A General Framework for Codes Involving Redundancy Minimization

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1 IEEE TRANSACTIONS ON INFORMATION THEORY A General Framework for Codes Involvng Redundancy Mnmzaton Mchael Baer, Member, IEEE Abstract A framework wth two scalar parameters s ntroduced for varous problems of fndng a prefx code mnmzng a codng penalty functon. The framework encompasses problems prevously proposed by Huffman, Campbell, Nath, and Drmota and Szpankowsk, sheddng lght on the relatonshps among these problems. In partcular, Nath s range of problems can be seen as brdgng the mnmum average redundancy problem of Huffman wth the mnmum maxmum pontwse redundancy problem of Drmota and Szpankowsk. Usng ths framework, two lnear-tme Huffman-lke algorthms are devsed for the mnmum maxmum pontwse redundancy problem, the only one n the framework not prevously solved wth a Huffman-lke algorthm. Both algorthms provde solutons common to ths problem and a subrange of Nath s problems, the second algorthm beng dstngushed by ts ablty to fnd the mnmum varance soluton among all solutons common to the mnmum maxmum pontwse redundancy and Nath problems. Smple redundancy bounds are also presented. Index Terms Huffman algorthm, mnmax redundancy, optmal prefx code, Rény entropy, unfcaton. I. INTRODUCTION A source emts symbols drawn from the alphabet X = {,,...,n}. Symbol has probablty p, thus defnng probablty mass functon vector p. We assume wthout loss of generalty that p > 0 for every X,andthatp p j for every >j(, j X). The source symbols are coded nto bnary codewords. Each codeword c correspondng to symbol has length l, thus defnng length vector l. It s well known that Huffman codng [] yelds a prefx code mnmzng P X p l gven the natural codng constrants: the nteger constrant, l Z +, and the Kraft (McMllan) nequalty []: X l. X Hu, Kletman, and Tamak [] and Parker [] ndependently examned other cases n whch Huffman-lke algorthms were optmal; ths work was later extended [5], [6]. Other modfcatons of the Huffman codng problem were consdered n analytcal papers [7] [9], although none of these proposed a Huffmanlke algorthmc soluton. In each paper, relatonshps between the modfed problem and the Huffman codng problem were Ths work was supported n part by the Natonal Scence Foundaton (NSF) under Grant CCR-997 and the Multdscplnary Unversty Research Intatve (MURI) under Grant DAAD M. Baer was wth the Department of Electrcal Engneerng, Stanford Unversty, Stanford, CA USA. He s now wth Electroncs for Imagng, 0 Velocty Way, Foster Cty, CA 90 USA (e-mal: Mchael.Baer@ef.com). explored. Parker proposed an algorthmcally-motvated twofuncton parameterzaton defnng varous Huffman codng problems; these two parameter functons are a weght combnaton functon and a tree cost functon []. Three problems, frst examned n [], [7], [], were consdered as a part of ths framework; here we show that a fourth [9] fts nto t as well. In addton, we fnd a smpler redundancy-motvated unfyng problem class that relates the four problems, one nvolvng two scalar parameters rather than two functonal parameters. Ths new framework reveals a unted analytcal structure, ncludng smple redundancy bounds and novel algorthmc results whch mprove upon the algorthm of [9]. In Secton II, background s gven on the codng problem ntroduced n [7]. In Secton III, the new framework, based on an extenson of ths problem, s ntroduced. The problem and the three other aforementoned problems are then put nto the context of ths framework. In Secton IV, the framework s used to help fnd lnear-tme algorthms for the problem n [9]. Redundancy bounds are presented n Secton V, wth concludng thoughts followng n Secton VI. II. BACKGROUND: EXPONENTIAL HUFFMAN CODING One partcular applcaton of a modfed codng problem was found by Humblet [0] for a problem nvolvng mnmzaton of buffer overflow n communcatons. In ths applcaton, the functon mnmzed s P X p βl for a gven β>0.ths s easly generalzed to negatve β by specfyng mnmzaton of the β-exponental average F β (p, l) = β log X p βl. () X Ths problem was orgnally proposed by Campbell [7] and a lnear-tme algorthm found ndependently by Hu et al. n [, p. 5], Parker n [, p. 