10 Lempel-Ziv Compression
\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\TextOrMath }[2]{#2}\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\renewcommand {\vec }{\boldsymbol }\)
\(\newcommand {\Edge }{\ensuremath {\,\textemdash \,}}\)
\(\newcommand \Const [1]{\text {\textsf {#1}}}\)
\(\DeclareMathOperator {\lerp }{lerp}\)
\(\DeclareMathOperator {\bitlen }{bitlen}\)
\(\DeclareMathOperator {\sign }{sign}\)
\(\newcommand {\I }{\mathrm {i}}\)
\(\newcommand \AND {\mathbin {\&}}\)
\(\newcommand \OR {\mathbin {|}}\)
\(\newcommand \XOR {\mathbin {{}^{\wedge }}}\)
\(\newcommand \shl {\ll }\)
\(\newcommand \shr {\ggg }\)
\(\newcommand \asr {\gg }\)
\(\newcommand \NOT {\ensuremath {\mathord {\sim }}}\)
\(\newcommand {\isep }{\mathrel {{.}\,{.}}}\)
\(\newcommand {\Id }[1]{\mathit {#1}}\)
\(\newcommand {\const }[1]{\mathsf {#1}}\)
\(\newcommand {\algorithmname }[1]{\text {\textsc {#1}}}\)
\(\newcommand {\bits }[1]{\text {#1}}\)
\(\newcommand {\hexa }[1]{\mathtt {0x#1}}\)
\(\newcommand {\num }[1]{#1}\)
\(\newcommand {\qed }{\quad \square }\)
\(\newcommand {\idiv }[2]{\lfloor #1/#2\rfloor }\)
\(\newcommand \attribdot {\ensuremath {\mkern 1.5mu.\mkern 1.5mu}}\)
\(\newcommand \attribxr [2]{#1\attribdot \text {#2}}\)
\(\newcommand \attribir [2]{\Id {#1}\attribdot \text {#2}}\)
\(\newcommand \attribii [2]{\Id {#1}\attribdot \Id {#2}}\)
\(\newcommand \textsc [1]{#1}\)
\(\require {colortbl}\)
\(\let \LWRorigcolumncolor \columncolor \)
\(\renewcommand {\columncolor }[2][named]{\LWRorigcolumncolor [#1]{#2}\LWRabsorbtwooptions }\)
\(\let \LWRorigrowcolor \rowcolor \)
\(\renewcommand {\rowcolor }[2][named]{\LWRorigrowcolor [#1]{#2}\LWRabsorbtwooptions }\)
\(\let \LWRorigcellcolor \cellcolor \)
\(\renewcommand {\cellcolor }[2][named]{\LWRorigcellcolor [#1]{#2}\LWRabsorbtwooptions }\)
\(\newcommand {\tcbset }[1]{}\)
\(\newcommand {\tcbsetforeverylayer }[1]{}\)
\(\newcommand {\tcbox }[2][]{\boxed {\text {#2}}}\)
\(\newcommand {\tcboxfit }[2][]{\boxed {#2}}\)
\(\newcommand {\tcblower }{}\)
\(\newcommand {\tcbline }{}\)
\(\newcommand {\tcbtitle }{}\)
\(\newcommand {\tcbsubtitle [2][]{\mathrm {#2}}}\)
\(\newcommand {\tcboxmath }[2][]{\boxed {#2}}\)
\(\newcommand {\tcbhighmath }[2][]{\boxed {#2}}\)
\(\newcommand {\toprule }[1][]{\hline }\)
\(\let \midrule \toprule \)
\(\let \bottomrule \toprule \)
\(\def \LWRbooktabscmidruleparen (#1)#2{}\)
\(\newcommand {\LWRbooktabscmidrulenoparen }[1]{}\)
\(\newcommand {\cmidrule }[1][]{\ifnextchar (\LWRbooktabscmidruleparen \LWRbooktabscmidrulenoparen }\)
