Partition Information and its Transmission over Boolean Multi-Access Channels

Transcription

1 Partition Information and its Transmission ovr Boolan Multi-Accss Channls Shuhang Wu, Shuangqing Wi, Yu Wang, Ramachandran Vaidyanathan and Jian Yuan 1 arxiv: v2 [cs.it] 19 Jul 2014 Abstract In this papr, w propos a novl partition rsrvation systm to study th partition information and its transmission ovr a nois-fr Boolan multi-accss channl. Th objctiv of transmission is not mssag rstoration, but to partition activ usrs into distinct groups so that thy can, subsquntly, transmit thir mssags without collision. W first calculat (by mutual information) th amount of information ndd for th partitioning without channl ffcts, and thn propos two diffrnt coding schms to obtain achivabl transmission rats ovr th channl. Th first on is th brut forc mthod, whr th codbook dsign is basd on cntralizd sourc coding; th scond mthod uss random coding whr th codbook is gnratd randomly and optimal Baysian dcoding is mployd to rconstruct th partition. Both mthods shd light on th intrnal structur of th partition problm. A novl hyprgraph formulation is proposd for th random coding schm, which intuitivly dscribs th information in trms of a strong coloring of a hyprgraph inducd by a squnc of channl oprations and intractions btwn activ usrs. An xtndd Fibonacci structur is found for a simpl, but nontrivial, cas with two activ usrs. A comparison btwn ths mthods and group tsting is conductd to dmonstrat th uniqunss of our problm. Indx Trms partitioning information, conflict rsolution, Boolan algbra, Fibonacci numbrs. 1 S. Wu, Y. Wang and J. Yuan is with Dpartmnt of Elctronic Enginring, Tsinghua Univrsity, Bijing, P. R. China, ( wsh05@mails.tsinghua.du.cn; wangyu, jyuan@mail.tsinghua.du.cn). S. Wi and R. Vaidyanathan ar with th School of Elctrical Enginring and Computr Scinc, Louisiana Stat Univrsity, Baton Roug, LA 70803, USA ( swi@lsu.du, vaidy@lsu.du).

2 2 I. INTRODUCTION On primary objctiv of many coordination procsss is to ordr a st of participants. For xampl, multiaccss can b viwd as (xplicitly or implicitly) ordring a st of usrs for xclusiv accss to a rsourc. Information intraction plays a ky rol in stablishing such an ordr. To formaliz this intractiv information and driv fundamntal limits on its transmission, w propos in this papr a novl partition rsrvation modl ovr a nois-fr Boolan multiaccss channl and us an information thortic approach in its analysis. For th simplst variant of th problm w study, lt N = {1,..., N} b a st of N usrs and lt G s = {i 1,..., i K } N b a st of K activ usrs. Th problm is to lt all usrs obtain a common ordrd K-partition 1 Π = (B 1,..., B K ) of N, so that ach group (or block) B i has xactly on activ usr from G s. Equivalntly, w us a vctor z = [z 1,..., z K ] to rprsnt th ordrd K-partition Π, whr z i K {1, 2,, K} is th id of th group that usr i blongs to, i.., i B k iff z i = k. Th dsird partition is dtrmind by a sris of transmissions and obsrvations ovr a channl, mor spcially, usrs ar connctd through a shard slottd Boolan multi-accss channl. Suppos that during slot t, ach activ usr i transmits bit x i,t {0, 1} on th channl. A common fdback y t = i G s x i,t will b obsrvd by all usrs, i.., if no activ usrs transmit bit 1 during slot t, y t = 1; if at last on activ usr transmits 1, y t = 0. Th goal is to prviously schdul T rounds of transmissions (dnotd by an accssing matrix X [x i,t ] 1 i N,1 t T ), and a common dcoding function g( ), so that aftr obsrving T rounds fdbacks y [y 1,..., y T ], a dsird ordrd K-partition of N, dnotd by z = g(y) can b obtaind by all usrs. Th objctiv is to find an achivabl lowrbound on th numbr of slots T, within which thr xists a matrix X and g( ) so that vry possibl activ st G s N can b partitiond. In th problm w considr, w do not sk to rstor th stats of all usrs (that is, dtrmin G s xactly), but to partition G s and to mak usrs know th partition z. Thus, a particular partition information that only prtains to th rlationship btwn activ usrs in G s, is transmittd through th Boolan multi-accss channl. W will formaliz this partition information, and focus on th achivabl bound of its transmission rat ovr Boolan multi-accss channls. This problm plays 1 An ordrd K-partition Π = (B 1,..., B K) of N is a squnc of K non-mpty substs of N that satisfis th following conditions: (a) for all 1 i < j K, B i B j = and (b) K i=1 Bi = N.