5], and Humblet n [, p. 5] (later publshed as [0, p. ]). Ths algorthm covers all of R; the case of β =0s consdered by notng that β 0 yelds the orgnal Huffman codng problem. Below s the procedure for the exponental extenson of Huffman codng wth parameter β. Note that t mnmzes () over l, even f the probabltes do not add to. We refer to such arbtrary postve nputs as weghts, often denoted by w = {w } nstead of p = {p }: Procedure for Exponental Huffman Codng ) Each tem m {m,m,...,m n } has weght w W X,whereW X s the set of all such weghts. (Intally, m =.) Assume each tem m has codeword c,tobe determned later. ) Combne the tems wth the two smallest weghts w j and w k nto one tem m j wth the combned weght w j = β (w j +w k ). Ths tem has codeword c j,tobe determned later, whle m j s assgned code c j = c j 0

2 IEEE TRANSACTIONS ON INFORMATION THEORY and m k code c k = c j. Snce these have been assgned n terms of c j, replace w j and w k wth w j n W to form W X. ) Repeat procedure, now wth the remanng n codewords and correspondng weghts n W, untl only one tem s left. The weght of ths tem s P X w βl.all codewords are now defned by assgnng the null strng to ths trval tem. Ths algorthm can be modfed to run n lnear tme (to nput sze) gven sorted weghts n the same manner as Huffman codng []. An example of exponental Huffman codng for β = log. s shown n Fgure. The resultng code s dfferent from that whch would be obtaned va Huffman codng (β =0). The output of a Huffman-lke algorthm mght be a code (and thus the mplct code tree, e.g., []) or merely codeword lengths; we assume the latter from here on, because vald codewords can be nferred from the lengths. Thus we can vew the problem such an algorthm solves as an nteger optmzaton problem. Ths s useful because many dfferent codes can correspond to the same set of codeword lengths and thus all be optmal for a gven problem. Consderng the codeword lengths alone as the soluton to a gven problem, we fnd that some problems have a unque optmzng set of lengths, whle others have more than one dstnct optmal soluton. Multple dfferent solutons manfest themselves n the algorthm as possble tes n the weght of (possbly combned) tems n the combnaton step (step above). Thus the algorthm, as wth Huffman codng, s nondetermnstc. Two determnstc varants are bottom-merge Huffman codng and top-merge Huffman codng []. Code trees yelded from the former method have been called, dependng on the propertes focused upon, best Huffman trees [], compact Huffman trees [5], mnmal Huffman trees [6], and mnmum varance Huffman trees [7], the last of these because varance s mnmzed among (ted) optmal code trees (codeword lengths). Gven b R and p, f we relax the nteger constrant on l, mnmzng F b (p, l) becomes a smple numercal optmzaton and provdes a lower bound for the nteger-valued problem. (Weuse b nstead of β from here on to refer to the parameter for the real-valued problem.) Campbell [7] noted that the optmal value of F b (p, l) for b (, + )\{0} s the Rény entropy of order α =(+b) : H α (p) = log α = +b b log P X pα P X p(+b). Ths should not be surprsng gven the relatonshp between Huffman codng and Shannon entropy, whch corresponds to b 0, H (p) []. Gven b (, + ) and p, the optmal deal real-valued () lengths achevng () are gven by l = +b log p X +log j X p j +b. () At the extremes of the (, + ) range, solutons are defned as the lmt of the solutons for b and b +, respectvely. For b<, there s no real-valued soluton, the problem beng optmzed by l =0and l =+ for every >. III. MINIMIZATION OF d-average b-redundancy We call the dfference between an nteger l and the optmal real-valued soluton l the pontwse b-redundancy r b () = l l, to emphasze ts dependence on b. The arthmetc average of pontwse 0-redundancy was the problem consdered by Huffman n hs orgnal paper, A Method for the Constructon of Mnmum-Redundancy Codes. Here we ntroduce a generalzaton of ths problem encompassng several cases of nterest. Suppose we wsh to mnmze d-average b-redundancy or DABR, R b,d (p, l) = d log X p drb(). () X Ths amounts to fndng l b,d(p) such that R b,d (p, l b,d(p)) = mn l R b,d (p, l) = mn l log d P X p d(l l ) +b+d = mn l log P +b p d X dl P j X p +b j (5) where l s restrcted to the ntegers and by the Kraft nequalty (mplct from here on). Ths reduces to an exponental Huffman codng problem. Then, gven sorted {p }, (5) s solvable n lnear tme; note that the normalzaton of the terms s optonal for the algorthm. For d <, the soluton s always the unary code l = (,,..., n, n ). Consderng the edges va lmtng (as we dd wth real-valued solutons), the range of nontrval cases for mnmal DABR codes for a gven probablty mass functon can thus be consdered to be parameterzed by b d [, + ] [, + ], as n Fgure. As ndcated n ths fgure, many nterestng codng problems ft wthn ths framework. These problems, whch we dscuss below, correspond to subsets of ths two-dmensonal extended quadrant ([, + ] [, + ]). On the set of ponts for whch b s +, for example, the mnmzaton reduces to exponental Huffman codng wth parameter β = d. For d = 0 (b (,+ ]) we have Huffman codng. A partcular type

3 IEEE TRANSACTIONS ON INFORMATION THEORY Codeword length Codeword Item Weght 00 m m 0 m m Fg.. Exponental Huffman codng for weghts w =(0.6, 0.0, 0.0, 0.) and β =log.. In the frst step, the two smallest tems (wth weghts w =0.0 and w =0.) are combned nto a compound tem wth weght 0.7 =. ( ); thus c and c end n 0 and, respectvely. At each addtonal step, the two smallest remanng tems are combned n a smlar fashon. In ths manner a code to optmze (and as a by-product calculate) the value of P X w βl s bult from the bottom up. In ths case, the mnmzed value s.. of Huffman codng occurs for b =+,d 0. In such a case, we note that R b,d (p, l) = X p l + d σ p(l)+o(d ) as d 0 (6) X where the second term on the rght-hand sde represents varance. Ths beng the te-breakng term, we have bottommerge Huffman codng. IV. MINIMIZATION OF MAXIMAL POINTWISE REDUNDANCY As average pontwse (0-)redundancy has been well understood for some tme, Drmota and Szpankowsk decded to explore the prevously overlooked mnmzaton of maxmal pontwse redundancy [9], [9]. We defne dth exponental redundancy as DABR for b =0. Note that the maxmal redundancy problem s equvalent to mnmzng dth exponental redundancy as d +. Thus, consderng d [0, + ], dth exponental redundancy s a subproblem wth a parameter that vares soluton values between mnmzng average redundancy (Huffman codng) and mnmzng maxmal redundancy; such a range of problems and solutons was sought n [9]. Ths was prevously derved axomatcally wthout regard to such a range and wthout soluton []. The verson of the mnmal DABR codng soluton applyng to the maxmal redundancy subproblem was found shortly thereafter [], although t was not generalzed to b 0 or to d =+. Drmota and Szpankowsk presented a smple method for fndng a code wth mnmum maxmal redundancy [9], [9]. However, ths soluton s defcent n the followng senses: Frst, tme complexty s O(n log n). Second, the Kraft nequalty s not necessarly satsfed wth equalty, meanng that the optmal code found n ths manner s often, n some sense, wasteful. Thrd, the code does not necessarly optmze dth exponental redundancy for any d<+. The method s also not generalzed to maxmal b-redundancy (b 0). In order to overcome the frst two defcences, we propose a reducton to a prevously-known algorthm wth lnear complexty prevously dscussed by Parker []. Ths problem was termed the tree-heght measure problem, though t was not prevously consdered n the context of the maxmal redundancy or DABR problems. The tree-heght measure problem mnmzes the maxmum value of w + c l gven c>0and weght vector w. Instead of usng w j = β (w j +w k )on the merge step of Huffman codng, the Huffman-lke tree-heght measure algorthm uses w j = c+max(w j,w k ). In order to use the tree-heght measure algorthm, assgn weghts accordng to w (b) = +b log p, p n whch s always nonnegatve, and let c =. Then ths modfed Huffman algorthm mnmzes max(w (b)+c l ) = max l + «+b log p = max r b ()+log X log p +b n. j X p n p j +b Thus ths lnear-tme algorthm returns a length vector mnmzng maxmum b-redundancy and satsfyng the Kraft nequalty wth equalty. Because tes can occur n selectng weghts to combne, the exponental Huffman algorthm mght yeld one of many possble optmal codes, ncludng codes not optmal for the lmt of dth exponental redundancy (as d + ). For example, consder p = (,,,, ). For dth exponental redundancy, l =(,,,,) and l =(,,,,) are both optmal for d +. These not only mnmze maxmal redundancy, but, among codes that optmze ths, these codes also have the lowest probablty of achevng ths maxmal redundancy, as ths s related to the second term of the expanson of R b,d (p, l)

