1. Quasi-Distinct Parsing and Optimal Compression Methods
Avivit Levy
Shenkar College
and
CRI, Haifa University
Joint work with:
Amihood Amir, Yonatan Aumann, Yuri Roshko

2. Motivation Analysis of compression schemes is a complex task.
We do not well compress everything that is theoretically compressible.
Conceptually different schemes are needed.
Theoretical tools capable of analysis of such schemes are needed.

3. Our Work
Provide a generalization of LZ78 optimality proof - New tool for proving optimality of compression schemes.
Present a new code:
Arithmetic Progressions Tree (APT)
Apply the new theorem to analyze APT as a compression scheme.

4. Parsing of Strings
Parsing divides the given string into phrases. Possibly a dictionary.
Example: LZ78 parsing

5. Distinct Parsing Distinct parsing: All phrases are distinct.
Allows no repetitions in the created dictionary.
Example 1
Consider the parsing:
The i-th phrase is the next i bits.
This is a distinct parsing.

6. Distinct Parsing 1,0,11,01,111,011,00,010,0110,10 Example 2
Consider LZ78 parsing:
This is a distinct parsing.
1,0,11,01,111,011,00,010,0110,10

7. Quasi-Distinct Parsing
Quasi-distinct parsing allows infinitely many repetitions in the created dictionary.
As long as overall repetitions are o(n/log n).
Gives more flexibility in the parsing definition.
Does it affect the efficiency?

8. Quasi-Distinct Parsing
Example
Consider the parsing:
Next i phrases defined by i times taking the next i bits.
This is a quasi-distinct parsing:
A phrase of length i cannot repeat more than i-1 times.
Overall repetitions bounded by O(n2/3).

9. LZ Optimality Theorem For any stochastic ergodic process, if
S distinct parsing of X1,…,Xn giving c(n) phrases
The total codeword length Len(X1,…,Xn)=c(n)(log c(n)+1)
Then
lim sup Len(X1,…,Xn)/n≤H().

10. Generalized Optimality Theorem
For any stochastic ergodic process, if
S quasi-distinct parsing of X1,…,Xn giving c(n) phrases
The total codeword length Len(X1,…,Xn)=c(n)(log c(n)+)
Then
lim sup Len(X1,…,Xn)/n≤H().

11. Correctness Where in LZ78 proof the distinctness condition is used ?
For bounding c(n)=O(n/log n).
To get Ziv’s inequality:
The relation between c(n)log c(n) and a Markov approximation of .

12. The Dictionary Size Bound
c(n)=O(n/log n) apply for quasi-distinct parsing:
Take one instance for each phrase – This set of distinct phrases is
O(n/log n) by LZ bound.
The multi-set of other phrases is o(n/log n) by quasi-distinctness definition.

13. Ziv’s Inequality We somehow get:
But what about the sum of probabilities?

14. Application of Theorem: APT
Arithmetic Progressions Tree (APT):
Recursive cover by arithmetic progressions.
Idea presented by AmirLevyReuveni (FI, 2008) for different purpose.
Example: S= 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 1 1 1 1 1

15. Application of Theorem: APT…19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 1 1 19 17 16 14 11 9 8 5 4 2 1 19 17 16 14 11 9 8 5 4 2 1 19 17 16 14 11 9 8 5 4 2 1 19 17 16 14 11 9 8 5 4 2 1 19 17 16 14 11 9 8 5 4 2 1 19 17 16 14 11 9 8 5 4 2 1 2 16 8 4 1 2 8 4 1 2 16 8 4 1 2 4 1 2 16 8 4 3 2 11 17 2 4 2 1 16

16. Application of Theorem: APT…
generate APT tree representing data
generate code representing APT tree
APT
tree
binary data
user
APT code
binary data
decode data to APT tree
reconstruct binary data from APT tree
APT code
APT
tree

17. Application of Theorem: APT…
How to analyze?
By theorem:
Show a QD parsing
Bound total phases representation
First step: parsing
By start positions of progressions. S=
The start positions in APT: 2,4,11,16 1 1 1 1

18. Application of Theorem: APT…
Is it quasi-distinct?
empirical evidence
no combinatorial proof yet
APT quasi-distinctness hypothesis.

19. Application of Theorem: APT…
APT parsing considers only leaves.
How to apply theorem?
use theorem for leaves storage analysis.
use APT structure to infer tree storage.
Idea:
Get bound on total number of APT nodes, based on leaves bound.
Property:
Should be enough 1’s,
but also enough 0’s to get APT nodes. Sum is n.

20. Application of Theorem: APT…
Second step: how to store?
The challenge is the references.
use APT reference string.
Example:
The APT reference string of S= R=
2,4,11,16 (leaf), 1,2,3,4,17 (internal)
reference string more compressible, has smaller entropy.

21. Conclusions Contribution 1: Theoretical tool
Two conditions sufficient for optimality:
Defined by quasi-distinct parsing.
Having total length c(n)(log c(n)+).
Contribution 2: The APT coding
First application of new theorem.
First presentation of APT as code.
First (weak) theoretical analysis of APT.