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Engineering the Compression of Massive Tables An Experimental Approach SIAM 2000.

Engineering the Compression of Massive Tables An Experimental Approach SIAM 2000.
Engineering the Compression of Massive Tables An Experimental Approach SIAM 2000.

To appear in Proc.11th ACM-SIAM Symp.on Discrete Algorithms,San Francisco,CA c SIAM2000.

Engineering the Compression of Massive Tables:An Experimental Approach Adam L.Buchsbaum Donald F.Caldwell Kenneth W.Church Glenn S.Fowler

S.Muthukrishnan

Abstract

We study the problem of compressing massive tables.We devise a

novel compression paradigm—training for lossless compression—

which assumes that the data exhibit dependencies that can be

learned by examining a small amount of training material.We

develop an experimental methodology to test the approach.Our

result is a system,pzip,which outperforms gzip by factors of

two in compression size and both compression and uncompression

time for various tabular data.Pzip is now in production use in an

AT&T network traf?c data warehouse.

1Introduction

We study the problem of compressing massive tables,which

arises naturally in corporate data warehouses.Our goal

is to provide a working system that can be put into pro-

duction use and achieve100:1compression,in particular,

one that can compress10s of TB of data into100s of GB.

We devise a novel compression strategy—training for loss-

less compression—which can leverage standard compression

methods,and we demonstrate its effectiveness experimen-

tally.In the process,we identify the requirements for our

compression application and design algorithmic solutions to

various technical problems.The system we built,pzip,can

compress1TB of data from AT&T’s network traf?c data

warehouse into about28GB,small enough to?t on one PC

disk and a two-fold improvement over existing solutions(in

time as well as space).Pzip is now in production use in the

warehouse.

Our motivating applications are tables of traf?c data

from telecommunication networks.For example,the AT&T

voice communications network generates a record of each

phone call it carries.A typical record consists of several

hundred bytes and depicts network-level information(e.g.,

endpoint exchanges),time-stamp information(e.g.,start

and end times),and billing-level information(e.g.,applied

tariffs).This application generates about1TB of data per

month and is just one example of the many different tables

that AT&T and other corporations generate.Others include

switch-and router-level traf?c data,equipment sensor data

(e.g.,alarm status),and credit card transaction records.

These data have certain unifying characteristics:they have

and maintained.Finally,it is preferable that any solution be general,applying to the many different tables,or portions of tables,that are processed.Thus we cannot exploit domain-speci?c syntactic or semantic information.

Studying massive tables is a new focus in compression research.To distinguish our context from extant ones,con-sider the related?eld of database compression,where rela-tional data may be viewed as tables.This differs from our table compression problem in many ways.First,the goals are different.Database compression stresses the preserva-tion of indexing—the ability to retrieve an arbitrary record—under compression[7].Table compression does not re-quire indexing to be preserved.Next,the data are different. Database records are often dynamic,unlike table data,which have a write-once discipline.Databases consist of hetero-geneous data,possibly with several string?elds of variable length;table data are more homogeneous,with?xed?eld lengths.Also,non-tabular databases are not routinely TBs in size.(An exception is NASA’s EOSDIS database[13], which anticipates processing1TB of satellite images every two weeks.)Finally,the approaches to database compression include lightweight techniques such as compressing each tu-ple by simple encodings[7,8]and tiling the entire table[8]. These approaches are not appropriate for table compression: the former is too wasteful,and the latter too expensive and cumbersome.

Our contribution is a novel approach for the table com-pression problem:lossless compression via training.The idea is to construct a compression plan for the table by study-ing a very small training set off line.To do so,we assume that the data can be modeled by an underlying source that can be learned from a small sample.We further assume and exploit dependencies in the columns of the data in one of two ways:(1)implicitly,by grouping the columns that compress well together;and(2)explicitly,by determining a depen-dency tree among the columns.We then employ the com-pression plan on the entire dataset.To test our assumptions, we implement algorithms to construct compression plans on some training sets,and we compress test sets with respect to the plans.We compare the resulting compression to the straightforward approach of treating the tables as text and applying Lempel-Ziv compression[20,21].It will be clear that comparable performance would falsify our assumptions about the data dependencies.In all cases,however,our algorithms provide substantial compression improvements. While training has been applied to lossy compression,e.g., in speech coding[12,15],ours is the?rst known instance of applying training to lossless compression.

