Specialized Tree

1

• Specialized Tree
• Structures
• By
• Anubhav Singh
• Sidharth Saboo

2
Agenda

• Threaded binary trees
• Splay trees
• Segment tree
• R-Tree

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Threaded Binary Trees

• Introduction
• Operations
• Insertion
• Deletion
• Traversal
• Application

4
TREE

• A tree is a collection of nodes.
• Each of these nodes are connected by directed
  edge from root node r.
• Tree consists of a
• distinguish node r,
• called root, and zero
• or more non-empty
• (sub)trees T1,T2,T3,.TK.

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• BINARY TREE
• A binary tree is a tree in which no nodes can
  have more than two children.
• The recursive definition is that a binary tree is
  either empty or consists of a root, a left tree,
  and a right tree.
• The left and right trees
• may themselves be empty,
• thus a node
• with one child could
• have a left or right child.

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THREADED BINARY TREE

• A binary tree is threaded by
• Making all right child pointers that would
  normally be null point to the inorder successor
  of the node.
• All left child pointers that would normally be
  null point to the inorder predecessor of the
  node.
• class threadedTree
• Cdata pData
• threadedTree pLeft, pRight
• bool leftFlag, rightFlag

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Representation of Threaded Tree
f
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E
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• Threading Rules
• A RightChild field at node p is replaced by a
  pointer to the node that would be visited after p
  when traversing the tree in inorder. (i.e., it is
  replaced by the inorder successor of p).
• A LeftChild link at node p is replaced by a
  pointer to the node that immediately precedes
  node p in inorder (i.e., it is replaced by the
  inorder predecessor of p).

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• Other Threaded Tree Structures

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Right Threaded Binary Tree
Left Threaded Binary Tree
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Inserting A Node In A Threaded Tree

• Inserting a node r as the right child of node s
• If s has an empty right subtree, then the
  insertion is simple.
• If the right subtree of s is not empty,
• right subtree is made the right subtree of r
  after insertion.
• r becomes the inorder predecessor of a node that
  has a LeftThreadTRUE field and consequently
  there is a thread which has to be updated to
  point to r.
• The node containing this thread was previously
  the inorder successor of s.

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Insertion of r As Right Child of s in Threaded
Binary Tree
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r
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Insertion of r As Right Child of s in Threaded
Binary Tree
s
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after
before
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Deletion of r As Right Child of s
s
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Before
After
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InOrder Traversal Of Threaded BinaryTree

• B-gtD-gtC-gtF-gtE-gtG-gtA

A
B
s
C
D
E
F
G
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Application (Timing Diagram Editor)

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Splay Tree

• Introduction
• Operations
• Splay
• Insertion
• Deletion
• Traversal
• Application

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Introduction

• Splay trees self-adjusting BST.
• Tree automatically reorganizes itself after each
  operation.
• For each operation on x, rotate x up to root
  using
• double rotations.
• Tree remains "balanced" without explicitly
  storing any balance information.
• Amortized guarantee any sequence of N operations
  takes O(N log N) time.

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SPLAY OPERATION

• Basic operation of splay tree
• When a node x is accessed, a sequence of splay
  steps are performed to move x to the root of the
  tree.
• In case x is not present in the tree, the last
  element on the search path for x is moved
  instead.
• There are 6 types of splay steps, each consisting
  of 1 or 2 rotations.

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RR Rotation
x lt y lt z
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LL Rotation

• z lt y lt x

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LR Rotation

• y lt x lt z

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RL Rotation

• z lt x lt y

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L Rotation

• y lt x

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R Rotation

• y gt x

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Basic Splay Tree Operations

• INSERTION
• When an item is inserted, a splay is performed.
• o As a result, the newly inserted item becomes
  the root of the tree.

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Basic Splay Tree Operations

• FIND
• The last node accessed during the search is
• splayed.
• If the search is successful, then the node that
  is
• found is splayed and becomes the new
  root.
• If the search is unsuccessful, the last node
  accessed prior to reaching the NULL pointer is
  splayed and becomes the new root.
• FindMin and FindMax perform a splay after
• the access.

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Basic Splay Tree Operations

• DELETE
• DeleteMin
• Perform a FindMin.
• This brings the minimum to the root, and by the
  binary search tree property, there is no left
  child.
• Use the right child as the new root and delete
  the node containing the minimum.

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Basic Splay Tree Operations

• DeleteMax
• Peform a FindMax.
• Set the root to the post-splay roots left child
  and delete the post-splay root.

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Basic Splay Tree Operations

• Delete
• Access the node to be deleted bringing it to the
  root.
• Delete the root leaving two subtrees L (left) and
  R (right).
• Find the largest element in L using a DeleteMax
  operation, and move the element to the root thus
  the root of L will have no right child.
• Make R the right child of Ls root.

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Basic Splay Tree Operations
Remove From Splay Tree Example
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Applications

• Network Router

• Cache Memory Algorithms

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SEGMENT TREE
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Outline

• Introduction
• Operations
• Insertion
• Allocation
• Deletion
• Properties
• Application

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Introduction

• Data Structure designed to handle intervals on
  the real line
• The extremes of the real line belong to fixed set
  of N abscissae/ ordinates
• It is a static data structure , one that does not
  support insertions or deletions of abscisae/
  ordinate
• It is a rooted binary tree.

