Binary Search Trees of Almost Optimal Height

simone tino
Algorithms &
Complexity evaluation
Discoverying
Binary Search Trees
of Almost Optimal
Height

@ Università di Catania #20/11/2012

so... what’s this all about?

Introducing
SBB(k) trees
Before getting to SBB(k) trees, we
really have a long way to go... What
about a shortcut?

AVL-Tree

(a,b)-Tree

SBB(k)
Tree

Binary
Search Tree

B-Tree

SBB
Tree

SBB-tree and SBB(k)-tree introduce
the concept of “pseudo-node”!
Presentation by Simone Tino - All rights reserved. Authored from October 2012 to November 2012 - University of Catania - Faculty of Computer Science - Algoritmi e Complessità

some definitions...

Matrioska-like definitions, let’s start
from the most inner one...

Optimally balanced
The difference between
its maximum and minimum
heights <= 1

Introducing important
concepts: Pseudo-Node
Minimum
height

Maximum
height


Let’s open the box and
see what’s inside...

Defining SBB(k)-trees
SBB(k)-tree is a binary-tree which
satisfies the following 3 criteria...
The

Pseudo-Root

is optimally

balanced and has a maximum
height <= (k+1).
All other

Pseudo-Nodes

have a

minimum height >= (k) and a
maximum height <= (k+1).
The number of

Pseudo-Nodes

on the path

from the root to a leaf is the same for all leaves.

47

The importance of k

Let’s draw some
trees...

35

64

19
9

50

40
28

38

44

48

69
52

66

77

47

35

64
40

k -> k++
19
9

38
28

44

50
48

69
52

66

77

What happens when k
increases?


Caging
SBB(k)-trees?

Height upper-bound
START

h ( ab-tree ) 1+ log a

h ( SBB(k)-tree )

n +1
2

= 1+

log ( n + 1) 1
log ( 2 k )

log ( n + 1) + h ( ab-tree )
log ( n + 1) + 1+
1+

1
k

log ( n + 1) 1
log ( 2 k )

log ( n + 1) + 1

1
k

h ( SBB(k)-tree )

1
1+
k

log ( n + 1)


Maintenance algorithms

ops.balance

ops.coBalance

Merging, cutting,
rotating... the bricks
of SBB(k)-trees

ops.split

ops.join


Update algorithms
ops.insertion
Inserting a node in the tree.

Analyzing algorithms for
insertion and deletion in
SBB(k)-trees

int ops.insertion(pn P, int c) {
return (c==0) ? ops.balance(P,null) :
/*(c==1)*/ ops.split(P,null);
}

ops.deletion
Deleting a node in the tree.
int ops.deletion(pn P, pn S, int c) {
return (c==0x00) ? ops.balance(P,S) :
ops.rotation() +
(c==0x10) ? ops.coBalance(P,S) :
/*(c==0x11)*/ ops.join(P,S);
}


Inspecting ops.balance
and most important issues

Restructuring Work
A very important
operation is
Restructuring.

It is needed to keep the
tree a SBB(k)-tree.
Restructuring is
performed by the
Balance operation.

Restructuring Work
=
ops.balance

insertion

2 k+2
2P

max ( ll ( P ) ,rl ( P ))

log ( P + 1)

2k 4

deletion

3 2 k+1 2k 7


Binding k to n and
seeing what happens

Improving height
log log ( n + 1)

log log ( 3n )

k

Recalling dynamic arrays,
we can create a range for k

At the begging, we proved the upper-bound. By
tuning k, we improve the upper-bound

h ( SBB(k)-tree )

k = O ( log log n )

O ( log n )

1+

1
k

1+

1
log log ( n + 1)

log ( n + 1)
log ( n + 1)

( )

O 2 k Restructuring
work


An efficient application

What’s this? Am I
seeing double ?!?

Keeping two SBB(k)-trees
synchronized
Ghosting one tree on
the other one
Hiding rebuilding
work


the end
thank you
by

simone
tino

Binary Search Trees of Almost Optimal Height

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