For our primary application,compression exploiting implicit dependencies outperformed that using explicit de-pendencies.We have implemented the corresponding algorithm—optimum partitioning—in pzip,a fully work-ing software system for compressing table data,which has been deployed in the AT&T network traf?c data warehouse. Pzip achieves factors of about2improvement in compres-sion size and both compression and uncompression time over gzip,the method previously used in this application.

In Section2,we discuss the table compression problem further and de?ne our assumptions regarding data dependen-cies.In Section3,we present technical problems that ex-ploit our assumptions,and we give algorithmic solutions to these problems.In Section4,we present our experimental results,and in Section5,we discuss the pzip system and some additional applications.In Section6,we summarize our contributions and present directions for future work.

2Problem Discussion

Our input consists of a table,,of a large number of rows,each of length bytes.We de?ne column to be the projection of the th byte of each row,for .(A byte is the smallest unit of data that can be easily and rapidly accessed;moreover,this level of granularity captures patterns among larger lexical units.)The table compression problem is to compress,such that the requirements discussed in Section1are satis?ed.

From an information-theoretic point of view,can be treated as a string,e.g.,of bytes in row-major order.It would thus suf?ce to perform Lempel-Ziv[20,21]or Huffman[9] compression,yielding provably optimal asymptotic perfor-mance in terms of certain ergodic properties of the source that generates the table.This does not,however,adequately solve the table compression problem.For speci?c classes of inputs,e.g.,tables of network traf?c data,the optimality re-sults may not necessarily hold.In particular,the optimality results hold only with respect to compression methods that likewise treat as a(byte)string;i.e.,methods that do not account for complex dependencies in.Some compression does result,however,and we use this method as our bench-mark in Section4.

We need a few technical de?nitions.Denote by the th column of.Denote by the interval of columns through of.Finally,denote by the size of the result of compressing some interval of columns,in row-major order,using an arbitrary but?xed compressor.

2.1Assumptions.Our approach to the table compression problem assumes that there are dependencies among the columns of.In particular,we will consider dependencies of two types:combinational and differential.

D EFINITION2.1.Two contiguous intervals of columns

and are combinationally dependent if

Combinational dependency is an implicit dependency between intervals.It merely expresses that intervals of

columns are dependent,without determining which columns are dependent on the others.

D EFINITION2.2.Column is differentially dependent on column if

where is the column formed by taking the row-wise difference between columns and.

Differential dependency is an explicit dependency be-tween columns,in that it determines which column is de-pendent on the other.In general,we might compress and by different methods,and we might consider other transformations.This does not affect the ensuing discussion.

Our approach also makes the important assumption that the data is generated by some source that is well behaved, in particular,that dependencies(such as those above)among columns,if they exist,can be captured by examining a small amount of data,independent of the size of.

3Algorithmic Issues

We design compression schemes based on the assumptions embodied in De?nitions2.1and2.2.

3.1Combinational Approach.We can exploit combina-tional dependencies as follows.Consider a partition,,of into intervals, such that and.We refer to an interval

in as a class.We de?ne the cost of to be

The goal is?nd an optimum partition,i.e.,such that

.

We can?nd an optimum partition as follows.De?ne to be the cost of an optimum partition of for ,and de?ne.Then for,

Assuming that the cost has been computed for all,we can compute(and the corresponding partition)in time by simple dynamic programming.

We call this optimum partitioning.This gives the following compression plan for compressing:compress each class in the optimum partition independently in row-major order.

We can further speed up the dynamic programming, under the assumption that there are many optimum or near optimum partitions for compressing,and that among these are some in which the classes are not too wide.To do so,we develop a“chunking”approach,in which,for some ,we divide the columns into pairwise-disjoint intervals of size at most each,and solve our general problem on each such interval.The running time becomes .We call this chunk partitioning,and it likewise returns a compression plan.

3.2Differential Approach.We can exploit differential dependencies as follows.Consider a partition of the columns of into two sets,and.We treat the columns in as source columns and those in as derived columns.Given a mapping,we de?ne the cost,to be

The goal is to?nd a pair of minimum cost.