3 3 a significant rol in undrstanding th fundamntal limits on th capability of stablishmnt of ordr in distributd systms. Th proposd problm has a clos rlationship to a typical slottd conflict rsolution problm [1], whr ach activ usrs must transmit without conflict at last onc during T slots, i.., if x i,t = 1 dnots a trial of transmission for activ usr i at slot t, thn thr xists a 1 t i T such that x i,ti = 1, and for all j G s {i}, w hav x j,ti = 0. To achiv this goal, mainly two typs of systms ar studid: dirct transmission systm and rsrvation systm with group tsting [2]. Dirct transmission focuss on dirctly dsigning an N T dr accssing matrix X dt (subscript dt is for dirct transmission), so that ach nod finds at last on slot for its xclusiv accss to th channl. Not that activ usrs ar implicitly partitiond during th transmission to nsur th succss of transmission, (mor spcially, if th succssful transmission tim t i for ach activ usr i is known, th dsird partition can b constructd by all th usrs), but th partition is not ncssary to b known. Rsrvation with group tsting has two stags. In th first rsrvation stag, an accssing matrix X g and dcoding function g( ) ar dsignd such that G s is xactly dtrmind by g(y), whr y is th channl fdback. That is, (activ or inactiv) stats of all usrs ar rstord and, subsquntly, activ usrs can transmit in a prdtrmind ordr without conflict in a scond stag. Th rsrvation stag is also calld group tsting [3] or comprssd snsing [4] in diffrnt filds. Th two stags can b of diffrnt tim scals. W can s th payload transmission is sparatd, but in th rsrvation stag, G s is known to all usrs, which is mor than w nd. Compard with group tsting and dirct transmission systm, our partitioning rsrvation systm provids a nw way to individually analyz th procss of partitioning, which is th ssnc of coordination in conflict rsolution problms. It can b usd as a rsrvation stag instad of group tsting in conflict rsolution problms, and holds th possibility of rquiring fwr rsourcs, sinc it sks only to partition N, rathr than rstor G s. (Notic that onc G s is rstord, obtaining a partition is straightforward.) Compard with dirct transmission, w obsrv that usually, th tim scal for rsrvation can b much smallr in partition/rsrvation than that in payload transmission, thus it may nd lss tim for conflict rsolution in practical us. Th proposd partition rsrvation systm has abundant applications in diffrnt aras. First, and formost, it can b dirctly applid to th rsrvation stag in conflict rsolution instad of

