Patent Application: US-13743702-A

Abstract:
a method for forming a maximal tree by searching statistical data of symbols from training data . a table is formed including a string having first and second ends , and a symbol of the training data in the first end and a minimal number of preceding symbols in the training data to make a context unique for the symbol . the contexts are sorted in lexical order with priority on the first end . a device for forming a maximal tree by searching statistical data of symbols from training data . the device forms a table including a string having first and second ends , and a symbol of the training data in the first end and a minimal number of preceding symbols in the training data to make the context unique for the symbol . the device sorts the contexts in lexical order with priority on the first end .

Description:
in the following the construction of the maximal tree machine out of a training set of data will be described . it will be done in terms of the non - restrictive example training data x n = λbbccacbabcaaaab , where the alphabet consists of the symbols { a , b , c }, sorted alphabetically together with a special symbol λ , [ 1 ], appended to the alphabet as the first symbol . the construction of the maximal tree machine can be performed by using a suitable training data and the tree machine can then be stored both to the encoder 1 ( fig8 ) and decoder 2 . it is obvious that the training data can be different in different embodiments of the present invention . in the prefered embodiment the alphabet is the set of the first 256 integers , each written in binary as an 8 - bit ‘ byte ’. the longest prefix of each symbol in x n , such as λbbcca of the first occurrence of symbol c , gives the context where this symbol occurs . there are so - called suffix algorithms , [ 2 ], which can sort all the n suffixes lexically in time linear in n . the following table illustrates the result of an analogous prefix algorithm , which sorts the prefixes lexically with preference to the right . the middle column shows indices ( or lengths ) of the prefixes , pointing to the string x n . the last column shows the symbols following their prefix . notice that the symbol following the entire string is taken as λ . the string defined by the last column , namely y n = bbaabacbλccabaca , is the burrows - wheeler transform of the original string x n . it has the crucial property that it determines x n and the entire table . because even the prefix sorted trees can be generated in time proportional to the length of the string n , very long strings can be processed . the maximal tree is obtained by retaining just enough of the rightmost symbols of each prefix to make the so - defined minimal prefixes , called from here on contexts , of the symbols in the last column distinct . the contexts define an up - down tree with the root at the top , and each context , when read from right - to - left , defines a path from the root to a leaf node . hence , in the example , illustrated in fig1 , the left - most leaf is the context λ , the next is the context aaaa and so on . in particular , the branches emanating from the root in left - to - right order are the right - most symbols of the contexts ; i . e ., the symbols λ , a , b and c in the alphabet . this is the maximal tree . there is exactly one symbol occurrence at each node of the maximal tree that is defined by a context . for instance , the symbol occuring in context aaaa is b and that occuring in context caaa is a and so on . fig1 shows the maximal tree for an example string , x n = λbbccacbabcaaaab . its representation in terms of the burrows - wheeler transformed string y n = bbaabacbλccabaca is given in table 1 . by collapsing adjacent rows into a common suffix a subtree of the maximal tree will be obtained . for instance , the leaves aaaa and caaa give rise to their father node aaa , in which both of the son - nodes &# 39 ; symbols occur , namely the set { a , b }. table 2 shows the maximal tree and the subtree defined by the contexts aa , bca , cca , cb and ac which may be identified with the root . the numbers in the last column are enough to describe the structure of the subtree : if the number is positive , there is one node in the subtree for the corresponding range of prefixes ; if the number is negative , there is no node for that range . the substrings in the third column falling between the horizontal lines give the occurences of the symbols at the contexts . note the crucial property that they partition the burrows - wheeler transformed string in the second column . the subtree is also shown in fig2 : the strings at the leaves together with the last symbol b at the root partition of the transformed string in fig1 . a node in the tree will be denoted by s , and n i | s for the number of times symbol i occurs at node s . hence n i | s = 0 if a symbol i does not occur at node s . write the maximal tree can be pruned by dropping the son nodes of a collection of father nodes while retaining the counts of the symbol occurrences at the father nodes . the father nodes then become leaves of the pruned tree . any such subtree t assigns an ideal code length