9 Encoding Information
\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\TextOrMath }[2]{#2}\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\renewcommand {\vec }{\boldsymbol }\)
\(\newcommand {\Edge }{\ensuremath {\,\textemdash \,}}\)
\(\newcommand \Const [1]{\text {\textsf {#1}}}\)
\(\DeclareMathOperator {\lerp }{lerp}\)
\(\DeclareMathOperator {\bitlen }{bitlen}\)
\(\DeclareMathOperator {\sign }{sign}\)
\(\newcommand {\I }{\mathrm {i}}\)
\(\newcommand \AND {\mathbin {\&}}\)
\(\newcommand \OR {\mathbin {|}}\)
\(\newcommand \XOR {\mathbin {{}^{\wedge }}}\)
\(\newcommand \shl {\ll }\)
\(\newcommand \shr {\ggg }\)
\(\newcommand \asr {\gg }\)
\(\newcommand \NOT {\ensuremath {\mathord {\sim }}}\)
\(\newcommand {\isep }{\mathrel {{.}\,{.}}}\)
\(\newcommand {\Id }[1]{\mathit {#1}}\)
\(\newcommand {\const }[1]{\mathsf {#1}}\)
\(\newcommand {\algorithmname }[1]{\text {\textsc {#1}}}\)
\(\newcommand {\bits }[1]{\text {#1}}\)
\(\newcommand {\hexa }[1]{\mathtt {0x#1}}\)
\(\newcommand {\num }[1]{#1}\)
\(\newcommand {\qed }{\quad \square }\)
\(\newcommand {\idiv }[2]{\lfloor #1/#2\rfloor }\)
\(\newcommand \attribdot {\ensuremath {\mkern 1.5mu.\mkern 1.5mu}}\)
\(\newcommand \attribxr [2]{#1\attribdot \text {#2}}\)
\(\newcommand \attribir [2]{\Id {#1}\attribdot \text {#2}}\)
\(\newcommand \attribii [2]{\Id {#1}\attribdot \Id {#2}}\)
\(\newcommand \textsc [1]{#1}\)
\(\require {colortbl}\)
\(\let \LWRorigcolumncolor \columncolor \)
\(\renewcommand {\columncolor }[2][named]{\LWRorigcolumncolor [#1]{#2}\LWRabsorbtwooptions }\)
\(\let \LWRorigrowcolor \rowcolor \)
\(\renewcommand {\rowcolor }[2][named]{\LWRorigrowcolor [#1]{#2}\LWRabsorbtwooptions }\)
\(\let \LWRorigcellcolor \cellcolor \)
\(\renewcommand {\cellcolor }[2][named]{\LWRorigcellcolor [#1]{#2}\LWRabsorbtwooptions }\)
\(\newcommand {\tcbset }[1]{}\)
\(\newcommand {\tcbsetforeverylayer }[1]{}\)
\(\newcommand {\tcbox }[2][]{\boxed {\text {#2}}}\)
\(\newcommand {\tcboxfit }[2][]{\boxed {#2}}\)
\(\newcommand {\tcblower }{}\)
\(\newcommand {\tcbline }{}\)
\(\newcommand {\tcbtitle }{}\)
\(\newcommand {\tcbsubtitle [2][]{\mathrm {#2}}}\)
\(\newcommand {\tcboxmath }[2][]{\boxed {#2}}\)
\(\newcommand {\tcbhighmath }[2][]{\boxed {#2}}\)
\(\newcommand {\toprule }[1][]{\hline }\)
\(\let \midrule \toprule \)
\(\let \bottomrule \toprule \)
\(\def \LWRbooktabscmidruleparen (#1)#2{}\)
\(\newcommand {\LWRbooktabscmidrulenoparen }[1]{}\)
\(\newcommand {\cmidrule }[1][]{\ifnextchar (\LWRbooktabscmidruleparen \LWRbooktabscmidrulenoparen }\)
\(\newcommand {\morecmidrules }{}\)
\(\newcommand {\specialrule }[3]{\hline }\)
\(\newcommand {\addlinespace }[1][]{}\)
\(\newcommand {\LWRsubmultirow }[2][]{#2}\)
\(\newcommand {\LWRmultirow }[2][]{\LWRsubmultirow }\)
\(\newcommand {\multirow }[2][]{\LWRmultirow }\)