4 IEEE TRANSACTIONS ON INFORMATION THEORY + maxmal b-redundancy that a suffcently large D s gven by D = log δ p n > [0, pp. 59-6]. However, fndng D requres sortng, so an algorthm derved from ths D would not be a lnear-tme algorthm. d 0 0 maxmal redundancy dth exponental redundancy (standard) Huffman codng b exponental (Huffman) codng bottom-merge Huffman codng + Fg.. The parameter space for mnmal d-average b-redundancy (DABR) codng wth the followng noted subproblems: the Huffman problem (the above lne at d = 0), Campbell s exponental codng problem (lne at b = + ), the problem solved va Schwartz s bottom-merge Huffman codng method (lmt pont at (+, 0) when approached from above), Nath s dth exponental redundancy problem (lne at b =0), Drmota and Szpankowsk s maxmal redundancy (pont at (0, + )), and maxmal b-redundancy (lne at d = + ). Each pont n the extended quadrant represents a dfferent (parameterzed) problem, as n Fgure 5. for d + : R b,d (p, l) = log d P X p dr b() = max r b () + log d P X [r b (X) =max j r b (j)] + O ` as d + d d Each term n the expanson has a dfferent asymptotc complexty. As wth mnmum varance (bottom-merge) Huffman codng (6), each addtonal term further restrcts the set of feasble codes to those that mnmze the current term gven the optmzaton of prevous terms. In the above example, all terms are mnmzed by both the aforementoned sets of lengths. In contrast, l =(,,,,), although also mnmzng maxmal redundancy, results n a code where codewords have a hgher probablty of achevng maxmal redundancy. Ths soluton, whch s n some sense nferor, can nevertheless be acheved by the tree-heght measure algorthm, specfcally the bottommerge verson. It s possble to fnd a D R such that, for every d D, dth exponental Huffman codng mnmzes maxmal redundancy. Let mn +,j γ,j denote the mnmum strctly postve value of γ,j,andlet x denote the fractonal part of x,.e., x = x x. Assgn δ =mn +,j l l j. It s possble to show (7) Fortunately, t s possble to arrve at a lnear-tme algebrac Huffman algorthm, that s, one that keeps D as a varable. Algebrac Huffman algorthms were ntroduced by Knuth [5]. The one proposed here uses a Huffman algorthm whch keeps track of both the frst- and second-order terms; tes between these pars of terms can occur only when all terms are ted, ths due to the manner n whch the Huffman procedure works. Before explanng why ths s the case, we present the algorthm. The aforementoned frst- and second-order terms are and w w = lm [w (b, d)] (d ) d + = lm d + [w (b, d)] [w ] d, respectvely, where leaf nodes have +b+d +b w (b, d) =p, as n d-average b-redundancy. One can thnk of w as representng an nvertble functon of maxmal b-redundancy, " n # X w +b = p j max rb(), j= where, at any gven pont of the algorthm, r b () =l l uses the depth of tem n ts nterm code tree as the value l. Note that only r b () s varable; the denomnator term of w s a result of not normalzng the weghts at the start of the algorthm. In a smlar manner, w represents the probablty of maxmal b-redundancy P X [r b (X) =max j r b (j)]. To mplement ths algorthm, we let w +b = p and w = p for the ntal case. In comparng tems j and k, we consder them as lexcographcally ordered pars e.g., w j =(w j,w j ) sothatw j w k f and only f ether w j > w k or f w j = w k and w j w k, as n [5]. In combnng tems j and k (where w j w k as descrbed), the new tem wll have w j =w j = max(w j,w k).ifw j >w k,then w j = w j. Otherwse, w j = w j + w k.thats, j (w j,w w j = j ) f w j >w k otherwse. (w j,w j + w k ) The reasons for ths are easly seen f we vew w as the representaton of maxmal redundancy and probablty ths maxmal redundancy s acheved. Take the maxmum and add for the addtonal bt of the codeword (multplyng w by ). Then, f the redundances are dentcal, add ther probabltes (w ). Otherwse, take the probablty of the maxmal redundancy.

