A Simpler Analysis of BurrowsWheeler Based Compression

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A Simpler Analysis of Burrows-Wheeler Based
Compression

• Haim Kaplan Shir Landau Elad Verbin

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Our Results

• Improve the bounds of one of the main BWT based
  compression algorithms
• New technique for worst case analysis of BWT
  based compression algorithms using the Local
  Entropy
• Interesting results concerning compression of
  integer strings

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The Burrows-Wheeler Transform(1994)

• Given a string S the Burrows-Wheeler Transform
  creates a permutation of S that is locally
  homogeneous.

S
BWT
S is locally homogeneous
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Empirical Entropy - Intuition

• H0(s) Maximum compression we can get without
  context information where a fixed codeword is
  assigned to each alphabet character (e.g.
  Huffman code )
• Hk(s) Lower bound for compression with order-k
  contexts the codeword representing each symbol
  depends on the k symbols preceding it
• Traditionally, compression ratio of compression
  algorithms measured using Hk(s)

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History

• The Main Burrows-Wheeler Compression Algorithm
  (Burrows, Wheeler 1994)

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MTF
Given a string S baacb over alphabet a,b,c,d
b a a c b
S
1
1
0
2
2
MTF(S)
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Main Bounds (Manzini 1999)

• gk is a constant dependant on the context k and
  the size of the alphabet
• these are worst-case bounds

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Now we are ready to begin
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Some Intuition

• The more the contexts are similar in the original
  string, the more its BWT will exhibit local
  similarity
• The more local similarity found in the BWT of the
  string the smaller the numbers we get in MTF
• ? We want a statistic that measures local
  similarity in a string and specifically in the
  BWT of the string
• ? The solution Local Entropy

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The Local Entropy- Definition

• We define given a string s
• s1s2sn
• The local entropy of s (Bentley, Sleator,
  Tarjan, Wei, 86)

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The Local Entropy - Definition

• Note LE(s) number of bits needed to write the
  MTF sequence in binary.
• Example
• MTF(s) 311
• ? LE(s) 4
• ? MTF(s) in binary 1111

In Dream world We would like to compress S to
LE(S)
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The Local Entropy Properties

• We use two properties of LE
• The entropy hierarchy
• Convexity

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The Local Entropy Property 1

• The entropy hierarchy
• We prove For each k
• LE(BWT(s)) nHk(s) O(1)
• ? Any upper bound that we get for BWT with LE
  holds for Hk(s) as well.

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The Local Entropy Properties 2

• Convexity
• ? This means that a partition of a string s does
  not improve the Local Entropy of s.

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Convexity

• Cutting the input string into parts doesnt
  influence much Only positions per part

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Convexity Why do we need it?

• Ferragina, Giancarlo, Manzini and Sciortino, JACM
  2005

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Using LE and its properties we get our bounds

• Theorem For every where

Our LE bound
Our Hk bound
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Our bounds

• We get an improvement of the known bounds
• As opposed to the known bounds (Manzini, 1999)

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Our Test Results
The files are non-binary files from the
Canterbury corpus. gzip results are also taken
from the corpus. The size is indicated in bytes.
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How is LE related to compression of integer
sequences?

• We mentioned dream world but what about
  reality?
• How close can we come to ?
• Problem
• Compress an integer sequence S close to its sum
  of logs
• Notice for any s

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Compressing Integer Sequences

• Universal Encodings of Integers prefix-free
  encoding for integers (e.g. Fibonacci encoding,
  Elias encoding).
• Doing some math, it turns out that order-0
  encoding is good.
• Not only good It is best!

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The order-0 math

• Theorem For any string s of length n over the
  integer alphabet 1,2,h and for any ,
• Strange conclusion we get an upper-bound on the
  order-0 algorithm with a phrase dependant on the
  value of the integers.
• This is true for all strings but is especially
  interesting for strings with smaller integers.

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A lower bound for SL

• Theorem For any algorithm A and for any ,
  and any C such that C lt log(?(µ)) there exists a
  string S of length n for which
• A(S) gt µSL(S) Cn

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Our Results - Summary

• New improved bounds for BWMTF
• Local Entropy (LE)
• New bounds for compression of integer strings

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Open Issues

• We question the effectiveness of .
• Is there a better statistic?

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Any Questions?
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Thank You!
Anybody want to guess??
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Creating a Huffman encoding

• For each encoding unit (letter, in this example),
  associate a frequency (number of times it occurs)
• Create a binary tree whose children are the
  encoding units with the smallest frequencies
• The frequency of the root is the sum of the
  frequencies of the leaves
• Repeat this procedure until all the encoding
  units are in the binary tree

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Example
Assume that relative frequencies are A 40 B
20 C 10 D 10 R 20
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Example , cont.
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Example, cont.

• Assign 0 to left branches, 1 to right branches
• Each encoding is a path from the root

A 0B 100C 1010D 1011R 11
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The Burrows-Wheeler Transform (1994)
Given a string S banana
banana
banan a
ananab
a bana n
nanaba
a naba n
anaban
nabana
abanan
banana
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Suffix Arrays and the BWT
The Suffix Array
Index of BWT
7 6 4 2 1 5 3
6 5 3 1 7 4 2