\(\newcommand {\mrowcell }{}\)
\(\newcommand {\mcolrowcell }{}\)
\(\newcommand {\STneed }[1]{}\)
\(\newcommand {\LWRldelimtwo }[1][]{\text {#1}~\LWRbigdelim }\)
\(\newcommand {\LWRldelimone }[2][]{\LWRldelimtwo }\)
\(\def \ldelim #1#2{\def \LWRbigdelim {#1}\LWRldelimone }\)
\(\newcommand {\LWRrdelimtwo }[1][]{\LWRbigdelim ~\text {#1}}\)
\(\newcommand {\LWRrdelimone }[2][]{\LWRrdelimtwo }\)
\(\def \rdelim #1#2{\def \LWRbigdelim {#1}\LWRrdelimone }\)
\(\let \symnormal \mathit \)
\(\let \symliteral \mathrm \)
\(\let \symbb \mathbb \)
\(\let \symbbit \mathbb \)
\(\let \symcal \mathcal \)
\(\let \symscr \mathscr \)
\(\let \symfrak \mathfrak \)
\(\let \symsfup \mathsf \)
\(\let \symsfit \mathit \)
\(\let \symbfsf \mathbf \)
\(\let \symbfup \mathbf \)
\(\newcommand {\symbfit }[1]{\boldsymbol {#1}}\)
\(\let \symbfcal \mathcal \)
\(\let \symbfscr \mathscr \)
\(\let \symbffrak \mathfrak \)
\(\let \symbfsfup \mathbf \)
\(\newcommand {\symbfsfit }[1]{\boldsymbol {#1}}\)
\(\let \symup \mathrm \)
\(\let \symbf \mathbf \)
\(\let \symit \mathit \)
\(\let \symsf \symsfit \)
\(\let \symtt \mathtt \)
\(\let \symbffrac \mathbffrac \)
\(\newcommand {\mathfence }[1]{\mathord {#1}}\)
\(\newcommand {\mathover }[1]{#1}\)
\(\newcommand {\mathunder }[1]{#1}\)
\(\newcommand {\mathaccent }[1]{#1}\)
\(\newcommand {\mathbotaccent }[1]{#1}\)
\(\newcommand {\mathalpha }[1]{\mathord {#1}}\)
\(\def\Alpha{\unicode{x1D6E2}}\)
\(\def\Beta{\unicode{x1D6E3}}\)
\(\def\Gamma{\unicode{x1D6E4}}\)
\(\def\Digamma{\mathit{\unicode{x03DC}}}\)
\(\def\Delta{\unicode{x1D6E5}}\)
\(\def\Epsilon{\unicode{x1D6E6}}\)
\(\def\Zeta{\unicode{x1D6E7}}\)
\(\def\Eta{\unicode{x1D6E8}}\)
\(\def\Theta{\unicode{x1D6E9}}\)
\(\def\Vartheta{\unicode{x1D6F3}}\)
\(\def\Iota{\unicode{x1D6EA}}\)
\(\def\Kappa{\unicode{x1D6EB}}\)
\(\def\Lambda{\unicode{x1D6EC}}\)
\(\def\Mu{\unicode{x1D6ED}}\)
\(\def\Nu{\unicode{x1D6EE}}\)
\(\def\Xi{\unicode{x1D6EF}}\)
\(\def\Omicron{\unicode{x1D6F0}}\)
\(\def\Pi{\unicode{x1D6F1}}\)
\(\def\Rho{\unicode{x1D6F2}}\)
\(\def\Sigma{\unicode{x1D6F4}}\)
\(\def\Tau{\unicode{x1D6F5}}\)
\(\def\Upsilon{\unicode{x1D6F6}}\)
\(\def\Phi{\unicode{x1D6F7}}\)
\(\def\Chi{\unicode{x1D6F8}}\)
\(\def\Psi{\unicode{x1D6F9}}\)
\(\def\Omega{\unicode{x1D6FA}}\)
\(\def\alpha{\unicode{x1D6FC}}\)
\(\def\beta{\unicode{x1D6FD}}\)
\(\def\varbeta{\unicode{x03D0}}\)
\(\def\gamma{\unicode{x1D6FE}}\)
\(\def\digamma{\mathit{\unicode{x03DD}}}\)
\(\def\delta{\unicode{x1D6FF}}\)
\(\def\epsilon{\unicode{x1D716}}\)
\(\def\varepsilon{\unicode{x1D700}}\)
\(\def\zeta{\unicode{x1D701}}\)
\(\def\eta{\unicode{x1D702}}\)
\(\def\theta{\unicode{x1D703}}\)
\(\def\vartheta{\unicode{x1D717}}\)
\(\def\iota{\unicode{x1D704}}\)
\(\def\kappa{\unicode{x1D705}}\)
\(\def\varkappa{\unicode{x1D718}}\)
\(\def\lambda{\unicode{x1D706}}\)
\(\def\mu{\unicode{x1D707}}\)
