one of the last major outstanding classical problems of information theory is the development of general - purpose , practical , efficiently implementable lossy compression algorithms .the corresponding problem for lossless data compression was essentially settled in the late 1970s by the advance of the lempel - ziv ( lz ) family of algorithms and arithmetic coding ; see also the texts .similarly , from the early- to mid-1990s on , efficient channel coding strategies emerged that perform close to capacity , primarily using sparse graph codes , turbo codes , and local message - passing decoding algorithms ; see , e.g. , , the texts , and the references therein .  for lossy data compression , although there is a rich and varied literature on both theoretical results and practical compression schemes , near - optimal , efficiently implementable algorithms are yet to be discovered . from rate - distortion theory we know that it is possible to achieve a sometimes dramatic improvement in compression performance by allowing for a certain amount of distortion in the reconstructed data .but the majority of existing algorithms are either compression - suboptimal or they involve exhaustive searches of exponential complexity at the encoder , making them unsuitable for realistic practical implementation .  until the late 1990s ,most of the research effort was devoted to addressing the issue of universality , see and the references therein , as well as ; algorithms emphasizing more practical aspects have been proposed in .in addition to many application - specific families of compression standards ( e.g. , jpeg for images and mpeg for video ) , there is a general theory of algorithm design based on vector quantization ; see and the references therein .yet another line of research , closer in spirit to the present work , is on lossy extensions of the celebrated lempel - ziv schemes , based on approximate pattern matching ; see .more recently , there has been renewed interest in the compression - complexity trade - off , and in the development of low - complexity compressors that give near - optimal performance , at least for simple sources with known statistics .the lossy lz algorithm of is rate - distortion optimal and of polynomial complexity , although , in part due the penalty paid for universality , its convergence is slow . for the uniform bernoulli source , present codes based on sparse graphs , and , although their performance is promising , like earlier approaches they rely on exponential searches at the encoder . in related work , present sparse - graph compression schemes with much more attractive complexity characteristics , but suboptimal compression performance .rissanen and tabus describe a different method which , unlike most of the earlier approaches , is not based on a random ( or otherwise exponentially large ) codebook .it has linear complexity in the encoder and decoder and , although it appears to be rate - distortion suboptimal , it is an effective practical scheme for bernoulli sources .sparse - graph codes that are compression - optimal and of subexponential complexity are constructed in .a simulation - based iterative algorithm is presented in and it is shown to be compression - optimal , although its complexity is hard to evaluate precisely as it depends on the convergence of a markov chain monte carlo sampler .the more recent work on the lossy compression of discrete markov sources also contains promising results ; it is based on the combination of a viterbi - like optimization algorithm at the encoder followed by universal lossless compression .  the present work is partly motivated by the results reported in by gupta - verd - weissman ( gvw ) .their compression schemes are based on the `` divide - and - conquer '' approach , namely the idea that instead of encoding a long message using a classical random codebook of blocklength , it is preferable to break up into shorter sub - blocks of shorter length , say , and encode the sub - blocks separately .the main results in state that , with an appropriately chosen sub - block length , it is possible to achieve asymptotically optimal rate - distortion performance with near - linear implementation complexity ( in a sense made precise in section [ s : hyb ] below ) .our starting point is the observation that there is a closely related , in a sense dual , point of view . on a conceptual as well as mathematical level ,the divide - and - conquer approach is very closely related to a pattern - matching scheme with a restricted database . in the divide - and - conquer setting ,given a target distortion level and an , each sub - block of length in the original message is encoded using a random codebook consisting of codewords , where is the rate - distortion function of the source being compressed ( see the following section for more details and rigorous definitions ) .to encode each sub - block , the encoder searches all entries of the codebook , in order to find the one which has the smallest distortion with respect to that sub - block .now suppose that , instead of a random codebook , the encoder and decoder share a random database with length , generated from the same distribution as the shannon - optimal codebook . as in , the encoder searches for the longest prefix of the message that matches somewhere in the database with distortion or less .then the prefix is described to the decoder by describing the position and length of the match in the database , and the same process is repeated inductively starting at . although the match - length is random , we know that , asymptotically , it behaves like , therefore , because the length of the database was chosen to be , in effect both schemes will individually encode sub - blocks of approximately the same length , and will also have comparable implementation complexity at the encoder .thus motivated , after reviewing the gvw scheme in section [ s : gvw ] we introduce a ( non - universal ) version of the lossy lz scheme in , termed llz , and we compare its performance to that of gvw .theorem 1 shows that llz is asymptotically optimal in the rate - distortion sense for compressing data from a known discrete memoryless source with respect to a single - letter distortion criterion .simulation results are also presented , comparing the performance of llz and gvw on a simple bernoulli source .these results indicate that for blocklengths around 1000 bits , gvw offers better compression than llz at a given distortion level , but it requires significantly more memory for its execution .[ the same findings are also confirmed in the other simulation examples presented in section [ s : sim ] . ]  in order to combine the different advantages of the two schemes , in section [ s : hyb ] we introduce a hybrid algorithm ( hyb ) , which utilizes both the divide - and - conquer idea of gvw and the single - database structure of llz . in theorems 2 and3 we prove that hyb shares with gvw the exact same rate - distortion performance and implementation complexity , in that it operates in near - linear time at the encoder and linear time at the decoder .moreover , like llz , the hyb scheme requires much less memory , by an unbounded factor , depending on the choice of parameters in the design of the two algorithms .experimental results are presented in section [ s : sim ] , comparing the performance of gvw and hyb .these confirm the theoretical findings , and indicate that hyb outperforms existing schemes for the compression of some simple discrete sources with respect to the hamming distortion criterion .the earlier theoretical results stating that hyb s rate - distortion performance is the same as gvw s are confirmed empirically , and it is also shown that , again for blocklengths of approximately 1000 symbols , the hyb scheme requires much less memory , by a factor ranging between 15 and 240 .after a brief discussion on potential extensions of the present results , some conclusions are collected in section [ s : con ] . the appendix contains the proofs of the theorems in sections [ s : gvw ] and [ s : hyb ] .after describing the basic setting within which all later results will be developed , in section [ s : gvw2 ] we recall the divide - and - conquer idea of the gvw scheme , and in section [ s : gvw3 ] we present a new , non - universal lossy lz algorithm and examine its properties .let be a memoryless source on some finite alphabet and suppose that its distribution is described by a known probability mass function on .the objective is to compress with respect to a sequence of single - letter distortion criteria , where is an arbitrary source string to be compressed , is a potential reproduction string taking values in a finite reproduction alphabet , and is an arbitrary distortion measure .we make the customary assumption that for any source letter there is a reproduction letter with zero distortion ,   the best achievable rate at which data from the source can be compressed with distortion not exceeding is given by the _ rate - distortion function _ , r(d ) = _ w(y|x):_x , yap(x)w(y|x)(x , y)d i(x;y ) , [ eq : rddefn ] where denotes the mutual information between a random variable with the same distribution as the source and a random variable with conditional distribution given .let $ ] ; in order to avoid the trivial case where is identically equal to zero , is assumed to be strictly positive .it is well - known and easy to check that , for all distortion values in the nontrivial range , there is a conditional distribution that achieves the infimum in ( [ eq : rddefn ] ) , and this induces a distribution on via , for all . with a slight abuse of terminology ( as may not be unique )we refer to as _ the optimal reproduction distribution at distortion level . recall also the analogous definition of the _ distortion - rate function _ of the source ; cf. .   the gvw algorithm . ]is a fixed - rate , variable - distortion code of blocklength and target distortion .it is described in terms of two parameters ; a `` small '' , and an integer so that .  