4 4 group tsting. Scond, sinc th obtaind common partition is known to all usrs in th partition rsrvation systm, mor complicatd coordination among activ usrs can b anticipatd othr than avoiding conflict in tim domain, which is not attainabl in traditional conflict rsolution schms. For xampl, cod-division multipl accssing cods could b assignd to usrs in diffrnt groups basd on th obtaind partition, so that activ usrs can claim accssing cod squncs from a common pool in a distributd way without coordination from a cntral schdulr. Othr xampls can b found in paralll and distributd computation [5 7], such as ladr lction [8], broadcasting [9]. In this papr, th systm is constraind to a cas with K activ usrs non-adaptivly accssing a noislss Boolan binary fdback channl. It is a fundamntal cas of th problm, but also has practical valus. Considr a systm with N usrs ach of which stays activ with a probability p. If p scals with N such that th numbr of activ usrs K Np is approximatd as a constant for larg N, th accssing of ths K activ usrs to a Boolan channl is th cas w ar tackling. In our proposd framwork, no adaptation is allowd ovr th Boolan channl, thrby rducing th xpnditurs on fdback ovrhad as compard with th adaptiv modls. It should b notd such non-adaptiv channl modl has also bn considrd in MAC or group tsting litratur (for xampl, [3, 10 12], tc.). Our study will hlp us undrstand th fundamntal limit on transmission rsourcs to attain a partitiond coordination among th activ usrs. To achiv this goal, w first us sourc coding to quantify th partition information. Thn two coding schms for th accssing matrix X, and dcoding function g( ) ar proposd. Th first is a brut forc mthod to dsign X and g( ) dirctly basd on rsults from sourc coding. Th purpos of sourc coding is to comprss th sourc information, mor spcifically, to find a st C of minimum numbr of partitions, so that for narly any possibl activ st s, thr is a valid partition in C. Thn th brut forc mthod tris to find th valid partition by chcking vry partition in C using th channl. Th scond schm, mploying random coding, gnrats accssing matrix lmnts x i,t i.i.d. a by Brnoulli distribution, thn th partition is rcovrd by optimal Baysian dcoding. Th two mthods can both work, and provid diffrnt viws of this problm. In particular, in th brut forc mthod, if T BF = KK+1 K! f(n) and f(n) is an arbitrary function satisfying lim N f(n) =, th avrag rror probability P (N) f(n) 0, as N. Whil for a simpl but non-trivial K = 2 cas, w prov in random coding, if for

5 5 any ξ > 0, logn T max 0 p 1 C(p) ξ, whr C(p) = (1 (1 p)2 ) log ϕ(p) (1 p) 2 log(1 p), ϕ(p) = p+ 4p 3p 2 2, w hav th avrag rror probability P (N) 1 N 0 for som > 0, i.., with polynomial spd. Th two achivabl bounds ar shown bttr than that of group tsting. Morovr, for th random coding approach, w us a framwork to solv th problm from th viw of strong coloring of hyprgraphs, namly, th partition objctiv can b transformd to th strong coloring problm of a rsulting hyprgraph, and th ffct of channl(s) is rflctd by a sris of oprations on hyprgraph dgs. Undr such a framwork, th partition information is rprsntd by typs of hyprgraphs in which hypr-dgs ar dtrmind by th intraction among a st of possibl activ nods. Th joint work btwn th ncodr and dcodr is to mak sur that th rsulting hyprgraphs bcom strong colorabl aftr transmissions by activ nods and intrvntion by channls. In a simpl, but nontrivial, cas with K = 2 activ usrs for a st of N usrs, a suboptimal odd cycl basd analysis is proposd, and a structur of xtndd Fibonacci numbrs is found, which shds lights on th inhrnt structur of th partition information and Boolan channl, and could b furthr xplord for K > 2 cass. As a summary, th contributions of this papr ar twofold. First, w formulat a novl partition rsrvation problm which capturs th transmission and rstoration of som rlationship information among activ usrs. This rlationship communication problm is also rprsntd in a hyprgraph basd framwork. Scondly, w propos two typs of coding approachs, and th corrsponding achivabl bounds on th communication priod, which provids th intuitiv xampls to study th rlationship information transmission ovr Boolan multi-accss channls. Part of our rsults has bn prsntd in [13]. In this papr, w provid a mor complt and comprhnsiv solution th formulatd problms. In particular, w discuss th sourc coding problm in Sction IV. Thn basd on sourc coding, w giv a brut forc coding mthod in Sction V to solv th partition problm. In Sction VII, a sub-optimal dcoding approach for th cas of K = 2 is providd which rquirs th rsulting graph without odd cycls (i.. two-colorabl). Dtaild proofs ar thn givn to both Lmma 2 and Thorm 3, which ar not includd in [13] du to spac limitation. Th rst of this papr is organizd as follows: in Sction II, w introduc th rlatd work. Th problm formulation appars in Sction III. In Sction IV, th partition information is illustratd by cntralizd sourc coding, thn a brut forc mthod dirctly inspird by sourc coding is

