ORDINAL COMPUTING WITH 2-ARY CODE IN COMBINATORICS CODING METHOD

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1 20 June 20. Vol. 52 No JATIT & LLS. All rights reserved. ISSN: E-ISSN: ORDINAL COMPUTING WITH 2-ARY CODE IN COMBINATORICS CODING METHOD RUSHENG ZHU, 2 ZHONGMING YONG, Department of Information Science and Technology, Heilongjiang University, Harbin,China 2 Department of Computer Science and Technology, Heilongjiang University, Harbin,China hlju202zrs@6.com, 2 yzm5757@yahoo.com.cn ABSTRACT All characters in any character sequence can constitute a dictionary space by permutation and combination and e character sequence can be replaced by e ordinal of is sequence in e dictionary space. This coding meod is named combinatorics coding meod. Combinatorics coding meod is expatiated firstly in is paper and ordinal computing in different data system is analysized. Combinatorics coding meod can be done by different code modes. In ese code modes, combinatorics coding meod wi 2-ary code has particular property. Ordinal computing meod wi 2-ary code is advanced and e characteristics of ordinal wi 2-ary code are researched. Combinatorics coding technology wi 2-ary code has reference to e intensive study of e combinatorics coding meod. Keywords: Data Coding, Source Coding, Combinatorics Coding, Ordinal, Max Ordina, Whole Max Ordinal. INTRODUCTION Today, e information technology is developing very much. As one of e basic tools and researched problems, data coding technology plays an important role in all kinds of studies and applications. It is well known about Huffman coding, algorim coding and dictionary coding [-]. There are ree embranchments in coding technology: source coding, channel coding and secrecy coding. The main task of source coding is to compress data. Channel coding is used to improve reliability of information transmission. Secrecy coding prevents e information from being filched and e used technology is encryption. Data compression technology is of great significance to information disposing, data storage, and data transmission. Combinatorics coding is a new data coding meod based on Combinatorics. At first, Combinatorics coding is only used in researching data compression [4-7]. In fact, is kind of coding technology can be used in many fields such as data encryption and error detection/error correction. It is of great significance to information disposal, information storage and information transmission. 2. THE BASIC THEORY OF COMBINATORICS CODING To e sequence which leng is n, if e number of e different elements is m and e number of e i element is n i. Thus e sum of e different permutation sequences at e different element in each sequence has e same number wi is sequence can be expressed by formula (): m P = n!/( ( ni!)) () i= All ese permutation sequences can constitute a permutation space. This space is an ordered dictionary space. Any permutation sequence in is dictionary space has its position number. This position number is named ordinal. The relation between e character sequence space and e corresponding ordinal space is utilized in combinatorics coding meod. It is to say at all characters in any character sequence can constitute a dictionary by full permutation. Then combinatorics coding meod comes into being. In fact, e ordinal of e sequence indicates e number of e oer sequences in front of is sequence in e dictionary space. 84

2 20 June 20. Vol. 52 No JATIT & LLS. All rights reserved. ISSN: E-ISSN: To e same sequence, e different ordinal can be obtained based on e different element order which has different elements. It is to say at e ordinal of e sequence is not exclusive. This element sequence depended on is named benchmark sequence. Max ordinal of e sequence space can be computed rough which e number of e sequences in e ordinal space subtracts one. When e number of each element in e sequence is equal, whole max ordinal is obtained. In e optimizing procedure of ordinal computing, ordinal computing is depended on e value of max ordinal and max ordinal computing is depended on e value of whole max ordinal[ ]. In combinatorics coding meod, a benchmark sequence needs to be agreed on in advance between e coder and e decoder. When e coder works, e ordinal of e sequence can be computed out based on is benchmark sequence. When e decoder works, e sequence can be deduced by e ordinal based on e same benchmark sequence. To e j element in e sequence which has n elements, if is element is e same as e i element in benchmark sequence, it is predicated at ere are existed permutation and combination sequences before e current sequence involved i- elements. To ese i- elements, e permutation and combination number of e x element (e element position in ese i- elements is x) occupied e j position can be expressed by S. It is shown in equation (2). 2 S = C C C w w w n j n j w n j ( w+ w2) C C C wx wx wx+ n j ( w+ w2+ + wx 2) n j ( w+ w2+ + wx ) n j ( w+ w2+ + wx) wii ii ii 2 ii C n j wq C n j wq q= q=... (2) w expresses e number of each element. To e j element in e sequence which has n elements, e sequence sum involved i- elements before j e element can be computed by e formula (). i S () x= At last, e position (ordinal) of e sequence which has n elements in e whole space can be expressed by e formula (4). n i S (4) j= x= The basic idea of combinatorics decoding is at suppose e element in e inspected position is a certain element, e corresponding permutation and combination value p is computed out based on benchmark sequence according to is suppose. Compared p wi ordinal, if ordinal is not less an p, en e permutation and combination value p of e next element is computed out based 2 on benchmark sequence. It needs to judge at ordinal is less an p + p 2 or not. Until it finds out at ordinal is less an p + p p r. This moment, e element corresponding p should be e element in e r inspected position. At last, e new ordinal needs to be computed in order to deduce e next position element according to e formula (5): new ordinal= old ordinal-( p + p + + p 2 r- ) (5) When e ordinal is 0, e algorim is over. If ere are some remainder data, ese remainder data can be appended to e decoded sequence based on benchmark sequence. In fact, e process of combinatorics decoding is e inverse process to combinatorics coding. Probing step by step meod is adopted to parse correlative character. The most important step of probing meod is also ordinal computing. III. THE RELATION BETWEEN THE SPEED OF ORDINAL COMPUTING AND DATA SYSTEM In e procedure of ordinal computing, e speed of ordinal computing can be affected by different data system. It adopts 2 6 system to computing ordinal in e figure. In is example, ere are 256 different elements in benchmark sequence ( m =256). The higher data are stored in e left nodes and e lower data are stored in e right nodes. There are nodes in e ordinal 6 of e sequence which leng is 200k in 2 system, is ordinal can be expressed by e figure : 85