to the training set as follows l t ⁡ ( x n ) = ∑ x ⁢ ⁢ log ⁢ n s ⁡ ( x ) n x | s ⁡ ( x ) ( 1 ) where x runs through the symbols in the training sequence x n and s ( x ) denotes the deepest node in the tree where symbol x occurs , except for the early symbols x 1 , x 2 , . . . whose contexts do not reach the leaves . the word ‘ ideal ’ is used because a real code length is an integer - valued number while the negative logarithm of a probability may not be so . however , the ideal can be achieved to a sufficiently good approximation in practice . there are other more important reasons why the ideal code length is not achievable in real coding . first , the decoder has not yet decoded the symbol x and does not know the steepest node where this symbol occurs . the second problem arises because some symbol counts may be zero . the first problem can be solved if the deepest node s ( x ) is replaced by the deepest node in the tree that can be found by ‘ climbing ’ the tree t thus : let x n = x 1 , . . . x t − 2 , x t − 1 , x t , x , . . . , where x = x t + 1 is the new symbol to be encoded . climb the tree by reading the prefix . . . x t − 1 , x t from right - to - left . hence , at the root take the branch marked by symbol x t . next , take the branch x t − 1 at this node and continue until a node is reached where the scanned symbol does not occur , or if it is the very first symbol x 1 in the string . the branches at the root are marked by all the symbols that occur in the training sample . while both the encoder 1 ( fig8 ) and the decoder 2 always find the same context by this climbing process it may well happen that the next symbol to be encoded and decoded does not occur at this context ; i . e ., its symbol count is zero . nevertheless we must assign a codeword even for such a symbol , which amounts to assigning a positive probability . this choice is very important in achieving a good compression , because it leaves smaller probabilities and hence longer code lengths for the symbols that do occur . assume that the number of symbols in the alphabet is d , and let the number of distinct symbols among the n s occurrences of symbols at node s be m . since each of these symbols must have been a new symbol at the time it occured first , m binary events “ the next symbol is a symbol not among the ns symbols ” have been observed in n s occurrences of this and the opposite binary event . a good way to assign probabilities to such binary events is to put the probability ( m + ½ )/( n s + 1 ) to a new symbol ; see for instance [ 5 ]. this reflects the belief that for small m and large n s the context s strongly favors repeated occurrences of the few symbols that already have been seen at this context . on the other hand , if m is close to d , then most of the symbols in the alphabet have already been seen in this context , which leaves few new symbols , and the probability of a symbol being new should again be small . in light of these considerations we set the probability of the unseen symbol i ; i . e . n i | s = 0 , as follows ; see also fig3 , in which the assignment of probabilities of symbols i is demonstrated , of which m distinct symbols are among the n symbols occuring at a node ( right branch ), or they are absent ( left branch ). for the already seen symbol i for which n i | s & gt ; 0 put these probability calculations will be illustrated for strings of length one and two over the alphabet { a , b , c }, where the occurrence counts at the node s are for n s = 2 , ( 2 , 0 , 0 ) and ( 1 , 1 , 0 ), with the examples : for n s = 1 : { circumflex over ( p )} a | s ={ circumflex over ( p )} b | s ={ circumflex over ( p )} c | s = 1 / d = ⅓ ( 4 ) p ^ b | s = p ^ c | s = 1 + 1 / 2 3 × 1 2 = 1 / 4 ( 5 ) { circumflex over ( p )} a | s =( 1 − ½ )= ½ ( 6 ) the maximal tree will be pruned to a subtree with two distinct strategies in mind . the first objective , which is the preferred one , is to search for a subtree such that its ideal code length does not exceed that of the shortest possible , obtainable with the best subtree of the maximal tree , by more than a given percentage p , taken as a parameter . the second objective is to require the subtree to assign the shortest ideal code length to the training data among all that take no more than a desired amount of storage space , determined by the number of nodes together with the symbol occurrence counts at the nodes . since not all the nodes of the subtree have the same number of symbol occurrences , the storage space is not determined by the number of nodes alone . both goals can be reached optimally by modifying the algorithms [ 3 ] and [ 5 ], which were designed for finding complete binary mdl ( minimum description length ) subtrees , [ 4 ]. these require calculation of a code length at the nodes which includes the description of the parameters needed to encode the tree itself . in a system according to the present invention the subtree will be communicated to the decoder , and its specification need not be a part of the encoded string unlike in mdl trees . a further problem arises from the fact that in a method according to the present invention the trees are incomplete with some symbols not occuring in all nodes , which change the probabilities even of the seen symbols as discussed above . finally , subtrees will be seeked with restrictions on either the size of the subtree or its performance . the first preliminary algorithm finds the overall optimal subtree , denoted by t 0 of the maximal tree without any restrictions placed on the subtrees . 