6 6 proposd in Sction V. In Sction VI, a random coding mthod is considrd and th problm is rformulatd in trms of a hyprgraph. Basd on this, a simpl, but non-trivial, rsult for K = 2 in random coding is analyzd in Sction VII. In Sction VIII, w compar our rsults with that of group tsting. W summariz our rsults and mak som concluding rmarks in Sction IX. II. RELATED WORK Although th proposd partition modl could b usful in many problm sttings, typical applications ar in conflict rsolution problms. Th works on conflict rsolution ar too xtnsiv to b includd in our rviw hr, and w thus only includ thos most rlvant to our problm sttings as dscribd arlir. To th bst of our knowldg, Pippngr [14] first xprsss th natur of conflict rsolution th mixtur of two stags: (a) partitioning activ usrs into diffrnt groups; (b) payload transmission. Hajak [15] furthr studis this problm. In thir modl, K usrs ar randomly distributd (uniform or Poisson) in th [0, 1] ral intrval, dnotd by U = (U 1,..., U K ), whr U k is th arrival tim of usr k; a valid K-partition of [0, 1], dnotd by A, should b don during th conflict rsolution so that activ usrs ar sparatd in diffrnt groups. This modl corrsponds to our modl whn N. By dirctly considring th mutual information btwn input U and valid output A (without considr th channl ffct), a bound of throughput is drivd to solv th conflict in an adaptiv schm ovr a (0,..., d)-channl (d 2), whr th fdback is 0 if no activ usrs transmittd; 1, if only on usr is activ;...; and d, if mor than d usrs ar activ. Minimum of I(A; U) (= log KK ) was calld th partition information by Hajak [15], which gav an achivabl K! bound of a probabilistic problm. Suppos th lmnts of U ar uniformly distributd in [0, 1], and for a st of K-partitions {A l } L l=1, lt P {A l }(K) b th probability of th vnt that at last on of th A l is a valid partition, and lt Q L (K) b th minimum of 1 P {Al }(K) for diffrnt choic of {A l } L l=1. Thn, Q L(K) (1 K! K K ) L. Th lowr bound of Q L (K) is discussd by Hajk, Körnr, Simonyi and Marton [16 18], and sking th tight lowr bound still rmains an opn qustion 2. This partition problm (without considring channl ffct) is also closly rlatd to prfct hashing, zro-rror capacity, list cods, tc. [19, Chap. V]. Th problm is formulatd in a 2 Körnr givs 1 L log 1 Q L (K) K! K K 1 by using graph ntropy in [17].

7 7 combinatorial way: a subst A of K L is calld K-sparatd if vry subst of A consisting of K squncs is sparatd, i.., if for at last on coordinat i, th ith coordinats of th said squncs all diffr. Lt A L = A L (K) dnot a maximal K-sparatd subst of K L. It can b sn that A L corrsponds to N usrs in our problm sttings, and st A can b viwd as a st of K partitions with siz L, so that for any activ st out of A L usrs, thr xists a valid K partition. Th rlationship btwn this combinatorial modl and th probabilistic modl is statd by Körnr [17]. Not that ths problms did not considr th channl ffct, thus, thy wr kind of sourc coding from information thortic prspctiv. For compltnss, w will stat th sourc coding problm furthr in Sction IV of this papr. In contrast, th problm w ar focusing on is th transmission problm, i.., construction of a valid partition rlationship among activ usrs by th fdback rsulting from thir xplicit transmission ovr a collision Boolan multi-accss channl. This problm has not bn addrssd prviously, to th bst of our knowldg. In addition to th conflict rsolution problms, thr hav bn xtnsiv works on dirct transmission and group tsting that considr channl ffcts from th combinatorics and probabilistic prspctivs. Ding-Zhu and Hwang provid in [3] an ovrviw; mor spcific approachs can b found on suprimposd cods for ithr disjunct or sparabl purposs [11, 20 23], on slctiv familis [9], on th broadcasting problm [24], and for othr mthods [10, 22, 25]. It should b notd that rcntly, group tsting has bn rformulatd using an information thortic framwork to study th limits of rstoration of th IDs of all activ nods ovr Boolan multipl accss channls [26]. W addrss in this papr th transmission of partition information (rathr than idntification information) ovr th channl, and it is thus, diffrnt from xisting work. III. SYSTEM MODEL A. Formulation In this papr, lowr-cas (rsp., uppr-cas) boldfac lttrs ar usd for column vctors (rsp., matrics). For instanc, w i is usd for th i-th lmnt of vctor w, and w i,t is usd for th (i, j)- th lmnt of matrix W. Logarithms ar always to bas 2. Th probability of a random variabl A having valu à is dnotd by pa(ã) Pr(A = Ã). Similarly, pa B(à B) Pr(A = à B = B). Whr thr is no dangr of ambiguity, w will drop th subscripts and simply writ p(a) or p(a B) to dnot th abov quantitis.