\(\def\nu{\unicode{x1D708}}\)
\(\def\xi{\unicode{x1D709}}\)
\(\def\omicron{\unicode{x1D70A}}\)
\(\def\pi{\unicode{x1D70B}}\)
\(\def\varpi{\unicode{x1D71B}}\)
\(\def\rho{\unicode{x1D70C}}\)
\(\def\varrho{\unicode{x1D71A}}\)
\(\def\sigma{\unicode{x1D70E}}\)
\(\def\varsigma{\unicode{x1D70D}}\)
\(\def\tau{\unicode{x1D70F}}\)
\(\def\upsilon{\unicode{x1D710}}\)
\(\def\phi{\unicode{x1D719}}\)
\(\def\varphi{\unicode{x1D711}}\)
\(\def\chi{\unicode{x1D712}}\)
\(\def\psi{\unicode{x1D713}}\)
\(\def\omega{\unicode{x1D714}}\)
\(\def\upAlpha{\unicode{x0391}}\)
\(\def\upBeta{\unicode{x0392}}\)
\(\def\upGamma{\unicode{x0393}}\)
\(\def\upDigamma{\unicode{x03DC}}\)
\(\def\upDelta{\unicode{x0394}}\)
\(\def\upEpsilon{\unicode{x0395}}\)
\(\def\upZeta{\unicode{x0396}}\)
\(\def\upEta{\unicode{x0397}}\)
\(\def\upTheta{\unicode{x0398}}\)
\(\def\upVartheta{\unicode{x03F4}}\)
\(\def\upIota{\unicode{x0399}}\)
\(\def\upKappa{\unicode{x039A}}\)
\(\def\upLambda{\unicode{x039B}}\)
\(\def\upMu{\unicode{x039C}}\)
\(\def\upNu{\unicode{x039D}}\)
\(\def\upXi{\unicode{x039E}}\)
\(\def\upOmicron{\unicode{x039F}}\)
\(\def\upPi{\unicode{x03A0}}\)
\(\def\upVarpi{\unicode{x03D6}}\)
\(\def\upRho{\unicode{x03A1}}\)
\(\def\upSigma{\unicode{x03A3}}\)
\(\def\upTau{\unicode{x03A4}}\)
\(\def\upUpsilon{\unicode{x03A5}}\)
\(\def\upPhi{\unicode{x03A6}}\)
\(\def\upChi{\unicode{x03A7}}\)
\(\def\upPsi{\unicode{x03A8}}\)
\(\def\upOmega{\unicode{x03A9}}\)
\(\def\itAlpha{\unicode{x1D6E2}}\)
\(\def\itBeta{\unicode{x1D6E3}}\)
\(\def\itGamma{\unicode{x1D6E4}}\)
\(\def\itDigamma{\mathit{\unicode{x03DC}}}\)
\(\def\itDelta{\unicode{x1D6E5}}\)
\(\def\itEpsilon{\unicode{x1D6E6}}\)
\(\def\itZeta{\unicode{x1D6E7}}\)
\(\def\itEta{\unicode{x1D6E8}}\)
\(\def\itTheta{\unicode{x1D6E9}}\)
\(\def\itVartheta{\unicode{x1D6F3}}\)
\(\def\itIota{\unicode{x1D6EA}}\)
\(\def\itKappa{\unicode{x1D6EB}}\)
\(\def\itLambda{\unicode{x1D6EC}}\)
\(\def\itMu{\unicode{x1D6ED}}\)
\(\def\itNu{\unicode{x1D6EE}}\)
\(\def\itXi{\unicode{x1D6EF}}\)
\(\def\itOmicron{\unicode{x1D6F0}}\)
\(\def\itPi{\unicode{x1D6F1}}\)
\(\def\itRho{\unicode{x1D6F2}}\)
\(\def\itSigma{\unicode{x1D6F4}}\)
\(\def\itTau{\unicode{x1D6F5}}\)
\(\def\itUpsilon{\unicode{x1D6F6}}\)
\(\def\itPhi{\unicode{x1D6F7}}\)
\(\def\itChi{\unicode{x1D6F8}}\)
\(\def\itPsi{\unicode{x1D6F9}}\)
\(\def\itOmega{\unicode{x1D6FA}}\)
\(\def\upalpha{\unicode{x03B1}}\)
\(\def\upbeta{\unicode{x03B2}}\)
\(\def\upvarbeta{\unicode{x03D0}}\)
\(\def\upgamma{\unicode{x03B3}}\)
\(\def\updigamma{\unicode{x03DD}}\)
\(\def\updelta{\unicode{x03B4}}\)
\(\def\upepsilon{\unicode{x03F5}}\)
\(\def\upvarepsilon{\unicode{x03B5}}\)
\(\def\upzeta{\unicode{x03B6}}\)
\(\def\upeta{\unicode{x03B7}}\)
\(\def\uptheta{\unicode{x03B8}}\)
\(\def\upvartheta{\unicode{x03D1}}\)
\(\def\upiota{\unicode{x03B9}}\)