\(\newcommand {\morecmidrules }{}\)
\(\newcommand {\specialrule }[3]{\hline }\)
\(\newcommand {\addlinespace }[1][]{}\)
\(\newcommand {\LWRsubmultirow }[2][]{#2}\)
\(\newcommand {\LWRmultirow }[2][]{\LWRsubmultirow }\)
\(\newcommand {\multirow }[2][]{\LWRmultirow }\)
\(\newcommand {\mrowcell }{}\)
\(\newcommand {\mcolrowcell }{}\)
\(\newcommand {\STneed }[1]{}\)
\(\newcommand {\LWRldelimtwo }[1][]{\text {#1}~\LWRbigdelim }\)
\(\newcommand {\LWRldelimone }[2][]{\LWRldelimtwo }\)
\(\def \ldelim #1#2{\def \LWRbigdelim {#1}\LWRldelimone }\)
\(\newcommand {\LWRrdelimtwo }[1][]{\LWRbigdelim ~\text {#1}}\)
\(\newcommand {\LWRrdelimone }[2][]{\LWRrdelimtwo }\)
\(\def \rdelim #1#2{\def \LWRbigdelim {#1}\LWRrdelimone }\)
\(\let \symnormal \mathit \)
\(\let \symliteral \mathrm \)
\(\let \symbb \mathbb \)
\(\let \symbbit \mathbb \)
\(\let \symcal \mathcal \)
\(\let \symscr \mathscr \)
\(\let \symfrak \mathfrak \)
\(\let \symsfup \mathsf \)
\(\let \symsfit \mathit \)
\(\let \symbfsf \mathbf \)
\(\let \symbfup \mathbf \)
\(\newcommand {\symbfit }[1]{\boldsymbol {#1}}\)
\(\let \symbfcal \mathcal \)
\(\let \symbfscr \mathscr \)
\(\let \symbffrak \mathfrak \)
\(\let \symbfsfup \mathbf \)
\(\newcommand {\symbfsfit }[1]{\boldsymbol {#1}}\)
\(\let \symup \mathrm \)
\(\let \symbf \mathbf \)
\(\let \symit \mathit \)
\(\let \symsf \symsfit \)
\(\let \symtt \mathtt \)
\(\let \symbffrac \mathbffrac \)
\(\newcommand {\mathfence }[1]{\mathord {#1}}\)
\(\newcommand {\mathover }[1]{#1}\)
\(\newcommand {\mathunder }[1]{#1}\)
\(\newcommand {\mathaccent }[1]{#1}\)
\(\newcommand {\mathbotaccent }[1]{#1}\)
\(\newcommand {\mathalpha }[1]{\mathord {#1}}\)
\(\def\Alpha{\unicode{x1D6E2}}\)
\(\def\Beta{\unicode{x1D6E3}}\)
\(\def\Gamma{\unicode{x1D6E4}}\)
\(\def\Digamma{\mathit{\unicode{x03DC}}}\)
\(\def\Delta{\unicode{x1D6E5}}\)
\(\def\Epsilon{\unicode{x1D6E6}}\)
\(\def\Zeta{\unicode{x1D6E7}}\)
\(\def\Eta{\unicode{x1D6E8}}\)
\(\def\Theta{\unicode{x1D6E9}}\)
\(\def\Vartheta{\unicode{x1D6F3}}\)
\(\def\Iota{\unicode{x1D6EA}}\)
\(\def\Kappa{\unicode{x1D6EB}}\)
\(\def\Lambda{\unicode{x1D6EC}}\)
\(\def\Mu{\unicode{x1D6ED}}\)
\(\def\Nu{\unicode{x1D6EE}}\)
\(\def\Xi{\unicode{x1D6EF}}\)
\(\def\Omicron{\unicode{x1D6F0}}\)
\(\def\Pi{\unicode{x1D6F1}}\)
\(\def\Rho{\unicode{x1D6F2}}\)
\(\def\Sigma{\unicode{x1D6F4}}\)
\(\def\Tau{\unicode{x1D6F5}}\)
\(\def\Upsilon{\unicode{x1D6F6}}\)
\(\def\Phi{\unicode{x1D6F7}}\)
\(\def\Chi{\unicode{x1D6F8}}\)
\(\def\Psi{\unicode{x1D6F9}}\)
