5/1/2018. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees

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1 /1/018 W usully no strns y ssnn -lnt os to ll rtrs n t lpt (or mpl, 8-t on n ASCII). Howvr, rnt rtrs our wt rnt rquns, w n sv mmory n ru trnsmttl tm y usn vrl-lnt non. T s to ssn sortr os to rtrs tt our mor otn. W must rul wn ssnn vrl-lnt os. For mpl, lt us no wt 0, wt 1, n t wt 01. How n w tn no t wor t? T non s Unortuntly, ts non s muous. It oul lso stn or t,, or tt. O ours ts on s unptl, us t rsults n loss o normton. 1 To vo su muts, w n us pr os. In pr o, t t strn or rtr nvr ours s t pr (rst prt) o t t strn or notr rtr. For mpl, t non o wt 0, wt 10, n t wt 11 s pr o. How n w now no t wor t? T non s Ts t strn s unqu, t n only no t wor t. W n rprsnt pr os usn nry trs, wr t rtrs r t lls o t lvs n t tr. T s o t tr r ll so tt n ln to lt l s ssn 0 n n ln to rt l s ssn 1. T t strn us to no rtr s t squn o lls o t s n t unqu pt rom t root to t l ll wt ts rtr. 3 T tr orrsponn to our mpl: t In tr, no l n t nstor o notr l. Tror, no non o rtr n pr o n non o notr rtr (pr o). To trmn t optml (sortst) non or vn strn, w rst v to n t rquns o rtrs n tt strn. Lt us onsr t ollown strn: It ontns 1, 1, 3,, 1, 1,, 3, 1, n. W n now us Humn s lortm to ul t optml on tr. 1

2 /1/018 For n lpt ontnn n lttrs, Humn s lortm strts wt n vrts, on or lttr, ll wt tt lttr n ts rquny. W tn trmn t two vrts wt t lowst rquns n rpl tm wt tr wos root s ll wt t sum o ts two rquns n wos two lrn r t two vrts tt w rpl. In t ollown stps, w trmn t two lowst rquns mon t snl vrts n t roots o trs tt w lry rt. Ts s rpt untl w otn snl tr

3 /1/ Fnlly, w onvrt t tr nto pr o tr: T vrl-lnt os r: (rq. 1): (rq. 1): (rq. 3): 0001 (rq. ): 011 (rq. 1): 1 (rq. 1): (rq. ): 0101 (rq. 3): 0100 (rq. 1): (rq. ):

4 /1/018 I w no t ornl strn usn -lnt o, w n our ts pr rtr (or tn rnt rtrs). Tror, t non o t ntr strn s 3 = 18 ts lon. Wt our vrl-lnt o, w only n = 101 ts. It n sown tt, or ny vn strn, Humn on trs lwys prou vrl-lnt o wt mnmum srpton lnt or tt strn. Sn Humn s lortm s not ovr n t ttook, pls tk look t: ttp:// Bktrkn n Dson Trs A son tr s root tr n w ntrnl vrt orrspons to son, wt sutr t ts vrts or possl outom o t son. Dson trs n us to mol prolms n w srs o sons ls to soluton (ompr wt t ountrt on n nry sr tr mpls). T possl solutons o t prolm orrspon to t pts rom t root to t lvs o t son tr. Bktrkn n Dson Trs Tr r prolms tt rqur us to prorm n ustv sr o ll possl squns o sons n orr to n t soluton. W n solv su prolms y onstrutn t omplt son tr n tn n pt rom ts root to l tt orrspons to soluton o t prolm. In mny ss, t ny o ts prour n rmtlly nrs y tnqu ll ktrkn. 1 Bktrkn n Dson Trs I: Strt t t root o t son tr n mov ownwrs, tt s, mk squn o sons, untl you tr r soluton or you ntr stuton rom wr no soluton n r y ny urtr squn o sons. In t lttr s, ktrk to t prnt o t urrnt vrt n tk rnt pt ownwrs rom tr. I ll pts rom ts vrt v lry n plor, ktrk to ts prnt. Contnu ts prour untl you n soluton or stls tt no soluton sts (tr r no mor pts to try out). 3 Bktrkn n Dson Trs Empl: T n-quns prolm How n w pl n quns on n n n ssor so tt no two quns n ptur otr? A qun n mov ny numr o squrs orzontlly, vrtlly, n onlly. Hr, t possl trt squrs o t qun r mrk wt n.

5 /1/018 Dson Trs Lt us onsr t -quns prolm. uston: How mny possl onurtons o ssors ontnn quns r tr? Answr: Tr r 1!/(1!!) = ( )/( 3 ) = 13 = 180 possl onurtons. Sll w smply try tm out on y on untl w nountr soluton? No, t s nrlly usul to tnk out sr prolm mor rully n sovr onstrnts on t prolm s solutons. Su onstrnts n rmtlly sp up t sr pross. Bktrkn n Dson Trs Ovously, n ny soluton o t -quns prolm, tr must tly on qun n olumn o t or. Tror, w n sr t soluton o ts prolm s squn o n sons: Dson 1: Pl qun n t rst olumn. Dson : Pl qun n t ourt olumn. For t -quns prolm, tr r = rnt squns. Wl ts s lry n ny n, w wll now s ow ktrkn n urtr mprov t. Bktrkn n Dson Trs mpty or pl 1 st qun pl n qun pl 3 r qun pl t qun