\(\def\upkappa{\unicode{x03BA}}\)
\(\def\upvarkappa{\unicode{x03F0}}\)
\(\def\uplambda{\unicode{x03BB}}\)
\(\def\upmu{\unicode{x03BC}}\)
\(\def\upnu{\unicode{x03BD}}\)
\(\def\upxi{\unicode{x03BE}}\)
\(\def\upomicron{\unicode{x03BF}}\)
\(\def\uppi{\unicode{x03C0}}\)
\(\def\upvarpi{\unicode{x03D6}}\)
\(\def\uprho{\unicode{x03C1}}\)
\(\def\upvarrho{\unicode{x03F1}}\)
\(\def\upsigma{\unicode{x03C3}}\)
\(\def\upvarsigma{\unicode{x03C2}}\)
\(\def\uptau{\unicode{x03C4}}\)
\(\def\upupsilon{\unicode{x03C5}}\)
\(\def\upphi{\unicode{x03D5}}\)
\(\def\upvarphi{\unicode{x03C6}}\)
\(\def\upchi{\unicode{x03C7}}\)
\(\def\uppsi{\unicode{x03C8}}\)
\(\def\upomega{\unicode{x03C9}}\)
\(\def\italpha{\unicode{x1D6FC}}\)
\(\def\itbeta{\unicode{x1D6FD}}\)
\(\def\itvarbeta{\unicode{x03D0}}\)
\(\def\itgamma{\unicode{x1D6FE}}\)
\(\def\itdigamma{\mathit{\unicode{x03DD}}}\)
\(\def\itdelta{\unicode{x1D6FF}}\)
\(\def\itepsilon{\unicode{x1D716}}\)
\(\def\itvarepsilon{\unicode{x1D700}}\)
\(\def\itzeta{\unicode{x1D701}}\)
\(\def\iteta{\unicode{x1D702}}\)
\(\def\ittheta{\unicode{x1D703}}\)
\(\def\itvartheta{\unicode{x1D717}}\)
\(\def\itiota{\unicode{x1D704}}\)
\(\def\itkappa{\unicode{x1D705}}\)
\(\def\itvarkappa{\unicode{x1D718}}\)
\(\def\itlambda{\unicode{x1D706}}\)
\(\def\itmu{\unicode{x1D707}}\)
\(\def\itnu{\unicode{x1D708}}\)
\(\def\itxi{\unicode{x1D709}}\)
\(\def\itomicron{\unicode{x1D70A}}\)
\(\def\itpi{\unicode{x1D70B}}\)
\(\def\itvarpi{\unicode{x1D71B}}\)
\(\def\itrho{\unicode{x1D70C}}\)
\(\def\itvarrho{\unicode{x1D71A}}\)
\(\def\itsigma{\unicode{x1D70E}}\)
\(\def\itvarsigma{\unicode{x1D70D}}\)
\(\def\ittau{\unicode{x1D70F}}\)
\(\def\itupsilon{\unicode{x1D710}}\)
\(\def\itphi{\unicode{x1D719}}\)
\(\def\itvarphi{\unicode{x1D711}}\)
\(\def\itchi{\unicode{x1D712}}\)
\(\def\itpsi{\unicode{x1D713}}\)
\(\def\itomega{\unicode{x1D714}}\)
\(\let \lparen (\)
\(\let \rparen )\)
\(\newcommand {\cuberoot }[1]{\,{}^3\!\!\sqrt {#1}}\,\)
\(\newcommand {\fourthroot }[1]{\,{}^4\!\!\sqrt {#1}}\,\)
\(\newcommand {\longdivision }[1]{\mathord {\unicode {x027CC}#1}}\)
\(\newcommand {\mathcomma }{,}\)
\(\newcommand {\mathcolon }{:}\)
\(\newcommand {\mathsemicolon }{;}\)
\(\newcommand {\overbracket }[1]{\mathinner {\overline {\ulcorner {#1}\urcorner }}}\)
\(\newcommand {\underbracket }[1]{\mathinner {\underline {\llcorner {#1}\lrcorner }}}\)
\(\newcommand {\overbar }[1]{\mathord {#1\unicode {x00305}}}\)
\(\newcommand {\ovhook }[1]{\mathord {#1\unicode {x00309}}}\)
\(\newcommand {\ocirc }[1]{\mathord {#1\unicode {x0030A}}}\)
\(\newcommand {\candra }[1]{\mathord {#1\unicode {x00310}}}\)
\(\newcommand {\oturnedcomma }[1]{\mathord {#1\unicode {x00312}}}\)
\(\newcommand {\ocommatopright }[1]{\mathord {#1\unicode {x00315}}}\)
\(\newcommand {\droang }[1]{\mathord {#1\unicode {x0031A}}}\)
