1. CSC 213 – Large Scale Programming

2. Today’s Goals  Review a new search tree algorithm is needed  What real-world problems occur with old tree?  Why does garbage collection make problem worse?  What was ideal approach? How could we force this?  Consider how to create other search tree types  Not limit nodes to 1 element & what could happen?  How to perform insertions on multi-nodes?  What about withdrawal? How can we remove data?  Can this sound dirtier? And do I hear banjos playing?

3. Dictionary ADT  Dictionary and Map maps keys to values  O(1) time with hash, but only if hash is good  Can guarantee better -- O(log n) with balanced BST  Assumes data fits in memory since locality will suck  But, honestly, how big can a tree be?

4. Dictionary ADT  Dictionary and Map maps keys to values  O(1) time with hash, but only if hash is good  Can guarantee better -- O(log n) with balanced BST  Assumes data fits in memory since locality will suck  But, honestly, how big can a tree be?  Library of Congress – 20 TB in text database  Amazon.com – 42 TB of combined data  ChoicePoint – 250 TB of data on everyday Americans  World Data Center for Climate – 4 PB of climate data

5. Dictionary ADT

6. Optimal Tree Partition

7. But no GC algorithm produces this!

8. Real-World Big Search Trees  Excellent way to test roommates system

9. Real-World Big Search Trees roommates  Excellent way to test roommates system

10. Real-World Big Search Trees roommates  Excellent way to test roommates system

11. (a,b) Trees to the Rescue!  General solution to frequent hikes to Germany  Linux & MacOS to track files & directories  MySQL & other databases use this to hold all the data  Found in many other places where paging occurs  Simple rules define working of any ( a, b ) tree  Grows upward so that all leaves found at same level  At least a children for each internal node  Every internal node has at most b children

12. What is “the BTree?”  Common multi-way tree implementation  Describe B-Tree using order (“BTree of order m ”)  m / 2 to m children per internal node  Root node can have m or fewer elements  Many variants exist to improve some failing  Each variant is specialized for some niche use  Minor differences only between each variant  Will just describe most basic B-Tree during lecture

13. BTree Order  Select order minimizing paging when created  Elements & references to kids in full node fills page  Nodes have at least m / 2 elements, even at their smallest  In memory guarantees each page is at least 50% full  How many pages touched by each operation?

14. Multi-Way Search Tree  Nodes contain multiple elements  Tree grows up with leaves always at same level  Each internal node:  At least 2 children  1 fewer Entry s than children  Entry s sorted from smallest to largest 11 24 2 6 81527 30

15. Multi-Way Search Tree  Children v 1 v 2 v 3 … v d & keys k 1 k 2 … k d  1  Keys in subtree v 1 smaller than k 1  Keys in subtree v i between k i  1 and k 2  Keys in subtree v d greater than k d-1 11 24 27 302 6 815

16. Inorder Traversal  Visit each child, v i, before visiting Entry e i  As with BST, visits keys in increasing order 11 24 2 6 81527 30 1237 4 6 5 8

17. Multi-Way Searching  Similar to BST treeSearch finding a key for i = 0 to numChildren – 1 do if k < e[i].getKey() then return search(child[i]) if k == e[i].getKey() then return e[i] endfor if k > e[e.length-1].getKey() then return search(child[child.length-1]) 11 24 2 6 81527 30

18. Multi-Way Searching for i = 0 to numChildren – 1 do if k < e[i].getKey() then return search(child[i]) if k == e[i].getKey() then return e[i] endfor if k > e[e.length-1].getKey() then return search(child[child.length-1])  Example: find(8) 11 24 2 6 81527 30

19. Multi-Way Searching for i = 0 to numChildren – 1 do if k < e[i].getKey() then return search(child[i]) if k == e[i].getKey() then return e[i] endfor if k > e[e.length-1].getKey() then return search(child[child.length-1])  Example: find(8) 11 24 2 6 81527 30

20. Multi-Way Searching for i = 0 to numChildren – 1 do if k < e[i].getKey() then return search(child[i]) if k == e[i].getKey() then return e[i] endfor if k > e[e.length-1].getKey() then return search(child[child.length-1])  Example: find(8) 11 24 2 6 81527 30

21. Multi-Way Searching for i = 0 to numChildren – 1 do if k < e[i].getKey() then return search(child[i]) if k == e[i].getKey() then return e[i] endfor if k > e[e.length-1].getKey() then return search(child[child.length-1])  Example: find(8) 11 24 2 6 81527 30