8 8 Assum th numbr of activ usrs K is known to all usrs. Th usrs ar also givn a common N T accssing matrix (or codbook) X, and a dcoding function g( ). W us a Boolan vctor s = [s 1,..., s N ] to rprsnt th activ or inactiv stats of usrs, whr s i = 1 iff usr i is activ (that is, i G s ). Activ usrs will us T slots to transmit according to codbook X and obsrv th fdback y = [y t : 1 t T ] ovr ths T slots. Thn usrs driv th partition z = g(y). Thr ar two dimnsions in this problm, th usr dimnsion of siz N and th tim dimnsion of siz T. An Exampl Our approach is illustratd by an xampl in Fig. 1 with four usrs from N = {1, 2, 3, 4}, of which th usrs of st G s = {1, 2} ar activ. Th N T codbook is X. In ach slot 1 t 3 = T, usr i writs to th channl iff i is activ and x i,t = 1. For xampl, in slot 1, that has x 1,1 = x 2,1 = 1 and x 3,1 = x 4,1 = 0, both activ usrs 1 and 2 writ to th channl, rsulting in a channl fdback of y 1 = 1. In slot 2, x 3,2 = 1, howvr, sinc usr 3 is not activ, thr is no writ and y 2 = 0. In slot 3, usrs 1 and 3 ar calld upon to writ, but only usr 1 writs as usr 3 is not activ. Th channl fdback ovr th thr slots is y = [y 1, y 2, y 3 ] = [1, 0, 1]. From this fdback, th knowldg of K = 2 and th accssing matrix X, th following conclusions can b drawn. Bcaus x 3,2 = 1 and y 2 = 0, it can b concludd that usr 3 is not activ. Bcaus x 1,3 = x 3,3 = 1 and y 3 = 1, it can b concludd that usr 1 is activ (as usr 3 is inactiv), also G s {2, 4}. Th intraction in slot 1 only says that G s {3, 4}. Sinc K is known to b 2, w conclud that xactly on of usrs 2 and 4 must b activ and th othr inactiv. Thus partition {{1, 3}, {2, 4}} of N sparats activ nods into diffrnt groups, and z = [ ] can b slctd as th rsult of dcoding y. Obsrv that (unlik th rstoration of G s ), w do not (and nd not) know which among usrs 2 and 4 is activ. Likwis although w happn to know that usr 1 is activ and usr 3 is not, this knowldg is coincidntal; th partition approach dos not invst rsourcs to sk this knowldg. To hav a mor gnral formulation, th problm can b tratd as a coding problm in multiaccss channls from th information thortic viw. Considr N usrs whos activ stats ar