\(\newcommand {\leftharpoonaccent }[1]{\mathord {#1\unicode {x020D0}}}\)
\(\newcommand {\rightharpoonaccent }[1]{\mathord {#1\unicode {x020D1}}}\)
\(\newcommand {\vertoverlay }[1]{\mathord {#1\unicode {x020D2}}}\)
\(\newcommand {\leftarrowaccent }[1]{\mathord {#1\unicode {x020D0}}}\)
\(\newcommand {\annuity }[1]{\mathord {#1\unicode {x020E7}}}\)
\(\newcommand {\widebridgeabove }[1]{\mathord {#1\unicode {x020E9}}}\)
\(\newcommand {\asteraccent }[1]{\mathord {#1\unicode {x020F0}}}\)
\(\newcommand {\threeunderdot }[1]{\mathord {#1\unicode {x020E8}}}\)
\(\newcommand {\Bbbsum }{\mathop {\unicode {x2140}}\limits }\)
\(\newcommand {\oiint }{\mathop {\unicode {x222F}}\limits }\)
\(\newcommand {\oiiint }{\mathop {\unicode {x2230}}\limits }\)
\(\newcommand {\intclockwise }{\mathop {\unicode {x2231}}\limits }\)
\(\newcommand {\ointclockwise }{\mathop {\unicode {x2232}}\limits }\)
\(\newcommand {\ointctrclockwise }{\mathop {\unicode {x2233}}\limits }\)
\(\newcommand {\varointclockwise }{\mathop {\unicode {x2232}}\limits }\)
\(\newcommand {\leftouterjoin }{\mathop {\unicode {x27D5}}\limits }\)
\(\newcommand {\rightouterjoin }{\mathop {\unicode {x27D6}}\limits }\)
\(\newcommand {\fullouterjoin }{\mathop {\unicode {x27D7}}\limits }\)
\(\newcommand {\bigbot }{\mathop {\unicode {x27D8}}\limits }\)
\(\newcommand {\bigtop }{\mathop {\unicode {x27D9}}\limits }\)
\(\newcommand {\xsol }{\mathop {\unicode {x29F8}}\limits }\)
\(\newcommand {\xbsol }{\mathop {\unicode {x29F9}}\limits }\)
\(\newcommand {\bigcupdot }{\mathop {\unicode {x2A03}}\limits }\)
\(\newcommand {\bigsqcap }{\mathop {\unicode {x2A05}}\limits }\)
\(\newcommand {\conjquant }{\mathop {\unicode {x2A07}}\limits }\)
\(\newcommand {\disjquant }{\mathop {\unicode {x2A08}}\limits }\)
\(\newcommand {\bigtimes }{\mathop {\unicode {x2A09}}\limits }\)
\(\newcommand {\modtwosum }{\mathop {\unicode {x2A0A}}\limits }\)
\(\newcommand {\sumint }{\mathop {\unicode {x2A0B}}\limits }\)
\(\newcommand {\intbar }{\mathop {\unicode {x2A0D}}\limits }\)
\(\newcommand {\intBar }{\mathop {\unicode {x2A0E}}\limits }\)
\(\newcommand {\fint }{\mathop {\unicode {x2A0F}}\limits }\)
\(\newcommand {\cirfnint }{\mathop {\unicode {x2A10}}\limits }\)
\(\newcommand {\awint }{\mathop {\unicode {x2A11}}\limits }\)
\(\newcommand {\rppolint }{\mathop {\unicode {x2A12}}\limits }\)
\(\newcommand {\scpolint }{\mathop {\unicode {x2A13}}\limits }\)
\(\newcommand {\npolint }{\mathop {\unicode {x2A14}}\limits }\)
\(\newcommand {\pointint }{\mathop {\unicode {x2A15}}\limits }\)
\(\newcommand {\sqint }{\mathop {\unicode {x2A16}}\limits }\)
\(\newcommand {\intlarhk }{\mathop {\unicode {x2A17}}\limits }\)
\(\newcommand {\intx }{\mathop {\unicode {x2A18}}\limits }\)
\(\newcommand {\intcap }{\mathop {\unicode {x2A19}}\limits }\)
\(\newcommand {\intcup }{\mathop {\unicode {x2A1A}}\limits }\)