22. Multi-Way Searching for i = 0 to numChildren – 1 do if k < e[i].getKey() then return search(child[i]) if k == e[i].getKey() then return e[i] endfor if k > e[e.length-1].getKey() then return search(child[child.length-1])  Example: find(8) 11 24 2 6 81527 30

23. Multi-Way Searching for i = 0 to numChildren – 1 do if k < e[i].getKey() then return search(child[i]) if k == e[i].getKey() then return e[i] endfor if k > e[e.length-1].getKey() then return search(child[child.length-1])  Example: find(8) 11 24 2 6 81527 30

24. Multi-Way Searching for i = 0 to numChildren – 1 do if k < e[i].getKey() then return search(child[i]) if k == e[i].getKey() then return e[i] endfor if k > e[e.length-1].getKey() then return search(child[child.length-1])  Example: find(8) 11 24 2 6 81527 30

25. Multi-Way Searching for i = 0 to numChildren – 1 do if k < e[i].getKey() then return search(child[i]) if k == e[i].getKey() then return e[i] endfor if k > e[e.length-1].getKey() then return search(child[child.length-1])  Example: find(8) 11 24 2 6 8 1527 30

26. Multi-Way Searching for i = 0 to numChildren – 1 do if k < e[i].getKey() then return search(child[i]) if k == e[i].getKey() then return e[i] endfor if k > e[e.length-1].getKey() then return search(child[child.length-1])  Example: find(8) 11 24 1527 30 2 6 8

27. (2,4) Trees  Multi-way search tree with 2 properties:  Node-Size Property Internal nodes have at most 4 children  Depth Property All external nodes at same depth  Nodes are either 2-node, 3-node or 4-node  Node’s number of children used as basis for name 10 15 24 2 81227 3218

28. Insertion  Start by searching for key k last  Entry added to last internal node searched  Depth property preserved by enforcing this  Example: insert(30) 10 15 24 2 81227 3218

29. Insertion  Start by searching for key k last  Entry added to last internal node searched  Depth property preserved by enforcing this  Example: insert(30) 10 15 24 2 81227 3218

30. Insertion  Start by searching for key k last  Entry added to last internal node searched  Depth property preserved by enforcing this  Example: insert(30) 10 15 24 2 81227 3218

31. Insertion  Start by searching for key k last  Entry added to last internal node searched  Depth property preserved by enforcing this  Example: insert(30) 10 15 24 2 81227 3218

32. Insertion  Start by searching for key k last  Entry added to last internal node searched  Depth property preserved by enforcing this  Example: insert(30) 10 15 24 2 81227 3218

33. Insertion  Start by searching for key k last  Entry added to last internal node searched  Depth property preserved by enforcing this  Example: insert(30) 10 15 24 2 81227 3218

34. Insertion  Start by searching for key k last  Entry added to last internal node searched  Depth property preserved by enforcing this  Example: insert(30) 10 15 24 2 81227 3218

35. Insertion  Start by searching for key k last  Entry added to last internal node searched  Depth property preserved by enforcing this  Example: insert(30) 10 15 24 2 81227 3218

36. Insertion  Start by searching for key k last  Entry added to last internal node searched  Depth property preserved by enforcing this  Example: insert(30) 10 15 24 2 81227 3218

37. Insertion  Start by searching for key k last  Entry added to last internal node searched  Depth property preserved by enforcing this  Example: insert(30) 10 15 24 2 81227 3218

38. Insertion  Start by searching for key k last  Entry added to last internal node searched  Depth property preserved by enforcing this  Example: insert(30) 10 15 24 2 812 27 32 18

39. Insertion  Start by searching for key k last  Entry added to last internal node searched  Depth property preserved by enforcing this  Example: insert(30) 10 15 24 2 812 27 30 32 18

40. Insertion  Insertion may cause overflow!  5-node created by the insertion Node-Size  This would make it violate Node-Size property 27 32 35 15 24 1218

41. Insertion  Insertion may cause overflow!  5-node created by the insertion Node-Size  This would make it violate Node-Size property 27 30 32 35 15 24 1218

42. In Case Of Overflow Split Node  Split 5-node into 2 new nodes  Entry s e 1 e 2 & children v 1 v 2 v 3 become a 3-node  2-node created with Entry e 4 & children v 4 v 5 15 24 1218 27 30 32 35