\(\def\Omega{\unicode{x1D6FA}}\)
\(\def\alpha{\unicode{x1D6FC}}\)
\(\def\beta{\unicode{x1D6FD}}\)
\(\def\varbeta{\unicode{x03D0}}\)
\(\def\gamma{\unicode{x1D6FE}}\)
\(\def\digamma{\mathit{\unicode{x03DD}}}\)
\(\def\delta{\unicode{x1D6FF}}\)
\(\def\epsilon{\unicode{x1D716}}\)
\(\def\varepsilon{\unicode{x1D700}}\)
\(\def\zeta{\unicode{x1D701}}\)
\(\def\eta{\unicode{x1D702}}\)
\(\def\theta{\unicode{x1D703}}\)
\(\def\vartheta{\unicode{x1D717}}\)
\(\def\iota{\unicode{x1D704}}\)
\(\def\kappa{\unicode{x1D705}}\)
\(\def\varkappa{\unicode{x1D718}}\)
\(\def\lambda{\unicode{x1D706}}\)
\(\def\mu{\unicode{x1D707}}\)
\(\def\nu{\unicode{x1D708}}\)
\(\def\xi{\unicode{x1D709}}\)
\(\def\omicron{\unicode{x1D70A}}\)
\(\def\pi{\unicode{x1D70B}}\)
\(\def\varpi{\unicode{x1D71B}}\)
\(\def\rho{\unicode{x1D70C}}\)
\(\def\varrho{\unicode{x1D71A}}\)
\(\def\sigma{\unicode{x1D70E}}\)
\(\def\varsigma{\unicode{x1D70D}}\)
\(\def\tau{\unicode{x1D70F}}\)
\(\def\upsilon{\unicode{x1D710}}\)
\(\def\phi{\unicode{x1D719}}\)
\(\def\varphi{\unicode{x1D711}}\)
\(\def\chi{\unicode{x1D712}}\)
\(\def\psi{\unicode{x1D713}}\)
\(\def\omega{\unicode{x1D714}}\)
\(\def\upAlpha{\unicode{x0391}}\)
\(\def\upBeta{\unicode{x0392}}\)
\(\def\upGamma{\unicode{x0393}}\)
\(\def\upDigamma{\unicode{x03DC}}\)
\(\def\upDelta{\unicode{x0394}}\)
\(\def\upEpsilon{\unicode{x0395}}\)
\(\def\upZeta{\unicode{x0396}}\)
\(\def\upEta{\unicode{x0397}}\)
\(\def\upTheta{\unicode{x0398}}\)
\(\def\upVartheta{\unicode{x03F4}}\)
\(\def\upIota{\unicode{x0399}}\)
\(\def\upKappa{\unicode{x039A}}\)
\(\def\upLambda{\unicode{x039B}}\)
\(\def\upMu{\unicode{x039C}}\)
\(\def\upNu{\unicode{x039D}}\)
\(\def\upXi{\unicode{x039E}}\)
\(\def\upOmicron{\unicode{x039F}}\)
\(\def\upPi{\unicode{x03A0}}\)
\(\def\upVarpi{\unicode{x03D6}}\)
\(\def\upRho{\unicode{x03A1}}\)
\(\def\upSigma{\unicode{x03A3}}\)
\(\def\upTau{\unicode{x03A4}}\)
\(\def\upUpsilon{\unicode{x03A5}}\)
\(\def\upPhi{\unicode{x03A6}}\)
\(\def\upChi{\unicode{x03A7}}\)
\(\def\upPsi{\unicode{x03A8}}\)
\(\def\upOmega{\unicode{x03A9}}\)
\(\def\itAlpha{\unicode{x1D6E2}}\)
\(\def\itBeta{\unicode{x1D6E3}}\)
\(\def\itGamma{\unicode{x1D6E4}}\)
\(\def\itDigamma{\mathit{\unicode{x03DC}}}\)
\(\def\itDelta{\unicode{x1D6E5}}\)
\(\def\itEpsilon{\unicode{x1D6E6}}\)
\(\def\itZeta{\unicode{x1D6E7}}\)
\(\def\itEta{\unicode{x1D6E8}}\)
\(\def\itTheta{\unicode{x1D6E9}}\)
\(\def\itVartheta{\unicode{x1D6F3}}\)
\(\def\itIota{\unicode{x1D6EA}}\)
\(\def\itKappa{\unicode{x1D6EB}}\)
\(\def\itLambda{\unicode{x1D6EC}}\)