given the target distortion level , let , and take , = r^-1 ( r(d)+/2 ) = d ( r(d)+/2 ) d. [ eq : dbargvw ] first a fixed - rate code of blocklength and rate is created according to shannon s classical random codebook construction .letting denote the optimal reproduction distribution at level , the codebook consists of i.i.d .codewords of length , each generated i.i.d . from .writing , as the concatenation of sub - blocks , each sub - block is matched to its -nearest neighbor in the codebook , and it is described to the decoder using bits to describe the index of that nearest neighbor in the codebook .this code is used times , once on each of the sub - blocks , to produce corresponding reconstruction strings , for .the description of is the concatenation of the descriptions of the individual sub - blocks , and the reconstruction string itself is the concatenation of the corresponding reproduction blocks , .the overall description length of this code is bits , so the ( fixed ) rate of this code is bits / symbol , and its ( variable ) distortion is .the llz algorithm described here can be seen as a simplified ( in that it is non - universal ) and modified ( to facilitate the comparison below ) version of the algorithm in .it is a fixed - distortion , variable - rate code of blocklength , described in terms of three parameters ; an integer blocklength , and `` small '' .the algorithm will be presented in a setting `` dual '' to that of the gvw algorithm , in the sense that was described in the introduction .the main difference is that the source sting will be parsed into substrings of _ variable _ length , not of fixed length .  given and a target distortion level , define , take , and let denote the optimal reproduction distribution at level . then generate a single i.i.d .database of length ,m = m()=2^r+-1 , [ eq : m ] and make it available to both the encoder and decoder .the encoding algorithm is as follows : the encoder calculates the length of the longest match ( up to -many symbols ) of an initial portion of the message , within distortion , in the database .let denote the length of this longest match , and let denote the initial phrase of length in then the encoder describes to the decoder :  * the length ; this takes bits ; * the position in the database where the match occurs ; this takes bits .  from and the decoder can recover the string which is within distortion of .alternatively , can be described with _zero _ distortion by first describing its length as before , and then describing itself directly using , l_,1|| [ eq : desc2 ] the encoder uses whichever one of the two descriptions is shorter .[ note that is not necessary to add a flag to indicate which one was chosen ; the decoder can simply check if is larger or smaller than . ] therefore , from  , , and ( [ eq : desc2 ] ) the length of the description of is , ( ( 1+))+\ { m , l_,1|| } .[ eq : desc4 ] after has been described within distortion , the same process is repeated to encode the rest of the message : the encoder finds the length of the longest string starting at position in that matches within distortion into the database , and describes to the decoder by repeating the above steps .the algorithm is terminated , in the natural way , when the entire string has been exhausted .at that point , has been parsed into distinct phrases , each of length , with the possible exception of the last phrase , which may be shorter .since each substring is described within distortion , also the concatenation of all the reproduction strings , call it , will be within distortion of .the distortion achieved by this code is , and it is guaranteed to be by construction . regarding the rate , if we write for the overall description length of , then from ( [ eq : desc4 ] ) , ( x_1^n)= _ k=1^ _ [ eq : ell - bound ] andthe rate achieved by this code is bits / symbol.  * remark .* as mentioned in the introduction , there are two main differences between the gvw algorithm and the llz scheme .the first one is that while the gvw is based on a shannon - style random codebook , the llz uses an lz - type random database .the second is that gvw divides up the message into fixed - length sub - blocks of size , whereas llz parses into variable - length strings of ( random ) lengths .but there is also an important point of solidarity between the two algorithms .recall ( * ? ? ?* theorem 23 ) that , for large , the match length behaves logarithmically in the size of the database ; that is , with high probability , where the second approximation follows by the choice of and of therefore , both algorithms end up parsing the message into sub - blocks of length symbols .our first result shows that llz is asymptotically optimal in the usual sense established for fixed - database versions of lz - like schemes ; see .specifically , it is shown that by taking large enough and small enough , the llz comes arbitrarily close to any optimal rate - distortion point .note that is a parameter that simply controls the complexity of the best - match search , and its influence on the rate - distortion performance is asymptotically irrelevant .* theorem 1 .* suppose the llz with parameters and is used to compress a memoryless source with rate - distortion function at a target distortion rate .for any , the parameter can be chosen small enough such that : + ( a ) for any choice of and any blocklength the distortion achieved by llz is no greater than + ( b ) taking large enough , the asymptotic rate of llz achieves the rate - distortion bound , in that , _ _n e\{|x_1^n } r ( ) = r(d)-/2 , [ eq : as - optimality ] where the expectation is over all databases . therefore , also , _ _n e\ { } r ( ) = r(d)-/2 [ eq : optimality ] with the expectation here being over both the message and the databases .  next ,the performance of llz is compared with that of gvw on data generated from a bernoulli source with parameter and with respect to hamming distortion .simulation results at different target distortions are shown in figure [ f : bern04a ] and table [ tab:04a ] ; see section [ s : sim ] for details on the choice of parameter values .it is clear from these results that , at the same distortion level , the gvw algorithm typically gives a better rate than llz . in terms of implementation complexity , the two algorithms have comparable execution times , but the llz uses significantly less memory . the same pattern gvw giving better compression but using much more memory than llz is also confirmed in the other examples we consider in section [ s : sim ] .note that , like for the case of gvw , more can be said about the implementation complexity of llz and how it depends on the exact choice of parameters and .but since , as we will see next , the performance of both algorithms is dominated by that of a different algorithm ( hyb ) , we do not pursue this direction further .bits generated from a bernoulli source with parameter .the solid line is the rate - distortion function , the rate - distortion pairs achieved by llz are shown as red stars and those of gvw as blue diamonds ., width=345,height=326 ]   & + & & & rate & memory & time + gvw & 0.05 & 0.07143 & 0.70095 & 26 mb & 27m53s + gvw & 0.08 & 0.10286 & 0.59143 & 23 mb & 21m11s + gvw & 0.11 & 0.12667 & 0.50381 & 27 mb & 20m48s + gvw & 0.14 & 0.15714 & 0.41619 & 31 mb & 19m52s + gvw & 0.17 & 0.18857 & 0.32857 & 36 mb & 18m48s + gvw & 0.2 & 0.20571 & 0.26286 & 46 mb & 19m18s + gvw & 0.23 & 0.22857 & 0.21905 & 57 mb & 18m42s + gvw & 0.26 & 0.26381 & 0.15333 & 79 mb & 19m46s + gvw & 0.29 & 0.31429 & 0.10952 & 113 mb & 20m29s + llz & 0.05 & 0.03238 & 1.00029 & 1.5 mb & 4m23s + llz & 0.08 & 0.07524 & 0.79129 & 1.28 mb & 6m15s + llz & 0.11 & 0.10571 & 0.6754 & 1.46 mb & 8m53s + llz & 0.14 & 0.1381 & 0.55171 & 1.69 mb & 11m18s + llz & 0.17 & 0.16952 & 0.41827 & 2.6 mb & 18m15s + llz & 0.2 & 0.2019 & 0.36381 & 3.6 mb & 20m09s + llz & 0.23 & 0.23333 & 0.27975 & 6.2 mb & 41m32s + llz & 0.26 & 0.26571 & 0.23102 & 13 mb & 63m56s + llz & 0.29 & 0.29714 & 0.1741 & 47 mb & 165m54s +   order to combine the rate - distortion advantage of gvw with the memory advantage of llz , in this section we introduce a hybrid algorithm and examine its performance .the new algorithm , termed hyb , uses the divide - and - conquer approach of gvw , but based on a random database like the llz instead of a random codebook .it is a fixed - rate , variable - distortion code of blocklength and target distortion , and it is described in terms of two parameters ; a `` small '' , and an integer so that .like with the gvw , given a target distortion level , let and take as in ( [ eq : dbargvw ] ) .now , like for the llz algorithm , let as in ( [ eq : m ] ) , and generate a random database , where the are drawn i.i.d.from the optimal reproduction distribution at level . the database is made available to both the encoder and the decoder , and the message to be compressed is parsed into non - overlapping blocks , .the first sub - block is matched to its -nearest neighbor in the database , where we consider each possible , as a potential reproduction word .then is described to the decoder by describing the position of its matching reproduction block in the database using bits , and the same process is repeated on each of the sub - blocks , to produce reconstruction strings .the description of is the concatenation of the descriptions of the individual sub - blocks , and the reconstruction string itself is the concatenation of the corresponding reproduction blocks .the overall description length of this code is bits .the following result shows that the hyb algorithm shares the exact same rate - distortion performance , as well as the same implementation complexity characteristics , as the gvw .let :   * theorem 2 .* consider a memoryless source with rate - distortion function , which is to be compressed at target distortion level .there exists an such that , for any , the hyb algorithm with parameters and as in ( [ eq : elldefn1 ] ) achieves a rate of bits / symbol , its expected distortion is less than , and moreover :   encoding time per source symbol is proportional to ,   decoding time per symbol is independent of and ,  where and are independent of and .