\(\newcommand {\upint }{\mathop {\unicode {x2A1B}}\limits }\)
\(\newcommand {\lowint }{\mathop {\unicode {x2A1C}}\limits }\)
\(\newcommand {\bigtriangleleft }{\mathop {\unicode {x2A1E}}\limits }\)
\(\newcommand {\zcmp }{\mathop {\unicode {x2A1F}}\limits }\)
\(\newcommand {\zpipe }{\mathop {\unicode {x2A20}}\limits }\)
\(\newcommand {\zproject }{\mathop {\unicode {x2A21}}\limits }\)
\(\newcommand {\biginterleave }{\mathop {\unicode {x2AFC}}\limits }\)
\(\newcommand {\bigtalloblong }{\mathop {\unicode {x2AFF}}\limits }\)
\(\newcommand {\arabicmaj }{\mathop {\unicode {x1EEF0}}\limits }\)
\(\newcommand {\arabichad }{\mathop {\unicode {x1EEF1}}\limits }\)
9.2 Implementing Prefix Codes¶
Now that we understand the basics of prefix codes, we can turn to their implementation. The PrefixCode class in Listing 9.1 stores a prefix code in two forms: The
codewords field is an array of Codewords indexed by the numeric value of the symbol, and root is the root of the corresponding binary tree. The nested type Codeword
represents a single codeword and stores the bit sequence in the least significant bits of bits and its length in a separate length field.
Listing 9.1ch9 / PrefixCode
// Representation of prefix codes.
public class PrefixCode {
public record Codeword(long bits, int length) {
}
public record Node(Node left, Node right, int symbol) {
}
private final Codeword[] codewords; // table of codewords
private final Node root; // associated binary tree
// ...
}
To represent the binary tree we use the nested type Node that has links to the left and right subtree and an optional field symbol for storing the symbol associated with a
leaf node. The Node type can represent both inner nodes and leaf nodes. If both left and right are null, the node is a leaf node and symbol holds the numeric value of the
associated symbol. Otherwise, it is an inner node and left and right point to the left and right child, respectively. To simplify the task of creating these two types of nodes, we define the two convenience methods makeLeaf() and makeInner(), as shown in Listing 9.2.
Listing 9.2ch9 / PrefixCode
// Create a leaf node for 'symbol'.
public static Node makeLeaf(int symbol) {
return new Node(null, null, symbol);
}
// Create an inner node.
public static Node makeInner(Node left, Node right) {
return new Node(left, right, -1);
}
The constructor PrefixCode() in Listing 9.3 takes the root of a finished binary tree as its
only argument and uses it to initialize the table of codewords. We first initialize codewords by allocating an array that is large enough to hold every symbol that occurs in the tree; the helper method maxSymbol() recursively finds the largest symbol in the tree. We then call computeCodewords()
to fill the table of codewords.