\(\def\itMu{\unicode{x1D6ED}}\)
\(\def\itNu{\unicode{x1D6EE}}\)
\(\def\itXi{\unicode{x1D6EF}}\)
\(\def\itOmicron{\unicode{x1D6F0}}\)
\(\def\itPi{\unicode{x1D6F1}}\)
\(\def\itRho{\unicode{x1D6F2}}\)
\(\def\itSigma{\unicode{x1D6F4}}\)
\(\def\itTau{\unicode{x1D6F5}}\)
\(\def\itUpsilon{\unicode{x1D6F6}}\)
\(\def\itPhi{\unicode{x1D6F7}}\)
\(\def\itChi{\unicode{x1D6F8}}\)
\(\def\itPsi{\unicode{x1D6F9}}\)
\(\def\itOmega{\unicode{x1D6FA}}\)
\(\def\upalpha{\unicode{x03B1}}\)
\(\def\upbeta{\unicode{x03B2}}\)
\(\def\upvarbeta{\unicode{x03D0}}\)
\(\def\upgamma{\unicode{x03B3}}\)
\(\def\updigamma{\unicode{x03DD}}\)
\(\def\updelta{\unicode{x03B4}}\)
\(\def\upepsilon{\unicode{x03F5}}\)
\(\def\upvarepsilon{\unicode{x03B5}}\)
\(\def\upzeta{\unicode{x03B6}}\)
\(\def\upeta{\unicode{x03B7}}\)
\(\def\uptheta{\unicode{x03B8}}\)
\(\def\upvartheta{\unicode{x03D1}}\)
\(\def\upiota{\unicode{x03B9}}\)
\(\def\upkappa{\unicode{x03BA}}\)
\(\def\upvarkappa{\unicode{x03F0}}\)
\(\def\uplambda{\unicode{x03BB}}\)
\(\def\upmu{\unicode{x03BC}}\)
\(\def\upnu{\unicode{x03BD}}\)
\(\def\upxi{\unicode{x03BE}}\)
\(\def\upomicron{\unicode{x03BF}}\)
\(\def\uppi{\unicode{x03C0}}\)
\(\def\upvarpi{\unicode{x03D6}}\)
\(\def\uprho{\unicode{x03C1}}\)
\(\def\upvarrho{\unicode{x03F1}}\)
\(\def\upsigma{\unicode{x03C3}}\)
\(\def\upvarsigma{\unicode{x03C2}}\)
\(\def\uptau{\unicode{x03C4}}\)
\(\def\upupsilon{\unicode{x03C5}}\)
\(\def\upphi{\unicode{x03D5}}\)
\(\def\upvarphi{\unicode{x03C6}}\)
\(\def\upchi{\unicode{x03C7}}\)
\(\def\uppsi{\unicode{x03C8}}\)
\(\def\upomega{\unicode{x03C9}}\)
\(\def\italpha{\unicode{x1D6FC}}\)
\(\def\itbeta{\unicode{x1D6FD}}\)
\(\def\itvarbeta{\unicode{x03D0}}\)
\(\def\itgamma{\unicode{x1D6FE}}\)
\(\def\itdigamma{\mathit{\unicode{x03DD}}}\)
\(\def\itdelta{\unicode{x1D6FF}}\)
\(\def\itepsilon{\unicode{x1D716}}\)
\(\def\itvarepsilon{\unicode{x1D700}}\)
\(\def\itzeta{\unicode{x1D701}}\)
\(\def\iteta{\unicode{x1D702}}\)
\(\def\ittheta{\unicode{x1D703}}\)
\(\def\itvartheta{\unicode{x1D717}}\)
\(\def\itiota{\unicode{x1D704}}\)
\(\def\itkappa{\unicode{x1D705}}\)
\(\def\itvarkappa{\unicode{x1D718}}\)
\(\def\itlambda{\unicode{x1D706}}\)
\(\def\itmu{\unicode{x1D707}}\)
\(\def\itnu{\unicode{x1D708}}\)
\(\def\itxi{\unicode{x1D709}}\)
\(\def\itomicron{\unicode{x1D70A}}\)
\(\def\itpi{\unicode{x1D70B}}\)
\(\def\itvarpi{\unicode{x1D71B}}\)
\(\def\itrho{\unicode{x1D70C}}\)
\(\def\itvarrho{\unicode{x1D71A}}\)
\(\def\itsigma{\unicode{x1D70E}}\)
\(\def\itvarsigma{\unicode{x1D70D}}\)