* remarks .theorem 2 is an exact analog of theorem 1 proved for gvw in , the only difference being that we consider average distortion instead of the probability - of - excess distortion criterion .the reason is that , instead of presenting an existence proof for an algorithm with certain desired properties , here we examine the performance of the hyb algorithm itself .indeed , the proof of theorem 2 can easily be modified to prove the stronger claim that there exists _ some _ instance of the random database such that , using that particular database , the hyb algorithm also has the additional property that the probability of excess distortion vanishes as .the same comments apply to theorem 3 below .\2 . in similar result is proved with the roles and and interchanged .in fact , it should be pointed out that the scheme we call `` the '' gvw algorithm here corresponds to the scheme used in the proof of ( * ? ? ? * theorem 1 ) .a slight variant ( having to do with the choice of parameter values and not with the mechanics of the algorithm itself ) is used to prove ( * ? ? ?* theorem 2 ) .having gone over the proofs , it would be obvious to the reader that , once the corresponding changes are made for hyb , an analogous result can be proved for hyb .the straightforward but tedious details are omitted .\3 . in terms of memory ,the gvw scheme requires reproduction symbols for storing the codebook , while using the same memory parameters the hyb algorithm needs symbols .the ratio between the two is , so that the gvw needs _ times _ more memory than hyb .moreover , the closer we require the algorithm to come to achieving an optimal point , the smaller the values of and need to be taken in theorem 2 , and the larger the corresponding value of ; cf .equation ( [ eq : elldefn1 ] ) .therefore , not only the difference , but even the ratio of the memory required by gvw compared to hyb , is unbounded .the next result shows that , choosing the parameters and in hyb appropriately , optimal compression performance can be achieved with linear decoding complexity and near - linear encoding complexity .it is a parallel result to ( * ? ? ?* theorem 3 ) .* theorem 3 .* for a memoryless source with rate - distortion function , a target distortion level , and an arbitrary increasing and unbounded function , the hyb algorithm with appropriately chosen parameters and , achieves a limiting rate equal to bits / symbol and limiting average distortion .the encoding and decoding complexities are and respectively .the actual empirical performance of hyb on simulated data is compared to that of gvw and llz in the following section .here the empirical performance of the hyb scheme is compared with that of gvw and llz , on three simulated data sets from simple memoryless sources .the following parameter values were used in all of the experiments . for the gvw and hyb algorithms , was chosen as in to be , where is the rate - distortion function of the source , and was taken equal to . similarly , for llz we took , and .note that , with this choice of parameters , the complexity of all three algorithms is approximately linear in the message length .all experiments were performed on a sony vaio laptop running ubuntu linux , under identical conditions .first we revisit the example of section [ s : gvw ] ; bits generated by a bernoulli source with parameter , are compressed by all three algorithms at various different distortion levels with respect to hamming distortion .figure [ f : bern04 ] shows the rate - distortion pairs achieved .bits generated from a bernoulli source with parameter .the solid convex curve is the rate - distortion function ; the rate - distortion pairs achieved by gvw are shown as blue diamonds ; by llz as red stars ; and by hyb as bold green dots . , width=403,height=364 ]  * rate - distortion performance .* it is evident that the compression performance obtained by gvw and hyb is near - identical , and better than that of llz .this example was also examined by rissanen and tabus in , where it was noted that it is quite hard for any implementable scheme to produce rate - distortion pairs below the straight line connecting the end - points of the rate - distortion curve corresponding to and .as noted in , the rissanen - tabus scheme produces results slightly below the straight line , and it is one of the best implementable schemes for this problem .   &+ & & & rate & memory & time + hyb & 0.05 & 0.06952 & 0.70095 & 0.79 mb & 2m45s + hyb & 0.08 & 0.11238 & 0.59143 & 0.6 mb & 3m06s + hyb & 0.11 & 0.12952 & 0.50381 & 0.59 mb & 3m33s + hyb & 0.14 & 0.15714 & 0.41619 & 0.56 mb & 4m06s + hyb & 0.17 & 0.19143 & 0.32857 & 0.52 mb & 4m40s + hyb & 0.2 & 0.22095 & 0.26286 & 0.53 mb & 5m21s + hyb & 0.23 & 0.23905 & 0.21905 & 0.51 mb & 5m26s + hyb & 0.26 & 0.27048 & 0.15333 & 0.53 mb & 6m27s + hyb & 0.29 & 0.29333 & 0.10952 & 0.53 mb & 6m56s +  * memory and complexity . * tables [ tab:04a ] and [ tab:04 ] contain a complete listing off all performance parameters obtained in the above experiment , including the execution time required for the encoder and the total amount of memory used . as already observed in section [ s : gvw ] , the llz scheme requires much less memory that gvw , and so does the hybrid algorithm hyb .in fact , while gvw and hyb produce essentially identical rate - distortion performance , the hyb algorithm requires between 32 and 213 _ times _ less memory than gvw . [ note that these figures are deterministic ; the memory requirement is fixed by the description of the algorithm and it is not subject to random variations produced by the simulated data . ] in terms of the corresponding execution times , the gvw and hyb share the exact same theoretical complexity in their implementation . nevertheless , because of the vastly different memory requirements , in practice we find that the execution times of hyb were approximately three to ten times faster than gvw .the second example is again on a bernoulli source with respect to hamming distortion , this time with source parameter .the corresponding simulation results are displayed in figure [ f : bern02 ] and table [ tab:02 ] .finally , in the third example is taken as a memoryless source uniformly distributed on , to be compressed with respect to hamming distortion .the empirical results are shown in figure [ f:4letter ] and table [ tab:4letter ] .  in both these cases ,the same qualitative conclusions are drawn .the rate - distortion performance of the gvw and hyb algorithms is essentially indistinguishable , while the compression achieved by llz is generally somewhat worse , though in several instances not significantly so . in the second example notethat the memory required by hyb is smaller than that of gvw by a factor that ranges between 44 and 242 , while in the third example the corresponding factors are between 16 and 218 . and again , although the theoretical implementation complexity of gvw and hyb is identical , because of their different memory requirements the encoding time of hyb is smaller than that of gvw by a factor ranging between approximately 3 and 9 in the second example , and between 1.25 and 1.5 in the third example .bits generated from a bernoulli source with parameter .the solid curve is the rate - distortion function ; the rate - distortion pairs achieved by gvw are shown as blue diamonds ; by llz as red stars ; and by hyb as bold green dots ., width=403,height=336 ]   & + & & & rate & memory & time + gvw & 0.04 & 0.05429 & 0.50381 & 25 mb & 19m05s + gvw & 0.055 & 0.07048 & 0.4381 & 28 mb & 18m13s + gvw & 0.07 & 0.08857 & 0.37238 & 35 mb & 19m50s + gvw & 0.085 & 0.10476 & 0.32857 & 42 mb & 20m14s + gvw & 0.1 & 0.12762 & 0.28476 & 49 mb & 19m55s + gvw & 0.115 & 0.12381 & 0.21905 & 59 mb & 20m03s + gvw & 0.13 & 0.12857 & 0.17524 & 73 mb & 19m57s + gvw & 0.145 & 0.14571 & 0.15333 & 90 mb & 19m08s + gvw & 0.16 & 0.16286 & 0.10952 & 126 mb & 19m38s + llz & 0.04 & 0.0381 & 0.64495 & 1.36 mb & 3m05s + llz & 0.055 & 0.05048 & 0.59165 & 2.02 mb & 7m45s + llz & 0.07 & 0.06857 & 0.54836 & 1.9 mb & 8m02s + llz & 0.085 & 0.08381 & 0.50616 & 2.4 mb & 13m38s + llz & 0.1 & 0.09714 & 0.42154 & 3.1 mb & 22m18s + llz & 0.115 & 0.11619 & 0.3083 & 5.2 mb & 24m03s + llz & 0.13 & 0.13048 & 0.26809 & 8.3 mb & 58m07s + llz & 0.145 & 0.14857 & 0.20223 & 21 mb & 132m30s + llz & 0.16 & 0.16571 & 0.1472 & 100 mb & 377m10s + hyb & 0.04 & 0.05429 & 0.50381 & 0.56 mb & 2m02s + hyb & 0.055 & 0.07048 & 0.4381 & 0.53 mb & 2m54s + hyb & 0.07 & 0.08952 & 0.37238 & 0.57 mb & 3m32s + hyb & 0.085 & 0.08286 & 0.32857 & 0.58 mb & 3m52s + hyb & 0.1 & 0.12 & 0.28476 & 0.57 mb & 4m46s + hyb & 0.115 & 0.12857 & 0.21905 & 0.56 mb & 5m21s + hyb & 0.13 & 0.13143 & 0.17524 & 0.55 mb & 5m45s + hyb & 0.145 & 0.14286 & 0.15333 & 0.52 mb & 6m30s + hyb & 0.16 & 0.17429 & 0.10952 & 0.52 mb & 7m11s +   symbols generated from the uniform source on .the solid curve is the rate - distortion function ; the rate - distortion pairs achieved by gvw are shown as blue diamonds ; by llz as red stars ; and by hyb as bold green dots ., width=403,height=336 ]   & + & & & rate & memory & time + gvw & 0.1 & 0.1419 & 1.41714 & 43 mb & 10m27s + gvw & 0.16 & 0.20095 & 1.16095 & 24 mb & 6m44s + gvw & 0.22 & 0.25238 & 0.92 & 31 mb & 8m19s + gvw & 0.28 & 0.31333 & 0.72286 & 44 mb & 11m12s + gvw & 0.34 & 0.36762 & 0.56952 & 43 mb & 9m45s + gvw & 0.4 & 0.42381 & 0.41619 & 65 mb & 12m29s + gvw & 0.46 & 0.47238 & 0.30667 & 92 mb & 13m59s + gvw & 0.52 & 0.53238 & 0.19714 & 124 mb & 14m12s + gvw & 0.58 & 0.58952 & 0.10952 & 229 mb & 17m30s + llz & 0.1 & 0.06857 & 1.97778 & 3.597 mb & 9m54s + llz & 0.16 & 0.1381 & 1.53794 & 1.79 mb & 7m46s + llz & 0.22 & 0.20381 & 1.25461 & 2.04 mb & 12m50s + llz & 0.28 & 0.26952 & 1.02841 & 2.61 mb & 18m51s + llz & 0.34 & 0.33524 & 0.76228 & 3.445 mb & 28m25s + llz & 0.4 & 0.4019 & 0.5393 & 3.49 mb & 30m37s + llz & 0.46 & 0.46381 & 0.3893 & 5.44 mb & 46m19s + llz & 0.52 & 0.52571 & 0.25807 & 14.6 mb & 105m56s + llz & 0.58 & 0.58571 & 0.17475 & 104 mb & 62m16s + hyb & 0.1 & 0.1419 & 1.41714 & 2.58 mb & 7m49s + hyb & 0.16 & 0.19714 & 1.16095 & 1.22 mb & 5m06s + hyb & 0.22 & 0.25429 & 0.92 & 1.26 mb & 6m37s + hyb & 0.28 & 0.30762 & 0.72286 & 1.39 mb & 8m48s + hyb & 0.34 & 0.37238 & 0.56952 & 1.05 mb & 7m42s + hyb & 0.4 & 0.42095 & 0.41619 & 1.18 mb & 9m39s + hyb & 0.46 & 0.47143 & 0.30667 & 1.15 mb & 10m34s + hyb & 0.52 & 0.52952 & 0.19714 & 1.01 mb & 10m14s + hyb & 0.58 & 0.58476 & 0.10952 & 1.05 mb & 11m43s +     starting point for this work was the observation that there is a certain duality relationship between the divide - and - conquer compression schemes of gupta - verd - weissman ( gvw ) in , and certain lossy lempel - ziv schemes based on a fixed - database as in . to explore this duality , llz ,a new ( non - universal ) lossy lz algorithm was introduced , and it was shown to be asymptotically rate - distortion optimal . to combine the low - complexity advantage of gvw with the low - memory requirement of llz , a hybrid algorithm , called hyb ,was then proposed , and its properties were explored both theoretically and empirically .  