1 / 31

B+ Trees

B+ Trees. Similar to B trees, with a few slight differences All data is stored at the leaf nodes ( leaf pages ); all other nodes ( index pages ) only store keys Leaf pages are linked to each other Keys may be duplicated; every key to the right of a particular key is >= to that key.

Olivia
Télécharger la présentation

B+ Trees

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. B+ Trees • Similar to B trees, with a few slight differences • All data is stored at the leaf nodes (leaf pages); all other nodes (index pages) only store keys • Leaf pages are linked to each other • Keys may be duplicated; every key to the right of a particular key is >= to that key 8/8/02

  2. B+ Tree Example 9, 16 2, 7 12 18 1 7 16 19 3, 4, 6 9 12 8/8/02

  3. B+ Tree Insertion • Insert at bottom level • If leaf page overflows, split page and copy middle element to next index page • If index page overflows, split page and move middle element to next index page 8/8/02

  4. B+ Tree Insertion Example 9, 16 Insert 5 2, 7 12 18 1 7 16 19 3, 4, 6 9 12 8/8/02

  5. B+ Tree Insertion Example 9, 16 Insert 5 2, 7 12 18 1 7 16 19 3, 4, 5,6 9 12 8/8/02

  6. B+ Tree Insertion Example 9, 16 Split page, copy 5 2, 5, 7 12 18 1 7 16 19 9 12 3, 4 5, 6 8/8/02

  7. B+ Tree Insertion Example 2 Insert 17 9, 13, 16 3, 4, 6 9 14 16, 18, 20 8/8/02

  8. B+ Tree Insertion Example 2 Insert 17 9, 13, 16 3, 4, 6 9 14 16, 17, 18, 20 8/8/02

  9. B+ Tree Insertion Example 2 Split leaf page, copy 18 9, 13, 16, 18 3, 4, 6 9 14 16, 17 18, 20 8/8/02

  10. B+ Tree Insertion Example 2 Split index page, move 13 13 9 16, 18 3, 4, 6 9 14 16, 17 18, 20 8/8/02

  11. B+ Tree Deletion • Delete key and data from leaf page • If leaf page underflows, merge with sibling and delete key in between them • If index page underflows, merge with sibling and move down key in between them 8/8/02

  12. B+ Tree Deletion Example Remove 9 13 9 16, 18 3, 4, 6 9 14 16, 17 18, 20 8/8/02

  13. B+ Tree Deletion Example Remove 9 13 9 16, 18 3, 4, 6 14 16, 17 18, 20 8/8/02

  14. B+ Tree Deletion Example Leaf page underflow, so merge with sibling and remove 9 13 16, 18 3, 4, 6 14 16, 17 18, 20 8/8/02

  15. B+ Tree Deletion Example Index page underflow, so merge with sibling and demote 13 13, 16, 18 3, 4, 6 14 16, 17 18, 20 8/8/02

  16. Threaded Trees • Binary trees have a lot of wasted space: the leaf nodes each have 2 null pointers • We can use these pointers to help us in inorder traversals • We have the pointers reference the next node in an inorder traversal; called threads • We need to know if a pointer is an actual link or a thread, so we keep a boolean for each pointer 8/8/02

  17. Threaded Tree Code • Example code: class Node { Node left, right; boolean leftThread, rightThread; } 8/8/02

  18. Threaded Tree Example 6 8 3 1 5 7 11 9 13 8/8/02

  19. Threaded Tree Traversal • We start at the leftmost node in the tree, print it, and follow its right thread • If we follow a thread to the right, we output the node and continue to its right • If we follow a link to the right, we go to the leftmost node, print it, and continue 8/8/02

  20. Threaded Tree Traversal Output 1 6 8 3 1 5 7 11 9 13 Start at leftmost node, print it 8/8/02

  21. Threaded Tree Traversal Output 1 3 6 8 3 1 5 7 11 9 13 Follow thread to right, print node 8/8/02

  22. Threaded Tree Traversal Output 1 3 5 6 8 3 1 5 7 11 9 13 Follow link to right, go to leftmost node and print 8/8/02

  23. Threaded Tree Traversal Output 1 3 5 6 6 8 3 1 5 7 11 9 13 Follow thread to right, print node 8/8/02

  24. Threaded Tree Traversal Output 1 3 5 6 7 6 8 3 1 5 7 11 9 13 Follow link to right, go to leftmost node and print 8/8/02

  25. Threaded Tree Traversal Output 1 3 5 6 7 8 6 8 3 1 5 7 11 9 13 Follow thread to right, print node 8/8/02

  26. Threaded Tree Traversal Output 1 3 5 6 7 8 9 6 8 3 1 5 7 11 9 13 Follow link to right, go to leftmost node and print 8/8/02

  27. Threaded Tree Traversal Output 1 3 5 6 7 8 9 11 6 8 3 1 5 7 11 9 13 Follow thread to right, print node 8/8/02

  28. Threaded Tree Traversal Output 1 3 5 6 7 8 9 11 13 6 8 3 1 5 7 11 9 13 Follow link to right, go to leftmost node and print 8/8/02

  29. Threaded Tree Traversal Code void inOrder(Node n) { Node cur = leftmost(n); while (cur != null) { print(cur); if (cur.rightThread) { cur = cur.right; } else { cur = leftmost(cur.right); } } } Node leftMost(Node n) { Node ans = n; if (ans == null) { return null; } while (ans.left != null) { ans = ans.left; } return ans; } 8/8/02

  30. Threaded Tree Modification • We’re still wasting pointers, since half of our leafs’ pointers are still null • We can add threads to the previous node in an inorder traversal as well, which we can use to traverse the tree backwards or even to do postorder traversals 8/8/02

  31. Threaded Tree Modification 6 8 3 1 5 7 11 9 13 8/8/02

More Related