5 IEEE TRANSACTIONS ON INFORMATION THEORY 5 l c m 9 p 9 w m (, ) (, ) (, ) (6, ) (, ) (, ) (, ) (, ) 000 m (, ) 00 m (, ) (, ) (, ) 00 m (, ) (, ) 0 m 5 (, ) Fg.. Algebrac maxmal redundancy codng, p = (,,,, ) (bottom-merge) 9 Ths combnng method s a Huffman algebra, satsfyng the propertes ntroduced n [5]. The Huffman combnng crteron s shown by example n Fgure. The remanng weght par after codng, ( 9, 9 log 9 and a probablty of 9 ), ndcates a maxmal redundancy of that ths redundancy s acheved. We now show that tes n the w pars mply tes n all terms of the expanson presented n (7), or, equvalently, for dth exponental redundancy for all d [D, + ) where D s some (unspecfed) constant. Theorem : If there s a te n the above w pars, there s a te n all terms of the correspondng d expanson. Proof: Consder two ted pars. Note that, n each, w w +b () because ths holds wth equalty n leaf nodes and the nequalty s preserved n the merge step, snce max(a, b) a + b max(a, b) for a, b 0. If nequalty () holds wthout equalty for the ted pars, nether node on the correspondng code tree can be a leaf node, and, due to orderng for the combnaton step, ther four chldren must be dentcally weghed. However, ths fact can be nvoked nductvely for ether par of chldren, also ted, and thus such a tree could not be fnte. Therefore, ted pars arse only n cases for whch the nequalty holds wth equalty. Thus, they must be leaf nodes or nodes wth two dentcally-weghted chldren. Inductvely, ths means the subtrees must be composed of leaf nodes that are relatvely dyadc, that s, are dyadc when multpled by a nontrval common constant. Thus they are equal n all terms, whch s what we set out to show. One can use bottom-merge or top-merge codng so that the algorthm s determnstc. If one uses top-merge codng that s, favorng combned tems over sngle tems wth dentcal weght [] one actually need not keep track of the second term; the top-merge algorthm behaves dentcally wthout consderng ths term. Ths varant, llustrated n Fgure, s actually a specal case of the tree-heght measure problem mentoned above. However, f we wsh to assure that the soluton has mnmum varance, the algebrac method s needed. V. BOUNDS One can easly see that f we relax the nteger constrant on length for mnmzng d-average b-redundancy, the real-valued soluton s not l, but some dfferent l. By substtutng the soluton n (), we fnd l = ω log X p +log where ω = j X +b+d = bd. (+b)(+d) (+b)(+d) Note that when the values of b and d are exchanged, the deal soluton remans the same. Ths problem thus has a hgh degree of symmetry. However, because the problem tself s not symmetrc, the symmetry of nteger solutons s not perfect, as we can see n Fgure 5. Usng ths and the Shannon code [] analogue l, we can fnd bounds for the optmal DABR when b,d, and b + d : 0 R b,d (p, l b,d(p)) αb(h ω (p) H α (p)) < where we recall ω = +b+d and α =, the subscrpt (+b)(+d) +b of Rény entropy n (). As wth exponental Huffman codng, equalty holds ff the deal soluton l has all nteger lengths. For b = + and d = 0, ths results n the well-known Shannon bounds. For b =0, t reduces to a normalzed verson of an nequalty n []. Wth a dfferent normalzaton, ths nequalty relates to Rény s gan of nformaton of order α, a generalzaton of relatve entropy []. Ths s not surprsng gven the relatonshp between relatve entropy and Huffman codng noted by Longo and Galasso []. Due to the reducton to exponental Huffman codng, more sophstcated redundancy results may be appled f desred. The bounds gven by Blumer and McElece [] apply to the exponental case but appear as solutons to related problems rather than n closed form. Taneja [] gave closed-form bounds usng an alternatve defnton of redundancy. p ω j