\(\def\ittau{\unicode{x1D70F}}\)
\(\def\itupsilon{\unicode{x1D710}}\)
\(\def\itphi{\unicode{x1D719}}\)
\(\def\itvarphi{\unicode{x1D711}}\)
\(\def\itchi{\unicode{x1D712}}\)
\(\def\itpsi{\unicode{x1D713}}\)
\(\def\itomega{\unicode{x1D714}}\)
\(\let \lparen (\)
\(\let \rparen )\)
\(\newcommand {\cuberoot }[1]{\,{}^3\!\!\sqrt {#1}}\,\)
\(\newcommand {\fourthroot }[1]{\,{}^4\!\!\sqrt {#1}}\,\)
\(\newcommand {\longdivision }[1]{\mathord {\unicode {x027CC}#1}}\)
\(\newcommand {\mathcomma }{,}\)
\(\newcommand {\mathcolon }{:}\)
\(\newcommand {\mathsemicolon }{;}\)
\(\newcommand {\overbracket }[1]{\mathinner {\overline {\ulcorner {#1}\urcorner }}}\)
\(\newcommand {\underbracket }[1]{\mathinner {\underline {\llcorner {#1}\lrcorner }}}\)
\(\newcommand {\overbar }[1]{\mathord {#1\unicode {x00305}}}\)
\(\newcommand {\ovhook }[1]{\mathord {#1\unicode {x00309}}}\)
\(\newcommand {\ocirc }[1]{\mathord {#1\unicode {x0030A}}}\)
\(\newcommand {\candra }[1]{\mathord {#1\unicode {x00310}}}\)
\(\newcommand {\oturnedcomma }[1]{\mathord {#1\unicode {x00312}}}\)
\(\newcommand {\ocommatopright }[1]{\mathord {#1\unicode {x00315}}}\)
\(\newcommand {\droang }[1]{\mathord {#1\unicode {x0031A}}}\)
\(\newcommand {\leftharpoonaccent }[1]{\mathord {#1\unicode {x020D0}}}\)
\(\newcommand {\rightharpoonaccent }[1]{\mathord {#1\unicode {x020D1}}}\)
\(\newcommand {\vertoverlay }[1]{\mathord {#1\unicode {x020D2}}}\)
\(\newcommand {\leftarrowaccent }[1]{\mathord {#1\unicode {x020D0}}}\)
\(\newcommand {\annuity }[1]{\mathord {#1\unicode {x020E7}}}\)
\(\newcommand {\widebridgeabove }[1]{\mathord {#1\unicode {x020E9}}}\)
\(\newcommand {\asteraccent }[1]{\mathord {#1\unicode {x020F0}}}\)
\(\newcommand {\threeunderdot }[1]{\mathord {#1\unicode {x020E8}}}\)
\(\newcommand {\Bbbsum }{\mathop {\unicode {x2140}}\limits }\)
\(\newcommand {\oiint }{\mathop {\unicode {x222F}}\limits }\)
\(\newcommand {\oiiint }{\mathop {\unicode {x2230}}\limits }\)
\(\newcommand {\intclockwise }{\mathop {\unicode {x2231}}\limits }\)
\(\newcommand {\ointclockwise }{\mathop {\unicode {x2232}}\limits }\)
\(\newcommand {\ointctrclockwise }{\mathop {\unicode {x2233}}\limits }\)
\(\newcommand {\varointclockwise }{\mathop {\unicode {x2232}}\limits }\)
\(\newcommand {\leftouterjoin }{\mathop {\unicode {x27D5}}\limits }\)
\(\newcommand {\rightouterjoin }{\mathop {\unicode {x27D6}}\limits }\)
\(\newcommand {\fullouterjoin }{\mathop {\unicode {x27D7}}\limits }\)
\(\newcommand {\bigbot }{\mathop {\unicode {x27D8}}\limits }\)