1 . initialize : for every leaf node ( a node without any son nodes ) compute from equations ( 2 ) and ( 3 ). set l ( s ′)= 0 for the other nodes s ′. 2 . consider a father node s and its son nodes sj . the idea is to calculate l ( s ) for each node as the shortest ideal code length for all the symbols occurrences at the son nodes and their descendants . for s ranging over all father nodes of the leaves calculate for each leaf sj the code length of the occurrences of the symbols at this node , obtained with use of the probabilities at the father node : the intent is to compare this with the optimal code length l ( sj ) for exactly the same symbol occurrences at the son node sj . if then the father node &# 39 ; s probability assignment is not worse , and we prune the son node sj ; else we leave it . replace l ( s ) by l ( s )+ i ( sj ), if the condition ( 9 ) is satisfied , and by l ( s )+ l ( sj ), otherwise . this process is repeated for the father nodes of all the so far processed father nodes until the root is reached . the subtree t 0 of the maximal tree resulting from algorithm a assigns the smallest ideal code length to the training data , because we pruned all the descendants where i ( sj )≦ l ( sj ) and hence which would have encoded the symbols worse than their father nodes . however , this subtree may have more nodes than desired . if we prune the tree further we would have to drop a leaf where i ( sj )& gt ; l ( sj ), and the ideal code length will necessarily increase . the question arises how to prune the maximal tree to the desired otm so as to lose the least amount of the smallest ideal code length . we describe first a pruning strategy that gives a subtree such that its ideal code length does not exceed that of the t 0 by more than a given percentage p , taken as a parameter . this is algorithm a except that the condition ( 9 ) is replaced by the following i ( sj )≦ max {( 1 + p ) l ( sj ), l ′ ( sj )} ( 10 ) where l ′( sj ) is the code length in node sj in the maximal tree . the parameter p is easy to select : either we want to select it once and for all regardless of the size of the final otm , or we simply increase or reduce an initial choice depending on whether the pruned otm is larger or smaller , respectively , than desired . a flow chart of the algorithm is given in fig4 , and the pruning process is illustrated in fig5 , where the various code lengths are obtained from equations ( 2 )–( 3 ); see also equations ( 4 )–( 8 ). in fig5 the son node c will be pruned if parameter p ≧ 0 . 099 this is just like algorithm b except that the condition ( 10 ) is replaced by the following where the threshold parameter t is somewhat more difficult to adjust . it can be found by a binary search in the range ( 0 , max { i ( s )− l ( s )}): first pick t as half the maximum range , then set t one quarter or three quarters of the maximum range , if the pruned tree machine is too small or too large , respectively , and continue until the desired size is reached . the final storage space of the optimal tree otm , which was determined from the training data by constructing the maximal tree and pruning it , can be reduced by a novel storage technique . this is very important , because it determines the coding speed of new data strings as described below , and its storage space represents the overhead that has to be paid both by the encoder and the decoder . the idea is to find a new data string , shorter than the training data string , and its burrows - wheeler transform , which in turn determines the structure of the otm in terms of pointers . it should be recalled that the substrings following the contexts partition the burrows - wheeler transform of the training data , so that the contexts can be defined in terms of pointers . such a string will be called a generating string . before giving the general algorithm it will be illustrated with the example subtree in table 2 . the new data string must include all the contexts from the subtree , that is , λ , aa , ba , bca , cca , λb , cb , λc and ac . the contexts λ , λb and λc are needed to store the counts of the pruned children of the non - leaf nodes ε ( root ), b and c , respectively . clearly , if any permutation of the contexts will be taken and concatenated , a string of twelve symbols that satisfy the condition will be formed . however , the resulting string is not the shortest possible . considering the contexts cb and bca , the shortest string containing both of the contexts is not cbbca — their concatenation — but cbca . any time when two contexts where the first one ends with the same substring as the latter begins are concatenated , it is not necessary to write that repeating substring twice . in the case of cb and bca , the substring b will be written only once , saving one symbol . the problem of finding the shortest superstring containing all the given strings is well known and solvable , see [ 8 ]. a shortest superstring for the strings λ , aa , ba , bca , cca , λb , cb , λc and ac is λccaacbcaλba . the burrows - wheeler transform of the superstring defines the generating string cbcλλaaccbaa , shown in table 3 , which also shows the prefixes of the data string , indices of the prefixes and the occurrence counts in columns one , two and four , respectively . 