Buffer Map Message Compression Based on Relevant Window in P2P Streaming Media System

Sub mit ting to ar Xiv.co m 1  Abstract —P op ul ar peer t o p eer s treaming media s ystems such as P PLive and UUSee r ely on perio dic buffer -m ap exchange between peers for pro per o peration. The b uffer-map exchange con tai ns redund ant infor mation wh ic h causes n on-negli gible o verhead. In t his paper we pres ent a theoretical framewo rk to study h o w the o verhead can be l owe red. Diff erentiati ng from t he traditi on al data co mpression ap proach, w e do no t treat each buffer- m ap as an isolated data blo ck, but consider the cor re lations between the sequenti ally exchan ge d bu f fer -maps. Under this frame work , two buffer-map comp ress ion schemes are pro posed a nd the correctness of t he schemes is prov ed mathem ati call y . Moreo ver, w e derive the theoretic al l imit o f com pressi o n gain based on probability t heo ry a nd i nfo rmation theory . Based o n the sy s tem parameters of U USee (a po pular P2P streaming platform ), our simulations s how that the buffe r-map siz es are reduced by 86% and 90% (from 456 bits down to o nly 6 6 bits and 4 6 bits) respectively a fter applying o ur schemes. Furthermore, b y co mbining with t he t raditi o nal co mpressi o n metho ds ( o n individual bl o cks), the siz es are decre ased by 91% and 9 5% (to 42 b i ts and 24 bit s) respectively. Our s tudy pro vides a guideli ne for developing practical com pres sio n algor i thms. Index Term s —buff er-map, bitmap, P2P , comp res sion, relevant windo w I. I NT RO DU CTI ON n recent y ear s, th e in creasing ly dev elop m ent of P2P str e am ing media sy stem ha s bee n at tr actin g mo r e a nd more att e n tion o f r esearch e r s. In the majo r i ty o f the p u b lic research reports, peo ple h ave tr ied to made use of the d i s co ve r ed buffe r-m ap mess a ge (B M) to c a rr y out sy stemat ic o b servations an d an alys is in diff erent aspec ts, such as b u f fer d escripti o n s [ 2] -[9], th e star tup perfo rm ance [3][4] [4], pe er off set an d of fse t lag [5], a nd d ata fetc h in g str ategies[ 12][ 13][14]. Ho we ve r , no o n e ha s paid atten tion to BM itself . Ge n erall y spe aki ng, BM is design e d to depict t he buff er f il lin g sta tes of a p e er, an d it is p e r iodic al ly ex cha ng e d be twee n Mnus crip t rece ived Octo be r 9, 2 011. Chunx i Li. A u tho r is with the Elec troni c a nd Inf ormation En gine erin g C olleg e, Beij ing Jia o tong Un ivers i ty , N o.3 Shang Yu an C un,Hai Dian D is trict Beij ing,Ch ina Post- Cod e 100 044 (pho ne: 86 -01- 5 168 47 59 ext. 115 ; fax: 86-01 - 5 1 6836 82 ; e-m ail: c hxli1@ b jtu.edu.c n). Chang jia C h en Author, is with the Elec tronic a nd Info rmation E n ginee ri n g Coll ege , Bei jing Jia o tong Uni versit y, No.3 Shang Yuan Cun,Hai Dian D is trict Beij ing,Ch ina Pos t- Code 100 04 4 (e -m a il: chang jiache n@si na.co m). Dahmi ng Chiu Author is with Dep a rtm ent of Inf ormatio n Engine ering , the Chin es e U nivers ity of H ong K ong, Ho S in Hang, Roo m 836, H o ng Kong , China (e- mail : dmc hiu@ie.c u hk.e du . hk). pair ed pe ers f or in fo r mi ng each o th er w h i c h data th e o t her c a n an d can ’t sha re. T he BM exchan ge be t w ee n peers pr oduc es so me n on-n eglec table overhead an d t ha t i s beco m in g an impor tant cause fo r co n cern . Acco r din g to our measurement on so me t o p p o pular P 2P s t r e am ing media sy ste ms in cludin g PPL ive an d U U See , th e B M ov erhead f or o n e pee r is at le a st abou t 30k bps an d 8kbps respectiv ely . Th is ov erh ead ma y n ot r e duce w ith th e decreasin g o f th e video play back ra te, such as in a nar r o wb a nd wireless environmen t. Moreo ver, it ma kes th ing s w orse in th e case o f e n co un teri ng unstable netwo r k co n ditions, b ecause a peer needs t o conn e ct w ith m uch more peers fo r e s c api ng th e bad situ ation. B esides th at, potentia l ov er h e a d in crease c a n al s o r esult fr o m th e fin ding s [1] t ha t decreasing t he time peri o d of BM ex cha nge c a n help s t r e a min g co n tent diff usion. On th e other h an d, o ur measur ement b ased an alys i s s how s t h ere i s much r e dun dan t informa tion need to b e r e mov ed from BM b ec a use th e f il ling s t ate of each piece of data w il l b e r e peat e dly repo r ted ma ny t imes t o a rec eive r pe er . Th eref ore, n o matt e r in w h ich cir cumstan c es, it is n ec essary to decrease t he BM ov erh ead. Obv i o usly , da ta comp r ess ion is th e m o st o p e r ati o n al way . Tr aditi onal loss less data compressio n m e th ods [16] -[22] can be appli e d to reduce th e BM siz e. In fact, i n 200 8 w e do f ir st f oun d a type of co mbinin g algorit hm o f Lempe l–Ziv (LZ) [ 17][18] an d r un-l e n gth encoding (RL E) [19] i s ad o pted in UU See th r o ugh o ur measurement study , w h ich dec r eases th e buf f er -map from 456 bits to 140 bits, a nd l ater w e go t t o kn ow th at P PL i ve ad o pted certai n 2-leve l al gorith ms of Huff ma n [16] algori th ms t o do t ha t. All th e tr adit ional meth ods treat each B M as a genera l an d in dependen t da ta p iece . How ev er, th e succ essiv e ex cha nged BM s ar e str o n gly co rr elated with more th an ha lf of th e in fo r mat ion in a BM being r edunda nt . E.g ., a peer n ee d s n ot to r e port a chun k buf fer state in it s B M if th e peer o n th e o th er side h as d o w n loaded th at chun k. Due to n ot r e co gn izi ng t he c h ar acteristic o f B M ex cha nge, much r e dun da nt in f or mat ion still rema in s in th e buf f er -map co mpr es sed by t r aditi o na l me t h o d. Ho w to remov e the r e dun da ncy info r mati o n as far as poss ible? Ho w to e valuate th e co mpr es sion eff iciency ? Wh at ar e th e fun dament al l i mits o f th e size of th e compresse d buff er -map? All suc h issues ar e very in terestin g to bot h r esearch e r