On the Usefulness of Fibonacci Compression Codes

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1 The Computer Journal Advance Access publshed May 14, 2009 The Author 2009 Publshed by Oxford Unversty Press on behalf of The Brtsh Computer Socety All rghts reserved For Permssons, please emal: do:101093/comjnl/bxp046 On the Usefulness of Fbonacc Compresson Codes Shmuel T Klen and Mr Kopel Ben-Nssan Department of Computer Scence, Bar Ilan Unversty, Ramat-Gan 52900, Israel Correspondng author: tom@csbuacl Recent publcatons advocate the use of varous varable length codes for whch each codeword conssts of an ntegral number of bytes n compresson applcatons usng large alphabets Ths paper shows that another tradeoff wth smlar propertes can be obtaned by Fbonacc codes These are fxed codeword sets, usng bnary representatons of ntegers based on Fbonacc numbers of order m 2 Fbonacc codes have been used before, and ths paper extends prevous work presentng several novel features In partcular, the compresson effcency s analyzed and compared to that of dense codes, and varous table-drven decodng routnes are suggested Keywords: text compresson; Fbonacc codes; dense codes Receved 7 June 2008; revsed 29 December 2008 Handlng edtor: Alberto Apostolco 1 INTRODUCTION In spte of the amazng advances n data storage technology, compresson technques are not becomng obsolete, and n fact research n data compresson s flourshng as can be seen by the large number of papers publshed recently on the topc In ths work we concentrate on very large textual databases as those found n large Informaton Retreval Systems Such systems could contan hundreds of mllons of words, whch should be compressed by some method gvng, n addton to good compresson performance, also very fast decodng and the ablty to search for the appearance of some strngs drectly n the compressed text Ths paradgm of compressed pattern matchng s a well establshed research topc that has generated a large lterature n recent years; see [1 4] to cte just afew Classcal Huffman codng, when appled to ndvdual characters, gves relatvely poor compresson, but when every word of a large textual database s consdered as an atomc element to be encoded, ths so-called Huffword varant may compete wth the best other compresson methods [5] However, the codewords of a bnary Huffman code are not necessarly algned on byte boundares, whch complcates both the decodng process and the ablty to perform searches n the compressed fle The next step was therefore to pass to 256- ary Huffman codng, n whch every codeword conssts of an ntegral number of 8-bt bytes [3] The loss ncurred n the compresson effcency, whch s only a few percent for large enough alphabets, s compensated for by the advantages of the easer processng When searches n the compressed text should also be supported, Huffman codes suffer from a problem of synchronzaton: denotng by E the encodng functon, the compressed form E(x) of an element x may appear n the compressed text E(T ), wthout correspondng to an actual occurrence of x n the text T, because the occurrence of E(x) s not necessarly algned on codeword boundares Ths problem has been overcome n [6], relyng on the tendency of Huffman codes to resynchronze quckly after errors, but the suggested soluton s probablstc and may produce wrong results A determnstc soluton has recently been gven n [7] As an alternatve, [3] propose to reserve the frst bt of each byte as tag, whch s used to dentfy the last byte of each codeword, thereby reducng the order of the Huffman tree to 128-ary These tagged Huffman codes have then been replaced by end-tagged dense codes (ETDC) n [8] and by (s, c)-dense codes (SCDC) n [9] The two last mentoned codes consst of fxed codewords that do not depend on the probabltes of the tems to be encoded Thus ther constructon s smpler than that of Huffman codes: all one has to do s to sort the tems by non-ncreasng frequency and then assgn the codewords accordngly, startng wth the shortest ones SCDC s based on a par of numbers (s, c), where the parameter s s chosen as 0 < s < 256 and c s defned as c = 256 s Each codeword conssts of a sequence of bytes b 1 b r, the last of whch satsfes b r <sand the others b s

2 2 ST Klen and M Kopel Ben-Nssan for <r ETDC s just the specal case of SCDC for whch s = c = 128 ETDC and SCDC compress less than Huffman codes, but are better than tagged Huffman codes as more codewords can be formed for every gven length In addton, codng s smplfed and the good searchng capabltes are mantaned We show n ths work that smlar propertes can be obtaned by Fbonacc codes, whch have been suggested n the context of compresson codes for the unbounded transmsson of strngs [10] and because of ther robustness aganst errors n data communcaton applcatons [11] They are also studed as a smple alternatve to Huffman codes n [12] The man contrbuton of ths paper s to show that Fbonacc codes are useful n ths context and to present several new propertes that have not been mentoned before, ncludng () compresson performance relatve to other statc methods lke ETDC and SCDC, () fast decodng technques and () support of compressed matchng Note that Fbonacc codes have applcatons not only as alternatves to dense codes for large textual word-based compresson systems They are n partcular mentoned n [13] as a good choce for compressng a set of small ntegers, and fast decodng as well as compressed searches may be mportant tools for such applcatons In the next secton, we revew the relevant features of Fbonacc codes of order m 2 Secton 3 then treats the new propertes mentoned above and compares ther performance wth that of other statc compresson codes 2 FIBONACCI CODES Fbonacc numbers of order m 2, denoted by F (m), are defned by the followng