This is precisely the facility location problem[17].We will assume that the differential cost is a metric.In general, this depends on the base compressor.We apply the simple, greedy algorithm for this problem[14].

At any time,we have a candidate pair.We determine the smallest cost solution,,obtained by

1.removing a column from,

2.adding a column to,or

3.substituting one of the columns in for one not in. (Ties are broken arbitrarily.)If,then we set and and iterate.Otherwise we are done.We call this greedy differential compression.The?nal solution is roughly-optimal under the metric assumption [14].Better approximations[3,4,5,11]are known,but the greedy algorithm suf?ces for our purpose of testing the presence of differential dependencies.

Greedy differential compression produces the follow-ing compression plan:compress each column in indepen-dently,and for each column,compress.

3.3Lossless Compression via Training.Our overall ap-proach is thus the following.

1.Select a small subset as training material.

https://www.doczj.com/doc/0614885682.html,ing,compute a compression plan,,for by

either the combinational or differential approach.

https://www.doczj.com/doc/0614885682.html,press with the compression plan.

Our assumption that the amount of training data needed is independent of the size of implies that,once we have

generated a compression plan,we can use it to compress future tables generated by the same source as.Training is thus an off-line procedure.

So far,we have abstracted the base problem of comput-ing and.Rather than develop our own base compression method,we decided to use one of the stan-dard programs,which have already been well optimized: e.g.,compress[18,21],gzip[20],and vdelta[10]. Each is fast,on-line,and well-suited to our application.Of other available compressors,we note that PPM[6,19],which exploits context sensitivity and thus seems applicable to ta-ble data,and bzip[1]are too slow for our environment, although attempts have been made to tune PPM for speed at the expense of compression size[16].We therefore do not use bzip and PPM in our compression scheme,but we do compare our scheme against bzip and PPM by themselves. We note but do not consider in this paper hybrid approaches, in which we pick the best compressor for a given interval. We can even nest the differential approach within the combi-national approach.

4Experiments

4.1Methodology.We summarize our assumptions as fol-lows.

1.Our data sets present combinational dependencies.

2.The combinational approach is likely to induce some

(near)optimum partition in which no class is wide.

3.Our data sets present differential dependencies.

4.The above dependencies can be detected with a small

amount of training data,independent of the size of.

We?x gzip as our underlying compression method. While this does not explore the range of possible base com-pressor options,it suf?ces to test our approach.As bench-mark R,we apply gzip to in row-major order,corre-sponding to the usage of gzip without off-line training;as benchmark C,we apply gzip to in column-major order, corresponding to the other extremal partition in which no combinational or differential dependencies exist.We thus designed experiments to compare the performance of

1.optimum partitioning to the benchmarks,

2.chunk partitioning to optimum partitioning,and

3.the greedy differential compression to both optimum

partitioning and benchmark C.

Each experiment has the potential to falsify one of our assumptions.If either benchmark outperforms optimum par-titioning(with respect to output size),then our data sets do not present combinational dependencies.If optimum parti-tioning signi?cantly outperforms chunk partitioning,then all (near)optimum partitions must have at least one wide class. If benchmark C outperforms the greedy differential compres-sion,then our data sets do not present differential dependen-cies.We discuss testing assumption(4)below.

For each experiment,we produced a compression plan by running the corresponding algorithm on a training data https://www.doczj.com/doc/0614885682.html,ing the resulting plan,we compressed a disjoint test data set,and we compared the compression performance (time and size)to that of the benchmark(s)for that goal. Although size of compressed output is the metric by which our assumptions can be falsi?ed,we also measured running times,to assess the practicality of our methods.This method-ology extends to assess other,similar compression systems.

We also varied the amount of training data available, to gauge the effect of training size on compression perfor-mance.This is only the?rst step in testing assumption(4). If we do not see compression performance stabilize at some point as we increase the amount of training data,then as-sumption(4)is likely falsi?ed.After observing this stabi-lization,however,a second test will be required:namely, to?x the training set size above the point at which we ob-served stability and increase the test set size arbitrarily.If the relative performance of the compression systems being compared does not remain stable,again assumption(4)is likely falsi?ed.Otherwise,we will have evidence supporting assumption(4).Although this second experiment remains to be conducted,based on our results we do not expect assump-tion(4)to be falsi?ed.