s an d sy stem designers. T o th e be st of our kn o wledge , t he study o n buff er -map co mpr essio n h as n e ver b ee n r epo r t ed in either pr actical o r th eo ret ical lev el, n o t t o menti o n th e serious and thorough study. R elev ant W in dow ba sed Buf fer -map Compression in P2P S treaming Media Sy st em Chunx i Li , Chang ji a Che n, Dahm in g Chi u I Sub mit ting to ar Xiv.co m 2 In th is pa per, we pr es ent an origin al and b i lateral f r am e w ork w hi ch opens a no th er doo r fo r BM c ompr e ssion. Tot a lly diff er ent f r o m th e tr adit ional d ata compressio n pr in c ip le, we don’t t reat e a c h buf fe r -map as a g eneral and in dependent data b lock , b ut r ec ogniz e t he correla tions betw ee n th e sequential e xch an ged bu ffe r - maps so a s to exclude majority r edun dant in fo r mat ion from a regular BM. Moreo ve r , our a pproach doe s not co n f l ict w ith tr ad itional d ata co mp ressio n p ri nciple but c an w ork t o gether with it. Our co nt ributions in th is pap e r i n clude : i) . we pr es ent a fundam ental f r am e wo rk of co mp ressio n based o n t w o crucial b ut easily o ve r loo ked compressio n p rin ciples disco v ered from BM exc h ang e; ii ) . U n der th e ori gin al f r amew ork, we pr esent t wo e ff icient BM c ompressio n s chemes, th e fe asi b ilit ies o f w h ich ar e prov ed fr o m mathemat ics v iewp oin t; iii ). The t heoretical limi t of ave r age siz e o f th e c ompr es sed B M i s deduced based on probab ilit y th eory a nd in fo r mat ion th eo ry. The n umeri c a l r es ul ts acco r din g to U U See’ s sy stem par ameters show th at, i f w it hout tran smission e r r ors, th e b uff er-map can b e reduce d by 86% an d 90% from 456 b it s d o w n to onl y 66 b it s an d 4 6 b its respe ctive ly acco r di ng to th e t w o schemes w e p resented; Fur th ermore, if comb i ni ng w ith th e t ra ditiona l dat a compr e ssio n pri ncipl e , i t can be dec r e a se d by 91% an d 95% to 42 bits an d 24 bits r espe ctively . In t h e remain in g o f th is paper, we h igh ligh t th e importan ce an d c ontr ibutions o f o ur research in th e co nt ext of an ov erview an d related w o r k abo ut P2 P streami ng media sy stem i n se ction II; Our co r e i dea an d th e fundament al t heo r etical fram e wo rk ar e presented i n section I II ; Sec t io n IV puts forw ar d two BM c ompr es sion schemes un der t he theoretical fra me wo r k and proof s th e f easibility t heo r etically ; In s ec tion V, we in -depth an aly z e the th eo r etical limi t o f the c ompr ess ed buf fer-m ap b ased on pr ob ability th e ory an d in fo r mat ion t heo r y , an d d is c uss th e si m ulati o n results w i th UU See’s sy stem par ameter. Sect i o n VI co n c ludes th e paper. II. A N OVERV IEW OF P 2P STREAMING ME D IA S YST EM R ef err i ng to Fig. 1, a ty pical P2P str e am ing media sy s tem uses fe w serv ers to serv e lar ge nu mb er of audi ence s (named as pe er ) w ith b oth liv e an d V oD progra ms by shar in g t he c a pacities of a ll t he in dividuals as a whole [6]-[10]. In suc h a sy st em, t h e see der w ill div i de t h e media str eamin g in to co n tin uo u s da ta blo cks c a lled chunk , a nd i njec t th em in to th e netw or k acc ordin g to pee r s’ requir ement. Each chun k has a un ique ID which is seq uen tiall y as si gned in asce nd ing order, i.e. th e ear lie r playe d c h un k h as th e sma ller ID. I n the oth er s ide, each pe er w i ll use a b uff er o r gan iz ed w it h chun k un its a s Fig.2 s h ow n , to cac he the c h un ks r eceiv ed from o th e r peer s in mo st cases or th e see d e r in fe w case s fo r smoo th pl ay b ack an d m o r e signi f icant ly , shar in g w it h o th er peers. Live pee r only caches a sm all f r act io n o f th e w h o l e video , whi le Vo D p e er may cache a lmost th e w hole video . Peer’s buf f er is usu a lly p ar tial ly filled due to th e in f lu e nce of man y fa c tors. Th e dow nl o aded c h un ks (th e shadow squar e in Fig.2) c an be sha red, w hi le the emp t y ar eas nee d to be filled by dow n loadin g from o th ers. T hi s is t he shar in g pri nciple play in g th e ke y role in th e simil ar P2P co n tent di s t ributi o n sy stem. Fo r enabling th e key sh ar in g pri nciple b etwe en P2P use r s, a buffe r-m ap message ( BM) is int r o duce d to exchan ge th e b uff er in fo r ma tion betw ee n t he p air ed peers. Re f er ri ng to Fig.2 , a peer’s B M conta in s tw o par ts, a n offset  an d a bitmap b . Th e offset  co r respo n ds to th e o l des t chunk , i.e., the s m al lest chun k ID i n t he buff er , an d th e bitmap is a { 0, 1 } se q uence b = ( b 0 ,…,b | b |  1 ), which repr ese n ts t he buf f er fillin g state. Th e lengt h | b | i n dicates t h e b u f fer size. I n bi tmap b , a bit v alue 1(0) at th e i th (0  i< | b | ) co mponent b i mean s th at th e pee r ha s(ha s n o t) th e chu nk wi t h ID  +i- 1 in i ts buf fe r . Fo r si mplicity , we c a ll such a buf f er-m ap m e ssage as a regular BM(  , b ). Fig . 2 . Buf fe r and b uff er- map Acc ordin g to our m easurement o n t he t o p popular P2P str eam m e dia sy stems in cludin g PPL ive a nd UU See, a p e er se n ds out a B M o f 250-byte or 8 0-byt e lo ng ev ery 4 or 5 seconds respective l y , an d eac h peer k e eps at least 30 conne ctions w ith other pe ers co ncur rent ly . Therefo r e, th e B M ov er h e a d f or one pee r is at least abo ut 30 kbps and 8k bps r e spec t iv ely . On th e o th er ha nd, since a peer c onsta ntl y r e mov es the e ve r play ed chu nk from i t s buf fe r b y r igh t shiftin g th e buffe r h e ad po in ter, and fetc h es new ch unk s to fill th e buf f er , th e chun k IDs at bo th e n ds of t he buf fe r w ill mo ve fo r ward with time. This w h ole pr ogress is r ef lecte d in t h e periodically reportin g BMs . H ow ev er, si n ce B M exc h ang e time period is m uch short er th an t he buff er size (measur ed in play back dura