1 . form the set of contexts s by the following rules . include the context of every leaf node of the tree . for each node s that has exactly one pruned child js , include the context js . for every node s that has more than one pruned children , include context λs . 2 . generate the shortest exact common superstring x k for the set of strings s ( see [ 8 ]). 3 . convert x k to the generating string y k by burrows - wheeler transform . exit . the storage consists of two parts , the first describing the structure of the optimal tree machine , and the second needed to store the counts of the symbols that occur in the leaves of the tree . the structure of any subtree can be conveniently stored by storing the generating string . it can be stored e . g . in the memory means 3 of the encoder and also in the memory means 4 of the decoder . at each leaf s of the optimal tree there is a sequence of positive integer - valued counts n 1 | s , . . . n m s | s , sorted in the increasing order , of the m s symbols that occur at this node . it will be necessary to store these symbols and their counts . the symbols themselves can be stored with m s log | a | bits , where a denotes the alphabet . hence , if the alphabet is the ascii symbols then the storage requirement is m s bytes and the total of then the increasing count sequence will be stored . using the same method as in storing the pointers the counts can be stored with log ⁡ ( n m s ❘ s + m s m s ) ≅ ( n m s ❘ s + m s ) ⁢ h ⁡ ( m s n m s ❘ s + m s ) ( 12 ) ∑ s ⁢ log ⁡ ( n m s ❘ s + m s m s ) the otm is determined from a large body of data , characteristic of the strings we wish to encode . the encoding is done recursively , symbol for symbol , as the string to be encoded is being scanned by the encoder 1 . the symbol occurrence counts at the states of the otm have been converted to a huffman code or they are used to define the addends of an arithmetic code . the encoding operations are as follows : 1 . encode the first symbol of the message string x 1 x 2 . . . with the huffman / arithmetic code at the root of otm . 2 . recursively , when t first symbols have been encoded , climb the otm until either a leaf s *= x t , x t − 1 , . . . , x t − k is met or s *= x t , x t − 1 , . . . , x 1 , in which case t is too small for a leaf to be reached . encode the next symbol x t + 1 with the huffman / arithmetic code at the node s *. continue until the entire string has been processed . 1 . decoding the first symbol x 1 from the code string at the root of the same otm . 2 . recursively , when t first symbols have been decoded , climb the otm until either a leaf s *= x t , x t − 1 , . . . , x t − k is met or s *= x t , x t − 1 , . . . , x 1 . decode the next symbol x t + 1 from the remaining code string at the node s *. the encoded string can be communicated from the encoder 1 to the decoder 2 by an appropriate communication medium , for example , by a communication network 5 ( fig8 ), or the encoded string can be stored to a storage medium 6 , such as a diskette , a cdrom , a dvd , a video tape , etc . the encoded string can then be read from the storage medium 6 by the decoder when the decoding takes place . in the following the encoding operations will be illustrated with an example in fig6 . for greater coding speed the otm may be converted to a finite state machine by viewing each context defined by a node as a state . then each symbol effects a state transition , which can be tabulated . the described method will be applied to english text as training data to obtain the otm , with which text files of different size will then be compressed . the results of the compression algorithm , called ds , was compared with some of the best known data compression techniques in fig7 . it can be seen that the compression method according to an advantageous embodiment of the present invention is better for files up to size about 100 kilobytes , and significantly better for shorter files . it is obvious that the training data need not to be english text but any text data can be used as the training data . although in the above description of the invention the training data has been handled from left to right , i . e . from the beginning to the end , it is also possible to handle the training data other way round , i . e . from right to left . the present invention is not limited solely to the above described embodiments , but it can be modified within the scope of the appended claims . balkenhol , b . and kurtz , s ., “ universal data compression based on the burrows - wheeler transformation ”, ieee trans . on computers , vol . 49 , no . 10 , october 2000 . mccreight , “ a space - economical 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