recurrence relaton: F n (m) = F (m) n 1 + F (m) (m) n 2 + +F n m for n>0, and the boundary condtons F (m) 0 = 1 and F n (m) = 0 for m<n<0 For fxed order m, the number F n (m) can be represented as a lnear combnaton of the nth powers of the roots of the correspondng polynomal P (m) = x m x m 1 x 1 P (m) has only one real root that s >1, whch we shall denote by φ (m) ; the other m 1 roots are complex numbers wth norm < 1 (for m = 2, the second root s also real and ts absolute value s < 1) Therefore, when representng F n (m) as such a lnear combnaton, the term wth φ(m) n wll be the domnant one, and the others wll rapdly become neglgble for ncreasng n For example, m = 2 corresponds to the classcal Fbonacc sequence and φ (2) = (1 + 5)/2 = s the well-known golden rato As a matter of fact, the entre Fbonacc sequence can be obtaned by F n (m) =[a (m) φ(m) n ], where a (m) s the coeffcent of the domnatng term n the above-mentoned lnear combnaton, and [x] ndcates that the value of the real number x s rounded to the closest nteger, that s, [x] = x + 05 Table 1 lsts the frst few elements of the Fbonacc sequences of order up to 6 The column headed General Term brngs the values of a (m) and φ (m) For larger n, the numbers a (m) φ(m) n are usually qute close to ntegers, eg a (2) φ(2) 10 = and a (3)φ(3) 13 = The standard representaton of an nteger as a bnary strng s based on a numeraton system whose bass elements are the powers of 2 If B s represented by the k-bt strng b k 1 b k 2 b 1 b 0, then B = k 1 =0 b 2 However many other possble bnary representatons do exst, and those usng the Fbonacc sequences as bass elements have some nterestng propertes We frst consder the standard Fbonacc numbers of order 2 Any nteger B can be represented by a bnary strng of length r, c r c r 1 c 2 c 1, such that B = r =1 c F (2) The representaton wll be unque f one uses the followng procedure to produce t: gven the nteger B, fnd the largest Fbonacc number F r (2) smaller or equal to B; then contnue recursvely wth B F r (2) For example, 45 = , so that ts bnary Fbonacc representaton would be As a result of ths encodng procedure, there are never consecutve Fbonacc numbers n any of these sums, mplyng that n the correspondng bnary representaton, there are no adjacent 1s Ths property can be exploted to devse an nfnte code whose set of codewords conssts of the Fbonacc representatons of the ntegers: to assure the code beng unquely decpherable (UD), each codeword s prefxed by a sngle 1-bt, whch acts lke a comma and permts to dentfy the boundares between the codewords The frst few elements of ths code would thus be {u 1,u 2,} = {11, 110, 1100, 1101, 11000, 11001,}, where the separatng 1 s put n boldface for vsblty A typcal compressed text could be , whch s easly parsed as u 6 u 3 u 4 u 1 u 4 Though beng UD, ths s not a prefx code, and so decodng may be somewhat more nvolved In partcular, the frst codeword 11, whch s the only one contanng no zeros, complcates the decodng because f a run of several such codewords appears, the correct decodng of the codeword precedng the run depends on the party of the length of the run Consder for example the encoded strng ; a frst attempt to parse t as = u 2 u 1 u 1 u 1 10 would fal, because the tal 10 s not a codeword; hence only when tryng to decode the ffth codeword do we realze that the frst one s not correct, and that the parsng should rather be = u 4 u 1 u 1 u 2 To overcome ths problem, [10,11] suggest to reverse all the codewords, yeldng the set {v 1,v 2,} = {11, 011, 0011, 1011, 00011, 10011,}, whch s a prefx code, snce all codewords are termnated by 11 and ths substrng does not appear anywhere n any codeword, except at ts suffx In addton, we show below that havng a reversed representaton, wth the bts correspondng to ncreasng bass elements runnng from left to rght rather than as usual, s advantageous for fast

3 Usefulness of Fbonacc Codes 3 TABLE 1 Fbonacc numbers of order m = 2, 3, 4, 5, 6 F (m) n General term m = (16180) n m = (18393) n m = (19275) n m = (19659) n m = (19836) n decodng Table 2 brngs a larger sample of ths set of codewords n the column headed Fb2 Note that the order of the elements s not lexcographc, eg precedes The generalzaton to hgher order seems at frst sght straghtforward: any nteger B can be unquely represented by the strng d s d s 1 d 2 d 1 such that B = s =1 d F (m) usng the teratve encodng procedure mentoned above In ths representaton, there are no consecutve substrngs of m 1s For example, the representatons of the ntegers 10, 11, 12 and 13 usng F (3) are, respectvely, 1011, 1100, 1101 and However smply addng now m 1 1s as commas and reversng the strngs does not yeld a prefx code for m>2, and n fact the code so obtaned s not even UD For example, for m = 3, the above numbers would gve the codewords {v 10,,v 13 }= {110111, , , }, but the encodng of the fourth element of the sequence would be v 4 = 00111, whch s a prefx of v 11 The strng could be parsed both as = v 4 v 5 and as = v 11 v 2 The problem stems from the fact that for m>2, there can be more than one leadng 1 n the representaton of an nteger, and hence addng m 1 1s may gve a strng of up to 2m 2 consecutve 1s The fact that a strng of m 1s appears only as a suffx s thus only true for m = 2 To turn the sequence nto a prefx code, the defnton has to be amended as follows: the set Fbm wll be defned as the set of bnary codewords of lengths m, such that every codeword contans exactly one occurrence of the substrng consstng of m consecutve 1s, and ths occurrence s the suffx of every codeword The frst elements of these codes for m 4 are gven n Table 2 For m = 2, ths last defnton s equvalent to the one above based on the representaton wth bass elements F n (2) ; for m > 2, only a subset of the correspondng codewords s taken There s nevertheless a connecton between the codewords and the hgher order Fbonacc numbers: for m 2, and n 0, the code Fbm conssts of F (m) n codewords of length n + m (1) Ths s vsualzed n Table 2, where for each code, blocks of codewords of the same length are separated by horzontal lnes Wthn each such block of lengths m+2 for Fbm, the prefxes of the codewords obtaned by removng the termnatng strng of 1s correspond to consecutve ntegers n the representaton based on F (m) For decodng, the Fbonacc representaton wll TABLE 2 Fbonacc codes of order m = 2, 3, 4 Index Fb2 Fb3 Fb thus be used to get the relatve ndex wthn the block, to whch the startng ndex of the gven block has to be added Ths s dentcal to the decodng procedure of canoncal Huffman