43. In Case Of Overflow Split Node  Split 5-node into 2 new nodes  Entry s e 1 e 2 & children v 1 v 2 v 3 become a 3-node  2-node created with Entry e 4 & children v 4 v 5  Promote e 3 to parent node  If overflow occurs in root node, create new root  Overflow can cascade when parent already was 4-node 15 24 1218 27 30 32 35 12 27 30 18 35 15 24 32

44. Parent Overflow  In case of cascade, repeat overflow process  Works identically to when children are external  Example: insert(29) 27 32 35121825 15 24 26

45. Parent Overflow  In case of cascade, repeat overflow process  Works identically to when children are external  Example: insert(29) 27 32 35121825 15 24 26

46. Parent Overflow  In case of cascade, repeat overflow process  Works identically to when children are external  Example: insert(29) 27 32 35121825 15 24 26

47. Parent Overflow  In case of cascade, repeat overflow process  Works identically to when children are external  Example: insert(29) 27 32 35121825 15 24 26

48. Parent Overflow  In case of cascade, repeat overflow process  Works identically to when children are external  Example: insert(29) 27 29 32 35121825 15 24 26

49. Parent Overflow  In case of cascade, repeat overflow process  Works identically to when children are external  Example: insert(29) 27 29 32 35121825 15 24 26

50. Parent Overflow  In case of cascade, repeat overflow process  Works identically to when children are external  Example: insert(29) 27 29 32 35121825 15 24 26

51. Parent Overflow  In case of cascade, repeat overflow process  Works identically to when children are external  Example: insert(29) 121825 15 24 26 32 27 2935

52. Parent Overflow  In case of cascade, repeat overflow process  Works identically to when children are external  Example: insert(29) 121825 15 24 26 32 27 2935

53. Parent Overflow  In case of cascade, repeat overflow process  Works identically to when children are external  Example: insert(29) 121825 15 24 27 2935 32 26

54. Parent Overflow  In case of cascade, repeat overflow process  Works identically to when children are external  Example: insert(29) 121825 15 24 27 2935 32 26

55. Adding to MultiWay Tree  Leaves all at same level, so trees grow upwards  Add to last internal node from initial tree search  Addition not done – check if node too large  Push 1 item up into parent and split into 2 nodes  May pass problem along, so check if parent too large  Traverse until overflow stops or made new root node

56. Deletion  Must first find Entry to be deleted  Remove Entry & an external child if it is on leaf  Example: delete(27) 10 15 24 2 8121827 32 35

57. Deletion  Must first find Entry to be deleted  Remove Entry & an external child if it is on leaf  Example: delete(27) 10 15 24 2 8121827 32 35

58. Deletion  Must first find Entry to be deleted  Remove Entry & an external child if it is on leaf  Example: delete(27) 10 15 24 2 8121827 32 35

59. Deletion  Must first find Entry to be deleted  Remove Entry & an external child if it is on leaf  Example: delete(27) 10 15 24 2 8121827 32 35

60. Deletion  Must first find Entry to be deleted  Remove Entry & an external child if it is on leaf  Example: delete(27) 10 15 24 2 8121827 32 35

61. Deletion  Must first find Entry to be deleted  Remove Entry & an external child if it is on leaf  Example: delete(27) 10 15 24 2 8121827 32 35 10 15 24 2 8121832 35

62. Deletion  If Entry 's child internal, replace with successor  Go 1 to right and then go left just like with a BST

63. Deletion  If Entry 's child internal, replace with successor  Go 1 to right and then as far left as possible; like BST  Example: delete(24) 10 15 24 2 8121827 32 35

64. Deletion  If Entry 's child internal, replace with successor  Go 1 to right and then go left just like with a BST  Example: delete(24) 10 15 24 2 8121827 32 35 10 15 27 2 8121832 35

65. 15 9 14 Underflow and Fusion  Entry deletion may cause underflow  Node becomes 1-node  Choice of solution depends on situation  Example: remove(15) 102 5 7

66. 15 9 14 Underflow and Fusion  Entry deletion may cause underflow  Node becomes 1-node  Choice of solution depends on situation  Example: remove(15) 102 5 7

67. 15 9 14 Underflow and Fusion  Entry deletion may cause underflow  Node becomes 1-node  Choice of solution depends on situation  Example: remove(15) 102 5 7

