paolo ferragina, università di pisa compressed rank & select on general strings paolo ferragina...

Paolo Ferragina, Università di Pisa

Compressed Rank & Select

on general strings

Paolo FerraginaDipartimento di Informatica, Università di Pisa


Generalised Rank and Select

Rank(c,i) = #c in L[1,i]

Select(c,i) = position of the i-th c in L

L = a b a a a c b c d a b e c d ...

Rank( a , 7 ) = 4Select( a , 2 ) = 3


Generalised Rank and Select

If is small (i.e. constant) Build binary Rank data structure for each symbol of

Rank takes O(1) time and small space

If is large (words ?) Need a smarter solution: Wavelet Tree data structure

Algorithmic reduction:

>> Reduce Rank&Select over arbitrary strings

... to Rank&Select over binary strings


The Wavelet Tree

a c

b r

d

abracadabra

(Alphabetic ?)Tree


The Wavelet Tree

a c

b r

d

abracadabra

aacaaa brdbr

brbr

rr?

aaaaa?

bb?

d?


The Wavelet Tree

a c

b r

d

abracadabra

aacaaa brdbr

brbr

abracadabra01100010110

aacaaa001000

brdbr00100

brbr0101

01100010110

001000 00100

0101

Fact. Given the tree and the binary strings,we can recover the original string !!

In any case, O(|| log ||) bits.

Easier Alphabetic order + Heap structure


brdbr00100


brbr0101

aacaaa001000

The Wavelet Tree

a c

b r

d

Rank(b,8)

Rank(b,3)

Rank(b,2)

Reducetorightsymbols

Reducetoleftsymbols

It’s binary

Every step can be turned to binary



Rank1(8)=3

Rank0(2) = 2 – Rank1(1)= 1 Rank0(3) =

3 – Rank1(3)= 2brbr0101

brdbr00100

aacaaa001000

The Wavelet Tree

a c

b r

d

Generalised R&S implemented with log || binary R&S

Rank(b,8)

Right move=Rank1

Left move=Rank0

Left move=Rank0

Select is similar


Representing Trees

Paolo FerraginaDipartimento di Informatica, Università di Pisa

Standard representationBinary tree: each node has twopointers to its left and right children

An n-node tree takes2n pointers or 2n lg n bits.

Supports finding left child or right child of a node (in constant time).

For each extra operation (eg. parent, subtree size) we have to pay additional n lg n bits each.

x

xxxx

x xx x

Can we improve the space bound? There are less than 22n distinct binary trees on n nodes.

2n bits are enough to distinguish between any two different binary trees.

Can we represent an n node binary tree using 2n bits?

Binary tree representation A binary tree on n nodes can be represented using

2n+o(n) bits to support:

parent left child right child

in constant time.

Heap-like notation for a binary tree

1

11 1

1 1

1

1

00 00

0 000

0

Add external nodes

Label internal nodes with a 1and external nodes with a 0

Write the labels in level order

1 1 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0

One can reconstruct the tree from this sequence

An n node binary tree can be represented in 2n+1 bits.

What about the operations?

Heap-like notation for a binary tree

1 1 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

8

5 764

32

1

9

17161514

13121110

1

87

654

32

1 2 3 4 5 6 7 8

parent(x) = On red (⌊x/2⌋)

left child(x) = On green(2x)

right child(x) = On green(2x+1)

x x: # 1’s up to x (Rank)

x x: position of x-th 1 (Select)

paolo ferragina, università di pisa compressed rank & select on general strings paolo ferragina...

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