the main contribution of this short paper is the introduction of memory considerations in the usual compression - complexity trade - off .building on the success of the gvw algorithm , it was shown that the hyb scheme simultaneously achieves three goals : 1 .its rate - distortion performance can be made arbitrarily close to the fundamental rate - distortion limit ; 2 .the encoding complexity can be tuned in a rigorous manner so as to balance the trade - off of encoding complexity vs. compression redundancy ; and 3 .the memory required for the execution of the algorithm is much smaller than that required by gvw , a difference which may be made arbitrarily large depending on the choice of parameters .moreover , empirically , for blocklengths of the order of thousands , the hyb scheme appears to outperform existing schemes for the compression of simple memoryless sources with respect to hamming distortion .lastly , we briefly mention that the results presented in this paper can be extended in several directions .first we note that the finite - alphabet assumption was made exclusively for the sake of simplicity of exposition and to avoid cumbersome technicalities .while keeping the structure of all three algorithms exactly the same , this assumption can easily be relaxed , at the price of longer , more technical proofs , along the lines of arguments , e.g. , in .for example , theorem 4 of which gives precise performance and complexity bounds for the gvw used with general source and reproduction alphabets and with respect to an unbounded distortion measure , can easily be generalized to hyb .similarly , theorem 5 of which describes the performance of a universal version of gvw can also be generalized to the corresponding statement for a universal version of hyb ( with obvious modifications ) , although , as noted in , the utility of that result is purely of theoretical interest .recall that , under the present assumptions , the rate - distortion function is continuous , differentiable , convex and nonincreasing . given and , assume without loss of generality that ; then we can choose according to , so that .[ as it does not change the asymptotic analysis below , we take fixed and arbitrary . ]then the distortion part of the theorem is immediate by the construction of the algorithm .  before considering the rate , we record two useful asymptotic results for the match - lengths .let , and as in ( [ eq : m ] ) .then ( * ? ? ?* theorem 23 ) immediately implies that , moreover , for any , the following more precise asymptotic lower bound on holds : as , ( m())\{l_,1 | x_1^n } 0 [ eq : tails ] the proof of ( [ eq : tails ] ) is a straightforward simplification of the proof of ( * ? ? ?* corollary 3 ) , and therefore omitted .now let arbitrary .the encoder parses the message into distinct words , each of length .we let and following we assume , without loss of generality , that is an integer and that the last phrase is complete , i.e. ,   to bound above the rate obtained by llz , we consider phrases of different lengths separately .we call a phrase _ long _ if its length satisfies , and we call _ short _ otherwise . recalling ( [ eq : ell - bound ] ) , the total description length of the llz can be broken into two parts as , ( x_1^n ) & & _ k : z^(k ) + & & + _ k : z^(k ) .[ eq : total - bound ] for the first sum we note that , by the choice of and the definition of a short phrase , each summand is bounded above by a constant times , at least for all large enough ; therefore , the conditional expectation of the whole sum given is bounded by , e\ { c_1n _k=1^ _ i_\{l_m , kn } |x_1^n } c_2 m ( ) n \{l_m,1  function of an event , and the inequality follows by considering not just all s , but all the possible positions in where a short match can occur .dividing by and letting , from ( [ eq : tails ] ) we get that this expression converges to zero w.p .1 , so that the conditional expectation of the first term in ( [ eq : total - bound ] ) also converges to zero , w.p.1 .  for the second and dominant term in ( [ eq : total - bound ] ) , let be the number of long phrases . since each long has length , we must have , so that .[ eq : bad ] also , by the definition of , for all large enough ( independently of ) , we have , therefore , the second sum in ( [ eq : total - bound ] ) can be bounded above by , combining this with the fact that the first term in ( [ eq : total - bound ] ) vanishes , immediately yields _ _ n e\{|x_1^n } ( r()+)(1 + ) , and since was arbitrary we get the first claim in the theorem .finally , the second claim follows from the first and fatou s lemma .the proof of the theorem is based on lemma 1 below , which plays the same role as ( * ? ? ? * lemma1 ) in the proof of ( * ? ? ?* theorem 1 ) .the rest of of the proof is identical , except for the fact that we do not need to invoke the law of large numbers , since here we do not claim that the probability of excess distortion goes to zero .  before stating the lemma , we define the following auxiliary quantities : , , and ,   * lemma 1 .* consider a memoryless source to be compressed at target distortion level .then for any , the hyb algorithm with parameters and = , [ eq : elldefn1 ] when applied to a single block achieves rate , and its expected distortion is less than ._ given , choose a positive such that , now follow the proof of ( * ? ? ?* lemma 1 ) with in place of , until the beginning of the computation of the probability of excess distortion .the key observation is that , for hyb , this probability can be bounded above by the excess - distortion probability with respect to a random codebook with words , by just considering possible matches starting at positions , making the corresponding potentially matching blocks in the database independent .therefore , following the same computation , the required probability can be bounded above as before by , 2(2 ^ -c(d)^2)+2 ^ -/4 .[ eq : prob ] the first term is bounded above by , as before , and in order to show that the expected distortion is less than it suffices to show that the last term satisfies , ( -d)2^(r(d)+)</3 .[ eq : target ] substituting the choice of from ( [ eq : elldefn1 ] ) , it becomes , and since is restricted to be less than one , this can in turn be bounded above , uniformly in , by its value at .[ to see that , note that the function is increasing for and decreasing for . by our choice of ,the maximum above is achieved at the point . ] therefore , noting also that , this term is bounded above by , which , after some algebra , simplifies to , and this is less than by the choice of .this establishes ( [ eq : target ] ) and completes the proof of the lemma .   taking arbitrary , we let , as in the proof of ( * ? ? ?* theorem 3 ) , for each we use hyb with the corresponding parameters ; the rate result follows from the construction of the algorithm , which , at blocklength , has rate no larger than , as .regarding the distortion , equation ( [ eq : prob ] ) in the proof of theorem 2 shows that that the probability of the event that the distortion of the block will exceed is bounded above by , it is easily seen that , for large , this is dominated by the second term , therefore , the distortion of any one -block is bounded above by , noting that the excess term goes to zero as , it will still go to zero when averaged out over all sub - blocks , and , therefore , the expected distortion over the whole message will converge to .finally , the complexity results are straightforward by construction ; see the discussion in ( * ? ? ?* section ii - a ) .we thank sergio verd for sharing with us preprints of and .d. arnaud and w. szpankowski . pattern matching image compression with prediction loop : preliminary experimental results . in _ proc .data compression conf . dcc 97 _ , los alamitos , california , 1997 .ieee , ieee computer society press .d.j.c . mackay and r.m .good codes based on very sparse matrices . in _ cryptography and coding .5th i m a conference , number 1025 in lecture notes in computer science _ , pages 100111 .springer , 1995 .h. morita and k. kobayashi .an extension of lzw coding algorithm to source coding subject to a fidelity criterion . in _4th joint swedish - soviet int .workshop on inform .theory _ , pages 105109 , gotland , sweden , 1989 .coding theorems for a discrete source with a fidelity criterion . ,part 4:142163 , 1959 . reprinted in d.slepian ( ed . ) , _ key papers in the development of information theory _ , ieee press , 1974 .wainwright and e. maneva .lossy source encoding via message - passing and decimation over generalized codewords of ldgm codes . in _ proc . of the ieee international symposium on inform .theory _ , pages 14931497 , adelaide , australia , sept .

