Lossless Data Compression


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1 Lossless Data Compresson Lecture : Unquely Decodable and Instantaneous Codes Sam Rowes September 5, 005 Let s focus on the lossless data compresson problem for now, and not worry about nosy channel codng for now. In practce these two problems are handeled separately,.e. we frst desgn an effcent code for the source (removng source redundancy) and then (f necessary) we desgn a channel code to help us transmt the source code over the channel (addng redundancy). Assumptons (for now):. the channel s perfectly noseless.e. the recever sees exactly the encoder s output. we always requre the output of the decoder to exactly match the orgnal sequence X. 3. X s generated accordng to a fxed probablstc model, p(x) We wll measure the qualty of our compresson scheme (called a code) by examnng the average length of the encoded strng Z, averaged over p(x). Recall: Mathematcal Setup Start wth a sequence of symbols X = X, X,..., X N from a fnte source alphabet A X = {a, a,...}. Examples: A X = {A, B,..., Z, }; A X = {0,,,..., 55}; A X = {C, G, T, A}; A X = {0, }. Encoder: outputs a new sequence Z = Z, Z,..., Z M (usng a possbly dfferent code alphabet A Z ). Decoder tres to convert Z back nto X. In compresson, the encoder tres to remove source redundancy. In nosy channel codng, the encoder tres to protect the message aganst transmsson errors. We almost always use A Z = 0, (e.g. computer fles, dgtal communcaton) but the theory can be generalzed to any fnte set. Encodng One Symbol at a Tme To begn wth, let s thnk about encodng one symbol X at a tme, usng a fxed code that defnes a mappng of each source symbol nto a fnte sequence of code symbols called a codeword. (Later on we wll consder encodng blocks of symbols together.) We wll encode a sequence of source symbols X by concatenatng the codewords of each. Ths s called a symbol code. E.g. source alphabet s A X = {C, G, T, A}. One possble code: C 0; G 0; T 0; A 0 So we would have CCAT We requre that the mappng be such that we can decode ths sequence, no matter what the orgnal symbols were.
2 Notaton for Sequences & Codes A X and A Z are the source and code alphabets. A + X and A+ Z denote sequences of one or more symbols from the source or code alphabets. A symbol code, C, s a mappng A X A + Z. We use c(x) to denote the codeword to whch C maps x. We use concatenaton to extend ths to a mappng for the extended code, C + : A + X A+ Y : c + (x x x N ) = c(x )c(x ) c(x N ).e., we code a strng of symbols by just strngng together the codes for each symbol. I ll sometmes also use C to denote the set of all legal codewords: {w w = C(a) for some a A X }. Unquely Decodable & Instantaneous Codes A code s unquely decodable f the mappng C + : A + X A+ Z s onetoone,.e. x and x n A + X, x x c + (x) c + (x ) A code s obvously not unquely decodable f two symbols have the same codeword e, f c(a ) = c(a j ) for some j so we ll usually assume that ths sn t the case. A code s nstantaneously decodable f any source sequences x and x n A + for whch x s not a prefx of x have encodngs z = C(x) and z = C(x ) for whch z s not a prefx of z. Otherwse, after recevng z, we wouldn t yet know whether the message starts wth z or wth z. Instantaneous codes are also called prefxfree codes or just prefx codes. What Codes are Decodable? We only want to consder codes that can be successfully decoded. To defne what that means, we need to set some rules of the game:. How does the channel termnate the transmsson? (e.g. t could explctly mark the end, t could send only 0s after the end, t could send random garbadge after the end,...). How soon do we requre a decoded symbol to be known? (e.g. nstantaneously as soon as the codeword for the symbol s receved, wthn a fxed delay of when ts codeword s receved, not untl the entre message has been receved,...) Easest case: assume the end of the transmsson s explctly marked, and don t requre any symbols to be decoded untl the entre transmsson has been receved. Hardest case: requre nstantaneous decodng, and thus t doesn t matter what happens at the end of the transmsson. Examples Code A Code B Code C Code D a b c 0 0 Code A: Not unquely decodable Both bbb and cc encode as Code B: Instantaneously decodable End of each codeword marked by 0 Code C: Decodable wth onesymbol delay End of codeword marked by followng 0 Code D: Unquely decodable, but wth unbounded delay: 0 decodes as accccccc 0 decodes as bcccccc
3 More Examples Code E Code F Code G a b c d Code E: Instantaneously decodable All codewords same length Code F: Not unquely decodable e.g. baa,aca,aad all encode as 0000 Code G: Decodable wth sxsymbol delay. (Try to work out why.) A Check for Unque Decodablty The SardnasPatterson Theorem tells us how to check whether a code s unquely decodable. Let C be the set of codewords. Defne C 0 = C. For n > 0, defne C n = Fnally, defne {w A + X uw = v where u C, v C n or u C n, v C} C = C C C 3 Theorem: the code C s unquely decodable f and only f C and C are dsjont. We won t both much wth ths theorem, snce as we ll see t sn t of much practcal use. A Check for Instantaneous Codes A code s nstantaneous f and only f no codeword s a prefx of some other codeword. (e f C s a codeword, C Z cannot be a codeword for any Z). Ths s a prefx code. Proof: ( ) If codeword C(a ) s a prefx of codeword C(a j ), then the encodng of the sequence x = a s obvously a prefx of the encodng of the sequence x = a j. ( ) If the code s not nstantaneous, let z = C(x) be an encodng that s a prefx of another encodng z = C(x ), but wth x not a prefx of x, and let x be as short as possble. The frst symbols of x and x can t be the same, snce f they were, we could drop these symbols and get a shorter nstance. So these two symbols must be dfferent, but one of ther codewords must be a prefx of the other. Exstence of Codes Snce we hope to compress data, we would lke codes that are unquely decodable and whose codewords are short. Also, we d lke to use nstantaneous codes where possble snce they are easest and most effcent to decode. If we could make all the codewords really short, lfe would be really easy. Too easy. Why? Because there are only a few possble short codewords and we can t reuse them or else our code wouldn t be decodable. Instead, makng some codewords short wll requre that other codewords be long, f the code s to be unquely decodable. Queston : What sets of codeword lengths are possble? Queston : Can we always manage to use nstantaneous codes?