tion), th e state of eac h u ni que chun k will be r e peatedly r epo r t e d by man y seque n tial BMs. E.g., i n g e n eral a U USe e pee r has a buf fe r 140s l o n g, a nd sends BM ev ery 5s, th us each chu nk w il l be r ou g hly reported up t o 28 t imes on average. Obv i ously , t here is too mu c h r edunda nt in f orm ation i n a r egula r BM. Th e p e r f orma nce d ir e ctly c on nects to th e B M e x chan ge t ime period [1] [ 24] [ 25]. Since t h e f aster b uff er- map excha ng e can leads to th e small er in iti al buff erin g dela y , but a lso th e exce ssive o verh ead, each P2P streami n g m e dia s y stem h as an ov er h e a d-delay t ra deo ff. For a gi ve n ove r h ead constrai nt, a n ef fic ien t co mpr essi on o n buff er- map m eans a perfo rm ance impr ov ement i n ter ms o f in itial b uff er delay a nd the a bility to ov er co me flush crowd i n r ev erse p roportion to comp r essio n Fig. 1 .. The s ystem st ru c tu re Sub mit ting to ar Xiv.co m 3 ra tio acco r din g to theorem 3 an d c orolla ry 1 in [1]. Tr aditi o na l lo ssle ss d ata compressio n me th od s can be appli e d to r educ e t he size of buff er-m ap. Ho w ev er, th e y a re in c a pab le o f r emovi n g th e r edundan t i n f ormat ion be t w ee n suc ce ssive exchan ged b uff er-m aps. In t his paper, we seek to e st abl i sh a n ew th e oretical fra mew o r k fo r buf fer-map c ompr es sion, w h ich b elongs to a to tal ly dif ferent sy ste m oth er th an that o f t r aditi o n al data co mpr es si o n appr oaches . The new c ompr es sion th eo ry can guide u s t o dev ise th e ef f icient a nd e ve n t he m ost pow er fu l pra ctical c ompr es sion algor ithm s if c omb in g trad itional da ta co m press i o n pr in ciple. III. T HE BA SICS IDEA LS A N D CO R E CONCEPT As w e know , m o st r edunda nt BM i nformation is ascr ib ed to th e lar ge n umbe r of repeat state r epo r ts of t he same buff er pos it ions in bitma p. M o r e spec ifically , o nce a b uff er po sition is f il led , i t w ill b e fill e d f orev er an d needs n ot t o be r epeate dly reported in l ater b u f fe r -maps. In the oth e r side, o n ly t hose b uff er positions with 0-v a lue i n curr ent bitmap m ay ch an ge th e ir values to ‘1’ i n the sub sequent b i tmaps. Such a se emin gly tri v ial ob servat io n on t he buff er-ma p excha nge leads to a nont rivial co mpr es si o n i nsi gh t: it is not n ece ssar y to in c lude all th e buff er p o sitions i n a bitmap. To f acilit ate figur ati v e under s t an din g th e b asic ideas, w e use two simple an al o gies to illust rate o ur point s. (a) BM c ompres si on in sin gle direc tion (b) BM co mpres sion i n b i-d irectio n Fig. 3. T he d em ons trations of BM c ompre ssi on Fo r sim plicity , w e ass umi ng an ideal c ommun ication situation without packe t los s an d t ran smission d elay . Suppo sin g a buff er -map e x c h an ge p r o gr es s from p ee r A t o p eer B acc ordi ng to th is idea as shown in Fig.3(a) , BM A ( i ) m e a ns th e i th BM of peer A to be sent to p e er B, a nd RW will be e xpl ain ed lat er. I n th e f ir st s tep, a ft er se n ding t he B M A (1), p ee r B gets to know w hi ch chunk s c a n b e dow nl o aded from pe er A. Th us, fo r B M A (2), pee r A only nee ds to r e port t hose po siti o n s w h ich have value 0 i n BM A (1) b ut e xcludes th o se w i th v a lue 1s in BM A (1), as w ell as the new positions a t {9, 10}. As a r e sult , w e get the co mpr esse d BM as C B M A (2) w hi ch w ill be sen t t o peer B . Furt herm o r e, once a chun k is fetched, a p eer w i ll nev er car e i f other p e ers h ave th e same ch unk or not. The excha ng e se q uence b ased on th e ideas is sh o wn in Fig. 3(b ). In th e fir st step, after sen ding th e BM A (1), pee r B g e t s to kn ow which chun ks can be d ow nl oade d from peer A, as w ell a s which chun ks are needed by peer A; When B M B (1) is pr o duced, peer B only extra cts those po s itions peer A needs , i.e. t he po siti o n s w h ich a re eith er value 0 or n eve r a nn o un ced i n BM A (1) but excludes th os e with v a lue 1s in BM A (1). As a r esult, peer B se n ds the comp r esse d BM CB M B (1) to p ee r A; In th e b eginn ing of step 3, peer A alr eady kn ow s w h ich o f i ts d ow nl o aded chun ks have ev er be en told to pee r B and which ch unks peer B does n ’ t car e d . At last , b y r emov i ng th o se positions with dete r mi na te state of value 1 f r om BM A (2), we o btai n a CBM A (2) only 2 bits lo n g. B a s e o n th ese o bservatio n s, w e co n clude two fix? ?(exclusi on) pri nci ple s on how t o c ompact th e b uff er-m ap. Principle 1 : A peer n ev er n ee ds t o r epo rt a b uff er position furth er in hi s b it map o nce sendi ng a value 1 i n t hi s pos it ion. Principle 2: A peer n e ve r need s to repo rt a b u f fer po sition to a receiv er pee r f ur ther onc e rec ei v ing a v alue 1 i n th is pos it ion from th at receiv er peer. It can be s een fr o m t he abov e illu strat ion, unli ke a r egu l ar buf f er -map, the c ompact b i tmap i tself cann ot f i x i ts locations. On ly by mapped t o t he rig ht position seq uen c e, can th e co mpr es sed bitmap be c orr ectly encode d an d deco d e d . Th us, w e in tr oduc e th e co n cept o f re levant wi ndo w to des cri b e th o se posi t io n s w h ic h are not e xcluded by abo ve pri nciples. In Fig.3, th e R W i s just shor t fo r t he conce ption. Ge ner ally speakin g, a relev ant w i ndow ( R W) is a set of un i qu e n onnegati ve int e ger s ar r an ged in ascending o r der, each of which eleme n t co r respo n ds to a n ID of chun k w h ich h as n e ve r be en r epo r ted w it h v a lue 1 in t he buf f er-ma p. Sin ce a relevant w i ndow ha s in finite elements in t heo r y , m at hemati c a lly , we express it a s RW } { 1 0 N l l l L      an d assume L l l l N     in abov e expr es si o n . By th e way , th e maximum excluded pos it ion (MEP) f o r a g iven releva nt w indow L