Abstract: the compression - complexity trade - off of lossy compression algorithms that are based on a random codebook or a random database is examined . motivated , in part , by recent results of gupta - verd - weissman ( gvw ) and their underlying connections with the pattern - matching scheme of kontoyiannis lossy lempel - ziv algorithm , we introduce a non - universal version of the lossy lempel - ziv method ( termed llz ) . the optimality of llz for memoryless sources is established , and its performance is compared to that of the gvw divide - and - conquer approach . experimental results indicate that the gvw approach often yields better compression than llz , but at the price of much higher memory requirements . to combine the advantages of both , we introduce a hybrid algorithm ( hyb ) that utilizes both the divide - and - conquer idea of gvw and the single - database structure of llz . it is proved that hyb shares with gvw the exact same rate - distortion performance and implementation complexity , while , like llz , requiring less memory , by a factor which may become unbounded , depending on the choice or the relevant design parameters . experimental results are also presented , illustrating the performance of all three methods on data generated by simple discrete memoryless sources . in particular , the hyb algorithm is shown to outperform existing schemes for the compression of some simple discrete sources with respect to the hamming distortion criterion .   lossy data compression , rate - distortion theory , pattern matching , lempel - ziv , random codebook , fixed database


a cellular automaton is a complex system defined by a local rule which acts synchronously and uniformly on the configuration space .these simple models exhibit a wide variety of dynamical behaviors and even in the one - dimensional case ( the focus of this article ) they are not well understood .formally , given a finite alphabet , a configuration is an element of the set which is compact for the product topology and a cellular automata is defined by a local function }\to { \mathcal a} ] .then as a consequence , any limit measure of is a convex combination of and .the cases correspond to deterministic automata and can be treated easily .the visual intuition suggests to consider interface defects according to the decomposition , where , ( monochromatic subshifts ) and ( checkerboard subshift ) , since those sfts are invariant under the action of both rules .the set of particles would be .however , as figure [ fig : trafmaj ] shows , the particle can `` explode '' and give birth to two particles and , contradicting the condition of coalescence . to solve this problem, we tweak the particle system by replacing each particle by one particle and one particle .cl &  ( 1.4 , -.8 ) node ; ( .8,-1 ) ( 0,-1.65 ) ; ( -.4,-2.2 ) node ; ( -.4,-2.1 ) ( -.15,-1.95 ) ; ( .2,-2.2 ) node ; ( .2,-2.1 ) ( -0.05,-1.95 ) ; ( 2.2,0.2 ) node ; ( 2.2,0 ) ( 2.65,-.5 ) ; ( 3.3,.8 ) node ; ( 3.5,.6 ) ( 4.35 , .15 ) ; ( 5.3,-2.1 ) node ;  the corresponding morphism is defined on the neighbourhood by the local rule : where the wildcards can take both values .  in the absence of interactions , the update function can be defined in the following manner .regardless of the rule that is applied , and move at a constant speed , and respectively . a particle move at speed if rule is applied at its position and at speed otherwise ( independent random walk with bias ) , except if a particle prevents its movement to the right , in which case it does not move. the particle behaves symmetrically .furthermore all interactions are of the form or ( when , by the last remark ) .a formal definition of the update function would be tedious , but it is entirely described by these remarks .the various conditions of locality , disjunction , particle control , surjectivity and coalescence are proved similarly to the previous examples .assume .we show that no particle can remain asymptotically by applying the main result on the sets : , , , and .we need only to show the clashes relative to the second and fourth sets since all other clashes are consequences of the speed of these particles .let and be such that and .since progresses at speed 1 , the distance can not increase , and it decreases by at least one with probability ( respectively ) .it is clear that the particles end up in interaction -almost surely . showing that and clash with is symmetric .let be such that and .as long as there are no interactions , the distance performs an independent random walk of bias , where a increasing step is sometimes replaced by a constant step .such a random walk reaches -almost surely , which shows that the particles end up in interaction .the clashes between and , and between and , are proved in a similar manner .the same proof holds for by exchanging the roles of and .  applying theorem [ prop : mainresultprob ], we conclude that only one particle can remain in the -limit set . however , if we consider , we notice that configurations in are of the form , where -\infty , k]} ] .we consider its action on any initial -ergodic measure satisfying )=\alpha ] , the sequence converges toward the -invariant measure supported by the -periodic orbit which correspond to the discrete line of slope .   where is chosen to the markov measure which maximizes the entropy of the subshift of finite type where the forbidden patterns are .,width=340 ]for some cellular automata with simple defect dynamics , the previous results can be refined with a quantitative approach : that is , to determine the asymptotic distribution of random variables related to the particles . in , p. kurka , e. formenti and a. dennunzio considered , the entry time after time on the initial configuration , which is the waiting time before a particle appears in a given position after time .they restricted their study to a gliders automaton , which is a cellular automaton on 3 states : a background state and two particles evolving at speeds 0 and -1 that annihilate on contact .thus , we have one entry time for each type of particle ( and ) .when the initial configuration is drawn according to the bernoulli measure of parameters , which means that each cell contains , independently , a particle of each type with probability , they proved that : they also called to develop formal tools in order to be able to handle more complex automata , starting with the symmetric case .  in section [ sec : entry ] , we extend this result to allow arbitrary values for the particle speeds and , and relax the conditions on the initial measure to some -mixing conditions .then , when and , we have : and symmetrically if we exchange and .the proof relies on the fact that the behavior of gliders automata can be characterised by some random walk process ; this idea was introduced by v. belitsky and p. ferrari in and was already used in and . in our case , a particle appearing in a position corresponds to a minimum between two concurrent random walks .the new tool here is that under -mixing conditions , we rescale this process and approximate it with a brownian motion . thus we obtain the explicit asymptotic distribution of entry times .this method , consisting in associating a random walk to each gliders automata and studying this random walk using scale invariance , is not limited to this particular conjecture concerning entry times .indeed , we see in the next two sections that it can be used to study the asymptotic behavior of two other , arguably more natural , parameters : the particle density at time and the rate of convergence to the limit measure .however , we obtain only an upper bound instead of an explicit asymptotic distribution .there is no doubt this method can be adapted to other parameters in a similar way .furthermore , these results can be extended to other automata with similar behavior , such as those in figure [ fig : particles ] , by factorising them onto a gliders automaton .this point is discussed in section [ sec : extensions ] .this method is more difficult to generalize when there is birth of particle , even in simple case as the -cyclic cellular automaton .our results apply on automata with simple defects dynamics , namely , automata admitting a particle system with and whose update function corresponds to a gliders automaton .we first prove our results for gliders automata before generalising them in section [ sec : extensions ] .let us introduce some tools that turn the study of the dynamics of a gliders automaton into the study of some random walk .[ lem : maralea ] let be the -gliders automaton . for all and , }{\mbox{argmin } } } s_{g(x ) } \longleftrightarrow j - v_+ = { \underset{[j - v_+,\ j+n - v_-]}{\mbox{argmin } } } { s_x},\\  j = { \underset{[j - n,\ j]}{\mbox{argmin } } } s_{g(x ) } \longleftrightarrow j - v_- = { \underset{[j - n - v_+,\ j - v_-]}{\mbox{argmin } } } s_x.\end{aligned}\ ] ]   base case .: :  }{\min } s_x.\end{aligned}\ ] ] induction .: :  assume both equivalences hold for some .