The Plan. Honey, I Shrunk the Data. Why Compress. Data Compression Concepts. Braille Example. Braille. x y xˆ

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1 h ln ony, hrunk th t ihr nr omputr in n nginring nivrsity of shington t omprssion onpts ossy t omprssion osslss t omprssion rfix os uffmn os th y 24 2 t omprssion onpts originl omprss o x y xˆ nor or omprss hy omprss onsrv storg sp u tim for t trnsmission no, sn, o is fstr thn sn osslss omprssion x = xˆ lso ll ntropy oing, rvrsil oing. ossy omprssion x xˆ lso ll irrvrsil oing. omprssion rtio = x y x is numr of its in x. th y 24 3 th y 24 4 rill ystm to r txt y fling ris ots on ppr (or on ltroni isplys). nvnt in 82s y ouis rill, rnh lin mn. z n th with mothr th h gh rill xmpl lr txt: ll m shml. om yrs go -- nvr min how long prisly -- hving \\ littl or no mony in my purs, n nothing prtiulr to intrst m on shor, \\ thought woul sil out littl n s th wtry prt of th worl. (238 hrtrs) r 2 rill in.,ll m,i\%ml4,``s y$>$s go -- n`` m9 h[ l;g prisly -- hv+ \\ ll or no m``oy 9 my purs \& no?+ ``piul$>$ 6 9t]/ m on \%or \\,i $?$``$ $,i w sil ll \& s! wt]y ``p (! \_w4 (23 hrtrs) omprssion rtio = 238/23 =.7 th y 24 5 th y 24 6

2 2 th y 24 7 ossy omprssion t is lost, ut not too muh. uio vio still imgs, mil imgs, photogrphs omprssion rtios of : oftn yil quit high fility rsults. roff twn omprssion rtio n fility ighr omprssion mns lowr fility jor thniqus inlu,, 3 th y 24 8 tio 58 : omprss th y 24 9 tio 24 : omprss th y 24 tio : omprss th y 24 tio 533 : omprss th y 24 2 tio 229 : omprss

3 3 th y 24 3 tio 68 : omprss th y 24 4 tio 26 : omprss th y 24 5 tio 3 : omprss th y 24 6 tio 7 : omprss th y 24 7 riginl th y 24 8 hy is omprssion ossil ost t from ntur hs runny hr is mor t thn th tul informtion ontin in th t. quzing out th xss t mounts to omprssion. owvr, unsquzing is nssry to l to figur out wht th t mns. nformtion thory is n to unrstn th limits of omprssion n giv lus on how to omprss wll.

4 osslss omprssion t is not lost - th originl is rlly n. txt omprssion omprssion of omputr inry fils omprssion rtio typilly no ttr thn 4: for losslss omprssion on most kins of fils. ttistil hniqus uffmn oing rithmti oing olom oing itionry thniqus, 77 quitur urrows-hlr tho tnrs - ors o, rill, nix omprss, gzip, zip, zip,,, osslss th y 24 9 ht is nformtion nlog t lso ll ontinuous t prsnt y rl numrs (or omplx numrs) igitl t init st of symols {, 2,..., m } ll t rprsnt s squns (strings) in th symol st. xmpl: {,,,,r} rr igitl t n pproximt nlog t th y 24 2 ymols impl rfix o omn lpht plus puntution symols inry - {,} n r ll its ll igitl informtion n rprsnt ffiintly in inry {,,,} fix lngth rprsnttion symol inry prfix o is fin y inry tr rfix o proprty no o is prfix of nothr inry tr for prfix o input output o th y 24 2 th y inry r rminology oing rfix o root no rpt strt t root of tr rpt if it = thn go right ls go lft until no is lf rport lf until n of th o lf. h no, xpt th root, hs uniqu prnt. 2. h intrnl no hs xtly two hilrn. xmpl th y th y

5 5 th y oing rfix o th y oing rfix o th y oing rfix o th y oing rfix o th y oing rfix o th y 24 3 oing rfix o

6 6 th y 24 3 oing rfix o th y oing rfix o th y oing rfix o th y oing rfix o th y ow oo is th o /3 /3 /3 vrg it rt = (/3)2 + (/3)2 + (/3) = 5/3 =.67 ps tnr o = 2 ps (ps = its pr symol) uppos tht ll symols r r qully likly. th y ttistil o ht if ll symols r not qully likly. xmpl: ttr prntgs in nglish

7 7 th y ttisitil oing : 7/ : / : / : / ix lngth o ril lngth o = 2 ps = (7/) + (/)2 + (/)3 + (/)3 = 5/ =.5 ps th y ttisistil oing rinipl f you know or n lrn th sttistis of your t, thn vn mor omprssion is possil. hr is n optiml prfix o ll th uffmn o. th y uffmn r lgorithm nitilly ll symols r sprt with thir on proilitis. oin two symols if thy hv th two lowst proilitis. thir proilitis. ontinu this pross until thr is singl symol. th y 24 4 xmpl () () =.4, ()=., ()=.3, ()=., ()= th y 24 4 xmpl (2) th y xmpl (3)

8 8 th y xmpl (4).4.6 th y uffmn o =.4 x +. x x 2 +. x 3 +. x 4 = 2. ps th y uffmn o for nglish th y onlusions uffmn oing ws invnt in 95 y grut stunt t. t is still us toy s prt of,, n othr ors. h thory of t omprssion uss proility thory n othr prts of mthmtis. th y sours ntroution to nformtion hory n t omprssion, on ition y rg rris, tr. ohnson, n rrl. nkrson ntroution to t omprssion, on ition y hli yoo th y 24 48