Finally,we compared the best results from the above experiments with isolated usages of compressors based on the Burrows-Wheeler transform[1]and prediction by par-tial match(PPM)[6,19].The goal was to assess empir-ically the bene?t of our training scheme,which leverages standard compression technology,versus other methods that claim improved compression via sophisticated analyses of the source data.

Data.We used100,000records from a network traf?c data warehouse.Each record is781bytes and pertains to an individual network event.The warehouse receives approximately one billion records per month,so effective compression is critical to this application.From the781 columns of bytes,we extracted the90with the highest frequency:i.e.,the number of times the value of the byte changed as the column was scanned top-down.We explain this in Section5;basically,in our real application,the other691columns were compressed more effectively using incomparable methods.

We divided the100,000(now90-byte)records into training and test sets.The training set was1/11th of the data;the test set was the rest.We chose the training set in two ways:(1)the?rst9091records,which we will call the ordered training set,and(2)a randomly chosen set of9091 records,which we will call the random training set.Each

way left the corresponding rest of the data as the test set. In all our experiments,we used the training sets to generate the corresponding compression plans,and we conducted the compression experiments using those plans and the test sets.

Software.To run the experiments,we implemented the following tools.

pin.Given a training set,pin computes a compression plan based on optimum partitioning.

pzip.Given a compression plan computed by pin,pzip compresses a test set with respect to the plan.It encodes enough of the plan into the output(which is included in the output size results reported)so that,given a com-pressed?le,pzip will uncompress it without needing the original plan.(Pin and pzip actually form our working system,and we discuss them in greater detail in Section5.)

colsel.Given a training set,colsel computes a com-pression plan based on the greedy differential algo-rithm.

cszip.Given a compression plan computed by colsel, cszip compresses a test set with respect to the plan.

It encodes enough of the plan into the output(which is included in the output size results reported)so that, given a compressed?le,cszip will uncompress it without needing the original plan.

Pzip and cszip use the zlib library,version1.1.3,1 to compress the intervals and columns,respectively.

System.All the training and experiments were run on one250MHz MIPS R10000processor on a16-processor SGI Origin2000running IRIX6.5,with10GB of main memory.Each time reported is the median of?ve runs, summing user and system time for each run.

4.2Experiments and Results.

Optimum Partitioning.To test assumption(1)and as-sess how optimum partitioning affects compression perfor-mance,we used pin to compute optimum partitions on pieces of the training data of increasing size.We ran pzip with each resulting compression plan on the test set and com-pared the compression time and resulting size to those of the benchmarks;we also compared uncompression times.We performed this experiment using both the ordered and ran-dom training sets.

Figures1and2displays the results,as a function of the amount of training material used.Training on the2%-size(w.r.t.the test set size)data set,at which we see the results stabilize,took about2.27CPU minutes.Because we anticipate using the same compression plan with multi-ple tables from a?xed source,though,training should be

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Figure 1:Results of optimum partition compression,as a function of amount of training material.Shown is the relative performance of optimum partitioning over benchmark R in terms of compressed size,compression time,and uncompression time.(a)Ordered training set;(b)random training set.in training time.

We offer a caveat:pzip has undergone signi?cantly more code optimization than cszip ,which partially ex-plains the relative difference in running times.We believe that we can improve the running time of cszip by combin-ing the column differencing and compression of the derived columns into a single pass.

4.3General Discussion.In all the experiments above,the difference between using ordered and random training sets was negligible,although the ordered sets did provide slightly better results,suggesting the need for future experiments to assess the effects of contiguity in the training data.

We did observe stabilization of compression perfor-mance in all the experiments.Perhaps most remarkable is that this stabilization occurred at training set sizes of 1–2%of the test set size.Again,further experiments with increas-ingly larger test sets and a ?xed training set size are required before assumption (4)can be assumed with con?dence.4.4Comparison to Other Methods.We compared the result of optimum partitioning (using the 2%training size)to Burrows-Wheeler [1]and PPM [6,19]compression in isolation.For Burrows-Wheeler,we used Seward’s bzip2,version 0.9.5d.2For PPM,we used Bloom’s ppmz ,version

Compress.Uncompress.Method gzip 6.340e-1 1.202e-0 1.129e-0col-major 7.768e-1

4.344e-1

2.165e-1

PPM

3https://www.doczj.com/doc/0614885682.html,/

bloom/src/ppmz.html

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Figure 2:Results of optimum partition compression,as a function of amount of training material.Shown is the relative performance of optimum partitioning over benchmark C in terms of compressed size,compression time,and uncompression time.(a)Ordered training set;(b)random training set.