4 4 ST Klen and M Kopel Ben-Nssan codes, for whch each block of codewords of a gven length conssts of the standard bnary representaton of consecutve ntegers; see [14] 3 COMPRESSION BY FIBONACCI CODES We now turn to an nvestgaton of some propertes of the Fbonacc codes and n partcular compare them wth the dense codes ETDC and SCDC The latter were ntroduced n [8,9]as alternatves to tagged Huffman codes, mprovng these because of the followng advantages: () better compresson ratos; () smpler vocabulary representaton; () smpler and faster decodng; (v) same searchng possbltes We show that on all these crtera, Fbonacc codes are a plausble alternatve to ETDC and SCDC, mprovng on the frst, beng equvalent on the second and nferor on the last two A smple mplementaton of decodng and searchng could be as much as 100 tmes slower than for the dense codes, and we show how to accelerate both procedures, reducng the advantage of SCDC to 2- to 3-fold We also add robustness as a ffth crteron, for whch the Fbonacc codes are preferable Our concluson s that overall, Fbonacc codes may be an attractve substtute for dense codes n certan applcatons A comprehensve study of the Fbonacc codes should possbly also nclude comparsons wth other alternatves, lke standard bnary Huffman codes or arthmetc codng We shall, however, keep the focus n ths paper on the comparson aganst the dense codes ETDC and SCDC, and menton the others only brefly, frst, because such comparsons can already be found n the papers on dense codes, and second, because the choce of an encodng scheme may be consdered as a package deal: f one had only a sngle applcaton n mnd, then dfferent codes could be chosen accordng to the ntended applcaton Therefore f the only crteron s compresson, one would use arthmetc codng, whch practcally acheves the entropy bound, and f speed s the only concern, one could use ETDC for example However the assumpton here, lke n [8,9], s that one chooses a sngle code that should be useful for varous applcatons and on several crtera Classcal 2-ary Huffman codng gves very good compresson when appled on the dfferent words of a large text, and the excess of the entropy, the nformaton theoretc lower bound, s below 03% on our examples; see Table 3 below Smple btwse decodng s slow, but several table-drven decodng methods have been devsed, achevng a sgnfcant speedup [15] Searchng n Huffman encoded texts has to deal wth synchronzaton problems, but can be done, as mentoned n the ntroducton Arthmetc codng gves optmal compresson, and examples of typcal compresson ratos are ncluded n Table 3 n the column headed Entropy, though the savngs relatve to Huffman codes are generally very low [16] However the processng for both encodng and decodng s much slower than for all the alternatves mentoned n ths paper, and searchng wthn the compressed text s mpossble, snce t s not true that dfferent occurrences of a word n the text are always encoded by the same bt strng 31 Compresson effcency We frst compare the number of codewords of each of the codes for a gven length As a rough approxmaton, ETDC utlze 7 of the 8 bts of each byte, so that the number of codewords grows lke 2 7k/8 = 1834 k, where k s the number of bts Ths should be compared wth the Fbonacc codes, for whch the number grows roughly lke φ(m) k Thus we expect the codes based on standard Fbonacc numbers to have less codewords than ETDC, but for m>2, the Fbm codes are denser than ETDC More precsely, the number of codewords of length up to M bytes for ETDC s 128 M = M 1 = 2 7M For Fbm, the last m bts of each codeword are reserved; hence usng equaton (1), the number of codewords wth up to 8M bts s approxmately 8M m =1 F (m) a (m) 8M m =1 φ (m) a (m) φ (m) 1 φ8m m+1 (m), whch should be compared wth 2 7M We get that the number of codewords for Fbm wll be larger f ( ) φ 8 M (m) > (φ (m) 1)φ m 1 (m) (2) 2 7 a (m) Nevertheless for m = 2, the constant φ 8 (2) /27 = 0367 s <1, whle the rght-hand sde of (2) s138 > 1, so that for any length M, ETDC has more codewords than Fb2 For m 3, we get from (2) that M> (m 1) log 2 φ (m) + log 2 (φ (m) 1) log 2 a (m), (3) 8 log 2 φ (m) 7 so that asymptotcally all the Fbm codes are denser, and ther number of codewords s larger than that of ETDC f M s at least 67, 7, 6 and 7, for m = 3, 4, 5, 6, respectvely These values derve from the approxmate lower bound of equaton (3) and concde wth the values obtaned usng precse nteger computatons Of course, the fact that Fbm codes are denser n the lmt has no mpact on real-lfe dstrbutons, as M = 4 suffces already for huge alphabets However, the codes should not be compared only on the bass of the number of ther elements Ths would correspond to a unform dstrbuton, for whch t makes no sense to use varable length codes anyway For non-unform dstrbutons, the advantage of the Fbonacc codes should be even more

