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Fast Parallel Algorithms for Universal Lossless Source Coding

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### Fast Parallel Algorithms for Universal Lossless Source Coding

Dror Baron

CSL & ECE Department, UIUC

dbaron@uiuc.edu

Ph.D. Defense – February 18, 2003

Overview

- Motivation, applications, and goals
- Background:
- Source models
- Lossless source coding + universality
- Semi-predictive methods
- An O(N) semi-predictive universal encoder
- Two-part codes
- Rigorous analysis of their compression quality
- Application to parallel compression of Bernoulli sequences
- Parallel semi-predictive (PSP) coding
- Achieving a work-efficient algorithm
- Theoretical results
- Summary

Motivation

- Lossless compression: text files, facsimiles, software executables, medical, financial, etc.
- What do we want in a compression algorithm?
- Universality: adaptive to a large class of sources
- Good compression quality
- Speed: low computational complexity
- Simple implementation
- Low memory use
- Sequential vs. offline

Why Parallel Compression ?

- Some applications require high data rates:
- Compressed pages in virtual memory
- Remote archiving + fast communication links
- Real-time compression in storage systems
- Power reduction for interconnects on a circuit board
- Serial compression is limited by the clock rate

Room for Improvement and Goals

- Previous Art:
- Serial universal source coding methods have reached the bounds on compression quality [Willems1998,Rissanen1999]
- Parallel source coding algorithms have high complexity and/or poor compression quality
- Naïve parallelization compresses poorly
- Parallel dictionary compression [Franszek et. al.1996]
- Parallel context tree weighting [Stassen&Tjalkens2001,Willems2000]
- Research Goals: “good” parallel compression algorithm
- Work-efficient: O(N/B) time with B computational units
- Compression quality: as good as best serial methods (almost!)

Main Contributions

- BWT-MDL (O(N) universal encoder):
- An O(N) algorithm that achieves Rissanen’s redundancy bounds on best achievable compression
- Combines efficient prefix tree construction with semi-predictive approach to universal coding
- Fast Suffix Sorting (not in this talk):
- Core algorithm is very simple (can be implemented in VLSI)
- Worst-case complexity O(N log0.5(N))
- Competitive with other suffix sorting methods in practice
- Two-Part Codes:
- Rigorous analysis of their compression quality
- Application to distributed/parallel compression
- Optimal two-part codes
- Parallel Compression Algorithm (not in this talk):
- Work-efficient O(N/B) algorithm
- Compression loss is roughly B log(N/B) bits

Source Models

- Binary alphabet X={0,1}, sequence x XN
- Bernoulli Model:
- i.i.d. model
- p(xi=1)=
- Order-K Markov Model:
- Previous K symbols called context
- Context-dependent conditional probability for next symbol
- More flexible than Bernoulli
- Exponentially many states

0

leaf

0

01

11

P(xn+1=1|01)=0.4

1

0

internal node

1

P(xn+1=1|11)=0.9

Context Tree Sourcesroot

- More flexible than Bernoulli
- More compact than Markov
- Particularly good for text
- Works for M-ary alphabet
- State= context + conditional probabilities
- Example: N=11, x=01011111111

Review of Lossless Source Coding

- Stationary ergodic sources
- Entropy rate H=limN H(x)/N
- Asymptotically, H is the lowest attainable per-symbol rate
- Arithmetic coding:
- Probability assignment p(x)
- Coding length l(x)=-log(p(x))+O(1)
- Can achieve entropy + O(1) bits

Universal Source Coding

- Source statistics are unknown
- Need probability assignment p(x)
- Need to estimate source model
- Need to describe estimated source (explicitly or implicitly)
- Redundancy: excess coding length above entropy

(x)=l(x)-NH

Redundancy Bounds

- Rissanen’s bound (K unknown parameters):

E[(x)] > (K/2) [log(N)+O(1)]