\(\newcommand {\bigtop }{\mathop {\unicode {x27D9}}\limits }\)
\(\newcommand {\xsol }{\mathop {\unicode {x29F8}}\limits }\)
\(\newcommand {\xbsol }{\mathop {\unicode {x29F9}}\limits }\)
\(\newcommand {\bigcupdot }{\mathop {\unicode {x2A03}}\limits }\)
\(\newcommand {\bigsqcap }{\mathop {\unicode {x2A05}}\limits }\)
\(\newcommand {\conjquant }{\mathop {\unicode {x2A07}}\limits }\)
\(\newcommand {\disjquant }{\mathop {\unicode {x2A08}}\limits }\)
\(\newcommand {\bigtimes }{\mathop {\unicode {x2A09}}\limits }\)
\(\newcommand {\modtwosum }{\mathop {\unicode {x2A0A}}\limits }\)
\(\newcommand {\sumint }{\mathop {\unicode {x2A0B}}\limits }\)
\(\newcommand {\intbar }{\mathop {\unicode {x2A0D}}\limits }\)
\(\newcommand {\intBar }{\mathop {\unicode {x2A0E}}\limits }\)
\(\newcommand {\fint }{\mathop {\unicode {x2A0F}}\limits }\)
\(\newcommand {\cirfnint }{\mathop {\unicode {x2A10}}\limits }\)
\(\newcommand {\awint }{\mathop {\unicode {x2A11}}\limits }\)
\(\newcommand {\rppolint }{\mathop {\unicode {x2A12}}\limits }\)
\(\newcommand {\scpolint }{\mathop {\unicode {x2A13}}\limits }\)
\(\newcommand {\npolint }{\mathop {\unicode {x2A14}}\limits }\)
\(\newcommand {\pointint }{\mathop {\unicode {x2A15}}\limits }\)
\(\newcommand {\sqint }{\mathop {\unicode {x2A16}}\limits }\)
\(\newcommand {\intlarhk }{\mathop {\unicode {x2A17}}\limits }\)
\(\newcommand {\intx }{\mathop {\unicode {x2A18}}\limits }\)
\(\newcommand {\intcap }{\mathop {\unicode {x2A19}}\limits }\)
\(\newcommand {\intcup }{\mathop {\unicode {x2A1A}}\limits }\)
\(\newcommand {\upint }{\mathop {\unicode {x2A1B}}\limits }\)
\(\newcommand {\lowint }{\mathop {\unicode {x2A1C}}\limits }\)
\(\newcommand {\bigtriangleleft }{\mathop {\unicode {x2A1E}}\limits }\)
\(\newcommand {\zcmp }{\mathop {\unicode {x2A1F}}\limits }\)
\(\newcommand {\zpipe }{\mathop {\unicode {x2A20}}\limits }\)
\(\newcommand {\zproject }{\mathop {\unicode {x2A21}}\limits }\)
\(\newcommand {\biginterleave }{\mathop {\unicode {x2AFC}}\limits }\)
\(\newcommand {\bigtalloblong }{\mathop {\unicode {x2AFF}}\limits }\)
\(\newcommand {\arabicmaj }{\mathop {\unicode {x1EEF0}}\limits }\)
\(\newcommand {\arabichad }{\mathop {\unicode {x1EEF1}}\limits }\)
10.4 The LZSS Encoding¶
To finish the implementation of LZ77, we have to convert the list of LZ77 tokens into a sequence of bits. We start with a simple encoding known as LZSS in this section and discuss a more advanced encoding that uses Huffman coding to compress the tokens themselves in Section 10.5.