68.  Entry deletion may cause underflow  Node becomes 1-node  Choice of solution depends on situation  Example: remove(15) 9 14 Underflow and Fusion 102 5 7

69.  Entry deletion may cause underflow  Node becomes 1-node  Choice of solution depends on situation  Example: remove(15) 9 14 Underflow and Fusion 102 5 7

70. Case 1: Transfer  Has adjacent 3- or 4-node sibling  Steal parent’s Entry closest to the 1-node  Prevent loneliness & promote sibling’s Entry  No further processing needed in this case  Example: remove(10) 4 9 6 8210

71. Case 1: Transfer  Has adjacent 3- or 4-node sibling  Steal parent’s Entry closest to the 1-node  Prevent loneliness & promote sibling’s Entry  No further processing needed in this case  Example: remove(10) 4 9 6 8210

72. Case 1: Transfer  Has adjacent 3- or 4-node sibling  Steal parent’s Entry closest to the 1-node  Prevent loneliness & promote sibling’s Entry  No further processing needed in this case  Example: remove(10) 4 9 6 8210

73. Case 1: Transfer  Has adjacent 3- or 4-node sibling  Steal parent’s Entry closest to the 1-node  Prevent loneliness & promote sibling’s Entry  No further processing needed in this case  Example: remove(10) 4 9 6 8210

74. Case 1: Transfer  Has adjacent 3- or 4-node sibling  Steal parent’s Entry closest to the 1-node  Prevent loneliness & promote sibling’s Entry  No further processing needed in this case  Example: remove(10) 4 9 6 82

75. Case 1: Transfer  Has adjacent 3- or 4-node sibling  Steal parent’s Entry closest to the 1-node  Prevent loneliness & promote sibling’s Entry  No further processing needed in this case  Example: remove(10) 4 9 6 82

76. Case 1: Transfer  Has adjacent 3- or 4-node sibling  Steal parent’s Entry closest to the 1-node  Prevent loneliness & promote sibling’s Entry  No further processing needed in this case  Example: remove(10) 4 6 829

77. Case 1: Transfer  Has adjacent 3- or 4-node sibling  Steal parent’s Entry closest to the 1-node  Prevent loneliness & promote sibling’s Entry  No further processing needed in this case  Example: remove(10) 4 6 829

78. Case 1: Transfer  Has adjacent 3- or 4-node sibling  Steal parent’s Entry closest to the 1-node  Prevent loneliness & promote sibling’s Entry  No further processing needed in this case  Example: remove(10) 4 8 629

79. Case 1: Transfer  Has adjacent 3- or 4-node sibling  Steal parent’s Entry closest to the 1-node  Prevent loneliness & promote sibling’s Entry  No further processing needed in this case  Example: remove(10) 4 8 629

80.  Emptied node has adjacent 2-node sibling  Merge node & sibling into one  Look to parent and steal Entry between siblings  May propagate underflow to parent!  Example: remove(15) Case 2: Fusion Mom 9 14 102 5 7

81.  Emptied node has adjacent 2-node sibling  Merge node & sibling into one  Look to parent and steal Entry between siblings  May propagate underflow to parent!  Example: remove(15) 9 14 Case 2: Fusion 102 5 7 Mom

82.  Emptied node has adjacent 2-node sibling  Merge node & sibling into one  Look to parent and steal Entry between siblings  May propagate underflow to parent!  Example: remove(15) 9 Case 2: Fusion 10 142 5 7 Mom

83.  Emptied node has adjacent 2-node sibling  Merge node & sibling into one  Look to parent and steal Entry between siblings  May propagate underflow to parent!  Example: remove(15) 9 Case 2: Fusion 10 142 5 7 Mom

84.  Emptied node has adjacent 2-node sibling  Merge node & sibling into one  Look to parent and steal Entry between siblings  May propagate underflow to parent!  Example: remove(15) 9 Case 2: Fusion 10 142 5 7 Mom

85. Deletion from MultiWay Tree  Removal like BST: swap element to legal node  If removal causes underflow, check its nearest siblings  If 3-node or 4-node as sibling, then solution is easy… … move sibling up and bring parent down into node  Merge with sibling & parent data if no big neighbor… … but must then check if parent has an underflow

86. For Next Lecture  Wednesday will be quiz on real-world stuff  Garbage collection, cache behavior & trees (oh, my)  End of day Wednesday lab project is due  Will have regular hour during lab, too  At end of day on Friday will have Project #3 due