1. Splay trees Go&Ta 10.3

2. How do you organize your world?

31. “move to front” heuristic …

32. splay trees A method that modifies a binary search tree (bst) Something like a rotation in AVL There are 3 operations: zig-zig, zig-zag, and zig

33. splay treeszig-zig T2 T3T4 T1 z y x x and y are right children (or left children, symmetry)

34. splay treeszig-zig T2 T3T4 T1 z y x T2 T3 T4 T1 z y x zig-zig

35. splay treeszig-zag T4 T2T3 T1 z y x one of x and y is a left children and the other is a right child (also symmetry)

36. splay treeszig-zag T4 T2T3 T1 z y x T4 T2T3 T1 zy x zig-zag

37. splay treeszig T2 T3T4 T1 y x w x has no grandparent (also symmetry)

38. splay treeszig T2 T3T4 T1 y x w zig T4 T2T3 T1 yw x

39. splay trees We repeat these splay steps until x becomes the root When do we splay? we insert k into the bstree x is the new node with key k we search for k in the bstree x is node with key k or node where we stopped search we delete k from the bstree delete method ultimately removes a leaf the leaf is x

40. splay trees Advantages: frequently accessed data moves nearer to root the “move to front” heuristic average performance is O(log(n)) simple no additional information/data required add-on to bstree

41. splay trees Disdvantages: worst case height can be linear O(n)

42. splay trees Question: in our experiments with bstree and avl tree we measured height is that measure meaningful? is there a more useful measure, such as average depth?

43. splay treesrefresher Nodes in a Complete Binary Tree

44. splay treesrefresher Nodes in a Complete Binary Tree depth #nodes totalNodes 0 1 1 2^1 - 1 1 2 3 2^2 – 1 2 4 7 2^3 – 1 3 8 15 2^4 -1 d 2^d 2^(d+1) - 1

45. splay treesrefresher Nodes in a Complete Binary Tree depth #nodes totalNodes 0 1 1 2^1 - 1 1 2 3 2^2 – 1 2 4 7 2^3 – 1 3 8 15 2^4 -1 d 2^d 2^(d+1) - 1 internal path length is the sum of the length of the paths to all nodes in the tree

46. splay treesrefresher Nodes in a Complete Binary Tree depth #nodes totalNodes 0 1 1 2^1 - 1 1 2 3 2^2 – 1 2 4 7 2^3 – 1 3 8 15 2^4 -1 d 2^d 2^(d+1) - 1 internal path length is the sum of the length of the paths to all nodes in the tree For a complete binary tree that would be: 0 + 1 + 1 + 2 + 2 + 2 + 2 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + …

47. splay treesrefresher Nodes in a Complete Binary Tree depth #nodes totalNodes 0 1 1 2^1 - 1 1 2 3 2^2 – 1 2 4 7 2^3 – 1 3 8 15 2^4 -1 d 2^d 2^(d+1) - 1 internal path length is the sum of the length of the paths to all nodes in the tree For a complete binary tree that would be: 0 + 1 + 1 + 2 + 2 + 2 + 2 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + …

48. splay treesrefresher Nodes in a Complete Binary Tree depth #nodes totalNodes 0 1 1 2^1 - 1 1 2 3 2^2 – 1 2 4 7 2^3 – 1 3 8 15 2^4 -1 d 2^d 2^(d+1) - 1 internal path length is the sum of the length of the paths to all nodes in the tree

49. splay treesrefresher Nodes in a Complete Binary Tree depth #nodes totalNodes 0 1 1 2^1 - 1 1 2 3 2^2 – 1 2 4 7 2^3 – 1 3 8 15 2^4 -1 d 2^d 2^(d+1) - 1 internal path length is the sum of the length of the paths to all nodes in the tree Average path length is then O(log(n))

50. splay trees Question: in our experiments with bstree and avl tree we measured height is that measure meaningful? is there a more useful measure, such as average depth?

51. splay trees Question: in our experiments with bstree and avl tree we measured height is that measure meaningful? is there a more useful measure, such as average depth? Answer: average depth would be more meaningful

52. splay trees Show me a demo