+  suppose  }{\mbox{argmin } } } s_{g(x)} ] , and  by induction hypothesis  }{\mbox{argmin } } } { s_x} ]  and we conclude ;  * otherwise , this means that   ( the walk can decrease by at most one at each step ) , and thus  }{\mbox{argmin } } } { s_x}.\ ] ]  by induction hypothesis ,  }{\mbox{argmin } } } { s_{g(x)}},\ ] ]  and in particular .  therefore , a  contradiction with the first assumption .+  the converse is proved in a similar manner .[ lem : min ] let be the -gliders automaton . for all and }{\mbox{argmin } } }s_x\\  g^t(x)_j= + 1&\longleftrightarrow j - v_+t = { \underset{[j - v_+t,\ j - v_-t+1]}{\mbox{argmin } } } s_x.\end{aligned}\ ] ] this is illustrated in figure [ fig : lemmin ] .   //in-2/0/0 , -1/0/0 , 0/0/1 , 1/1/1 , 2/1/1 , 3/1/1 , 4/1/1 , 5/1/2 , 6/2/2 , 7/2/2 , 8/2/2 , 9/2/2 , 10/2/2 , 11/2/1 , 12/1/1 , 13/1/1 , 14/1/1 , 15/1/2 , 16/2/2 , 17/2/2 , 18/2/3 , 19/3/3 , 20/3/3 , 21/3/2 , 22/2/2 , 23/2/2 , 24/2/2 , 25/2/2 , 26/2/2 , 27/2/2 , 28/2/1 , 29/1/1 , 30/1/1 , 31/1/1 , 32/1/1 , 33/1/0 , 34/0/0 , 35/0/0 , 36/0/0 ( -3.25+.21*,-1.55+.4 * ) ( -3.25+.21*+.23,-1.55+.4 * ) ; ( -3.7,-0.75 ) node ; ( -3.35,-1.8 ) node ; ( 3.35,-1.8 ) node ; ( -4 , 0 ) ( 5,0 ) ; ( -4.2,0 ) node ; ( -3.5 , 0 ) ( -3.5,3.25 ) ; ( -3.7,1.625 ) node ; ( 0,-0.1 ) ( 0,0.1 ) ; ( 0 , -0.3 ) node ; ( -0.13,3.2 ) rectangle ( 0.07 , 3.4 ) ; ( -0.12,3.21 ) ( -3.2,0.13 ) ( -3.2 , -1.55 ) ; ( 0.06,3.21 ) ( 3.13,0.13 ) ( 3.13,-1.55 ) ; ( -3.21 , -1.55 ) ( 3.14,-1.55 ) ; ( -0.5,3.7 ) node ;    base case : :  by definition of ,   .induction : :  assume that both equivalences hold for a given itme .  by applying the induction hypothesis on ,  }{\mbox{argmin } } } s_{g(x)} ] or the opposite . when ) = \mu([-1]) ] after time at position 0 .we define in a symmetrical manner .the size of the considered window is such that any particle `` passing through '' the column 0 appears in this window exactly once ( see figure [ fig : entry ] ) .of course entry times for particles of speed 0 make no sense . from now on ,we only consider for simplicity , all the results being valid for .( 0,4.2 ) ; ( 0.42,0 ) ( 0.42,4.2 ) ; ( 0,1 ) rectangle ( 0.42,1.14 ) ; ( 0,1 ) ( 0.42,1.14 ) ( 0.42,1 ) ( 0,1.14 ) ; ( 0.02,1.16 ) rectangle ( 0.40,3.09 ) ; ( -0.2,1.14 ) ( -0.2,3.1 ) ; ( -0.7 , 2.12 ) node ; ( -0.2,1 ) ( -0.2,0 ) ; ( -0.5 , 0.5 ) node ; ( -4,-0.01 ) ( 5,-0.01 ) ( -4,0.13 ) ( 5,0.13 ) ; ( -4.25,0.2 ) node ;      in the same article , they conjectured that this result could be extended to any initial bernoulli measure of parameters for by replacing the right - hand term by .we will prove that this conjecture is actually incorrect .to state our result , we introduce two particular subclasses of .we recall the definition of the -mixing coefficients of a measure : -\infty,0 ] } , b\in \mathcal { \mathfrak{b}}_{[n,+\infty[}\}.\ ] ] define :  * the set of bernoulli measures on and parameters for some ; * the set of measures satisfying : * * ; * * converges absolutely to a real ( asymptotic variance ) ; * * .       the third hypothesis for is chosen so that the large - scale behavior of the partial sums can be approximated by a brownian motion .this invariance principle is the core of our proofs .the first and second conditions ensures that the brownian motion obtained this way have no bias and nonzero variance , respectively .we now state some invariance principles , which consists in appoximating rescaled random walks by brownian motion .we use a strong version , which guarantees an almost sure convergence by considering a copy of the process in a richer probability space .          1 . has distribution ; 2 .every is a brownian motion of mean 0 and variance ; 3 .  for any , denoting the piecewise linear function defined by and for all , }\left|\frac{s_{x'}(nt)}{\sqrt{n } } - b_n(t)\right| = o\left(n^{-\frac 14 + \varepsilon}\right)\quad\mathbb p\mbox{-almost surely}.\ ] ]  we apply theorem [ thm : invariance ] on , where has distribution . because is -invariant , this is a stationary process .the first and third conditions are satisfied by definition of . for the second condition , by stationarity .we obtain two processes and on a richer probability space such that has the same distribution as , is a brownian motion of mean 0 , variance , and : since the variables take value in , we have for any  ( a staircase and piecewise linear function having the same values on ) . therefore :  for any , taking the sup for  ] and  ] . furthermore , since and are independent , so are } b^l_x ] . denote the law of the minimum of a brownian motion on  ] -almost surely , and where the third line is obtained by symmetry of the brownian motion .      in this section, we estimate the rate of convergence to the limit measure . for that we fix a distance on the space of -invariant measures , which induces the weak topology : )-\nu([u])\right|.\ ] ]      * lower bound when : * \right) ] for and . by lemma [ lem : min ] , }s_x.\ ] ] therefore : \right)&\leq \mu\left(\min_{[0,\ell ] } s_x < \min_{[\ell+1,t]}s_x\right)\\  & \leq \mu\left(\min_{[0,t]}s_x \geq -\ell\right)\\  & \leq \mu\left(\min_{[0,1]}s^t_x \geq -\frac \ell{\sqrt t}\right)\end{aligned}\ ] ] by corollary [ thm : brown ] , using the same notations as in the previous proofs : }s^t_{x ' } \geq -\frac \ell{\sqrt t}\right)\\  & \leq \mathbb p\left(\min_{[0,1]}b_t \geq -\frac \ell{\sqrt t } - c_t\right ) \quad\quad\textrm { where } c_t = o\left(t^{-\frac 14+\varepsilon}\right)\\  & = o\left(t^{-\frac 14+\varepsilon}\right)\end{aligned}\ ] ] for any , following the same calculations as in section [ sec : density ] . the case of particles is symmetrical , and we conclude .       in this section ,we extend the theorems [ thm : entry ] and [ thm : density ] to automata that factorise onto a gliders automaton , and discuss conditions for the extension of theorem [ thm : rate ] . in section [ sec : defects ] , we exhibited a general method to find such a factor using experimental intuition when such a factor is not obvious .   in order to extend the theorem to such cas , starting from an initial measure , we must first ensure that .we show that the third condition in the definition of is invariant under morphism .we keep the notations from the definition of . is defined by a local rule with neighbourhood  ] exponentially decreasing in   .  even though we showed that the asymptotic distributions of entry times are known for some class of cellular automata and a large class of measures , this covers only very specific dynamics .it is not known how these results extend for more than 2 particles and/or other kind of particle interaction .in particular , there is no obvious stochastic process characterising the behavior of such automata that would play the role of in our proofs .nino boccara , jamil nasser , and michel roger .particlelike structures and their interactions in spatiotemporal patterns generated by one - dimensional deterministic cellular - automaton rules . , 44(2):866875 , 1991 .benjamin hellouin de menibus and mathieu sablik .entry times in automata with simple defect dynamics . in enrico formenti ,proceedings 18th international workshop on cellular automata and discrete complex systems and 3rd international symposium journes automates cellulaires _ , volume 90 of _ electronic proceedings in theoretical computer science _ , pages 97109 .open publishing association , 2012 .benjamin hellouin de menibus and mathieu sablik .characterisation of sets of limit measures after iteration of a cellular automaton on an initial measure .arxiv:1301.1998 , accepted to _ergodic theory and dynamical systems _ , 2013 .damien regnault and eric rmila .lost in self - stabilization . in _ mathematical foundations of computer science 2015 - 40th international symposium , mfcs 2015 , milan , italy , august 24 - 28 , 2015 , proceedings , part i _ , pages 432443 , 2015 .

Abstract: this article introduces new tools to study self - organisation in a family of simple cellular automata which contain some particle - like objects with good collision properties ( coalescence ) in their time evolution . we draw an initial configuration at random according to some initial -ergodic measure , and use the limit measure to descrbe the asymptotic behaviour of the automata .   we first take a qualitative approach , i.e. we obtain information on the limit measure(s ) . we prove that only particles moving in one particular direction can persist asymptotically . this provides some previously unknown information on the limit measures of various deterministic and probabilistic cellular automata : and -cyclic cellular automata ( introduced in  ) , one - sided captive cellular automata ( introduced in  ) , n. fats candidate to solve the density classification problem  , self stabilization process toward a discrete line    in a second time we restrict our study to to a subclass , the gliders cellular automata . for this class we show quantitative results , consisting in the asymptotic law of some parameters : the entry times ( generalising  ) , the density of particles and the rate of convergence to the limit measure .


