2-3 Tree

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2-3 Tree

  1. 1. Algorithms 2 - 3 Tree 2 – 3 – 4 Tree Balanced Trees
  2. 2. 2-3 Trees  each internal node has either 2 or 3 children  all leaves are at the same level Features
  3. 3. 2-3 Trees with Ordered Nodes 2-node 3-node • leaf node can be either a 2-node or a 3-node
  4. 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. 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. 6. Example of 2-3 Tree
  7. 7. What did we gain? What is the time efficiency of searching for an item?
  8. 8. Gain: Ease of Keeping the Tree Balanced Binary Search Tree 2-3 Tree both trees after inserting items 39, 38, ... 32
  9. 9. Inserting Items Insert 39
  10. 10. Inserting Items Insert 38 insert in leaf divide leaf and move middle value up to parent result
  11. 11. Inserting Items Insert 37
  12. 12. Inserting Items Insert 36 insert in leaf divide leaf and move middle value up to parent overcrowded node
  13. 13. Inserting Items ... still inserting 36 divide overcrowded node, move middle value up to parent, attach children to smallest and largest result
  14. 14. Inserting Items After Insertion of 35, 34, 33
  15. 15. Inserting so far
  16. 16. Inserting so far
  17. 17. Inserting Items How do we insert 32?
  18. 18. Inserting Items  creating a new root if necessary  tree grows at the root
  19. 19. Inserting Items Final Result
  20. 20. 70 Deleting Items Delete 70 80
  21. 21. Deleting Items Deleting 70: swap 70 with inorder successor (80)
  22. 22. Deleting Items Deleting 70: ... get rid of 70
  23. 23. Deleting Items Result
  24. 24. Deleting Items Delete 100
  25. 25. Deleting Items Deleting 100
  26. 26. Deleting Items Result
  27. 27. Deleting Items Delete 80
  28. 28. Deleting Items Deleting 80 ...
  29. 29. Deleting Items Deleting 80 ...
  30. 30. Deleting Items Deleting 80 ...
  31. 31. Deleting Items Final Result comparison with binary search tree
  32. 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. 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. 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. 35. Deletion Algorithm IV If merging process reaches the root and root is without item  delete root
  36. 36. Operations of 2-3 Trees all operations have time complexity of log n
  37. 37. 2-3-4 Trees • similar to 2-3 trees • 4-nodes can have 3 items and 4 children 4-node
  38. 38. 2-3-4 Tree Example
  39. 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. 40. 2-3-4 Tree: Insertion Inserting 60, 30, 10, 20, 50, 40, 70, 80, 15, 90, 100
  41. 41. 2-3-4 Tree: Insertion Inserting 60, 30, 10, 20 ... ... 50, 40 ...
  42. 42. 2-3-4 Tree: Insertion Inserting 50, 40 ... ... 70, ...
  43. 43. 2-3-4 Tree: Insertion Inserting 70 ... ... 80, 15 ...
  44. 44. 2-3-4 Tree: Insertion Inserting 80, 15 ... ... 90 ...
  45. 45. 2-3-4 Tree: Insertion Inserting 90 ... ... 100 ...
  46. 46. 2-3-4 Tree: Insertion Inserting 100 ...
  47. 47. 2-3-4 Tree: Insertion Procedure Splitting 4-nodes during Insertion
  48. 48. 2-3-4 Tree: Insertion Procedure Splitting a 4-node whose parent is a 2-node during insertion
  49. 49. 2-3-4 Tree: Insertion Procedure Splitting a 4-node whose parent is a 3-node during insertion
  50. 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. 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. 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. 53. 2-3-4 Tree: Deletion Practice Delete 32, 35, 40, 38, 39, 37, 60
  54. 54. 2-3-4 Insert Example 6, 15, 25 2,4,5 10 18, 20 30 Insert 24, then 19
  55. 55. Insert 24: Split root first 6 2,4,5 10 18, 20,24 30 15 25
  56. 56. Insert 19, Split leaf (20 up) first 6 2,4,5 10 18, 19 30 15 20, 25 24

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