is d ef in ed as } : max{ ) ( N l p L p L me p    . Th e r e leva nt w indow can be in terp reted a s al l th e c h un ks w hi c h have not be en dow nl oade d, w h ile th e M EP mean s t he largest c hunk-id w h i c h h as eve r be en fe tched. Th e update sequence o f releva nt wi ndo w in t he se n der i s show n in Fig.3 . Compari n g to th is p r o g ress, i t i s no t diff i cult to deduce th e upda te seq uence i n th e r ece iver. When send ing a buf f er -map, the sender f ir st e n cod es t he bitmap acc ordin g t o it s R W, an d th en remov es th o se p o sitions which h av e s tat e valu e 1 from its RW. Wh e n a co m pact buff er-map ar r ive s, th e receiv er first deco des it a c co r din g t o t he co r r espo ndin g R W, an d th en Sub mit ting to ar Xiv.co m 4 delete s th ose pos it ions from RW i f th e sender repo rt s t hem w i th state value 1. Th us, b ased on the i llustr ati o n in Fig.3 , if without data tr an smissio n err o r and delay , and assuming th e same in itial RW s i n b oth si des , th e RW s i n bot h sides ar e alway s sy n chr o n ous . Therefo r e, in the new co mp ressio n mec h an ism, simil ar t o the b itma p in a r egular BM, t he co m pr ess ed b it map ha s t he sam e fo r m at of a {0,1 } seque n ce a s v = ( v 0 ,…, v |v|  1 ), w hi le th e diff erence lies on th at t he v al ue o f v i is expl ain ed as th e fillin g state o n th e b u f fer pos it ion l i i n th e relevant w indow L . In sum mar y , th e pr i nciples disco v ered from BM exchan ge lays a solid foundat ion fo r us t o estab li sh a rele vant wi n dow based f r am e w or k f or buf fer-map co m press ion . Th e ov erh ead f or BM exchan ge can be sign ificantl y r educe d if we can use b uff er-map on relev ant w indow in s t ead of on the regular b uff er w in dow . IV. T HE BM C OMPR ESSION BAS ED ON RELEVA NT WINDO W Giv en th e same releva nt w indow L, t he sender p eer can un iquely extra ct the co mpa c t sta tes fr o m t he r egular b uff er-map an d the r ece i v er peer c an lo ssless l y dec ode t he c ompact bitmap . Th e key po in ts i n the BM co mpression based o n rele vant w indow ar e i) h ow to loo k fo r th e pr o per r ele vant w i ndow L , a nd i i) how t o k e ep th e c onsistency o f releva nt w i ndow s betw een sender and recei ver. In li gh t o f th e exc lusion pr in ciples , th e rel e van t w in dow s b ased B M co mp ressio n can b e a pplied in to eith e r a singl e pee r o r a pair of peers. If using p rin ciple 1 alone to co nstr uct th e relev ant w i ndow , w e h ave th e B M c ompression sch e me based o n sin gle pee r ’s rele vant w ind ow (BMC S_SRW ); I f b oth pri nciples ar e c onsider e d, w e ha v e th e BM compr e ssio n scheme b a sed on pai red peers’ common releva nt window (BM CS_CR W). T he l att e r i s a l ittl e more c omplex th an t he f orm e r but it h as much b etter c ompr e ssion ef ficiency . In th e f ollow i ng theoretical d isc ussions, w e assume a n i deal n etw ork situation without packet los s an d tr an smission err or. Mo r e ov er, beca use t hese schemes a re esse n tia l diff erent f r om t r aditional m ethods li ke H uff ma n, L Z an d R L, there are so me r ooms to in v ent cert ain algor ithms to m ake u s e of t he jo int fo r ces . E.g., tr aditi o na l methods can r einfo r c e th e c ompr es sion after ou r me th o ds an d v i c e versa. A. B M compression scheme based o n single peer’s relevant w i ndow Co n siderin g a peer, say p e er A , is sendi ng t he compact buffe r f il lin g s tat e s v ( t ) base d on its relev an t w ind o w L A ( t ) t o its neigh bo r peers a t ti me t un der princ iple 1, fo r c orr ectly e n cod in g an d deco d in g th e b it map v ( t ) i n t he sen der A an d receiv er B respec tive ly, certa in exc h ang e mechan ism must b e des ign ed to ensur e t he co n sistency o f the rel evant w i ndow s in b oth sides at a ny t ime. Fo r th at , i t in tuit ive l y r e qui res each pair ed connectio n should kee p th e co mm unicat ion f r om th e ve r y b egin n ing in th eo r y , w hi le i n pr actice suc h a requir emen t can be met b y eith e r periodically an noun cing th e co m pr ess ed B M or by r ep or ting th e curr ent r elev a nt w in dow . After that, th e r e l ev an t windows w ill b e correctly an d seq uen tiall y upda ted acco r di ng to each BM e xch an ge. Str ictly speakin g, f ollo win g lemma giv es th e def in it e an swe r . Lemma 1 : In th e BM sendin g seque n ce from one peer to an other a c co r din g t o th e BM co mpr es si o n scheme base d on sin gle peer’s relev a nt wi n dow (BMCS_SR W), t he co n sistency of bo th t he rel eva nt window s in p e er A f o r e n co di ng a nd in peer B fo r deco d in g, as w ell as th e c orr ec t ness of the d eco ded buf f er -map correspo nd in g to th e e n cod ed buf fer-m ap, can be fully assur e d. Pro of : Let ’s consider a pr oc ess o f B MCS_S RW in cludin g a se n din g pr oto co l a nd a rece i v in g protoc ol. In the sendin g s ide, a ssu mi ng at time t , p eer A h as a relevant w indow L A (t) ={l 0  i + N o r Z( c , iT ) } (11) As sume peer A has an of fset  i at the time t = iT , th en  i +1 =  i + rT . (12) If c ont aini ng th e o ff set  i +1 in th e compresse d buf f er ma p, th e a ve ra ge co m pr ess ed b it map l ength is                                 1 0 1 1 1 1 1 1 rT N i N c i N c SRW_ i S N c) N- S r T N rT Z(c,iT) p rT W i i i i      (13) If not i ncludin g t h e o ffs et  i+1 in the compr e ssed buff er map, th e a ve ra ge co m pr ess ed b it map l ength is                             1 0 1 0 1 0 SRW _2 ) 1 ) , ( W N i N c i N c i S N r T c N S N rT iT c Z p rT i i    (14) ■ The compress ed bitmap leng th of th e latt er is a lit tle larger th an th e f ormer . Ho w eve r , fo r t h e former, a co mplete buff er map m e ssage n eed s to i nclude t he o ffset f ield w h ich can b e des ign ed to a relat ive off set W  o cc u py in g sev era l few bits ( W   8) in mo st ca