4 McMllan s Inequalty There s a unquely decodable bnary code wth codewords havng lengths l,..., l I f and only f I = l E.g. there s a unquely decodable bnary code wth lengths,, 3, 3, snce / + /4 + /8 + /8 = An example of such a code s {0, 0, 0, }. There s no unquely decodable bnary code wth lengths,,,,, snce /4 + /4 + /4 + /4 + /4 > We Can Always Use Instantaneous Codes Snce nstantaneous codes are a proper subset of unquely decodable codes, we mght have expected that the condton for exstence of a u.d. code to be less strngent than that for nstantaneous codes. But combnng Kraft s and McMllan s nequaltes, we conclude that there s an nstantaneous bnary code wth lengths l,..., l I f and only f there s a unquely decodable code wth these lengths. Implcaton: There s probably no practcal beneft to usng unquely decodable codes that aren t nstantaneous. Happy consequence: We don t have to worry about how the encodng s termnated (f at all) or about decodng delays (at least for symbol codes; for block codes ths wll change). Kraft s Inequalty There s an nstantaneous bnary code wth codewords havng lengths l,..., l I f and only f I = l Ths s exactly the same condton as McMllan s nequalty! E.g. there s an nstantaneous bnary code wth lengths,, 3, 3, snce / + /4 + /8 + /8 = An example of such a code s {0, 0, 0, }. There s an nstantaneous bnary code wth lengths,,, snce /4 + /4 + /4 < An example of such a code s {00, 0, 0}. Provng the Two Inequaltes We can prove both Kraft s and McMllan s nequalty by provng that for any set of lengths, l,..., l I, for bnary codewords: A) If I = / l, we can construct an nstantaneous code wth codewords havng these lengths. B) If I = / l >, there s no unquely decodable code wth codewords havng these lengths. (A) s half of Kraft s nequalty. (B) s half of McMllan s nequalty. Usng the fact that nstantaneous codes are unquely decodable, (A) gves the other half of McMllan s nequalty, and (B) gves the other half of Kraft s nequalty. To do ths, we ll ntroduce a helpful way of thnkng about codes as...trees!
5 Vsualzng Prefx Codes as Trees We can vew codewords of an nstantaneous (prefx) code as leaves of a tree. The root represents the null strng; each level corresponds to addng another code symbol. Here s the tree for a code wth codewords 0,, 00, 0: NULL Constructng Instantaneous Codes Suppose that Kraft s Inequalty holds: I = l Order the lengths so l l I. Q: In the bnary tree wth depth l I, how can we allocate subtrees to codewords wth these lengths? A: We go from shortest to longest, =,..., I: ) Pck a node at depth l that sn t n a subtree prevously used, and let the code for codeword be the one at that node. ) Mark all nodes n the subtree headed by the node just pcked as beng used, and not avalable to be pcked later. Let s look at an example... Extendng the Tree to Maxmum Depth We can extend the tree by fllng n the subtree underneath every actual codeword, down to the depth of the longest codeword. Each codeword then corresponds to ether a leaf or a subtree. Prevous tree extended, wth each codeword s leaf or subtree crcled: NULL 0 Short codewords occupy more of the tree. For a bnary code, the fracton of leaves taken by a codeword of length l s / l Buldng an Instantaneous Code Let the lengths of the codewords be {,,3,3}. Frst check: Our fnal code can be read from the leaf nodes: {,00,00,0} NULL
6 Constructon Wll Always Be Possble Q: Wll there always be a node avalable n step () above? If Kraft s nequalty holds, we wll always be able to do ths. To begn, there are l b nodes at depth l b. When we pck a node at depth l a, the number of nodes that become unavalable at depth l b (assumed not less than l a ) s l b l a. When we need to pck a node at depth l j, after havng pcked earler nodes at depths l (wth < j and l l j ), the number of nodes left to pck from wll be j l j l j j l = l j = = > 0 j Snce / l < I / l, by assumpton. = = Ths proves (A). UD Codes Must Obey the Inequalty Let l l I be the codeword lengths. Defne K = I = l. For any postve nteger n, we sum over all possble combnatons of values for,..., n n {,..., I}. K n =,..., n l l n We rewrte ths n terms of possble values for j = l + + l n : K n nl I N j,n = j= j N j,n s the # of sequences of n codewords wth total length j. If the code s unquely decodable, N j,n j, so K n nl I, whch for bg enough n s possble only f K. Ths proves (B). (For nstantaneous codes, the ntuton s that short codes use up ther subtree.)
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