the bering mission is an autonomous mission , with the purpose of making sample observations of the inner asteroid populations . in particular , bering will travel through most of the space between venus and the outer parts of the asteroid belt at 3.5 au , and will thus be able to observe members of the near - earth asteroid ( nea ) population , objects in the asteroid main belt , and objects _ en route _ from the main belt toward the neas . the instruments on board the spacecrafts are the advanced stellar compass ( asc ) that allows bering to detect and follow moving objects , with the purpose of orbit determination of the objects .in addition , the asc can control a small telescope , so that observations of the objects can be obtained . in this way , bering is able to provide both an orbit as well as a physical characterisation of the objects .the key point is the autonomy of the asc and the telescope , that enables bering to systematically detect and follow objects down to object diameters at the meter level .the scientific objectives have been discussed in detail elsewhere .one of the key issues to address is the two - fold question of how many objects bering will be able to observe .first , this question is important to answer as a basis for the scientific objectives , and with the need to describe the scientific feasibility .second , the number of objects detected will depend on the limiting magnitude of the asc , and is thus vital for the technical requirements to the mission , also in terms of the number of objects that needs to be processed by the autonomous spacecraft platform .it is however not trivial to answer this question , hence we have a constructed a simulator capable of examining the detailed aspects of the detections .there already exists a number of simulations of the detection of neas , however these simulations differs from our needs in several ways .first of all , they are made for ground based surveys for nea discovery and follow - up . for bering ,we need to be able to simulate observations made by a spacecraft in an interplanetary orbit . also , the simulations are normally strongly restricted in the size of the objects included . for the bering mission, we need to have detailed knowledge of for how long time an object can be observed , as well as the angular velocity .for instance , the trailing losses experienced by ground based surveys due to fast moving objects across the field of view during the exposure , are addressed by the bering asc capabilities to handle fast moving objects , however we need to quantify the requirements to the asc .thus , we in general need to understand how bering will perform when inserted into a given asteroid population , and with the possibility of adjusting parameters , like the limiting detection magnitude and the orbit . we shall here focus on synthetic populations of the neas and the main belt asteroids .the synthetic objects are treated as massless test particles in the simulation , and after an initial sorting , the objects are numerically integrated .this allows a careful examination of the requirements to the asc when probing members of these populations down to the meter - range .as discussed , one of the challenges when observing asteroids are their rapid and drastic variations in magnitude .the magnitude of an asteroid does not only depend on physical parameters like the size and the albedo , but also on the distances to the sun and the observer , and the phase angle .depending on the geometry , there can thus be a strong time dependency on the magnitude variations .the magnitudes of asteroids are normally described by a two - parameter model .the absolute magnitude of an asteroid is defined as the v - magnitude at unit distance to the sun ( ) and the earth ( ) , and at phase angle .the second parameter is the slope parameter , which is an expression for the geometric albedo .thus the relation between the observed v - magnitude and and is  \end{split}\ ] ] where and are functions of the phase angle . and are measured in au .  only for a few objects has the slope parameter and the albedo been measured .the desire however persists to translate the absolute magnitude into a diameter of the object .this can be done by an _ a priori _ assumption on the albedo of the object .the diameter can then be expressed as  where is in units of km .care is needed when evaluating the sizes of the smallest sub - kilometer asteroids , as the actual albedo of these objects is currently not known .during recent years , many publications have been made in the attempt of establishing the size distribution of near - earth asteroids .bottke and collaborators have constructed a de - biased estimate of the nea size distribution , for the range up to absolute magnitude , by simulating a ground based nea survey , and taking into account the nature of the source regions of the neas . for this range, they find a best fit of the form , where is the accumulated number of objects and and are constants .werner provides an estimate for the distribution up to ( m ) , derived from the lunar crater size - frequency distribution .their distribution corresponds well to the estimates obtained from ground based neo surveys .werner obtains a distribution similar to that of bottke , however at around the distribution shows a turning point , with a much steeper slope . for the even smaller end ,brown and collaborators reports satellite observations of bolide detonations in the earth s atmosphere , for objects in the range of 1 m to 100 m .the distribution derived from these observations is consistent with the work of werner .it should be stressed that the estimated populations are based on indirect methods , and that various factors may influence the actual , present size distribution .one of the factors is the translation from to , as the albedo of the nea population is more or less unknown .there are indications , for instance in the work of werner , that the smallest neas have larger albedo .this is compatible with our hypothesis that the smallest asteroids are young collisional fragments without surface dust , and with minimum exposure to cosmic radiation , hence the objects have larger albedo .the distribution we have adopted for the nea population is a combination of the estimation by bottke , combined with the work by werner , supported by the measurements by brown .we shall thus split the distribution into three parts , with bins of 1 magnitude , as   \label{plateau } \\\log n & = 0.54 h -8.5 & ; \ \ & 24 < h \leq 32.7 \label{werner}\end{aligned}\ ] ]  [ neapopulation ]  the distribution can be seen in fig .[ fig : distrplot ] .  the first part is thus adopted from bottke , followed by a plateau .the last part of the distribution is taken from the low - albedo distribution of werner .the has been chosen , so that with an albedo of , this corresponds to a diameter of m .the total number of objects in the distribution amounts to , or approximately objects . for the slope parameter we initially assumed , a low - albedo population ,but later we made tests with a high - albedo population using , corresponding to ( see the discussion of the results below ) .  besides the distribution in terms of , the de - biased distribution by bottke also suggests a spatial distribution of the objects , in terms of the eccentricity , the semi - major axis and the inclination .however , for the above synthetic population , we have selected a uniform distribution of the orbital elements ( within the definitions of a near - earth object ) . in principle , this is a failure of the distribution , relative to the current best estimates . however , the suggested distribution of the orbital elements is only valid for objects with .there is only vanishing statistical material for smaller objects ( larger ) , hence there is no basis to assume a continuation of the distribution in the orbital elements for . in any case , the number of objects with amounts to approx .18600 , a vanishing number compared to the total number of objects .the main belt population is very much different from that of the neas .first of all , our observational knowledge is strongly restricted to only the very largest objects , due to the far distances to the objects .durda and collaborators have made numerical models of the collisional evolution of the main belt asteroids . the observed distribution shows two bumps relative to a linear fit , for objects with km and km . by adjusting the parameters of their models , durda is able to reproduce these bumps , and thus to obtain an estimate of the main belt distribution down to 1 m in size .other estimates for the distribution exists ( see for an overview ) , and they tend to agree for objects down to 1 km .  for these simulations we shall adopt the distribution by durda , with some modifications .the focus of our simulations are on the sub - kilometer objects , hence for the range  ] in steps of 0.5 magnitudes . using the albedo referred to by durda of , the corresponds to an object size of m .however , for the slope parameter we use  , a slightly larger albedo than used by durda , so that the smallest object in our distribution in fact has m .this distribution has two flaws .first of all , it does not take the mentioned known bumps of the large objects into account .however , the number of objects among the large sizes neglected in this manner is completely negligible in comparison with the total number of objects in this synthetic population .the large objects are as well not the main objective of this work , but should under all circumstances be treated separately , taking into account the already known objects . the works by durda and by davis seems to indicate an _ increase _ in the slope of the size distribution for m .whether this holds true or not is one of the scientific objectives of bering .however , it means that relative to these estimates , our synthetic main belt population is heavily underestimating the number of 10 m objects , with a factor of 10 or more .this must be taken into account , when reviewing the results of the simulations .the total number of objects in the main belt population is thus , or around objects .the orbital elements are assumed to be uniformly distributed .with the large number of objects involved in estimating the detection rate , a direct numerical integration is not feasible due to the very heavy demands on cpu time and disk space .instead , we adopted a solution that provide a rough sorting of the objects , discarding those that would not become observable from bering within a reasonable time .a simple idea is to calculate the minimum orbital intersection distance ( moid ) between the bering orbit and every object .if the moid distance is sufficiently small , so that in a best case situation the object is observable within the magnitude limits , the object is selected for subsequent detailed analysis , otherwise it is discarded . instead of a purely numerical approach for determining the moid , for instance by integrating the orbits and calculating the smallest distance between bering and the objects , we use a semi - analytic approach , implementing the moid method suggested by sitarsky . for two keplarian orbits ,the distance function can be written as where are the true anomalies of the objects of concern , and are the cartesian coordinates relative to some origin . by examining the derivatives of this function ,sitarsky derives analytical equations for the minimum of the function , the moid .these equations can , for the general case , not be solved analytically , hence we apply a version of the newton - raphson method that is suited for non - linear cases ( press et al . ) , and in particular ensures convergence toward some minimum .this problem is however not trivial .often , the solution must be sought in very rough terrain , fig .