PPM,the relative speed difference was orders of magnitude.

The results suggest that,for our table application,opti-mum partitioning using gzip as the underlying compression method outperforms isolated usage of bzip and PPM ,which by themselves purport to outperform gzip .Since bzip did out-compress gzip with only a slight time penalty,it is worth future experimentation to assess the performance of optimum partitioning using bzip as the underlying com-pression method.

5Partition Compression System and Applications

Pin and pzip actually form our production compression system.Recall that the experiments in Section 4used only the 90highest frequency columns from the original data set.Prior to determining an optimum partition,pin calculates

column frequencies.It actually computes the optimum par-tition only on the projection of the high frequency columns.(How it determines low from high is a heuristic outside the scope of this paper.)Furthermore,before computing the par-tition,it employs another heuristic to reorder the high fre-quency columns to improve compression size further.Again,this heuristic is outside the scope of this paper and was turned off for the experiments in Section 4.

Pzip then compresses the low frequency columns by differential encoding,additionally gzip ping the output of of that phase,and the high frequency columns with respect to the (reordered)partition.On the low frequency columns of the full network traf?c data set,this method outperformed the gzip benchmark (R)by two orders of magnitude in com-pression size and almost an order of magnitude in compres-sion time.To measure how the full system works on the original data set,we repeated the structure of the optimum partitioning experiment,allowing pin to detect the low fre-quency columns and pzip to use differential encoding on them.The system setup was as in Section 4,and again we ?xed gzip as the underlying compression method.We com-pared to benchmark R,corresponding to off-the-shelf use of gzip ,which was the method of choice in the AT&T net-work traf?c data warehouse prior to our work.The results are shown in Figure 5.For this experiment,we used only the ordered training set,as the random training set destroys the frequency information.

The results indicate overall improvements relative to straight gzip of 55%in compression size and 40–50%in both compression and uncompression time,supporting the argument that the space improvement by optimum partition-ing is worth the extra time.The time savings for the low frequency columns more than paid for the extra time needed to compress the high frequency columns via optimum parti-tioning.Again,training sets of 1–2%of the test set size suf-?ced to achieve these results.Applied to the network traf-?c warehouse,pin/pzip compresses the raw data for an entire month from the original 1TB to about 28GB,small enough to ?t on a large PC disk.By comparison,gzip com-pressed the data only to about 65GB.

Chunk size 0

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Figure 3:Results of chunk partition compression,as a function of chunk size.Shown is the relative performance of chunk partitioning over optimum partitioning in terms of compressed size,compression time,uncompression time,and training time.(a)Ordered training set;(b)random training set.5.1Additional Test Sets.

The AT&T network switch statistics project.In this project,statistics are collected at 15-minute intervals from ATM switches in a network.A record corresponds to a re-placeable circuit component in one of the switches and con-sists of a 16-byte hardware identi?er,a 4-byte statistic iden-ti?er,a 4-byte time stamp,and a 4-byte count value.Each 15-minute interval produces a ?le with about 80,000records for a 6-switch network.The items are sorted by the 16-byte hardware identi?er;sorted identi?ers usually differ by one byte from one item to the next.The ?le format is determined by the switch manufacturer and has an irregular structure:variable length headers and interspersed sequencing records make the records variable length in general.The average ?le size is about 2.2MB and gzip s to about 192kB,making the daily space requirement 18.4MB.

To use pzip ,each record was padded to a ?xed 32bytes.This expanded the average ?le size to about 2.6MB.Training data produced 10high frequency columns,for which an optimum (reordered)partition was generated.The resulting average pzip ped ?le size was 10.3kB,for a daily space requirement of 1MB,a 95%improvement over straight gzip .Compression and uncompression time improvement was only about 20%.

U.S.census data.We took a portion of the United States 1990Census of Population and Housing Summary Tape File 3A (a.k.a.STF3A )[2].The data format is ?xed length ASCII records.We used ?eld group 301,level 090,for all states.