The LZSS encoding converts each LZ77 token into a bit sequence that consists of two parts:
The first bit identifies the token’s type (0 for Byte and 1 for Match tokens) and the remaining bits describe the tokens contents. For Byte tokens we encode the uncompressed byte in a block of eight
bits, and for Match tokens we encode the token’s offset and length fields in a fixed number of bits each. The example shown above uses 14 bits for the offset field and 8 bits for the length field.
In the worst case, every byte of the input must be encoded as a Byte token because no suitable match could be found in the dictionary region. Since LZSS encodes every Byte token as a 9-bit sequence, the output ends up being
12.5 % larger than the input — in this case, the algorithm doesn’t compress the data but expands it! This worst case isn’t as rare as it may seem. Many types of files, including digital photographs, audio files, and files that have already been compressed, contain little redundancy
that can be exploited by LZ77. To handle this kind data more gracefully, most compression programs have the ability to store blocks of incompressible data in its original form.
Conversely, the best case for LZSS occurs when almost all tokens are Match tokens. (We always require a few Byte tokens at the beginning of the file to populate the dictionary.) The compression rate in this case depends on the
number of bits we use to encode the offset and length fields. In many types of files — including English text or source code — it’s mostly individual words or short phrases that are repeated frequently, whereas longer character sequences or entire sentences tend to occur only once. As a
result, we generally achieve better compression rates if we allocate more bits to the offset field than to the length field.
Let’s define a few constants before we embark on the actual implementation of LZSS; since these constants are used by both the encoder and the decoder, we collect them in a separate class LZSS shown in Listing 10.13. OFFSET_BITS and LENGTH_BITS are the number of bits that are used to encode the offset and length fields, respectively. We use 14 bits for the offset field and only
4 bits for the length field, which is a trade-off that works well in practice. With this choice, a single Match tag is encoded using 19 bits and a single Byte tag using 9 bits. It is therefore slightly more efficient to encode a
match of length 2 as a sequence of two Byte tokens. We require that all matches are at least three bytes long by setting the MIN_LENGTH field to 3. The two constants BYTE_TAG and MATCH_TAG contain the value of bit at the beginning of each encoded token.
// Constants used by the LZSS implementation.
public class LZSS {
public static final int OFFSET_BITS = 14;
public static final int LENGTH_BITS = 4;
public static final int MIN_LENGTH = 3;
public static final int BYTE_TAG = 0;
public static final int MATCH_TAG = 1;
public static final int MAX_OFFSET = (1 << OFFSET_BITS) - 1;
public static final int MAX_LENGTH =
(1 << LENGTH_BITS) + MIN_LENGTH - 1;
}
The last two constants in Listing 10.13 are MAX_LENGTH and MAX_OFFSET, which specify the largest values of the length and offset fields that can be encoded in a Match token. The largest offset value is simply the largest integer that can be encoded using OFFSET_BITS bits, which is
\(\seteqnumber{0}{10.}{0}\)
\begin{equation*}
\mathtt {MAX\_OFFSET}=2^{\mathtt {OFFSET\_BITS}}-1.
