree: Deletion Turning a 2-node into a 3-node ... Case 1: an adjacent sibling has 2 or 3 items "steal" item from sibling by rotating items and moving subtree 30 50 10 20 40 25 20 50 10 30 40 25 "rotation"
- 52. 2-3-4 Tree: Deletion Turning a 2-node into a 3-node ... Case 2: each adjacent sibling has only one item "steal" item from parent and merge node with sibling (note: parent has at least two items, unless it is the root) 30 50 10 40 25 50 25 merging 10 30 40 35 35
- 53. 2-3-4 Tree: Deletion Practice Delete 32, 35, 40, 38, 39, 37, 60
- 54. 2-3-4 Insert Example 6, 15, 25 2,4,5 10 18, 20 30 Insert 24, then 19
- 55. Insert 24: Split root first 6 2,4,5 10 18, 20,24 30 15 25
- 56. Insert 19, Split leaf (20 up) first 6 2,4,5 10 18, 19 30 15 20, 25 24

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