5 Usefulness of Fbonacc Codes 5 TABLE 3 Average codeword lengths and excesses over the entropy Excess n % over the entropy lower bound Sze Entropy Language words bts Huffman Fb2 Fb3 Fb4 Fb5 Fb6 ETDC (s, c) Englsh (189,67) 1412 French (185,71) 1310 Hebrew (176,80) 1390 Artfcal (171,85) 1332 Codeword length Wall Street Journal ETDC encodng Fb2 Fb Index of element FIGURE 1 Codeword lengths evdent, as because of the possblty of usng a larger set of dfferent lengths, the codes can approach the optmal codeword lengths more closely, and n partcular assgn codewords that are shorter than 1 byte to the elements of hghest frequency The rgdty of ETDC, whch assgns 1 byte codewords only to the 128 most frequent elements and then already passes to codewords of length 2 bytes, was the man motvaton for the development of the (s, c)-codes; Fbonacc codes also have codewords shorter than 1 byte Fgure 1 plots the lengths of the codewords as a functon of the ndex of the element on a logarthmc scale for ETDC, Fb2 and Fb3 In addton, there s also a (seemngly contnuous) curve correspondng to a real-lfe dstrbuton, 500 MB (87 mllon words) of the Wall Street Journal [17]; the plot corresponds to an deal encodng of the dfferent words, usng log 2 p bts to encode an element wth probablty p, as could be approxmated by an arthmetc encoder We see how the Fbonacc curves approach the real dstrbuton by a seres of small steps much closer than the larger steps of ETDC To compare the codes analytcally on more realstc dstrbutons than the unform one, consder Zpf s law [18], whch s beleved to govern the dstrbuton of the words n a large natural language text It s defned by the weghts p = 1/( H N ) for 1 N, where H n = n j=1 (1/j ) s the nth harmonc number 1 It s well known that H n ln n γ, where γ = s Euler s constant Snce Fbm and ETDC are step functons, t s convenent to have a notaton for the ndces of the last elements of every codeword length Let L E (k) and L m (k) denote, respectvely, these ndces for ETDC and Fbm, m 2 Thus L E (8) = 128, L E (16) = 16512, L E (24) = ; L 2 (k) = 1, 2, 4, 7, 12, 20, 33,, L 3 (k) = 1, 2, 4, 8, 15, 28, and L 4 (k) = 1, 2, 4, 8, 16, 31,, for k m, as can be seen n Table 2 Smlarly, let M E (N) and M m (N) denote the number of dfferent codeword lengths, whch s the number of blocks of codewords of equal length for an alphabet of sze N Snce approxmately (128) ME(N) = N and a (m) φ M m(n) (m) /(φ (m) 1) = N, we obtan that M E (N) ln N ln 128 and M m (N) ln(φ (m) 1) + ln N ln a (m) ln φ (m) Denotng the th codeword by C, the average length of a codeword usng a Zpf dstrbuton s then N =1 1 C 1 M E (N) 8k H N H N However, L(k) L(k 1)+1 k=1 L E (k) L E (k 1)+1 1 M m (N) (k + m 1) H N k=1 1 = H L(k) H L(k 1) 1 for ETDC L m (k) L m (k 1)+1 ln L(k) γ (ln L(k 1) γ) L(k) = ln L(k 1), 1 for Fbm 1 Actually, a more precse defnton of the law s p = 1/ θ H N (θ) for some constant θ>1, where H n (θ) = n j=1 (1/j θ ), but we shall stck to the smpler defnton; accordng to Wkpeda, θ s just slghtly >1

6 6 ST Klen and M Kopel Ben-Nssan Average codeword length Alphabetsze FIGURE 2 Zpf averages ETDC Fb2 Fb3 Fb4 Fb5 Fb6 and the rato of consecutve L values s roughly 128 for ETDC and φ (m) for Fbm We thus get that the average s 4 ln 128 ln N γ M E(N)(M E (N) + 1) and for Fbm, t s, after smplfcaton, for ETDC, ln φ (m) 2(ln N γ) M m(n)(m m (N) + 2m 1) Fgure 2 plots these average values for N up to 10 8 for ETDC (the bold lne) and Fbm, wth 2 m 6 As all the functons are roughly proportonal to log N, we get almost straght lnes on the dsplayed logarthmc scale We see that ETDC has a consstently longer average than the Fbonacc codes, at least for m 5 and up to alphabet szes of 10 8 The addtonal overhead may reach 33% for alphabets as small as 1000 elements, but even for N = , the overheads of ETDC relatve to Fb2, Fb3 and Fb4 are 1, 10 and 7%, respectvely For a comparson of real data and dfferent languages, the followng sets were used, besdes Wall Street Journal: the data for French was collected from the Trésor de la Langue Françase, a database of 680 MB of French language texts (115 mllon words) of the 17th 20th centures [19], and for Hebrew, the data was a part of the Responsa Retreval Project, 100 MB of Hebrew and Aramac texts (15 mllon words) wrtten over the past ten centures [20] All these texts have been encoded as sequences of words Table 3 lsts the szes of the alphabets (number of words) and the entropy of each dstrbuton n bts per codeword, whch gves a lower bound on the possble average codeword lengths, as could be approached by an arthmetc encoder Then follow the values for 2-ary Huffman, Fbm, ETDC and SCDC codes, all gven as the excess, n percent, over the entropy, the nformaton theoretc lower bound For SCDC, the best (s, c) par was chosen for each of the dstrbutons, and ths optmal par appears also n the last column The table also gves values for an artfcal Zpf dstrbuton of sze 10 6 Among the statc codes consdered here, the best results are consstently gven by Fb3, savng addtonal 6 9% over ETDC or the best SCDC, and exceedng the entropy or a Huffman code only by 4 7% Even Fb2 encodng mproves by 1 4% on the Englsh, French and artfcal texts, and only for Hebrew, Fb2 s margnally nferor (by 12%) to the best SCDC 32 Vocabulary representaton There are many dfferent Huffman codes for a gven probablty dstrbuton Even f canoncal Huffman codes are used, one stll needs to store the number of codewords of every length The sze of ths addtonal storage s of course neglgble relatve to the szes of the other fles nvolved, but the fact that the set of codewords wll vary accordng to the dstrbuton at hand puts an addtonal burden on both encoder and decoder On the other hand, ETDC, SCDC and the Fbm codes are fxed sets, not dependng on the frequences The codewords are thus stored once and for all, and from that pont of vew, ETDC, SCDC