- Worst-case redundancy for Bernoulli sequences (K=1): (x*)=maxxXN {(x)} 0.5 log(N/2)
- Asymptotically, (x)/N 0
- In practice, e.g., text, the number of parameters scales almost linearly with N
- Low redundancy is still essential

Phase I

Phase II

y

MDL Estimator

x

Encoder

Semi-Predictive Approach- Semi-predictive methods describe x in two phases:
- Phase I: find a “good” tree source structure S and describe it using codelength lS
- Phase II: encode x using S with probability assignment pS(x)
- Phase I: estimate minimum description length (MDL) tree source model S*=arg min {lS –log(pS(x))}

Determine s

p(xi|s)

s

xi

y

S*

Assign p(xi|s)

Semi-Predictive Approach - Phase II- Sequential encoding of x given S*
- Determine which state s of S* generated symbol xi
- Assign xi a conditional probability p(xi|s)
- Arithmetic encoding
- p(xi|s) can be based on previously processed portion of x, quantized probability estimates, etc.

node 0

node 11

node 01

unique

sentinel

Context Trees- We will provide an O(N) semi-predictive algorithm by estimating S* using context trees
- Context trees arrange x in a tree
- Each node corresponds to

sequence of appended arc

labels on path to root

- Internal nodes correspond

to repeating contexts in x

- Leaves correspond to unique contexts
- Sentinel symbol x0=$ makes sure

symbols have different contexts

or

MDL structure for 1s

additional structure

0s

1s

10s

s

s

00s

Context Tree Pruning(“To prune or not to prune…”)- The MDL structure for state s yields the shortest description for symbols generated by s
- When processing state s:
- Estimate MDL structures for states 0s and 1s
- Decide whether to keep 0s and 1s or prune them into state s
- Base decision on coding lengths

1

1

1

0

0

0

0

Phase I with Atomic Context Trees- Atomic context tree:
- Arc labels are atomic (single symbol)
- Internal nodes are not necessarily branching
- Has up to O(N2) nodes
- The coding length minimization of Phase I processes each node of the context tree [Nohre94]
- With atomic context trees, the worst-case complexity is at least O(N2) ☹

1

0

0

Compact Context Trees- Compact context tree:
- Arc labels not necessarily atomic
- Internal node are branching
- O(N) nodes
- Compact representation of the same tree
- Depth-first traversal of compact context tree provides O(N) complexity
- Theorem: Phase I of BWT-MDL requires O(N) operations performed with O(log(N)) bits of precision

Determine s

p(xi|s)

s

xi

y

S*

Assign p(xi|s)

Phase II of BWT-MDL- We determine the generator state using a novel algorithm that is based on properties of the Burrows Wheeler transform (BWT)
- Theorem: The BWT-MDL encoder requires O(N) operations performed with O(log(N)) bits of precision
- Theorem:[Willems et. al. 2000]: redundancy w.r.t. any tree source S is at most |S|[0.5 log(N)+O(1)] bits

Arithmetic Encoder 1

Arithmetic Encoder B

Assign p(xi(B))

xi(1)

p(xi(1))

x(1)

y(1)

Encoder 1

…

…

…

x

Splitter

xi(B)

p(xi(B))

x(B)

y(B)

Encoder B

Distributed/Parallel Compression of Bernoulli Sequences- Splitter partitions x into B blocks x(1),…,x(B)
- Encoder j{1,…,B} compresses x(j); it assigns probabilities p(xi(j)=1)= and p(xi(j)=0)=1-
- The total probability assigned to x is identical to that in a serial compression system
- This structure assumes that is known; our goal is to provide a universal parallel compression algorithm for Bernoulli sequences

y

k{1,…,K}

rk

ML(x)

x

Determine ML(x)