\end{equation*}
We use a slightly different expression for MAX_LENGTH. Since length values below MIN_LENGTH are unused, we can subtract MIN_LENGTH from every length value before encoding it and thereby
represent values of up to
\(\seteqnumber{0}{10.}{0}\)
\begin{equation*}
\mathtt {MAX\_LENGTH} = 2^{\mathtt {LENGTH\_BITS}}-1 + \mathtt {MIN\_LENGTH}.
\end{equation*}
Listing 10.14 shows the complete compression algorithm with LZSS encoding. The compress() method first uses the constants
we just defined to create a suitable instance of LZTokenizer and then converts each token to a sequence of bits. For each Byte token we write a BYTE_TAG followed by eight bits, and for each Match token we write a MATCH_TAG followed by the offset and
length fields. The value of the length field is reduced by MIN_LENGTH before encoding it. The compressed data is always terminated with a Match
token whose offset and the length are both zero. This token can never occur as part of the compressed data since offset is always positive; we use it to terminate the bit stream and let the decoder know that no more data follows.
Listing 10.14ch10 / LZSSEncoder
public static void compress(InputStream in, BitOutputStream out)
throws IOException {
var lz = new LZTokenizer(in, LZSS.MAX_OFFSET, LZSS.MAX_LENGTH,
LZSS.MIN_LENGTH);
while (lz.hasRemaining()) {
long token = lz.nextToken();
if (LZToken.isByteToken(token)) {
out.writeBit(LZSS.BYTE_TAG);
out.writeBits(LZToken.getByte(token), 8);
} else {
out.writeBit(LZSS.MATCH_TAG);
out.writeBits(LZToken.getOffset(token), LZSS.OFFSET_BITS);
out.writeBits(LZToken.getLength(token) - LZSS.MIN_LENGTH,
LZSS.LENGTH_BITS);
}
}
out.writeBit(LZSS.MATCH_TAG);
out.writeBits(0, LZSS.OFFSET_BITS);
out.writeBits(0, LZSS.LENGTH_BITS);
out.flush();
}
The decompression algorithm for LZSS mirrors the operation of the compression algorithm. The decompress() method in Listing 10.15 takes a bit stream produced by compress() as its input and outputs the original, uncompressed sequence of bytes. The function
starts by defining two variables: a circular buffer dict that we use to store the contents of the dictionary region, and an integer pos that holds the current position in the decompressed file. The while loop then
reads tokens until it encounters the end marker Match(offset=0, length=0). The two types of tokens are distinguished by reading a single bit. If the next token is of type Byte, we copy one byte from the bit stream to the
output and also append it to dict. If the next token is of type Match, we first read the fixed-length offset and length fields from the stream. If both fields are zero, the end of the compressed data has been reached and
decompress() stops immediately. Otherwise, we copy the specified number of bytes from dict to the output and append it to dict.
Listing 10.15ch10 / LZSSDecoder
public static void decompress(BitInputStream in, OutputStream out)
throws IOException {
var dict = new CyclicBuffer(LZSS.MAX_OFFSET);
int pos = 0; // position in uncompressed data
while (true) {
if (in.readBit() == LZSS.BYTE_TAG) {
byte b = (byte) in.readBits(8);
out.write(b);
dict.set(pos, b);
pos++;
} else {
int offset = (int) in.readBits(LZSS.OFFSET_BITS);
int length = (int) in.readBits(LZSS.LENGTH_BITS);
if (offset == 0 && length == 0)
break; // end of compressed data reached
for (int i = 0; i < length + LZSS.MIN_LENGTH; i++) {
byte b = dict.get(pos - offset);
out.write(b);
dict.set(pos, b);
pos++;
}
}
}
}