6 IEEE TRANSACTIONS ON INFORMATION THEORY 6 l c m 9 p 9 w m 6 0 m 00 m 0000 m 000 m 5 Fg.. Top-merge maxmal redundancy codng, p = (,,,, ) (sngle varable) 9 d () () consdered by Parker; the top-merge verson of the algorthm n partcular addtonally optmzes dth exponental redundancy for large d. A better soluton one mnmzng codeword length varance among such optmal codes s suggested by and s developed from the two-dmensonal framework ntroduced here. All algorthms dscussed are Huffman-lke and thus lnear-tme gven sorted nput, unlke the orgnal algorthm proposed for maxmal redundancy. (5) 0 () 0 () () Fg. 5. The parameter space for mnmal DABR codng for p = (0.5, 0., 0., 0., 0.09). Each regon represents a set of problems wth the same soluton. On the transton curves (sold), multple solutons are optmal. The fve dstnct soluton regons are () l = (,,,, ), ()l=(,,,,), ()l=(,,,,), ()l= (,,,, ), and(5)l=(,,,,). The dotted lnes wthn the parameter space ndcate the (+ ) asymptotc behavor of the lmts between regons. Note that b+ d + = 0 dvdes nonncreasng length vectors from nondecreasng. Also note the hgh degree of symmetry; the mperfecton of ths symmetry s best llustrated by the dfferent asymptotes. VI. CONCLUSION A two-dmensonal framework s demonstrated to encompass examples consdered by Parker [] classcal Huffman codng [], the exponental varant proposed by Campbell [7], and the dth exponental redundancy problem proposed by Nath []. These examples, along wth all problems wthn the framework, are solvable by Huffman-lke algorthms. The maxmal redundancy problem proposed by Drmota and Szpankowsk [9], [9] s shown to be optmzed by ts equvalence to another example b () It s unclear whether all nontrval problems wthn Parker s more general framework are covered by ths seemngly more specfc framework and trval extensons thereof. Such analyss, buldng upon Parker s work, could be a bass for further research. Extendng ths algorthm to alphabetc codes (alphabetc search trees) could also be explored. For nonnegatve exponents (d 0), ths framework s a trval extenson of [], but negatve exponents mght provde more of a challenge. ACKNOWLEDGEMENTS The author wshes to thank Thomas Cover, John Gll, Andrew Brzeznsk, and the two anonymous revewers for ther helpful comments and encouragement on ths paper. REFERENCES [] D. Huffman, A method for the constructon of mnmumredundancy codes, Proceedngs of the IRE, vol. 0, no. 9, pp. 09 0, Sep. 95. [] B. McMllan, Two nequaltes mpled by unque decpherablty, IRE Transactons on Informaton Theory, vol. IT-, no., pp. 5 6, Dec [] T.C. Hu, D.J. Kletman, and J.K. Tamak, Bnary trees optmum under varous crtera, SIAM Journal of Appled Mathematcs, vol. 7, no., pp. 6 56, Apr [] D.S. Parker, Jr., Optmalty of the Huffman algorthm, SIAM Journal on Computng, vol. 9, no., pp. 70 9, Aug. 90, Erratum: vol. 7, no., Feb. 99, p. 7. [5] D.E. Knuth, Huffman s algorthm va algebra, Journal of Combnatoral Theory, Ser. A, vol., pp. 6, 9. [6] C. Chang and J. Thomas, Huffman algebras for ndependent random varables, Dscrete Event Dynamc Systems, vol., no., pp. 0, Feb. 99. [7] L.L. Campbell, Defnton of entropy by means of a codng problem, Zetschrft für Wahrschenlchketstheore und verwandte Gebete, vol. 6, pp., 966.