[ fig : exdist ] ,  hence one set of initial conditions may not be enough to ensure that the global minimum is found .thus we solve the equations for several sets of initial conditions , and from the resulting local minima , we accept the smallest value as the moid .we verified the implementation of the moid method against a number of different orbit scenarios , and compared with the minimum distance between the two orbits , obtained by direct numerical integration .after the moid has been calculated , we apply a selection criteria for the sorting .we calculate the -magnitude of the asteroid by using the moid as the distance to the observer .the perihelion distance , the smallest distance to the sun , is used as the asteroid - sun distance .further , the phase angle is set to zero .this is truly a best case configuration of the asteroid .if the calculated magnitude is , where is the bering detection limit , the object is saved , otherwise it is discarded .this moid - sorting is done for all the objects .it is computationally heavy , but allows focus on only the most interesting objects . as the sorting can be quite aggressive , depending on , it can drastically decrease the demands for computing time and disk space for a detailed analysis of the bering detection rate .the objects that were selected by the moid sorting for further analysis , are endured to a numerical integration . for each time step ,the v - magnitude of the objects is calculated , and compared against , disregarding the direction to the object , thus assuming that bering is able to monitor the whole sky .this is a good approximation , as the only directions bering will not cover are toward the sun , in which direction only the night side of the objects will be seen , thus being too faint under all circumstances . during the integration ,book keeping is made of which objects had , the smallest of the objects and the corresponding tangential velocity during the minimum . by running a small filter after the integration, a complete list of the observable objects could be generated from the book keeping data .as integrator , we used the swift package .the package was modified to allow handling of the magnitude calculations and of the book keeping data . as the primary integrator we used the symplectic rmvs3 method , however for selected objects we made a refined check with the classic bulirsch - stoer method , without any noticeable difference in the results of the two integrators .  due to the very transient variations in brightness of the smallest objects ,the integrations were typically made over 50 days , with a data dump and magnitude calculation every 0.02 days ( 28.8 minutes ) .the moid implementation and the numerical integrations were parallelized at the run - level , and ran on the ibm rs-6000 cluster of the center for scientific computing aarhus ( csc - aa ) .the first key problem to deal with was to assess the value of .this value is critical for the mission feasibility in terms of the the detection rate of the bering advanced stellar compass , i.e. the number of objects that would be observed per unit of time .if is too small , only a small number of objects would be found during the mission life time , and the science telescope would remain idle most of the time . if is too large , it would exhaust the onboard system , for instance requiring faster cpus with larger power consumption , violating the power budget .  due to the heavy computations involved of the full asteroid populations, we decided to initially work on a small subset of the neas , by extrapolating the bottke distribution , eq ., to an object size of around 1 m , , for a total of exactly objects .many of the ground based surveys are able to reach , so this value we used as the initial in the moid and the integration . as the orbit of bering, we simply used the earth orbit , for comparison with the ground based surveys .the results are shown as run 1 in table [ tab : results ] .[ cols="<,^,^,^,^,^ " , ]   of the initial 100 million objects , only 7 million , or 7% , remains after the moid sorting .of these objects , 619 were seen by bering over 50 days .the detection rate would thus be 12.4 objects / day .for each object we know the circumstances of the detection , for instance the smallest object observed had m , and the object had for 144 minutes . at the brightest point , andthe object was moving with 25.2 " /sec .for each of the 619 objects we thus know the maximum brightness of the objects during the 50 days , and we can then determine the threshold of where the detection rate goes to zero .in this manner , we find that for , 17 objects could be observed , corresponding to a detection rate of 0.3 objects / day . taking into account that this run is a factor of 14 short in the number of objects , we would expect a detection rate of approximately 1 object / day using the full population eq .[ neapopulation ] with .run 2 utilizes the full nea population eq .[ neapopulation ] with . from the original approx . objects , around 10 million , or 0.7% , were left after the moid sorting .we find that 16 objects could be seen over 50 days , amounting to a detection rate of 0.3 objects / day .this number is the same as for the test population , run 1 .however for run 1 three objects have m ( , ) whereas of the 16 objects in run 2 , 8 have m , and 3 have m .these results we can compare to the observations by brown and collaborators . according to their work ,30 objects / yr collides with the earth , with diameters m .to `` collide with the earth '' would in the case of bering mean that the object is detected ( ) and encounters bering within a distance equal to the earth radius . on the other hand , an object within that distance is not necessarily seen by bering , as the magnitude of the object depends on the phase angle and the distance to the sun and bering ( an object does not need to have in order to collide with the earth ) .hence , in the best case , for an object with m at 1 au distance to the sun and at , the detection radius of bering would be 43260 km at and 6856 km at .for , we find three objects in that range , at during the 50 days time span .this amounts to 22 objects / year , comparable to the 30 objects / yr by brown et al .if we strictly look at the diameter , for m , the earth is hit one time per year according to brown et al .hence the earth is hit by 29 objects / year with . in that size range , we have from the simulation that 6 obj/50d were detected , equal to 44 obj / yr . with the differences between `` colliding with the earth '' and `` observing an object '' , and the short integration time span in mind ( brown et al.is based on 8.5 yr of observations ) , we are quite convinced that our simulations are consistent with the observations .  for objects smaller than 1 mit has been suggested that the earth is hit by one object per day with cm . for bering , such an object would have within a distance of 6548 km .an even more detailed integration of the smallest object of run 2 reveals that the object would be visible for 43 minutes , with a close approach distance of 19500 km .the angular velocities of the smallest of the objects were in the range 100 - 300 `` /sec . in comparison, a 15 cm object would be moving with 619 ' ' /sec at the maximum detection distance .  from these considerations ,we find that a detection limit of would provide high feasibility of the bering mission . concerning the large angular velocities , there may however be a conflict between the detection limit and the fastest moving objects , due to trailing losses .the capabilities of the asc to handle fast moving objects at the suggested limiting detection magnitude remains to be studied .as it was mentioned previously , there are strong indications that the albedo of the smallest asteroids is relatively large . to check the influence of the albedo on the detection rate , we for run 3 simply changed the slope parameter of the moid - sorted objects from run 2 , for a direct comparison . , the slope parameter , was changed from 0.15 to 0.32 , corresponding to a change in albedo from approx .0.17 to 0.25 . in this run ,23 objects were detected , 7 more than in run 2 , corresponding to a detection rate of 0.46 objects / day .14 objects were smaller than ( . in run 2 , 8 objects had , hence 6 of the 7 additional objects belongs to the small end of the size range .this indicates that the albedo may influence the detection rate by a factor of two .this , on the other hand , may mean that the detection rate measured by the bering mission will not only provide a constraint on the size / absolute magnitude distribution , but also on the albedo . particularly in the case where the bering observations are consistent with the ground based observations ,the albedo can be determined directly from the count statistics .this question should be address by a more detailed analysis , involving integrations over longer time spans .with the value of found suitable for the nea population , we continued to examine the capabilities of the mission with such a limiting magnitude in the asteroid main belt .initially , bering was kept at a circular orbit in the ecliptic , with a semi - major axis of 2.5 au . this places bering in the inner part of the main belt .a first run , run 4 , was made to test the behavior of the simulations .of the 400 million objects , 11 million or 2.7% survived the moid .the integration was made over 356 days , with a resolution of 1 day .647 objects were seen , i.e. a detection rate of 1.8 object / day .of these , 204 were smaller than 10 km ( ) , on the other hand , none of the smallest objects were seen , the smallest having ( , .hence only a fraction of the largest objects could be seen over one year .it remains to be examined how many of the largest objects that can be observed during the mission life time .the smallest objects were presumably not observable due to the transient brightness behavior upon a close encounter , and were thus not found as the time between each magnitude calculation ( ) is too large .hence we made a second run , run 5 , with a time span of 50 days and with a resolution of 0.02 days .this time , 352 objects could be seen , or 7 objects / day .of these , 55 have , so at least on object smaller than 10 km could be seen per day .the smallest object found had , and was visible for 6 hours , so none of the objects could be seen .as mentioned , this may be due to the underrepresentation in the number of these objects , in combination with the short integration time span .it should be examined whether any of the objects can be seen by extending the integration time span .the tangential velocities of the encountered objects were typically in the range of 5 - 20 " /sec .we have analyzed the scientific feasibility of the bering mission , when applied to the near - earth asteroids and the asteroid main belt .we have found that with a limiting magnitude of the asc of approx . , the scientific feasibility can be sustained .this would allow the detection of around one object per day , in both the nea and main belt populations .this initial detection by the asc will allow the science telescope to be pointed toward the object , and detailed observations can be initiated , both in terms of physical characterisation , but also in terms of following the object toward faint magnitudes .  for the main belt population , we find that there is room for lowering if e.g. required by the smaller power budget .some issues however still remains open , in particular the capabilities of the asc to follow very fast moving objects at faint magnitudes .it also remains to be examined how bering will perform when placed in the proposed eccentric orbit ranging from 0.7 au at perihelion to 3.5 au at aphelion .in addition to the already outlined scientific objectives , we also find that the albedo of the object populations has an influence on the detection rate , and the question of whether the albedo can be derived directly from detections among the nea population needs a further close analysis , as this is critical for the physical characterisation , including the object size .computing facilities were provided by the aarhus branch of the danish center for scientific computing ( csc - aa ) . this work was supported by the danish natural science research council through a grant from the center for ground - based observational astronomy ( ijaf ) .

Abstract: we determine the possible detection rate of asteroids with the bering mission . in particular we examine the outcome of the bering mission in relation to the populations of near - earth asteroids and main belt asteroids . this is done by constructing synthetic populations of asteroids , based on the current best estimates of the asteroid size - distributions . from the detailed information obtained from the simulations , the scientific feasibility of bering is demonstrated and the key technical requirement for the scientific instruments on bering is determined .   [ 2001/04/25 1.1 ( pwd ) ]