3 20 June 20. Vol. 52 No JATIT & LLS. All rights reserved. ISSN: E-ISSN: bytes Fig. Ordinal Number in 6 2 System If it adopts 2 2 system to computing e ordinal of e same sequence and e value range in each node of e ordinal is from 0~ to 0~ 2 2 -, en e number of e nodes is halved. If e number is odd, for example, e nodes number of e ordinal in figure is and it is an odd, so e value of e highest node is less an 256, is node can be extended to 4 bytes. The oer nodes which e leng of each node is 2 bytes can be merged into one node in pairs. Thus, in 2 2 system, e number of e nodes of e ordinal is 5. It is shown in figure bytes Fig.2 Ordinal Number in System The experiment manifests at e speed of ordinal computing is advanced after adopting 2 2 system, because e number of e nodes decreases and e leng of e ordinal is halved, ese mean at e computing times are halved in e computing procedure. To e main procedure of e computing, e assistant computing in e algorim can be ignored. So e computing speed is advanced double. IV. ORDINAL COMPUTING WITH 2-ARY CODE IN COMBINATORICS CODING METHOD Sequence ordinal can be computed by e different code system. At present, e research on combinatorics coding meod is mainly wi 256- ary code. Because e smallest stored unit of e file information is byte. Then e number of e different elements in benchmark sequence is 2 8. It is 256-ary code. In is paper, combinatorics coding is researched wi 2-ary code. It is to say at e number of e different elements in benchmark sequence is only two (e value of m is two and e elements are 0 and ). Thus ere are only two benchmark sequences: 0, and, 0. To e sequence 00, ere are two different elements: e number of 0 is 2 and e number of is 4. The number of sequences which are all composed by 0(e number is 2) and (e number is 4) can be computed by e formula (): m P = n!/( ( n!)) i = i = 6!/(2!* 4!) =5 According to e order of 0 first and last, e space of dictionary can be obtained as follows: 0) 00 ) 00 2) 00 ) 00 4) 00 5) 00 6) 00 7) 00 8) 00 9) 00 0)00 )00 2) 00 ) 00 4)00 In above instance, e number of e whole permutations about 0(e number is 2) and (e number is 4) is 5. Because it is counted from 0, e max ordinal is 4. The ordinal of 00 can be computed as follows: Step : The first element of e sequence is, e number of sequences at have e form of 0* and at are in front of is sequence can be computed by e formula (2). Of course, e number of 0 is 2 and e number of is 4 in ese sequences. 4 CC = Because e ordinal computing is based on 2- ary code, it only needs to compute once. This means at e formula () becomes vestigial wi 2-ary code. Step 2: The second element of e sequence is 0. There is no oer element in front of is element in benchmark sequence. So ere is no computation. Step : The rd element is, en each sequence in front of is sequence has e form of 00*. In each sequence, e number of 0 is 2 and e number of is 4 too. The number of ese sequences is C =. Step 4: The four element is 0. There is no element in front of is element in benchmark sequence. So ere is no computation. Step 5: The fif element and e six element are. But now ere is no 0 element in e sequence. So it doesn t need to handle. 86

