1. CSC 213 – Large Scale Programming

2. Today’s Goal  Review Map & Dictionary implementations  What do they do well? When would they be used?  Why do they suck so much another is needed?  Discuss how we can get it all NOW with BST s  Why greedy approach could be useful & helpful  Seeing how idea of splay tree not balanced or Zen  What & when to splay node in a BST

3. Map & Dictionary ADT ImplementationSearchingAddingRemoving Ordered List O(log n)O(n)O(n)O(n)O(n) Unordered List O(n)O(n)O(n)/O(1)O(n)/O(1)O(n)O(n) Hash O(n)O(n)O(n)O(n)O(n)O(n) if lucky/good O(1)O(1)O(1)O(1)O(1)O(1) BST O(n)O(n)O(n)O(n)O(n)O(n) AVL / balanced O(log n)

4. Map & Dictionary ADT ImplementationSearchingAddingRemoving Ordered List O(log n)O(n)O(n)O(n)O(n) Unordered List O(n)O(n)O(n)/O(1)O(n)/O(1)O(n)O(n) Hash O(n)O(n)O(n)O(n)O(n)O(n) if lucky/good O(1)O(1)O(1)O(1)O(1)O(1) BST O(n)O(n)O(n)O(n)O(n)O(n) AVL / balanced O(log n)

5. Ideal: Hash Without Withdrawal  Want O(1) times that hash provides  At the same time scared of slowdowns from bad hash  Or lack hash function that can work with your data  BST s provide safety that array-based hash lacks  Worst-case only O(log n) as long as tree is balanced  Requires way to keep tree balanced (e.g., AVL) and  Need O(1) access to tree’s nodes, but how?

6. Implementing Map with Tree  Accessing root much faster than going to leaves  In real-world, really want important data near root  Which key best at root of Tree of NHL teams?  BST : Key for Entry added first

7. Implementing Map with Tree  Accessing root much faster than going to leaves  In real-world, really want important data near root  Which key best at root of Tree of NHL teams?  BST : Key for Entry added first  AVLTree : Random Entry near midpoint

8. Implementing Map with Tree  Accessing root much faster than going to leaves  In real-world, really want important data near root  Which key best at root of Tree of NHL teams?  BST : Key for Entry added first  AVLTree : Random Entry near midpoint  SplayTree : Most recently used key

9. Building a SplayTree builds upon BST  Another approach which builds upon BST  Not an AVLTree  Not an AVLTree, however, but a new BST subclass

10. Concept Behind Splay Trees  Splay trees do NOT maintain balance  Recently used nodes clustered near top of BST  Most recently accessed nodes take O(1) time  Other nodes may need O ( n ) time to find, however

11. Concept Behind Splay Trees  Splay trees do NOT maintain balance  Recently used nodes clustered near top of BST  Most recently accessed nodes take O(1) time  Other nodes may need O ( n ) time to find, however

12. SplayTree Complexity  Without balancing, keeps BST 's O ( n ) complexity  Worst-case performance is like all unbalanced trees  But splaying gives expected O (log n ) complexity  Averages complexity of O(1) & O(n) operations  If work concentrated on small subset, time is faster  Worst-case hard to create without knowing tree

13. Be Kind: Splay Your Tree  Assumes nodes reused soon after initial use  At end of each method, moves node up to root  Using node now O(1) and will only slowly drop in tree  Splay  Splay tree with each find, insert & remove  AVL-like restructuring to reorganize nodes in tree  But

14. Be Kind: Splay Your Tree

15. How To Splay

16.  Uses trinode restructuring but not like AVL does nodeparentgrandparent  Not balancing: selects node, parent, & grandparent  Always move node to root of subtree when splaying  When splaying, new types restructures also exist  Node & parent always used in these rotations  Rotations will also use grandparent, if it exists  Moving node to tree root is goal of splaying  May get a balanced tree, but WANT IT ALL, ASAP

17. When To Use Splay Tree  What applications are good for a splay tree?  Where would splay trees be a BAD IDEA ?

18. Splay Node Rotations  Uses different nodes  Uses different nodes than AVL tree rotations  AVLTree moves node down to balance tree's tao  Hedonistic ideal moves node to SplayTree 's root

19. Splay Node Rotations  Uses different nodes  Uses different nodes than AVL tree rotations  AVLTree moves node down to balance tree's tao  Hedonistic ideal moves node to SplayTree 's root AVLTree

20. Splay Node Rotations  Uses different nodes  Uses different nodes than AVL tree rotations  AVLTree moves node down to balance tree's tao  Hedonistic ideal moves node to SplayTree 's root AVLTreeSplayTree

21. Zig-Zag Zig-Zag When Splaying a Tree  When node median of parent & grandparent  Just like in AVL tree, perform trinode restructuring  Use 7(+1) variables to set node's parent & children parent node T2T2 T3T3 T4T4 grandparent T1T1 parent node T2T2 T3T3 T4T4 grandparent T1T1

22. Zig-Zig Zig-Zig When Splaying a Tree parent node grandparent T1T1 T4T4 T2T2 T3T3

23. Zig-Zig Zig-Zig When Splaying a Tree parent node T2T2 T3T3 T4T4 grandparent T1T1 parent node grandparent T1T1 T4T4 T2T2 T3T3

24. Right Splaying Right Child Of Root  Simplest process is when parent is the root  Single rotation completes splaying process  Splaying does not stop until you reach the top  Rotation not always used, may only need restructures root node T3T3 T1T1 T2T2

25. Right Splaying Right Child Of Root  Simplest process is when parent is the root  Single rotation completes splaying process  Splaying does not stop until you reach the top  Rotation not always used, may only need restructures root node T3T3 T1T1 T2T2

26. Right Splaying Right Child Of Root  Simplest process is when parent is the root  Single rotation completes splaying process  Splaying does not stop until you reach the top  Rotation not always used, may only need restructures root node T2T2 T3T3 T1T1 root node T3T3 T1T1 T2T2

27. Left Splaying Left Child Of Root  Simplest process is when parent is the root  Single rotation completes splaying process  Splaying does not stop until you reach the top  Rotation not always used, may only need restructures T2T2 T3T3 T1T1 root node T3T3 T1T1 T2T2 root

28. Which Node Gets Splayed? MethodNode to splay find If found, splay node containing match If not found, splay last node in treeSearch add Splay node for new Entry remove If found, splay parent of external node removed If not found, splay last node in treeSearch

29. Which Node Gets Splayed? MethodNode to splay find If found, splay node containing match If not found, splay last node in treeSearch add Splay node for new Entry remove If found, splay parent of external node removed If not found, splay last node in treeSearch  Only internal nodes  Only internal nodes can be splayed  When operation end with leaf, splay its parent

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