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The Segment Tree T (4, 15)
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Outline

• Introduction
• Operations
• Insertion
• Allocation
• Deletion
• Properties
• Application

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Operations

• Insertion Operation
• Procedure Insert( b, e v)
• begin if ( b lt Bv ) and ( E(v)lt e )
• then allocate b,e to v
• else begin if ( b lt ( Bv Ev ) / 2 )
• then INSERT(b,e,LSONv)
• if (( Bv) Ev ) / 2 lt e )
• then INSERT(b,e,RSONv)
• end
• end

Here b begin e end v root Bv begin of
root Ev end of root
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Insertion of interval 5, 8 into T 4 , 15
4,15
9,15
4,9
12,15
6,9
9,12
4,6
4,5
5,6
6,7
7,9
8,9
7,8
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Allocation of an interval to a node

• Allocation of an interval to a node v depends
  upon the requirement
• Generally , it is managed using a parameter Cv
  a non negative integer, referred to as
  cardinality
• Allocation of b, e to v becomes
• Cv Cv 1
• In addition, we append to each node the
  identifiers of the intervals

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Delete Operation

• Procedure Delete( b, e v)
• begin if ( b lt Bv ) and ( E(v)lt e )
• then Cv Cv - 1
• else begin if ( b lt ( Bv Ev ) / 2 )
• then DELETE(b,e,LSONv)
• if (( Bv) Ev ) / 2 lt e )
• then DELETE(b,e,RSONv)
• end
• end

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Outline

• Introduction
• Operations
• Insertion
• Allocation
• Deletion
• Properties
• Application

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Some Properties

• It is a balanced tree
• Depth of tree is log2(r-l)
• Can be built in O(nlogn) time
• All intervals containing a query interval can be
  reported in O(logn k) time

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Some Properties

• Extremely versatile data structures
• Used extensively in computational geometry
  problems and geographic information systems
• Example Used by MC2 developers in isothetic
  polygon merging algorithm

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Outline

• Introduction
• Operations
• Insertion
• Allocation
• Deletion
• Properties
• Application

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Application

• Merged Using Segment Tree

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R - Tree
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Outline

• Introduction
• Operations
• Insertion
• Deletion
• Search
• Properties
• Application

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Introduction

• Tree data structures similar to B-trees
• Height Balanced Tree
• Leaf nodes contain pointers to data objects
  containing spatial data. (MBR)
• Rectangles at each node may be overlapping.

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Example of an R- Tree for 2-D Rectangles
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Motivation

• Masks used in the fabrication of integrated
  circuits are frequently expressed as collection
  of rectangles with sides parallel to orthogonal
  directions
• The number of such rectangles is huge requiring
  efficient data structures to store them
• R Tree is one such data structure

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Minimum Bounding Rectangle

• Objects are added to an MBR that will lead to
  the smallest increase in its size
• In R Tree Objects (shapes, lines and points)
  are grouped using the minimum bounding rectangle
  (MBR)
• MBR is represented by (min(x) min(y)) and max(x)
  max(y)

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Outline

• Introduction
• Operations
• Insertion
• Deletion
• Search
• Properties
• Application

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Operations

• Insertion Operation
• To Insert a new index entry E into an R tree
• Step 1 Find position for new record Invoke
  ChooseLeaf to select a leaf node L in which to
  place E.
• Step 2 Add record to leaf node If L has room,
  insert E. Otherwise invoke SplitNode to obtain L
  and LL containing E and all old entries of L

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Insertion Operation

• Step 3 Propagate changes upwards Invoke
  AdjustTree on L, also passing LL if split was
  performed
• Step 4 Grow Tree taller If node split
  propagation caused the root to split, create a
  new root whose children are the resulting nodes.

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T 2
R 4
T 1
R 2
R 1
R 3
R 5
Rectangular Data Objects
R Tree Insertion
R 1
R 2
R 3
T 1
T 2
R 1
R 2
R 3
R 4

R 5
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Deletion Operation

• To Remove index record E from R Tree
• Step 1 Find node containing record Invoke
  FindLeaf to locate the leaf node L containing E.
  Stop if the record was not found.
• Step 2 Delete record Remove E from L
• Step 3 Propagate changes Invoke CondenseTree,
  passing L

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Deletion Operation

• Step 4 Shorten Tree If the root node has only
  one child after the tree has been adjusted, make
  the child the new root

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Search Operation

• To find all index records whose rectangles
  overlap a search rectangle S in a given R
  Tree with root node T
• Step 1 Search subtrees if T is not a leaf,
  check each entry E to determine whether E
  overlaps S. For all overlapping entries, invoke
  Search on the tree whose root node is pointed by
  child pointer of E

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Search Operation

• Step 2 Search leaf node If T is a leaf, check
  all entries E to determine whether E overlaps S.
  If so, E is a qualifying record

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Outline

• Introduction
• Operations
• Insertion
• Deletion
• Search
• Properties
• Application

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Some Properties

• To split nodes when they become too full,
  different algorithms are used resulting in the
  "Quadratic" and "Linear" R-tree sub-types.
• Not the best worst case performance but generally
  perform well with real-world data
• Easy to add to any relational database system

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Some Properties

• Extensively used for spatial access methods i.e.,
  for indexing multi-dimensional information for
  example, the (X, Y) coordinates of geographical
  data.
• Real world example Used by Interras MC2 to
  store pins of a memory

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Outline

• Introduction
• Operations
• Insertion
• Deletion
• Search
• Properties
• Application

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View of a memory
courtesy MC2 Development Team
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Zoomed - in View
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• Overlapped Geometries

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• A Single Geometry

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Thank You !!