Succinct Trees

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M. S. Ramaiah School of Advanced Studies 1

Succinct Trees

Sivaramaiah.B.V

CJB0910006

M. Sc. (Engg.) in Computer Science andNetworking

Module Leader: N.D.Gangadhar

Presentation


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Max marks score

Technical content 10

Grasp and

Explanation

10

Slides and Delivery 10

Q &A 10


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Contents

Introduction to Trees

Succinct Trees

Representation of Trees LOUDS Representation

Balanced Parentheses

DOUDS Representation


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Introduction to Trees

Trees are the most fundamental objects which stores a

large amount of data.

Standard representation of the binary trees on n nodes

using pointers take O(n log n) bits of space.

The extra space 2n is needed in order to support constant

time navigation between tree nodes.


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Succinct trees

Succinct trees takes only 2n+O(n) bits and support

constant time computation and a set of Query operations:

Parent(x): returns the parent of node x.

Degree(x): returns the degree i.e. number of children of

node x.

Child(x,i): Ith child at node x.

Depth(x): height or distance from root x.

LevelAncestor(x,d): return the ancestor of x with depth d.


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.

Binary tree representation

A Binary Trees on n nodes can be represented using

2n+O(n) bits to support:

1) Parent

2) Left child

3) Right child

in constant time.


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Ordered trees

A rooted ordered tree of n nodes:

Navigational operations are

- Parent(x)=a

- first child(x)=b

- next sibling(x)=c

Other operations are

- degree(x)

- sub tree size(x)

x

a

c

b



LOUDS Representation

Louds is Level Order Unary Degree Sequence.

This is the name because of traversing the tree in level

order and writing the sequence of each node in unary

order.

The node with 3 children is written as 1110.

The parent and child can be formulated as

parent(x)=1+rank0(select1(x))

child(x)=rank1(select0(x-1)+k)



LOUDS contd..

The example for LOUDS is given below



LOUDS contd..

Now computing parent(7), for the first we need to calculateselect1(7).

Select1(7)=10, which is one corresponding position of the

node g.

Next is Rank0(10)=3, which gives the number of 0spreceding the unary degree of g parent.

Gs parent position is 1+Rank0(10)=4 and the node is D.



Balanced Parentheses Representation

This representation is obtained by traversing the tree by depth

first order writing a left parentheses when node is encounteredfirst and a closing parentheses when the same node is

encountered again while going up after traversing the sub tree.

The directions of parentheses is shown

in figure.( )

( )( ) ( )

( ) ( )

( )

( )

( )




The operations are

- Parent(x)=Rankopen(enclose(selectopen(x)))

-subtreeSize(x)=(Findclose(selectopen(x))-selectopen(x)+1/2)

-firstchild(x)=x+1

-Rightsibling(x)=Rankopen(Findclose(selectopen(x))+1)




The example is




For the computation of parent(6), selectopen(6) should be

computed.

Selectopen(6)=8, which is the position of open parentheses

associated to node j.

Then, enclose(8)=3, which is the position of the closestopen parenthesis associated to js parent.

At last Rank(3)=3, we get the identifier js parent.




Another computation is finding the SubtreeSize(9).

Selectopen(9)=16, which is the position of the ( of the

node d.

Findclose(16)=23, which is the position of ) associated

to d.

Then the SubtreeSize=(23-16+1)/2=4.



DFUDS Representation

DFUDS is Depth-First Unary Degree Sequence.

DFUDS combines both LOUDS and BP representation and

benefits of both.

Due to the combination of both representation, it is able to

support all the query operations with in the (2n+O(n)) bits.



DFUDS Representation

The example is shown below



Reference

[1] Mehta, D.P and S. Sahani (Eds.),Handbook of DataStructures and Applications, Chapman & Hall/CRC, 2005

[2] Jiri Fiala, Jan Kratochvil and Mirka Miller (Eds.),

Combinatorial Algorithms, Springer-Verlag Berlin

Heidelberg, 2009


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Succinct Trees