9 9 usr status: s transmission matrix: X fdback: y partition: z usr dimnsion s 1 s 2 s 3 s group #1 group #2 Fig. 1. Exampl of th formulation. (N = 4, K = 2, G = {1, 2} indicats that usrs 1 and 2 ar activ; th total numbr of tim slots is T = 3. ) givn in a vctor s S K;N {s {0, 1} N : s i = K}. Th i-th row of X, dnotd by x i b viwd as a codword of usr i (not that x i is a column vctor, w would lik to us a row vctor x i snds s i x i to rprsnt th codword according to th tradition). It is also asy to s that usr i on th channl. Th channl fdback is y = N i=1 s ix i [ N i=1 s ix i,t ] T t=1. Thn th dcodd output is a partition 3 z Z K;N, whr: Z K;N = { z K N : 1 k K, z i = k } is th st of all possibl K-ordrd partition. A distortion function is dfind for any status vctor s S K;N and a partition vctor z Z K;N as follows: 0, if i, j G s, (i j) = (z i z j ) d(s, z) =. (1) 1, othrwis Th objctiv is to dsign a propr matrix X and a corrsponding dcoding function z = g(y), so that d(s, g(y)) = 0 for narly all s S K;N. To simplify th notation, w writ y = X s, whr dnots Boolan matrix multiplication in which th traditional arithmtic multiplication and addition oprations ar rplacd by logical AND and OR, rspctivly. For any givn s, w dnot th st of all possibl dsird z as Z K;N (s) = {z Z K;N : d(s, z) = 0}. Th st of all possibl vctors s that ar compatibl with can 3 In th traditional viw [27], a K ordrd partition of N is a K-tupl of substs of N, dnotd by (B 1, B 2,..., B K), whr K 1 K 1 < K 2 K, B K1, B K1 B K2 = and B k = N. Our notation hr is quivalnt. For xampl, for a partition dnotd by z = [ ] Z 3;5, it rprsnts a partition ({2, 3}, {4, 5}, {1}). k=1

10 10 s s 1 s 2... s N Encod d(s, z)=0 T s 1 x 1 T s 2 x 2 Channl T s N x N... y Dcod z=g(y) Fig. 2. Encoding-channl-dcoding systm with distortion critrion a givn z to produc 0 distortion is dnotd by S K;N (z) = {s S K;N : d(s, z) = 0}. In som situations, w will nd to know th numbr of usrs, n k, in a givn group k K. Th st of K all possibl z with group sizs (n 1,..., n K ), whr n k = N, is dnotd by: { ( N ) } Z K;N (n 1,..., n K ) z Z K;N : 1(z i = k) = n k, 1 k K, hr th indicator function 1(A), which accpts a Boolan valu A as input, is 1 if A is tru, and 0 if A is fals. i=1 k=1 B. Prformanc Critria In this papr, w us a probabilistic modl and considr an avrag rror. Assum ach input s S K;N is with qual probability, i.., s U(S K;N ), whr U(A) mans th uniform distribution in any st A. Thus s S K, p s ( s) Pr(s = s) = 1/ ( N K). For a givn X, th avrag rror probability is dfind: P (N) (X) p(s)pr(d(s, g(y)) 0 s, X) s S K;N = ( 1 N ) K s S K;N 1(d(s, g(y)) 0)1(y = X s) (2) y Not that w us p(s) instad of p s ( s) for simplification. Th first trm 1(d(s, g(y)) 0) rvals th ffct of dcoding, and th scond trm 1(y = X s) th ffct of channl. W dfin a numbr of slots T (N) c to b achivabl, if for any T > T c (N), thr xists a N T matrix X (N) and a dcoding function g (N) ( ) for a givn N, such that Th aim is to find T (N) c, whn N. lim P (N) (X (N) ) = 0. N