Listing 9.3ch9 / PrefixCode
// Create a prefix code from a binary tree.
public PrefixCode(Node root) {
this.root = root;
this.codewords = new Codeword[maxSymbol(root) + 1];
computeCodewords(root, 0L, 0);
}
// Find the largest symbol in the subtree of node.
private int maxSymbol(Node node) {
if (node == null)
return Integer.MIN_VALUE;
if (node.left == null && node.right == null)
return node.symbol;
return Math.max(maxSymbol(node.left), maxSymbol(node.right));
}
The implementation of computeCodewords() is shown in Listing 9.4. The method recursively traverses the binary
tree while keeping track of the path that leads from the root to the current node. The path is stored as a string of up to 64 bits in bits and its length in length. The length starts at 0 is incremented
every time the method descends into the left or right subtree of a node. If the current node is a leaf node, which is the case when it contains a non-negative symbol, we create a new entry in the table of codewords from the bit sequence in bits.
Otherwise, if the current node is an inner node, we process its left and right subtrees recursively. For the left subtree we append a 0-bit to bits and increment length, and for the right subtree we append a 1-bit.
Listing 9.4ch9 / PrefixCode
private void computeCodewords(Node node, long bits, int length) {
if (node == null)
return;
if (node.symbol >= 0) {
codewords[node.symbol] = new Codeword(bits, length);
} else {
computeCodewords(node.left, bits << 1, length + 1);
computeCodewords(node.right, (bits << 1) | 1, length + 1);
}
}
Exercises
Exercise 9.7. Write a function that constructs a binary tree from an array of codewords. If the codewords don’t describe a valid prefix code, the function should throw an exception.
Encoding and Decoding with Prefix Codes
Let’s turn to the problem encoding and decoding data using prefix codes. In the following we will make use of two classes for reading and writing bits: BitOutputStream and BitInputStream. BitOutputStream write groups of bits to some kind of output stream (say, a file or a network connection). It is is defined as follows:
public class BitOutputStream {
public void writeBits(long bits, int n);
public void writeBit(long bits) {writeBits(bits, 1);}
// ...
}
Calling writeBits() writes the n least significant bits of its first argument bits to the output; writeBit() is provided for convenience and writes a single bit. The corresponding functionality for reading bits is provided by BitInputStream:
public class BitInputStream {
public long readBits(int n);
public long readBit() {return readBits(1);}
// ...
}
Calling readBits() returns the next n bits from the input and returns them in the least significant bits of a long integer; readBit() reads a single bit. We will discuss the implementation of these two classes in detail in Exercise 9.13.
We can now extend the PrefixCode class with methods for encoding and decoding individual symbols. To output a single symbol, we look up its entry in the codewords table and then append the corresponding bit
sequence to the output (Listing 9.5). If the symbol is invalid, the writeSymbol() method throws an exception.
Listing 9.5ch9 / PrefixCode
public void writeSymbol(int symbol, BitOutputStream out)
throws IOException {
if (symbol < 0 || symbol > codewords.length
|| codewords[symbol] == null) {
throw new RuntimeException("Invalid symbol");
}
Codeword codeword = codewords[symbol];
out.writeBits(codeword.bits, codeword.length);
}
The reverse operation decodes a single symbol by finding the codeword that matches the bit sequence at the current position in a BitInputStream. Since every codeword corresponds to a path from root to one of the
leaf nodes, the simplest way to decode the next symbol is to read one bit at a time and move left in the tree if it is 0 and right if it is 1 (Listing 9.6). Once we reach a leaf node, we have identified the next codeword
and return the associated symbol. If decodeSymbol() encounters a null reference while traversing the tree, it stops and throws an exception. This can only happen if
the input doesn’t start with a valid codeword, for example because it was encoded using a different code or because one of the bits was corrupted during transmission. In either case, there is nothing we can do to recover the original message, so we give up immediately.
Listing 9.6ch9 / PrefixCode
public int decodeSymbol(BitInputStream in) throws IOException {
Node node = root;
while (node != null) {
if (node.left == null && node.right == null)
return node.symbol;
node = in.readBits(1) == 0 ? node.left : node.right;
}
throw new IOException("Invalid bit stream");
}
Encoding the Code Itself
In our discussion so far, we have assumed that the decoder already knows the code that was used to create the bit stream. This isn’t always the case: In many applications, the prefix code must be tailored to the data being encoded and is therefore not known in advance. In this case, decoding is only
possible if we transmit the code along with the encoded bit sequence — in other words, we have to encode the code itself. In the case of prefix codes there are two options: We can encode either the table of codewords or its tree representation.