ses. A c co r din g t o th e o ptimi zed implementa tio n o f B MCS_S RW usin g bo t h t w o t y pes o f BM, the a v era ge B M lengt h s h ould be withi n [ W 2 , W 1 + W  ] . C. B itmap Length of BMCS_CRW We a ssume both peer A a nd peer B e xch an ge th eir b uff er-map to each o th er. Pee r A sends out its buff er- map at time t = iT a nd t =  +iT i s th e time fo r peer B to se n d. R ef err i ng to Fig.6 , at time t = i T , peer A in itial ly an n o un ce s its compress ed buf fe r ma p CBM(  i , v i ) an d updat e s i ts r elev an t w in dow t o L ’ AB ( i ) when fin ishin g th e sendin g. Acc or ding to pri nci ple 1, th e relev an t window L ’ AB ( i ) onl y in cludes th ose chun k posi t io n s w hi ch ar e either n ev er a nn ounced y et or ev er an n o un c ed a value 0 at ti me t = iT by pee r A. U p o n r ecei vin g CBM (  i , v i ), pee r B updates its r e levant w indow b y remo vin g th e positio n s which ar e l ess t han  i a nd th e p o sition s which fi lling stat es equal v alu e 1 in v i dur ing th e dec oding pr o cess . Fig.6. B M Exc hange mod el for BMCS_CRW Short ly af ter a t time t =  +iT, pee r B sen ds o ut its CBM (  i , u i ). T hen , peer B will updat e it s r elev a nt win do w to L ’ BA ( i ) by dele t ing t hose positions whi c h ar e less t h an  i or th ose posi t io n s w h ich fillin g states is value 1 in u i . U p o n receiv in g B M (  i , u i ), pee r A upd ate i ts rele vant window to L AB ( i ) by r e mov in g the posi t io n s w hi ch are less th an  i an d the po sit ions w it h fi llin g states e qual to 1 in u i o ut o f L ’ AB ( i ) dur in g th e deco d in g proce ss. Acc ordin g to pri nciples 1 a nd 2 , L AB ( i ) only in cludes those chun k p o siti o n s which ar e n eve r r epo r t e d w i th value 1 y et by b oth peer A an d B ti ll tim e t =  + i T , i. e . , L AB ( i ) = { c : c > max(  i ,  i )+ N or (Z( c , i T ) and Z( c ,  +iT ))} (15) In th e n e x t r o un d at time t = ( i+1 ) T, th e p e er A w i ll sen d a CBM w ith n ew of fse t:  i+1 =  i + rT . (16) To fa c il itate th e a na ly sis, w e a ssumin g th e same playb ack delay for b oth pair ed peers, i. e. th ey h ave th e same of fsets at an y tim e t . U nd e r t ha t a ssumptio n , w e h ave  i =  i + r  . Th eref ore, th e a v erage compressed bitmap len gth of p ee r A should be                               . 1 1 1 1 1 1 ) 1 ) , ( ) , ( ) , ( ) ( 1 0 i 1 0 1 1 1 1 1 1                                                      rT N r i N c i i N N c i N c N N c CRW _AB i r S i S i S rT c N S c N S c N S rT iT c Z p iT c Z p iT c Z p T r W i i i i i i i i                 (17) Simil ar ly , by substitutin g th e  with T-  , w e g e t t he ave r age co mpr es sed bitmap length of pee r B                            rT N i τ) r(T i CRW_B A . i τ) r(T S i S i S rτ W 1 0 1 0 1 1 ( 18) ■ Sub mit ting to ar Xiv.co m 9 Thu s w e pr o of t he fo llow i n g th eorem: Theorem 2 : In BMC S_CRW, th e average size o f t he c ompr es sed bitmap i s t o ta lly d eter min e d by th e di ff usion S c ur ve . Speci fically , if giv en t he d if fusio n function S( x) , un der th e same pla y back delay co n dition f or both pair ed p e ers, t he ave r age bitmap s i zes W CRW in each dir ec tion ar e , r es pective ly                          rT N r i CRW _A B i r S i S i S r T W 1 0 i 1 0 1 1   (19) an d                            rT N T r i CRW_BA i T r S i S i S r W 1 0 i 1 ) ( 0 . ) ( 1 1    (20) w her e N , T, r a nd  a re th e sy stem p ar ameters b uff er len gth, B M exc h an ge t ime peri o d, video play back ra te a nd th e B M sendin g tim e in terval be t w e en th e tw o peers respe ct iv ely . Ther ef o r e, th e av era ge bitma p length o ve r th is c on n e ction is W CRW =( W CRW_AB +W CRW_BA )/2 . (21) D. Bi tmap Length of Traditi onal Compression The idea of tr adit ional lossle ss co mpr ess ion i s dee ply c onn ected with stati stical in fe r en c e, a nd C laude Shan n o n la y s th e th eo ret ical fo un dati o n [21]. Acc ordin g to i nfo r mati o n th e or y [15], the r e gul ar b it map c an b e rega rd as th e c omb in ation o f N independent b in ar y s our ce s { h k , 0  k  N-1 }. As each chun k’s dow n loadin g can be aff ect ed by man y factors in c lu ding n etw ork co n di tions, p e er sele ction an d data f etchi ng pol i cy , for pee r in st able condi tion, it is r easo na ble to a ssu me th e N bi na ry sources ar e in dep end e n t t o each o th er. Fo r c onvenience, we u s e th is operat o r in t he fo llow in g discussio ns H( x)=xlog 2 x+( 1-x ) l og 2 ( 1-x) . ( 22) Ther ef o r e, with t radi tional co mpr e ssion, fo r an y regular B M(  , b ), we h ave t he compress ed bitma p size W Trad           . , 1 0 1            N i N c Trad i S H iT c Z p H W i i   (23) E. B itmap Length o f BMCS_ SRW w it h Traditional Compression We bo r r ow Fig .5 t o expl ain thi s derivation proce ss. Obv i o usly , an y tw o bina ry sources of h k an d h k-r T in t he t w o adjac ent BM i an d BM i +1 respectiv ely a re the s am e , i.e. bo th of th e m correspond t o th e same chu nk . As in BMCS _ S RW , if h k sends v al ue 1 in BM i , th e n h k-rT i n BM i +1 must be 1; whi le if h k =0 i n BM i , th e n h k-rT in BM i+1 may be 1 with certain probab ili ty . Fo r simplicity, we use s ymb ol q i, j (c) , j=i+1 to repr es ent t he c ondit ion pr ob abilit y that the chunk c in BM i is not dow n loaded but d o w n l o aded a t n e xt BM i +1 . For a st able proce ss of b uff er fi l lin g, w e h ave fo llow in g equation: 1-p( Z(c,t i )) + p(Z( c ,t i ) ) q i ,j ( c)=1- p(Z (c,t j )) (24) Th en, we ha v e ) ( 1 ) ( ) ( ) , ( ( ) , ( ( ) , ( ( ) ( , c s S c s S c s S t c Z p t c Z p t c Z p c q i i j t t t i j i j i         (25) i j t t s c N s    wh ere . Th e r ef ore, the th e oretical limi t of bitmap s i ze W JFS in th is jo i n t fo r c e scheme is                             . 