and the Fbm codes are equvalent 33 Fast decodng A major reason for abandonng the optmal bnary Huffman codes and usng nstead 256-ary Huffman codes or SCDC s to obtan faster decodng, snce each codeword conssts of an ntegral number of bytes Though fast methods for the decodng of general 2-ary Huffman codes have been devsed [14,15,21], they cannot be faster than the decodng of byte-algned Huffman codes, whch do not need tme-consumng bt manpulatons and can perform the whole process at the byte level On the other hand, the decodng of byte-algned Huffman codes has extensvely been compared wth that of ETDC and SCDC n [22], and found to be roughly equvalent: Huffman decodng s reported to be up to 3% faster The Fbm codes are also of varable lengths but not necessarly byte-algned, and so the nave approach to decodng, nvolvng many bt manpulatons, wll be more costly than the alternatves Ths secton deals wth ways to accelerate the decodng process As mentoned above, ths s mportant for any applcaton usng Fbonacc codes, not only as alternatves to dense codes, and we show a sgnfcant mprovement over the standard decodng procedure In comparson wth ETDC and SCDC, decodng s stll slower, but the nferorty of the Fbm codes s greatly reduced As a decodng baselne, consder the smple btwse decodng procedure as follows: the Fbm encoded text s denoted T 1 T 2 T 3, and an array Start m [j] gves the ndex of the frst element of length j n the Fbm code Thus Start 2 [] = 1, 2, 3, 5, 8,, Start 3 [] = 1, 2, 3, 5, 9, 16,, Start 4 [] = 1, 2, 3, 5, 9, 17, 32,, as can be seen n Table 2 Note also that usng the notaton of the prevous secton, where the ndex of the last element for each length was needed, we

7 Usefulness of Fbonacc Codes 7 have Start m [j + 1] = L m (j) + 1 for j m Let Word[j] be the element encoded by the jth codeword; ponts to the bt precedng the frst bt of a codeword, l s the number of bts decoded so far wthn the current codeword and rel_ nd s the relatve ndex of the current codeword wthn the lst of those of length l; F (m) [j] s the jth Fbonacc number of order m,as n Table 1 The formal nave decodng s then gven by: 0 l 0 rel_ nd 0 whle < length of encoded text n bts f T +l+1 T +l+m = 11 1 /* m consecutve 1s */ output Word[ Start m [l + m]+rel_ nd ] + l + m l 0 rel_ nd 0 else l l + 1 rel_ nd rel_ nd + T +l F (m) [l] In words, the nput s processed bt by bt If the last m bts are a strng of 1s, a codeword has been detected and can be sent to output, otherwse the relatve ndex of the current codeword s ncrementally calculated by addng the correspondng Fbonacc numbers 331 Partal decodng tables To mprove the btwse approach, we adapt, n a frst attempt, a method orgnally suggested for the fast decodng of Huffman codes, whch suffer from the same problem as the varable length Fbm codes The method uses a set of partal decodng tables that are prepared n advance and depend only on the code, not on the actual text to be decoded, by means of whch the decodng s then performed by blocks of k bts at a tme, rather than bt per bt Typcally, k = 8 or 16 One therefore gets a tme/space tradeoff, where faster decodng comes at the cost of storng larger tables The range of the parameter k s nevertheless restrcted, because too large tables, even f they ft nto the avalable RAM, may cause more page faults and cache msses, whch may cancel the speed advantages The method has frst been presented n [23] and has been renvented several tmes, eg [24] The basc scheme s as follows The number of entres n each table s 2 k, correspondng to the 2 k possble values of the k-bt patterns Each entry s of the form (S, R), where S s a sequence of characters and R s the label of the next table to be used The dea s that entry, 0 <2 k, of the frst table contans the longest possble sequence S of characters that can be decoded from the k-bt block representng the nteger (S may be empty when there are codewords of more than k bts); usually some of the last bts of the block wll not be decpherable, beng the prefx P of more than one codeword; there wll be one table for each of the possble prefxes P, n partcular, f P =, where denotes the empty strng, the correspondng table s the frst one The table for P = s constructed n a smlar way except for the fact that entry wll contan the analyss of the bt pattern formed by the prefxng of P to the bnary representaton of More formally, n the table correspondng to P, the th entry, Tab[, P ], s defned as follows: let B be the bt strng composed of the juxtaposton of P to the left of the k-bt bnary representaton of Let S be the (possbly empty) longest sequence of characters that can be decoded from B, and R the remanng undecpherable bts of B; then Tab[, P ]=(S, R) Table 4 brngs a part of these partal decodng tables for m = 3 and k = 8, showng certan lnes for selected values of P The frst columns are the ndces to be decoded n both decmal and bnary notaton; the decodng tables are labeled by the correspondng prefxes P The general decodng routne s thus extremely smple: denotng the th byte of the encoded text by Text[], one performs R for 1 to length of text n bytes (output,r) Tab[Text[],R] As an example, consder the nput strng dsplayed n Fg 3 The dfferent codewords appear n alternatng grey tones, the decmal value of the bytes s dsplayed above and the output generated by the decodng procedure appears below For the gven example, the bt strng n the frst byte s the bnary representaton of 227, and therefore one accesses the frst table at entry 227, yeldng as output C 1 and a remander suffx The next table accessed s thus that labeled 00011, at entry 221, whch s the value of the next byte; ths table contans at entry 221 the partal decodng of the strng , whch yelds the characters C 5 C 4 and the remander P = 01, etc