Encoder

Two-Part Codes- Two-part codes use a semi-predictive approach to describe Bernoulli sequences:
- First part of code:
- Determine the maximum likelihood (ML) parameter estimate ML(x)=n1/(n0+n1)
- Quantize ML(x) to rk, one of K representation levels
- Describe the bin index k with log(K) bits
- Second part of code encodes x using rk
- In distributed systems:
- Sequential compressors require O(N) internal communications
- Two-part codes need only communicate {n0(j),n1(j)}j{1,…,B}
- Requires O(B log(K)) internal communications

r2

rk

rK

bk

bK

b0

b1

b2

bk-1

Jeffreys Two-Part Code- Quantize ML(x)

Bin edges bk=sin2(k/2K)

Representation levels rk=sin2((2k-1)/4K)

- Use K 1.772N0.5 bins
- Source description:
- log(K) bits for describing the bin index k
- Need –n1 log(ML(x))-n0log(1-ML(x)) for encoding x

Redundancy of Jeffreys Code for Bernoulli Sequences

- Redundancy:
- log(K) bits for describing k
- N D(ML(x)||rk) bits for encoding x using imprecise model
- D(a||b) is Kullback Leibler divergence

- In bin k, l(x)=-ML(x)log(rk )-[1-ML(x)] log(1-rk )
- l( ML (x)) is poly-line
- Redundancy = log(K)+ l(ML(x))– N H(ML(x)) log(K) + L
- Use quantizers that have small L distance between the entropy function and the induced poly-line fit

Redundancy Properties

- For xs.t. ML(x) is quantized to rk, the worst-case redundancy is

log(K)+N max{D(bk||rk),D(bk-1||rk)}

- D(bk||rk) and D(bk-1||rk)
- Largest in initial or end bins
- Similar in the middle bins
- Difference reduced over wider range of k for largerN (larger K)
- Can construct a near-optimal quantizer by modifying the initial and end bins of the Jeffreys quantizer

Redundancy Results

- Theorem: The worst-case redundancy of the Jeffreys code is 1.221+O(1/N) bits above Rissanen’s bound
- Theorem: The worst-case redundancy of the optimal two-part code is 1.047+O(1) bits above Rissanen’s bound

y(1)

n0(1),n1(1)

ML(x)

rk

x(1)

…

ML(x) [jn0(j)] / [j n0(j)+j n1(j)]

Quantizer

…

n0(B),n1(B)

y(B)

x(B)

k

x(B)

Encoder 1

Encoder B

Determine n0(B) and n1(B)

Determine n0(1) and n1(1)

Parallel Universal Compression for Bernoulli Sequences- Phase I:
- Parallel units (PUs) compute symbol counts for the B blocks
- Coordinating unit (CU) computes and quantizes the MDL parameter estimate ML(x) and describes k
- Phase II: B PUs encode the B blocks based on rk

y(1)

x

…

…

…

Splitter

Compressor 1

x(B)

y(B)

Compressor B

Why do we need Parallel Semi-Predictive Coding?- Naïve parallelization:
- Partition x into B blocks
- Compress blocks independently
- The redundancy for a length-N/B block is O(log(N/B))
- Total redundancy is O(B log(N/B))
- Rissanen’s bound is O(log(N))
- The redundancy with naïve parallelization is excessive!

y(1)

x(1)

symbol counts 1

S*

…

…

symbol counts B

y(B)

x(B)

Phase I

Phase II

x(B)

S*

Compressor 1

Compressor B

Coordinating Unit

Statistics Accumulator B

Statistics Accumulator 1

Parallel Semi-Predictive (PSP) Concept- Phase I:
- Bparallel units (PUs) accumulate statistics (symbol counts) on the B blocks
- Coordinating unit (CU) computes the MDL tree source estimate S*
- Phase II:-- B PUs compress the B blocks based on S*

Coordinating Unit

Describe bin indices {ks}sS*

S*

{ks}sS*

Determine p(xi(b)|s)

s

p(xi(b)|s)

Arithmetic Encoder

xi(b)

Determine s

y(b)

Parallel unit b

p(xi(b)|s)1{xi(b)=1}rks+1{xi(b)=0}(1-rks )