1 - 1 , , 1 0 1 0 1 1 , 1 1 1                                      rT N i rT i N c i i N N c JFS i S i S rT i S H i S i S H c q H iT c Z P T iT c Z p H W i i i i     ( 26) ■ Th us we pr o of t he fo llow in g t heo r em: Theorem 3 : Fo r t he sch e me w it h t he jo in t fo r ce of B M CS _SR W and tr aditi onal c ompr es sion, if g i v en th e chun k diffu sion fun c ti o n S(x) , un der th e sa me p lay ba ck delay co n dition for bo th pai red pee r s, th e siz e of t he compr e ssed bitmap ha s a t heo r etical l imit v a lue               . 1 - 1 1 0 1 0                      rT N i rT i JFS i S i S rT i S H i S i S H W , (27) Wher e t he p aramet e r s N , T an d r a re th e buf fer lengt h, BM exchan ge time peri o d a nd vi deo playb a c k r at e respect ively . F. Bitmap Leng th of BMCS_CRW w ith Traditi onal Compression Le t ’s r ecall t ha t i n B MCS_CRW onl y th e sta tes o f th o se posi t io n s w hi ch h ave not be en b uff ered in both pair ed pee r s should be r epo rt ed in a co mpresse d bu ffe r - map. We u se Fig.6 to explain th is deri v at ion proce ss h e r e . W e assume peer A sends it s B M to pee r B at time i T a nd peer B se n d s to peer A at time  +iT . For pee r A at ti me t = (i+1) T , t he proba bility to sen d th e stat e abo u t a chun k c i s p(Z( c,i T)  p(Z (c,i T+  ) ; acco r din g t o (25), th e co ndi tion proba bility th at chun k c i s n ot dow n loaded at tim e t = iT b ut dow nl oade d a t t ime t = (i+1)T is q i ,i +1 (c) . Th eref ore, i n t he di rectio n of p e er A to peer B , th e t heo r etical li mit W JFC_AB of t he c ompr e ssed bitmap siz e in this j oin t force is                                                       . 1 - 1 - 1 - 1 , , , , , 1 0 1 0 1 ) ( 0 1 , 1 1 1 _ 1 1                                                       i S i S rT i S H r i S i S r rT i S H i S i S H c q H iT c Z p iT c Z p T iT c Z p H iT c Z p T iT c Z p H W rT N i r i T r i i i N c N N c N N c A B JFC i i i i i i             ( 28) Clear ly , th e resul t co nsi sts of t hr ee part s: t he n ewly in cr e a s ed par t, th e o ve r lapped par ts of bo th BM A (i+1) an d BM B ( i) , an d Sub mit ting to ar Xiv.co m 10 th e ove rl apped part of th e th ree B Ms ( BM A (i+1) , BM B (i) a nd BM A ( i) ). Co r r e spondin gly , b y substitutin g th e  w i th T-  , w e get t he th e or e tical limi t W JFC _ BA o f co mpr e ssed bitmap sized in t he rev er se direction from peer B to pee r A                           . 1 - 1 - 1 - 1 1 0 1 ) ( 0 1 0 _                              i S i S rT i S H r rT i S i S r i S H i S i S H W rT N i T r i r i BA JFC     (29) Ther ef o r e, th e average b i tmap l ength ov er t hi s conn ec tion b etwe en peer A and pee r B is W JFC =( W JFC _ AB +W JFC _ BA ) /2 (30) ■ Thu s w e pr o of t he fo llow i n g th eorem: Theorem 4 : F o r t he scheme with th e joi n t f orce of B MCS_CRW a nd t r aditi o n al co m pressio n , if g ive n the chunk diff u sio n fun c tion S(x) , und e r t he same pla y back delay c ond ition f or bo t h p aired pee r s, the av era ge s ize of t he c ompr es sed bi t map ha s a th eo r etical li mit,                                                     . 1 - 1 - 1 - 1 1 - 1 - 1 - 1 2 1 1 0 1 ) ( 0 1 0 1 0 1 0 1 ) ( 0                                                                                i S i S rT i S H r r T i S i S r i S H i S i S H i S i S rT i S H r i S i S r rT i S H i S i S H W rT N i T r i r i rT N i r i T r i JFC         ,(31) w her e N , T, r an d  are the sy stem p aram ete r s of buff er len gth, B M exc h an ge t ime peri o d, video play back ra te a nd th e B M sendin g tim e in terval be t w e en pai red pee r s r esp ectiv ely . G. S imulation w it h UUSee B y sub stit utin g the r e a l pa r ameters of U USee , i .e. T =5 s, r =3.37 ch unk s/s , N =456 b i ts, an d S(x) as show n i n Fig .4, we c a lculate th e av er age bitmap l e n gth s of all t he compr e ssio n schemes , i ncludin g the sin gle peer’s relevant w i dow b ased scheme (BM CS_SRW ), t he common relev ant w i dow base d scheme (BM CS_CRW ), th e jo in t f orce sc h eme (BM CS _JFS) o f co mbing BMCS_SR W w it h the tr adit ional meth od, an d the jo int force scheme ( B MCS _JFC) o f co mbin g B MCS_CRW w ith tradit io n al me th od. The num e r ical results show t ha t in an ideal situ ation, th e bitma p size can be r educe d by 86% an d 90% f r om 456 bits dow n t o o n ly 66 b its and 46 b it s b y BM CS_SRW an d BM CS_CRW r e spec t iv ely . Fur th ermore, by co m b in g th e tra ditional c ompr es sion appr o ach , the size c an b e de creased b y 91% a nd 95% to 4 2 bits an d 24 bits r esp ective l y . The impr o vement from BMCS_SR W to B M CS _JFS an d from B MCS_CRW to B MCS _JFC is ascrib ed to t he tr aditi o n al probab ili ty in f erence actin g on th e new co m press i o n t heo r i e s w e pr es ent . Th e detailed n umerical results ar e listed in t abl e I. B y a dj ustin g par a me ter o f t he BM excha ng e peri o d T , we figure out t he d iff e r ent siz e of t he comp r ess ed bitmap . Th e r e sult i s sh o w n in Fig .7 an d l isted i n table I. Ev en th ough d ue to adjustin g th e exchan ge peri o d T t he r eal n etwo r k sh arin g environ me n t may chan ge to some e xtent so as to influence th e diffu sion S curve, we b elieve o u r r esu l ts r ef l ec t t he ove ra ll tren ds. Fig.7(a) show s all curve s o f b i tmap siz e vs e xch ange period T of t hese co m pr e ssio n schemes. Both curve s of BMCS _ SR W (th e cur v es w it h th e la rg e st slope) correspondin g to (9) an d (10) of theorem 1 ar e near ly ov erlapp ing an d both curve s of B M SC _CRW (th e c ur ve s with th e second larg e st s lope) co r respo n ding t o (19) an d (20) o f t heo r em 2 a re id e nt ical t o each o t her u n der co nd ition  =T/2 . Th e two c urves on th e bo t tom with t ri an gle an d i n v erse-tri an gle mar kers ar e f or B M CS _JFS a nd BMCS _ CF S r espe ctivel y. It can be se en t ha t th e a ve r age bitmap siz e l in e a rly in crease s with t he excha ng e period T fo r BMCS _SR W a nd BMCS_ CRW, w h ile t he latter can b r in g mo r e th an 30% g ain ov er t he former due t o co n siderin g pr in ciple 2 bes ides p rin ciple 1 in th e la tter s cheme. Moreo ver, t he encod ed bitma p of eith er BM CS_SR W or B M CS _CRW can be furth er co mpr esse d bec a use th e r e is diffe r en t fill ing pr o bability on each positio n in th e en c oded bitmap. Therefo r e by co mbin in g t he tra dition al d ata co mpr es sion pri ncipl e , more th an 36% a nd 46% redund an t in fo r ma tion can b e fur th er r uled out from th e encode d bi t map of BMCS_ SRW an d BM CS_CRW by B MCS _JFS and BMCS _JFC respect ively . As a refe r ence, the b itm ap li mit of th e bitma p co mpr e ssed by tra dit ional a pp r oach is al so in cluded i n Fig.7(a) and ta b l e I . B ecause i t t reats e a ch BM a s a normal dat a blo ck w ith out co n siderin g the BM exchan ge featur e i n P2P sy ste m , th e th eo r e ti c a l l imit is a co ns tant value (77 bits) sh o w n a s a h o r izonta l lin e in Fig.7 (a). We note bo th c u rve s of B M CS _SR W and B MCS_CRW int ersect th e line of th e tra dit ional a pproach a t time ab out T = 8s an d T =18s res pective ly . On t h e lef t o f th e cr os s-po in t, our scheme ha s smal ler bitmap size. Consid e r in g th e fac t t h at the B M e x chang e u sually has a much h igh f r e quency suc h as 500ms in P PS team an d 4s in PPL ive an d a faster B M exchang e can speed up ch unk dif fusio n , th e ex cha ng e ti me T at t h e cross -point is large e n o u gh t o in dicate th e import an c e of our n ew c ompressio n TABLE I T HE TH E O RE TI CAL LIMI T OF TH E BIT M AP SI ZE IN DEF ERENT COMPRESSI ON SCH E MES s ize u n it: bit Sc heme s BM s en ding p eriod T (s) 5 10 1 5 20 2 5 T raditio nal co mp ress io n 77 77 77 77 77 BMCS_SRW 66 83 100 116 133 BMCS_ C RW 46 57 68 78 87 Joint for ce 1 (JFS) 42 52 60 64 66 Joint for ce 2 (JFC) 24 33 40 45 50 Join t forc e 1 means the s che me with the join t fo rce of BMCS_SRW and traditio nal co mpre ssion ; Join t forc e 2 me ans the sche me with the join t fo rce of BMCS_CRW and traditio nal co mpre ssion ; For bo th BMCS_CRW and Joint fo rce 2, w e us e  =T/2 in c alculation Sub mit ting to ar Xiv.co m 11 (a) T he B M le ngth v .s. pe r iod T (b) The bit rate v.s . pe riod T (c) the BM len gth v. s. i nterval  Fig. 7 . Numer ical r esults w i th UU See appr o ach e s. Moreov er, becau se bo t h our n ew and tr aditi o n al meth o ds ar e n ot m utuall y exclusiv e but can w o r k to gether , b y th e j oin t fo r ces , b oth curve s of BM C S _JFS an d BMCS _CFS ar e f a r belo w a ll other curves, i .e., t hey a re m uch mo r e pow erful th an an y sin gle scheme . In addit ion, we sh o w th e r esu l t i n an other f orm in Fig.7 (b ), w here th e v er tic a l a xis is th e bit r at e of b i tmap sendin g i. e. th e bitmap size dividin g by excha nge ti me p er iod T , an d the horiz o nt al axi s is T . B esides th e simil ar co n clusio n a s dr awn f r om Fig.7(a), w e c a n see t ha t the smaller ti me period T alway s leads to th e h igher b it ra te of sendin g. An o ther i nter e stin g issue is w hen to sen d t h e B M a f t er r e ceiving on e BM in BMS C_CRW is more pr e fe r able to mak e th e o verall o ve r hea d smaller on bo th dir ections. Ref er e n ce t o Fig.8, a ssu mi n g peer B se n ds out its BM at t ime t=t 0 +iT , we n ee d t o an swe r wha t i s the m o st sui tab l e ti me  behind t ime t f or peer A to send out i ts BM. Fo r an aly zi ng th is pr o ble m, we adjust th e se n d i ng in terval  w i th in [ 0 , T ] w h ile ke ep t h e se n din g pe r iod st ationar y ( T =5s) in the BM length calculati o n . Th e r e sult i s show n in Fig.7(c). We can s ee th at th e ave r age bitmap size o n th e bo th dir ectio n s i s n e ar ly in vari able n o ma tter w h at th e se n din g in terval  is cha nged to. I t s uggests a designer n ee d not to th ink ov er th e s elec t io n of s e n din g in terval  at al l. Th e same conc lu s i o n is a lso applied to BMC S_JFC. As a r esult, th e ori gin al compressio n a pproaches an d th e th eo r e ti c a l results c a n g uide us t o l o ok fo r pow erful engin eering d e sign . In fact, base d o n th e basic scheme s we present, ma ny eff i c ien t pra ctical so lut ions can be dev i se d to appr oach th e ide a l b it map sizes. H ow ev er, for th e spe cific engin eering de sign s, man y f a c tors need t o be consider e d. For example, bec ause we need bo th peers (who e xcha nge BM ) h ave exactly the same un derst andi ng o f what i s r ec eive d or n ot, ce r tai n typ e of r eliable pr o toco l sh ould be design e d to implemen t t he BM ex cha ng e . Detailed study on t he e ng in ee r in g design issues is bey ond t h e discus si o n of th is paper, an d w e w ill fo cus o n t hese w or ks, in cluding th e en gin ee ri ng implementa tio n , the ad ditiona l ov er h e a d evalua tio n a nd co mplexity d isc ussion, i n futur e r e sear ch wo rk s . VI. C ONCLUSI O N In thi s pa per, we present an origin al t heo r etical f ra mew o r k fo r buf fer-m ap mess age co m press ion in li ght o f th e disco v ered B M exchan ge p rin ciples i n P 2P sy stem a nd th e in tr od uced impor tant co n c ept of relev a nt window . Diff er ent fr o m th e existing gener al d ata co mpr es sion p r in ciple, we don’t tr eat each BM a s a gen e r al and i ndependent data bloc k , but r e co gn ize th e co rr elati o n s betw e en th e s e quent ial exchan ged B M s . In o th er w ords, a pee r n eve r n ee ds t o r epo r t a b uff er posi t io n furt her in h is b it map once send ing a v a lue 1 in t hi s posi t io n , an d moreo ver, a peer neve r needs to report a b u f fer posi t io n to a r ece iver p e er furth er once r e ce i v in g a v a lue 1 in th is p osition f rom tha t r ece i v er p eer. Under t he th eo r etical framewo r k, two ef ficie n t buf fer-map c ompr e ssion schemes ar e presented and t h e feasibi l ity o f t