Refer to Table 4, where the entres used n ths decodng example are boxed The man problem wth ths partal decodng approach s that the number of requred tables may be prohbtvely large A smlar problem occurs n the Huffman case, whch has led to varous attempts to reduce the number of tables [21,23] There s one table for each possble prefx of a codeword, and so for Huffman codes, the number of tables s N 1, where N s the sze of the alphabet Ths s true because a Huffman tree s a full bnary tree, each prefx corresponds to an nternal node and the number of nternal nodes s one less than the number of leaves The Fbm codes are also prefx codes, but the correspondng bnary trees are not full, e there are nternal nodes havng only one chld An upper bound for the number of prefxes can be obtaned as follows: let l be the length of the longest of the N codewords, that s, usng equaton (1), N l m, the number of =0 F (m) prefxes s then bounded by l m (m) =0 ( + m 1)F Ths bound s not tght, as prefxes shared by more than one codeword are counted more than once Every codeword, except the frst, of

8 8 ST Klen and M Kopel Ben-Nssan TABLE 4 Partal decodng tables Index Decmal Bnary S R S R S R S R C 8 00 C 5 C 4 00 C 2 C 2 00 C C 8 01 C 5 C 4 01 C 2 C 2 01 C C 8 10 C 5 C 4 10 C 2 C 2 10 C C 8 11 C 5 C 4 11 C 2 C 2 11 C C C C C C C C C C C C C C C C C C C C C C C FIGURE 3 Example of decodng by partal decodng tables oldsv oldsl PV PL 111 S S SV SL FIGURE 4 Decodng a sngle byte Fbm s of the form α01 m, where α s some bnary strng, so that the set of dfferent prefxes ncludes at least the strngs α01 j, for 0 j < m The number of tables s thus at least mn 1 For example, for m = 3 and N = 4, the codewords are 111, 0111, and 10111, and the set of dfferent prefxes s {1, 11, 0, 01, 011, 00, 001, 0011, 10, 101, 1011} Form = 3 and an alphabet of N = codewords, even f each table entry needs only 4 bytes and one uses a low value of k such as k = 8, the space for the tables would be more than 300 MB The partal decodng approach s thus not feasble for the large values of N for whch the Fbm encodng schemes are ntended as alternatves to the dense codes, but t could be an attractve varant for smaller alphabets 332 Reducng the number of tables It s possble to reduce the number of requred tables usng the propertes of the Fbonacc numeraton systems on whch the Fbm codes are based Consder the th byte of the text to be decoded, schematcally represented n Fg 4 The byte can be parttoned nto three zones: there s frst a prefx of length PL, contanng the bts necessary to complete a codeword, the frst bts of whch appeared already n the prevous byte Ths prefx may be empty (n case the prevous codeword ended at the last bt of byte 1), or t may, n the other extreme case, extend to the end of the th byte and possbly even beyond Whether a codeword ends wthn the byte or not can be decded by the (non-)occurrence of m consecutve 1s The second zone conssts of zero or more complete codewords, depcted n grey n Fg 4 The thrd zone s the suffx of length SL of byte and contans the prefx of a codeword, the completon of whch appears only n the next byte or even later; agan, ths zone may be empty The partal decodng tables deal easly wth the full codewords of the second zone and defer the treatment of the suffx n the thrd zone to the next teraton The man problem s thus to relate the PL bts of the frst zone to the oldsl bts of the thrd zone of the prevous byte, so as to evaluate the ndex of ths codeword that s splt by the byte boundary In contrast to the decodng of Huffman codes, for whch there s no connecton between dfferent parts of a codeword, the codewords of Fbm codes are n fact representatons of ntegers, whch can be exploted as follows For ease of descrpton, we restrct the dscusson and the examples n the sequel to the case m = 3, but all the statements can be easly generalzed to other values of m The codewords of Fb3 are defned as C 1 = 111, and for > 1, C = d 1 d 2 d s 0111, where rel_ ndex = s j=1 d j F (3) j s the relatve ndex of the codeword C wthn the ordered set of codewords of length s + 4; the absolute ndex s gven as = rel_ ndex+start[s +4], where Start[l] s the ndex of the

9 Usefulness of Fbonacc Codes 9 frst codeword of length l, as mentoned above n the descrpton of the nave decodng algorthm Suppose that the codeword C = d 1 d 2 d s 0111 s splt after the tth bt nto two parts that are treated separately: D = d 1 d 2 d t and E = d t+1 d s 0111 Denote by DV the value of the strng D, that s, DV = t j=1 d j F (3) j, and by EV the value of the strng E from whch the termnatng 0111 has been strpped, consdered ndependently of D, so that EV = s j=t+1 d j F (3) j t For example, f C = and t = 5, we get D = and E = , and the correspondng values are DV = 19 and EV = 12 To recover the relatve ndex, V = 273, of the orgnal strng wthn the sequence of codewords of length 13, EV has frst to be shfted so that the bts of E correspond to the Fbonacc numbers F (3) t+1,f(3) t+2,,f(3) s, rather than to F (3) 1,F (3) 2,,F (3) s t as gven n the defnton of EV However, here we can use the fact that F (3) j =[a (3) φ j (3)], so that roughly F (3) t+j φt (3) F (3) j One would thus expect the requested value to be V = DV + EV φ(3) t, rounded to the nearest nteger However, n our example, DV + EV φ(3) t = 27160, and hence even after roundng the target value would be mssed by 1 Ths can be explaned by the fact that though t s true that the jth Fb3 number can be obtaned by rasng φ (3) to the power j, multplyng by a (3) and then roundng, ths property s not addtve: when substtutng, n our approxmaton, F (3) j by a (3) φ j (3), the cumulatve error for several such substtutons s not necessarly <1/2, so that even after roundng, one may get wrong values To salvage the