Table 2:Summary of results.Size and times are ratios of pzip values to the corresponding gzip values.

Size time time .45.62.54Ntwk.switch

.56

.50

.33

This generated a 342MB ?le with 932-byte records.Gzip compressed the ?le to 31.5MB.

Pin determined that 186columns were high frequency.In the optimum partition generated,the largest class was 56bytes wide,indicating high combinational dependence.Pzip compressed the ?le to 17.5MB,a 44.4%improvement over gzip .The compression time improvement was 50%,and uncompression time improvement was 67%.

5.2Discussion.Table 2summarizes the results in this section.

Based on its performance on the network traf?c data,pzip has been put into production use in the AT&T network traf?c warehouse,using a compression plan generated by pin on about 100,000records.Although not in a controlled setting,this will provide an “in-production”experiment that

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Figure 4:Results of greedy differential compression,as a function of amount of ordered training material.(a)Shows the relative performance of greedy differential compression over optimum partitioning in terms of compressed size,compression time,uncompression time,and training time.(b)Shows the relative performance of greedy differential compression over benchmark C in terms of compressed size,compression time,and uncompression time.

can help assess assumption (4),because going forward,pzip will be compressing an arbitrary amount of data based on the ?xed-size training set.6Concluding Remarks

We have presented massive tables as a new focus in data compression research.We have given a systematic approach for solving the problem,based on the experimental vali-dation of data dependency assumptions.The result is a new compression paradigm:training for lossless compres-sion.By exploiting data dependencies,our scheme outper-forms standard methods based on information theoretic re-sults,e.g.,Lempel-Ziv [20,21].We tested two such depen-dencies.For our application,optimum partitioning is better,and it is in production use within AT&T,in the pzip sys-tem.We anticipate instances for which the differential ap-proach will outperform the combinational approach and also instances that favor a hybrid approach.We leave as an open problem to ?nd other data dependencies.

Our results demonstrate the utility of training for loss-less compression.Given multiple tables from a common

source,training becomes an off-line operation,suitable for computationally expensive optimizations.The bottleneck in our dynamic programming algorithm for optimum partition-ing is the computation of for all ,which requires running the base compressor (gzip ,in our case)on intervals of columns.A quick way to esti-mate the compressed size of an interval of columns,such as providing a suitable lower bound on their joint entropy—a fundamental problem of independent interest—would there-fore be valuable in speeding the overall algorithm.

Two aspects of our work that are now only heuristic are as follows.Permuting the columns before partitioning them effects greater compression.The problem of optimally per-muting the columns can be abstracted in combinatorial op-timization terms as versions of the Hamiltonian path prob-lem or clustering.We suspect that these formulations will prove to be hard,but proving their hardness is non-trivial.Any reduction must capture required costs by constructing columns whose compressed size using a particular program (such as gzip )will match required costs in the reduction.From a practical point of view,an ef?cient heuristic with good performance is desirable.Our second heuristic involves the choice of low frequency columns that are removed prior to training.In our data sets,simple rules of thumb suf?ced

to identify such columns,but a formal approach would be desirable.

In the differential approach,we focused on the case in which derived columns can be assigned only to source columns.In general,however,we can build a tree of derivations,which implies an interesting variation of the facility location problem that can be solved exactly by a reduction to minimum spanning trees.We also leave as open problems to explore the effect on compression plans

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Figure 5:Results of using pin/pzip with all heuristics and optimum partition compression,as a function of amount of ordered training material.Shown is the relative performance of pzip over benchmark R in terms of compressed size,compression time,and uncompression time.

of using other approximations to the metric facility location problem [3,11],and to explore hybrid approaches in which we apply optimum partitioning and differential compression to disjoint intervals of .

Our experimental methodology—assuming dependen-cies,deriving algorithms based on them,and testing to sup-port or falsify them—may be applied to other compression-based scenarios.It remains to conduct the second test of as-sumption (4)—that the amount of training material needed is independent of the size of the test set—by ?xing a compres-sion plan for an arbitrarily large amount of test material.The production use of pzip is providing an uncontrolled version of this experiment that supports the assumption.Finally,as-sessing the impact of data contiguity on training remains to be studied rigorously.

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