Source Description in PSP- Phase I: the CU describes the structure of S* and the quantized ML parameter estimates {ks}sS*
- Phase II: each of B PUs compresses block x(b) just like Phase II of the (serial) semi-predictive approach

2Dmax=O(N/B)

Complexity of Phase I- Phase I processes each node of the context tree [Nohre94]
- The CU processes the states of a full atomic context tree of depth-Dmax, where Dmax log(N/B)
- Processing a node:
- Internal node: requires
- O(1) time
- Leaf: CU adds up block
- symbol counts to compute
- each symbol count, i.e., ns=b ns(b), where {0,1}
- The CU processes a leaf node in O(B) time
- With O(N/B) leaves, the aggregate complexity is O(N), which is excessive

Phase I in O(N/B) Time

- We want to compute ns=b ns(b) faster
- An adder tree incurs O(log(B)) delay for adding up B block symbol counts
- Pipelining enables us to generate a result every O(1) time
- O(N/B) nodes, each requiring O(1) time

Determine p(xi(b)|s)

s

p(xi(b)|s)

Arithmetic Encoder

xi(b)

Determine s

y(b)

S*

{ks}sS*

Phase II in O(N/B) Time- The challenging part in Phase II is determining s:
- Define the context index for a length-Dmax context s preceding xi(b) as the binary number that represents s
- The length-2Dmaxgenerator table g satisfies gj=sS* if s is a suffix of the context whose context index is j
- We can construct g in O(N/B) time (far from trivial!)
- Compute context indices for all symbols of x(b) and determine the generating states via the generator table g

Decoding Unit 1

Decoding Unit B

x(1)

DEMUX

y(1)

Reconstruct S*,{ks}sS*

…

…

…

bus y

x(B)

y(B)

Decoder- An input bus is demultiplexed to multiple units
- The MDL source and quantized ML parameters are reconstructed
- The B compressed blocks y(B) are decompressed on B decoding units

Theoretical Results

- Theorem: With computations performed with 2 log(N) bits of precision defined as O(1) time:
- Phase I of PSP approximates the MDL coding length within O(1) of the true optimum
- The PSP algorithm requires O(N/B) time
- Theorem: The PSP algorithm uses a total of O(N) words of memory = a total of O(N log(N)) bits
- Theorem: The pointwise redundancy of PSP w.r.t. S* is (x) < B[log(N/B)+O(1)]+|S|*[log(N)/2+O(1)]

parallelization overhead

Main Contributions

- BWT-MDL (O(N) universal encoder):
- An O(N) algorithm that achieves Rissanen’s redundancy bounds on best achievable compression
- Combines efficient prefix tree construction with semi-predictive approach to universal coding
- Fast Suffix Sorting (not in this talk):
- Core algorithm is very simple (can be implemented in VLSI)
- Worst-case complexity O(N log0.5(N))
- Competitive with other suffix sorting methods in practice
- Two-Part Codes:
- Rigorous analysis of their compression quality
- Application to distributed/parallel compression
- Optimal two-part codes
- Parallel Compression Algorithm (not in this talk):
- Work-efficient O(N/B) algorithm
- Compression loss is roughly B log(N/B) bits

More…

- Results have been extended to |X|-ary alphabet
- Future research can concentrate on:
- Processing broader classes of tree sources
- Problems in statistical inference
- Universal classification
- Channel decoding
- Prediction
- Characterize the design space for parallel compression algorithms

Generic Phase I

- if (s is a leaf) {
- Count symbol appearances ns0 and ns1
- MDLslength(ns0, ns1)
- } else { /* s is an internal node */
- Recursively compute MDL length and counts for 0s and 1s
- ns0 n0s0+n1s0, ns1 n0s1+n1s1
- MDLslength(ns0, ns1)
- if (MDLs >MDL0s +MDL1s )
- Keep 0s and 1s
- } else {
- Prune0s and 1s, keeps
- }
- }

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