evaluaton procedure, the approxmate values should be used already n DV and EV More precsely, nstead of defnng EV = s j=t+1 d j F (3) j t, we defne EV = s j=t+1 d j φ j t (3), and smlarly for DV, wthout roundng to an nteger value The roundng wll only be performed at the end, after havng shfted EV by multplyng t wth φ(3) t For our example, one gets thus DV = and EV = 12063, so that DV + EV φ(3) t = ; now the roundng gves the correct answer Summarzng, the revsed verson of the partal decodng tables stll uses tables of 2 k entres for each of the possble k-bt bytes, each entry ncludng the followng nformaton: () PL: the length of the prefx of the byte termnatng a codeword, not ncludng the fnal 111, or f no codeword ends n ths byte (no occurrence of 111), then PL= k; () PV: the approxmate value of ths prefx, usng φ (3) ; () S: a sequence of zero or more full codewords that could be decoded from the byte; (v) SL: the length of the suffx of the byte that has not been decoded; (v) SV: the approxmate value of ths suffx, as above; (v) p: number of rghtmost 1-bts n the SL-bt suffx of the byte, 0 p<m The last tem, p, s needed to decde where the separatng 111 strngs occur Indeed, suppose a byte starts wth the bts 10111; f the prevous byte ended wth p = 2 non-decoded 1-bts, then only the leadng 1 of the current byte s needed to complete a codeword and the byte contans, n addton, the codeword 0111; f the prevous byte had p<2, then the fve frst bts of the current byte are used to complete a codeword Smlarly, f n the prevous byte p = 1, the current byte needs specal treatment f t starts wth 11 The decodng process thus uses three tables, one for each possble value of p The formal code s gven n Fg 5 The procedure uses a precomputed array Ph, defned by Ph[] =φ(3) ; oldsl and oldsv hold the SL and SV values of the prevous byte The fclause n lnes 4 6 deals wth the specal case when no codeword ends n ths byte (there s no occurrence of the substrng 111), that s, the current codeword started ether at the begnnng of ths byte or even earler and extends nto the next byte(s) In ths case, one has only to update the length and value of the prefx of the current codeword Otherwse, the ndex of the frst codeword n ths byte s evaluated n lne 8 A specal case has to be dealt wth when the termnatng 111 of a codeword s splt by a byte boundary, that s, for the prevous byte p 1, and the current byte starts wth at least 3 p ones PL has been defned as the length of the prefx n the frst zone, not ncludng the fnal 111, and hence to be consstent n our specal case, PL should be defned as p Ths also corrects the ndex to be used n the Start table, snce n the prevous byte, all the oldsl bts were used to calculate the value oldsv, and only after readng the current byte dd t turn out that the last p bts of the prevous byte were a part of the separator 111, and thus should not partcpate n evaluatng oldsv Moreover, the value of PV should be zero n ths case, but a correcton term s needed to subtract the weght added by these last p bts to oldsv Ifp = 1, the erroneously added amount s a (3) φ oldsl (3), and therefore f one defnes PV= a (3), the defnton of ndex n lne 8 also apples n ths case For p = 2, the amount to be FIGURE 5 Revsed decodng procedure wth partal decodng tables

10 10 ST Klen and M Kopel Ben-Nssan subtracted s ( a (3) φ oldsl 1 (3) + a (3) φ(3) oldsl = a (3) ) φ(3) oldsl φ (3) = φ oldsl (3), and hence PV should be defned as Table 5 brngs some sample lnes of the revsed partal decodng tables for k = 8 and m = 3 To estmate the sze of the tables, consder frst k = 8 Just 4 bts are then needed for the PL and SL felds, and 2 bts for p (whch s enough also for m = 4) S wll store the ndex of the decoded codeword rather than the codeword tself, and hence the number of bts allocated to S depends on the sze N of the alphabet If N<4 mllon, and n most cases t wll be, one can pack S and p together n 3 bytes Of course, there s the possblty of more than one codeword beng decoded from a sngle byte (see, eg the last lne n Table 5), but for k = 8, ths can only happen for the par C 1 C 1 Instead of reservng space for two codewords wthn the strng S, one may rather deal wth ths specal case by treatng the par C 1 C 1 as f t were another symbol, ndexed, eg N + 1 The values PV and SV are stored as 4-byte float numbers Addng t up, one needs 12 bytes per entry, and so for m = 3 and k = 8, the total sze s , <10 K Such a low value suggests that t mght even be reasonable to use k = 16, processng twce as many bts n each teraton The sze of each entry may ncrease to 13 bytes (there are more possble values for PL and SL, and more specal cases have to be taken care of f one restrcts S to a sngle ndex), but the overall sze s stll , <25 MB, whch s often acceptable TABLE 5 Sample lnes of revsed partal decodng tables Index p = 0 p = 1 p = 2 Decmal Bnary PL PV S SL SV p PL PV S SL SV p PL PV S SL SV p C C C C C C C C C C C C C C C C C C C C 1 C C C 1 C C 1 C