The writeTree() function in Listing 9.7 demonstrates how to encode the tree representation of a prefix code. The
function traverses the tree recursively and writes a binary representation of node and all its subtrees to out. The last argument bitsPerSymbol specifies the number of bits that should be used to
encode the symbol stored in the leaf nodes. The bit encoding of each node starts with two bits that indicate which of its subtrees are present. For leaf nodes, the two bits are 00 and are followed immediately by the bit representation of the node’s symbol field. For inner nodes, the two bits are
10, 01, or 11, depending on whether the node has a left child, a right child, or both. These two bits are followed by the bit representations of the left and right subtrees, which are generated by calling writeTree() recursively.
Listing 9.7ch9 / PrefixCode
// Write bit representation of 'node' and its children to 'out'.
void writeTree(BitOutputStream out, Node node, int bitsPerSymbol)
throws IOException {
if (node.left() == null && node.right() == null) {
out.writeBits(0b00, 2);
out.writeBits(node.symbol(), bitsPerSymbol);
} else {
out.writeBit(node.left() != null ? 1 : 0);
out.writeBit(node.right() != null ? 1 : 0);
if (node.left() != null)
writeTree(out, node.left(), bitsPerSymbol);
if (node.right() != null)
writeTree(out, node.right(), bitsPerSymbol);
}
}
Ideally, we would like to set the bitsPerSymbol parameter to the smallest number that covers all symbols of this prefix code. If \(n\) is the largest symbol, how many bits are needed to encode \(n\)? With \(k\) bits we can represent numbers between 0 and \(2^k-1\),
so we have to find the smallest integer \(k\) that satisfies the inequality \(n\le 2^k-1\). By taking the base-2 logarithm and rounding up to the nearest integer, we obtain the bit length of \(n\)
\(\seteqnumber{0}{9.}{0}\)
\begin{equation}
\label {eq:bitlen} \bitlen (n)=\bigl \lceil \log _2 (n+1)\bigr \rceil .
\end{equation}
We already met the expression on the right-hand side in Section 5.3, where we discussed a way to compute it without having to use logarithms or floating-point arithmetic. According to Eq. (5.8), we have
\[ \bitlen (n) = w - \operatorname {nlz}(n), \]
where \(w\) is the bit width of the variable that holds \(n\) and \(\operatorname {nlz}(n)\) the number of leading 0-bits of that variable. An implementation of the bitlen function is shown in Listing 9.8.
// The number of bits required to represent a non-negative integer.
public static int bitLength(int value) {
assert value >= 0;
return 32 - Integer.numberOfLeadingZeros(value);
}
For a given PrefixCode, the largest symbol is simply the length of the codeword array minus 1. The writeCode()
method in Listing 9.9 first writes this length as an 8-bit integer to the output stream, and then appends the bit representation of the entire tree. This completes the problem of encoding the prefix code.
Listing 9.9ch9 / PrefixCode
// Write representation of this prefix code to 'out'.
public void writeCode(BitOutputStream out) throws IOException {
int bitsPerSymbol = IntUtil.bitLength(codewords.length - 1);
out.writeBits(bitsPerSymbol, 8);
writeTree(out, root, bitsPerSymbol);
}
To decode a prefix code from a bit stream, we first read 8 bits to determine the value of bitsPerSymbol and then call readTree() to recursively decode the actual tree
(Listing 9.10). The implementation of readTree() reverses the operation of writeTree() by first reading two bits to determine whether to decode an inner node or a leaf node. In the case of an inner node, the function calls itself recursively to read the left and right child nodes. In the case of a leaf node, it
simply reads the next bitsPerSymbol bits to recover the symbol stored in the node.
Listing 9.10ch9 / PrefixCode
public static PrefixCode readCode(BitInputStream in)
throws IOException {
int bitsPerSymbol = (int) in.readBits(8);
return new PrefixCode(readTree(in, bitsPerSymbol));
}
static Node readTree(BitInputStream in, int bitsPerSymbol)
throws IOException {
boolean hasLeft = in.readBit() != 0;
boolean hasRight = in.readBit() != 0;
if (hasLeft || hasRight) {
Node left = hasLeft ? readTree(in, bitsPerSymbol) : null;
Node right = hasRight ? readTree(in, bitsPerSymbol) : null;
return makeInner(left, right);
} else {
return makeLeaf((int) in.readBits(bitsPerSymbol));
}
}