he scheme s is pr ov ed in t heo r y . Bo th th e n ew meth o d w e prese n ted an d t r aditi o n al d ata Fig.8. th e B M exc h an ge ti me se quence in BMS C_CRW off set  1  2 T t 0 +T t 0 +2T t 0 +3T t ime  1  2 T B  A B  A B  A A  B A  B Sub mit ting to ar Xiv.co m 12 c ompr es sion m ethod be long t o diff erent th eo retical sy stems in na ture, and they don’t conf l ict with each o th er but can wo rk tog eth er. T he t heoretical sizes of th e co mp ressed bitmap fo r b oth n e w schemes w e pr esented a s we l l a s t he sch e mes of c omb in ing t h e tradi tional co mp ressio n prin ciple a r e de r ived in math ematics. A t l a st, the n umeri cal r esu l ts cal c ula ted w i th sy stem par ameters of UU See v ali date t h e e ff iciency o f our meth o ds. T he co mpr es si o n r at io i s a b out 14% an d 10% fo r sin gle pee r ’s r elev an t window b a se d scheme and commo n relevant w in dow based scheme r e spec t ive ly . Mo r eo v er, if c omb in ing with th e g e n eral d ata comp r ess ion a pproach, c ompr es sion r at ios can be f urt her im prov ed t o abo ut 9% a nd 5%. The m o st im po rt an ce of the study in th is pap e r i s t ha t we e st abl i sh a n ew th e oretical fra mew o r k fo r buf fer-map c ompr es sion, which is d iff e r ent from tr ad itional da ta c ompr es sion t heory . T he n ew frame can guide us to de vise th e e ffic i e nt engin ee r i ng so l utions, an d enl ight e n s us to d ev elo p th e m o st pow er fu l solutions by co mb i ng tra ditiona l da ta c ompr es sion pr in c i ple. We will co n duct the study o n th e algorit hms an d pr o toco l s in eng in e er in our futur e r e search. R EFERENCES [1 ] C. Feng, B .C Li, B. Li. " Und ersta nding the Performa nce G ap be tw een Pull-based Mes h Stre aming Protoc ols an d F un da mental Limits " , in the Proce eding s of IEEE INFOCOM 20 09 , R io de Janeiro, B razi l, A pril 19-2 5, 2009 . [2 ] Y . S. Che n, C. J. Chen a nd C. X. Li, “ Meas ure a nd Mode l P 2P Stre aming Syste m by Buffer Bitmap ”, HPCC ’0 8, Sep t. 200 8, [3 ] C.X Li, C .J. Che n, “ Initial Offset Placemen t in P 2 P L i v e Stream ing Sys tems “, http ://arxiv.o rg/f tp/arxiv /pape rs/08 10/08 10 . 2 063 .pdf [4 ] Zhou, YIPe ng Ch iu, Dah Ming Chiu, Jo hn C. S., “A Simp le Mo de l fo r An a ly zing P 2P Stre a ming Proto co ls”, ICNP 2007 [5 ] X.J. Hei, Y.L i u , K. Ros s, "Inf erring Networ k -W ide Qu ality in P2P Live Streamin g Sys tems ", IEEE Journ al on Selec ted Area s in Com munica tion s , v ol. 25, no. 10, Dec. 2007. [6 ] X. H ei, C. L iang, J . L iang , et a l. “A meas u re ment st u dy of a large s cale P2P IPTV sys tem ” , IEEE Transa ction s on Multimedi a , 2007 , 9( 8):16 72- 1 687 [7 ] X. He i, C. Liang, J . Liang , Y. Liu , a n d K. W . Ros s, “I n sights into PPLiv e: A meas uremen t study of a la rge - scale P2 P I P T V s yst em,” IPT V worksh op in co njunction with W W W 2 006 , May 20 06. [8 ] S. Al i, A . Mathur, and H . Zhang, “M easureme nt of comm ercial Pe er- to-Pe er li ve video str eamin g,” F irst W or ksho p on Rec ent Adv anc es in P eer -to-P e er Strea ming, Aug. 2006. [9 ] L. Vu, I. Gupta, J. Liang, K. Nahrst ed t, “MapPing th e PPL ive Netw ork: Studying the Imp a cts o f Med ia Str eaming o n P2P Ove rla y s”, UIUC Tech repo rt , A ugust 200 6 [1 0] X. Zh ang, J. L iu, B. Li , et al. “Coo lst reaming/ donet : a da t a-driv en ov erlay net work for pee r- to-p eer li ve media strea ming”, In: Pr o cee dings of IEEE/I NFOCOM' 0 5 , Miami, US A, 200 5:210 2-211 1 [1 1] Y . James Hall, P. Pi em onte, M. We yant, Jo ost : a meas ur e me nt study . . [1 2] A. V lavianos , M. Iliofo tou, and M. F a louts os, “BiToS : Enhanc ing BitT o rrent for S upp orting St reamin g Ap plic a t ions ” , in Gl o bal Internet Workshop in conjunc tion with IEEE INFOCOM 2 006 , Apri l 20 06. [13] S . Tewari and L . Kle inrock, “An alytic a l Mo del fo r BitTorren t- base d Liv e V ideo S t re a m ing”, Con sume r Commu nica tions a nd Networking Confe rence , 2007 [14] Zho u , YIPen g Chi u , D ah Min g Lu i , John C . S., “A Si mp le Mo de l fo r A nalyz ing P 2 P Str eamin g Pro tocols ” , ICNP 2007 [15] THOMAS M . COVE R a nd JOY A. THOMAS , “Ele ments of Inf orm ation T heor y”, s e cond edit ion, publis hed by John W il ey & Sons, Inc. Hob oken , New J e rsey , July 200 6, ISBN: 0-471-241 95-4, p p 13-33 . [16] D.A . Huff man, "A M eth od f or th e Con str u ction of Min imum- Redund ancy Codes ", P roc eed ings of the I.R. E. , Se pte mber 1952 , pp 109 8 –110 2. [17] J. Z iv, A. L e mpe l; “A U nivers al Alg orith m f or Seq uenti a l Data Com pres s ion”, IEEE Tran saction s on Info rmation Theor y , 23(3) , pp .33 7–34 3, May 197 7 [18] J. Z iv, A. L e mpe l: “Compr ess ion of indiv idua l seq u e nce s v ia va riab le rate co ding”. I EEE T ransa ctions on Info rmation Theo ry . Vol . IT¨C 2 4, No. 5 , Sept. 197 8 , pp 5 30 ¨ C53 5. [19] Gol omb S W, "Run -le ngth encod ings ", IEEE Transa ction s on Informa tion The ory , 196 6 pp 1 40-14 9 [20] G.Q De ng, Y.F . Zhan g, C. X. Li, C.J. Che n, " A bitmap co ding me thod for P 2P streamin g pro tocols " ,in Pr oc. of CAR, 201 0 [21] Ris san en, J.J.; L angd on, G.G. " Arit hmeti c co ding" , IBM Journ al of Resea rch a nd Deve lop ment 23 (2 ): 149 –162 . March 19 79 [22] Ian H .; Neal, Radf ord M.; Cle ar y , Jo hn G. "Arit hme tic Cod ing for Data Com pres s ion". Com munic ations o f the ACM 30 (6): 520 –540 , June, 1987 [23] S hannon CE, Weaver W, “T he m a th ematic a l theory of co mmunic ation”, Unive rsity of Illinois Pr e ss , U rbana IL, 1949 . [24] M. Zhang , Q. Zhang , L. Sun, and S. Yang, “ U nde rstandin g the Po we r o f Pull- base d Streaming Protoco l: Ca n We Do Better ? ” IE EE J. on S el. Areas in Com munic ation s , De ce mber 2 007. [25] M. Zhang, Y. X iong, Q. Zhang, and S. Yang , “ A Pee r-to -Pe er Ne twork f or Liv e Med ia Streamin g: Us ing a Pus h- Pu ll A pproach ,” in P r oc. of ACM M ultimedia 200 5 , Nove mb er 200 5.