1.8 splay tree

Splay Trees

https://docs.google.com/presentation/d/1J8sIJDKi9DfnZJ5-VMeGtPqQ8nFfJ_mQVxz4oPEj-KA/edit#slide=id.p

Fundamental Concept

Carefully rearrange (using tree rotation operation) the tree so that the recently accessed elements are brought to the top and at the same time, the tree is also approximately balanced.

Splay Tree

Splay is a binary search tree in which recently accessed keys are quick to access again.

When an element is accessed, it is moved to the root using series of rotation operations so that it is quick to access again.

The tree is also approximately balanced while rearranging.

The process of rearranging is called splaying.

Operations

• Splay tree includes all the operations of BST along with splay operation.

• Splay operation rearranges the tree so that the element is placed at the top of the tree.

• Let x be the accessed node and let p be the parent node of x.

Three types of splay steps:

• Zig step (one left rotation or one right rotation)o carried out when p is the root (last step in the splay operation)

• Zig-Zig step (left-left rotation or right-right rotation)o carried out when p is not the root and both x,p are either right

or left children.

• Zig-Zag step (left-right rotation or right-left rotation)o carried out when p is not the root and x is left and p is right or

vice versa

Operations (contd)

Insertion

•Insert the node using the normal BST insert procedure

•splay the newly inserted node to the root

Deletion

•delete using normal BST delete procedure

•splay the parent of removed node to the root

Search

•Search using normal BST search procedure

•splay the accessed item to the root

Splay Operation - Pseudo codevoid splay(struct node *x){ struct node *p,*g;

/*check if node x is the root node*/ if (x->parent == NULL) { root = x; return; }

/*Performs Zig step*/ else if ( x->parent == root) { if(x == x->parent->left) rightrotation(root); else

leftrotation(root); }

else { p = x->parent; /*now points to parent of x*/ g = p->parent; /*now points to grand parent of x*/ /*when x is left and x's parent is left*/ if(x==p->left and p==g->left) //Zig-zig { rightrotation(g); rightrotation(p); } /*step when x is right and x's parent is right*/ else if (x==p->right and p==g->right) //Zig-zig { leftrotation(g); leftrotation(p); } /*when x's is right and x's parent is left*/ else if (x==p->right and p==g->left) // Zig-zag { leftrotation(p); rightrotation(g); } /*when x's is left and x's parent is right*/ else if (x==p->left and p==g->right) //Zig-zag { rightrotation(p); leftrotation(g); } splay(x); }}

Splay Trees and B-Trees - Lecture 98

• Let X be a non-root node with 2 ancestors.• P is its parent node.• G is its grandparent node.

P

G

X

G

P

X

G

P

X

G

P

X

Splay Tree Terminology

Zig-Zig and Zig-ZagSplay Trees and B-Trees - Lecture 9

9

4

G 5

1 P Zig-zag

G

P 5

X 2

Zig-zig

X

Parent and grandparentin same direction.

Parent and grandparentin different directions.

Zig at depth 1 (root)“Zig” is just a single rotation, as in an AVL treeLet R be the node that was accessed (e.g. using Find)

ZigFromLeft moves R to the top faster access next time


10

ZigFromLeft

root

Zig at depth 1

Suppose Q is now accessed using Find

ZigFromRight moves Q back to the top


11

ZigFromRight

root

Zig-Zag operation

“Zig-Zag” consists of two rotations of the opposite direction (assume R is the node that was accessed)


12

(ZigFromRight) (ZigFromLeft)

ZigZagFromLeft

Zig-Zig operation

“Zig-Zig” consists of two single rotations of the same direction (R is the node that was accessed)


13

(ZigFromLeft) (ZigFromLeft)

ZigZigFromLeft

Decreasing depth - "autobalance"


14

Find(T) Find(R)

Splay Tree Insert and Delete

Insert x Insert x as normal then splay x to root.

Delete xSplay x to root and remove it. (note: the node

does not have to be a leaf or single child node like in BST delete.) Two trees remain, right subtree and left subtree.

Splay the max in the left subtree to the rootAttach the right subtree to the new root of the left

subtree.


15

Example Insert

Inserting in order 1,2,3,…,8 Without self-adjustment


16

1

2

3

4

5

6

7

8

O(n2) time for n Insert

With Self-Adjustment


1

2

1 2

1

ZigFromRight

2

1 3

ZigFromRight2

1

3

1

2

3

With Self-Adjustment


ZigFromRight2

1

34

4

2

1

3

4

Each Insert takes O(1) time therefore O(n) time for n Insert!!

Example Deletion


10

155

201382

96

10

15

5

2013

8

2 96

splay

10

15

5

2013

2 96

remove

10

15

5

2013

2 9

6

Splay (zig)

attach

(Zig-Zag)

Advantages

Simple implementation (self optimizing) Average case performance is as efficient as other trees No need to store any bookkeeping data Allows access to both the previous and new versions after an

update(Persistent data structure) Stable sorting

How many squares can you create in this figure by connecting any 4 dots (the corners of a square must lie upon a grid dot?

TRIANGLES:

How many triangles are located in the image below?

There are 11 squares total; 5 small, 4 medium, and 2 large.

27 triangles. There are 16 one-cell triangles, 7 four-cell triangles, 3 nine-cell triangles, and 1 sixteen-cell triangle.

GUIDED READING

ASSESSMENT

After inserting 56,87,56,43,98 and 77 the

element 22 is inserted. Then the root node of

the resultant splay tree is1. 87

2. 56

3. 22

4. 77

CONTD..

The Zig-zig corresponds to

1.left rotation followed by right rotation

2.right rotation followed by left rotation

3.left rotation followed by left rotation

4.only left rotation.

CONTD..

After inserting, 7,6,5,4,3,2 and 1 into a normal

binary search tree, a splay operation is

performed at node 1. Then, the left and right

child of the root node of the resultant tree are

1.6 and null respectively

2.null and 6 respectively

3.5 and null respectively

4.null and 5 respectively

CONTD..

Let x be the accessed node and p be the parent

node of x. Then Zig-Zag operation is performed

When

1.p is the root

2.p is not the root and both x,p are left child nodes

3.p is not the root and both x,p are right child nodes

4.p is not the root and x is left and p is right or vice versa

CONTD..

After inserting, 7,6,5,4,3,2 and 1 into splay tree,

a search operation is performed at

node1. Then, the left and right child of the root

node of the resultant tree are1. null and 3 respectively

2. null and 4 respectively

3. null and 6 respectively

4. 2 and 5 respectively

1.8 splay tree

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