11 Usefulness of Fbonacc Codes 11 It should be noted that the valdty of the above decodng procedure depends on the precson used to represent floatng pont numbers n the program If sngle precson s used (32 bts of whch 23 bts are the mantssa), the accumulated error when rasng φ (3) to the power j wll yeld wrong values for j 24 For a gven number, a growng number of dfferent Fbonacc numbers s used n ts representaton, and here agan the absolute value of the error ntroduced by lower precson adds up untl t may exceed 05, so that roundng wll not gve the requred nteger We have checked all the numbers up to the frst error and found that t occurs for = If the alphabet to be dealt wth s smaller, then sngle precson s enough However, t seems that for larger alphabets (such as all the examples n Table 3), double precson (64 bts, 52- bt mantssa) must be used However, double precson wll not only ncrease substantally the sze of the tables, t s also much more tme consumng A compromse could be a hybrd approach, usng double precson only for all the offlne evaluatons, that s, for the all values n the tables Thus up to φ(3) 8, we shall use double precson to calculate the value, but actually store them n the table n four bytes only; all the onlne evaluatons (lnes 5 and 8) are done wth sngle precson Table 6 brngs the ndces of the frst wrong elements for the gven precson, as well as the accordng values of a (3) and φ (3) The mddle lne, headed 64/32, corresponds to the hybrd case For double precson, we checked all the values up to F (3) 39 > 1296 bllon, and dd not fnd any roundng error In all these cases, 0392 < error < 0247, where error s the dfference between the correct (nteger) value and the value based on powers of φ (3) 333 Elmnatng multplcatons One may stll object that the use of multplcatons, floatng pont numbers and roundng can have a negatve mpact on the processng tme One can get rd of all these at the cost of some addtonal tables The dea s to replace the multplcatons wth Ph[oldSL] n lnes 5 and 8 by pre-calculated values of shfted Fbonacc numbers One thus prepares a two-dmensonal table Fb3[ndex,shft], ndex runnng over all the possble PV values, and shft over the possble shfts, that s, from zero to the length of the longest possble prefx of a codeword Accordng to our earler notaton, the bt strng x 1 x k represents the nteger TABLE 6 Influence of precson on the correctness Index of frst Precson a (3) φ (3) wrong codeword / k =1 x F (3) Fb3[ndex,shft] wll contan the correspondng value obtaned by shftng all the bts by shft bts to the rght, e Fb3[ndex,shft] = k =1 x F (3) +shft Table 7 dsplays some sample lnes and columns of the Fb3[ndex,shft] table The last two lnes are a specal case, to be explaned below Usng ths table, the algorthm of Fg 5 can be modfed by replacng lnes 5 and 8, respectvely, by 5 oldsv oldsv + Fb3[PV, oldsl] and 8 ndex oldsv + Fb3[PV, oldsl] + Start[3 + oldsl + PL] Note that all the values are ntegers now; therefore no roundng s necessary and there are no multplcatons The tables Tab[] can be smplfed and all float numbers be replaced by the correspondng ntegers, usng for each only 1 nstead of 4 bytes To adapt the modfed algorthm also to the specal cases when the separatng 111 s splt by byte boundares, note that when p = 1 and the current codeword starts wth 1, then the rghtmost 1-bt of the prevous byte contrbuted the amount of F (3) oldsl to oldsv, but ths addton was erroneous and should now be corrected For p = 2, n case the current codeword starts wth 1, the erroneous addton s F (3) oldsl 1 + F (3) oldsl As above, the correct value of PL n these cases s p The correcton factors wll be accommodated n the Fb3 table, snce t s used n lnes 5 and 8 Two new lnes are added to the TABLE 7 Sample lnes and columns of the Fb3[ndex,shft] table Index Shft Decmal Fb-bn p = p =

12 12 ST Klen and M Kopel Ben-Nssan tables, ndexed M + 1 and M + 2, where M s the ndex of the hghest of the PV values For k = 8, the hghest number that can be represented n the Fb3 representaton s , correspondng to M = 148 and for k = 16, the largest number s , representng M = The new entres are defned as Fb3[M + 1,s]= F (3) s Fb3[M + 2,s]= F (3) s 1 F (3) s All that remans to be done s then to defne, n the Tab tables, the value PV as M + p n case PL= p Table 8 shows a few sample lnes of the updated Tab tables The reducton n processng tme and n the sze of the Tab tables comes at the prce of storng the addtonal Fb3 tables, the number of whch depends on the sze of the alphabet, N There s one table for each possble shft, and the number of shfts s the sze n bts of the largest nteger, whch s 16, 24 and 28 for N = 10 4,10 5 and 10 6, respectvely Table 9 summarzes the space needed for the varous decodng methods for dfferent szes N of the alphabet, and for k = 8or 16 Full tables refers to the basc partal decodng tables of Secton 331 and the values for t are lower bounds As the sze s at least proportonal to N, ths method s not feasble for larger alphabets The columns headed Mult correspond to the algorthm of Fg 5 usng multplcatons and floatng pont numbers n the tables Ther space does not depend on N Fnally, the last column for each value of N s for the method usng n addton to the partal decodng tables, also the Fb3 tables The addtonal space s O(log N) As can be seen, the last two methods have reasonable space requrements, and for low values of N, the last method, whch s also faster than the prevous one, may even requre less space For larger alphabets, the revsed partal decodng tables wth floats or ntegers offer a tme/space tradeoff To emprcally compare tmng results, we chose the followng nput fles of dfferent szes and languages: the Bble (Kng James verson) n Englsh, and the French verson of the European Unon s JOC corpus, a collecton of pars of questons and answers on varous topcs used n the arcade evaluaton project [25] The dctonary of the Bble was generated after strppng the text of all punctuaton sgns, whereas the French text has not been altered and every strng of consecutve nonblanks was consdered as a dctonary term; for example, TABLE 8 Sample lnes of updated, nteger only, partal decodng tables Index p = 0 p = 1 p = 2 Decmal Bnary PL PV S SL SV p PL PV S SL SV p PL PV S SL SV p C C C C C C C C C C TABLE 9 Requred storage space for dfferent decodng methods N = N = N = Full tables Mult Fb3 Full tables Mult Fb3 Full tables Mult Fb3 k = 8 30 M 10 K 10 K 300 M 10 K 15 K 3 G 10 K 214 K k = G 25 M 19 M 75 G 25 M 26 M 750 G 25 M 34 M

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