4 20 June 20. Vol. 52 No JATIT & LLS. All rights reserved. ISSN: E-ISSN: At last, all data are added and e ordinal of e sequence is obtained. The ordinal is 6. The ordinal of e sequence 00 is computed out based on e benchmark sequence (0 first and last). If e order of e benchmark sequence is contrary, 5 kinds of permutations can be obtained still. Of course, e order of ese permutations is contrary compared wi e case at e benchmark sequence is 0 and. It is shown as follows. 0) 00 ) 00 2) 00 ) 00 4) 00 5) 00 6) 00 7) 00 8) 00 9)00 0) 00 ) 00 2) 00 ) 00 4) 00 In above instance, e ordinal of e sequence 00 is 8 in e set of e whole permutations based on e benchmark sequence wi and 0. Its computing procedure is similar to e case at e benchmark sequence is 0 and. To e same sequence 00, e ordinal is 6 based on e benchmark sequence of 0 and, but e ordinal is 8 based on e benchmark sequence of and 0. Luckily e sum of two ordinals equals to e max ordinal 4. Thus it can be seen at, to e same sequence, its ordinal value is not unique. There are e close relation to e ordinal and e selection of benchmark sequence. To e same sequence, if e orders of two benchmark sequences are contrary, e sum of eir ordinals is max ordinal. The original sequence can be deduced based on frequency(such as e number of 0 is 2 and e number of is 4) benchmark sequence(such as 0 and ) and ordinal (such as 6). The procedure of decoding is described as follow: Step : Decode e first element of e sequence. () If e first element of e sequence is 0, en ere are one 0 and four in e follow five elements. So e number of us sequence can be computed as follow: 4 CC = Because 5 is less an 6, e supposition is not right based on decoding idea. It is to say at e first element is not 0. (2) If e first element of e sequence is, en ere are two 0 and ree in e follow five elements. So e number of us sequence can be computed as follow: 2 CC = =5,e sum is more an 6, so e first element of e sequence is. Adjust ordinal: 6-5=. In fact, it can be directly judged at e first element must be based on e result of () in e step. Because ere are only two elements in benchmark sequence, if it is not 0, it must be. It is to say at each step needs only computing once in 2-ary code of combinatiorics coding technology. It means at e speed of decoding in 2-ary code can be advanced. Step 2: Decode e second element of e sequence. () If e second element of e sequence (its form is *) is 0, en ere are one 0 and ree in e follow four elements. So e number of us sequence can be computed as follow: CC = is more an, e supposition is right based on decoding idea. It is to say at e second element is 0. This moment e ordinal doesn t need to be adjusted. Step : Decode e ird element of e sequence. () If e ird element of e sequence (its form is 0*) is 0, en ere are no 0 and ree in e follow ree elements. So e number of us sequence can be computed as follow: C = is not more an, e supposition is not right based on decoding idea. It is to say at e ird element is not 0. It can be directly judged at e ird element must be based on e result of () in e step. Adjust ordinal: -=0. This moment e ordinal is 0. Three elements in e front of e sequence are 0, ere are remained elements: one 0 and two. Based on e order of benchmark sequence, 0 is filled to 87

5 20 June 20. Vol. 52 No JATIT & LLS. All rights reserved. ISSN: E-ISSN: e tail of e decoded sequence. So e final decoded sequence can be obtained: 00. Thus e original sequence can be successfully decoded based on frequency benchmark sequence and ordinal. V. CHARACTERISTIC OF ORDINAL WITH 2-ARY CODE IN COMBINATORICS CODING METHOD At present, e research on combinatorics coding is mainly aimed at 256-ary code [8-0]. But in relation to 256-ary code or oer coding meod, 2- ary code has its distinctive character. At first, to e computing speed in combinatorics coding meod, decoding speed is slower an coding speed. The more e number of different elements is, e slower e speed of decoding (relation to coding) is. It is apparent at e speed difference of coding and decoding is e smallest wi 2-ary code. Coding technology can be optimized in oer existed coding meod. The paper [] and e paper [2] expatiate how to make combinatorics coding reduce computing speed from exponent level to n 2 level. In is paper, combinatorics computing becomes proportion computing. But at present, ere is no good optimization meod for decoding in combinatorics coding technology. Because decode meod is to probe step by step. It means at if one element doesn t be decoded, en all elements follow is element can t be decoded. Thus merging computing can t be done. Compared e code/decode procedure wi 256-ary code to e code/decode procedure wi 2-ary code in is paper, it can be seen at e speed of decoding is almost equal to e speed of coding wi 2-ary code,while e speed of decoding is slower more an e speed of coding wi oer code meod. Secondly, e result of existed experiment makes clear at e difference of e ordinal leng and e sequence leng wi 2-ary code is smaller an e difference of e ordinal leng and e sequence leng wi oer code. This conclusion can be estimated rough e formula (). For example, wi 2-ary code, when e sequence leng n is 8, it is to say at each element occupies bit and e sequence occupies one byte, e whole max ordinal of e sequence can be computed by e follow procedure: P = n!/(( n/ 2)!* ( n/ 2)!) = 8!/(4!* 4!) = 8!/(4!* 4!) =70 But wi 4-ary code, each element of e sequence occupies 2 bits, while e sequence itself still occupies one byte. Now, e leng of e sequence is 4 and its whole max ordinal can be computed: P = n!/(( n / 4)!* ( n / 4)!* ( n / 4)!* ( n / 4)!) = 4!/((4 / 4)!* (4 / 4)!* (4 / 4)!* (4 / 4)!) =24 p > p, log( p) >= log( p ), so e leng of p is not less an e leng of p. In e most case, e leng of p is more an e leng of p. VI. CONCLUSION The ordinal computing meod wi 2-ary code in combinatorics coding technology is advanced and e characteristic of 2-ary code ordinal is researched in is paper. The research makes clear at e difference between coding speed and decoding speed is e smallest in binary combinatorics coding. Furermore, e difference between ordinal leng and sequence leng wi 2- ary code is smaller an at wi oer code. 2-ary code/decode technology is significant to research intensively on combinatorics code meod. 88

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