7 IEEE TRANSACTIONS ON INFORMATION THEORY 7 [] P. Nath, On a codng theorem connected wth Rény entropy, Informaton and Control, vol. 9, no., pp., Nov [9] M. Drmota and W. Szpankowsk, Precse mnmax redundancy and regret, IEEE Transactons on Informaton Theory, vol.it- 50, no., pp , Nov. 00. [0] P.A. Humblet, Generalzaton of Huffman codng to mnmze the probablty of buffer overflow, IEEE Transactons on Informaton Theory, vol. IT-7, no., pp. 0, Mar. 9. [] P.A. Humblet, Source codng for communcaton concentrators, Ph.D. thess, Massachusetts Insttute of Technology, 97. [] J. van Leeuwen, On the constructon of Huffman trees, n Proceedngs of the rd Internatonal Colloquum on Automata, Languages, and Programmng, 976, pp. 0. [] E.S. Schwartz, An optmum encodng wth mnmum longest code and total number of dgts, Informaton and Control, vol. 7, no., pp. 7, Mar. 96. [] G. Markowsky, Best Huffman trees, Acta Informatca, vol. 6, pp. 6 70, 9. [5] Y. Horbe, Remarks on compact Huffman trees, Journal of Combnatorcs, Informaton and System Scences, vol. 9, no., pp. 7 0, 9. [6] G. Forst and A. Thorup, Mnmal Huffman trees, Acta Informatca, vol. 6, no. 9/0, pp. 7 7, Apr [7] L.T. Kou, Mnmum varance Huffman codes, SIAM Journal on Computng, vol. 9, no., pp., Feb. 9, Orgnal as Mnmal Varance Huffman Codes, Research report RC, Internatonal Busness Machnes Corporaton, 90. [] C.E. Shannon, A mathematcal theory of communcaton, Bell System Techncal Journal, vol. 7, pp. 79, Jul. 9. [9] M. Drmota and W. Szpankowsk, Generalzed Shannon code mnmzes the maxmal redundancy, n Proceedngs of the Latn Amercan Theoretcal Informatcs (LATIN) , pp. 06, Sprnger-Verlag. [0] M. Baer, Codng for General Penaltes, Ph.D. thess, Stanford Unversty, 00. [] A. Rény, Some fundamental questons of nformaton theory, Magyar Tudományos Akadéma III. Osztalyanak Közlemene, vol. 0, no., pp. 5, 960. [] G. Longo and G. Galasso, An applcaton of nformatonal dvergence to Huffman codes, IEEE Transactons on Informaton Theory, vol. IT-, no., pp. 6, Jan. 9. [] A.C. Blumer and R.J. McElece, The Rény redundancy of generalzed Huffman codes, IEEE Transactons on Informaton Theory, vol. IT-, no. 5, pp. 9, Sep. 9. [] I.J. Taneja, A short note on the redundancy of degree α, Informaton Scences, vol. 9, no., pp. 6, Sep. 96. PLACE PHOTO HERE Mchael Baer receved the Bachelor s degree n electrcal engneerng and computer scence from the Unversty of Calforna, Berkeley, CA, n 997 and the M.S. and the Ph.D. degrees n electrcal engneerng from Stanford Unversty, Stanford, CA, n 000 and 00, respectvely. From 00 to 005, he was a Research Scentst wth UtopaCompresson, Los Angeles, CA. Snce 005, he has been wth Electroncs for Imagng, Foster Cty, CA, where he s a Lead Scentst n the Imagng Group. Hs research nterests nclude source codng trees, optmzng prefx codes for nontradtonal codeword length functonals, and mage, vdeo, and data sgnal processng for compresson and other applcatons.

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