11 11 d Encod f N s Dcod g s N f N s Fig. 3. Sourc coding part with distortion critrion Rmark: In group tsting, th objctiv is to rstor vry usr s stat, i.., th output should b z g S K;N, and corrct rstoration mans z g = s. If by th dfinition of distortion d g (s, z g ) = 1(z g = s), (3) th problm abov is xactly a noislss group tsting problm. Thus th main diffrnc btwn our partition problm and group tsting problm lis in th diffrnt dfinitions of distortion functions, mor importantly, lis in th diffrnt forms of information to transmit. Furthrmor, sinc knowing G s will always induc a corrct partition of N by distortion dfinition (1), th partition problm rquirs no mor information transfrrd than that in th cas of group tsting. In th nxt sction, w rigorously analyz th amount of th information usd to solv th partition problm. IV. SOURCE CODING In this sction, w first focus on th inputs and outputs of th systm without considring channl ffcts, i.., a cntralizd sourc coding schm illustratd as in Fig. 3, to find th amount of information ndd for dscribing th sourc with th purpos of partition. In othr words, th purpos is to find a st of partitions C with minimum siz, so that for narly vry possibl s S K;N, thr is a partition z C and d(s, z) = 0. With th hlp of sourc codbook C, for any unknown input s, w can utiliz th channl to chck vry partition in C to find th valid partition; dtails appar in th nxt sction. For group tsting, th objctiv is to rstor all stats of usrs, if w us a sourc codbook C g {s 1,..., s L (N)} to rprsnt all s S K;N, th siz L (N) should b S K;N = ( N K). Howvr, in th partition rsrvation systm, for a givn z Z K;N, thr can b mor than on s so that d(s, z) = 0. Actually whn z Z K;N (n 1,..., n K ), w hav S K;N (z) = Π K k=1n k numbr of possibl activ vctors so that d(s, z) = 0 for s S K;N (z). Thus, w can us codbook with

12 12 siz smallr than S K;N to rprsnt th inputs. Strictly spaking, for s U(S K;N ), if thr xists a sourc ncoding function: and a sourc dcoding function: f s N : S K;N {1, 2,..., L (N) }, g s N : {1, 2,..., L (N) } Z K;N, so that w can map s to a dcoding output z = gn s (f N s (s)), and th avrag sourc coding rror: P s,(n) p(s)1(d(s, gn(f s N(s))) s 0) (4) s S K;N approachs 0 whn N, w will call (L (N), fn s, gs N ) an achivabl sourc cod squnc for th uniform sourc s U(S K;N ), th rang of gn s ( ) is dfind as th sourc codbook. Th minimum of log L (N) for all achivabl sourc cod squncs will b calld th partition information for s U(S K;N ). In this sction, w first comput th minimum constraind mutual information btwn s and valid partition z, dnotd by WN I, in Lmma 1, and thn prov th xistnc of an achivabl sourc cod squnc (L (N), f s N, gs N ) for thos L(N) > 2 W I N in Thorm 1. Not that whn N, W I N in Sction II. quals to min I(A; U), and th minimum of P s,(n) quals to Q L (K), as introducd Constraind mutual information is always rlatd to th rat distortion problm [28, 29]. Thus, w first calculat th constraind minimum mutual information for s U(S K;N ) and valid z, i.., whr th constraint is: W I N min I(s, z) (5) p(z s) P z s P z s {p(z s) : p(z s) = 0, if d(s, z) = 0}, (6) which mans only valid partition z can b chosn for givn s. Th rsult corrsponds to Hajak [15], whn N. Lmma 1: W I N min I(s, z) = log p(z s) P z s ( N K) K k=1 n k (7)

13 13 whr W I N subjct to can b achivd by choosing z s U (n 1,..., n K) = arg max n k K n k, k=1 K n k = N, and k K, n k 1. (8) k=1 ( Z K;N (n 1,..., n K) ) Z K;N (s). (9) Eq. (9) mans for any givn s, th partition z should b chosn from th corrct st Z K;N (s) sinc th constraint P z s, also w rquir that z Z K;N (n 1,..., n K ) to minimiz th mutual information, which mans thr ar n k usrs assignd to th group k. Th partition z can b chosn uniformly from th st satisfying ths two conditions. Th proof is in Appndix A, in which w first partition Z K;N to (n 1,...,n K ) Z K;N(n 1,..., n K ), and thn for ach st of partitions, log sum inquality is usd to obtain th lowr bound. For th achivability, a dirct construction of th optimal p(z s) is introducd by (9). Dnot L (N) as th siz of a codbook, w hav Thorm 1 (Sourc coding): Thr xists a codbook {z l } L(N) l=1 of siz L (N), and a sourc coding squnc (L (N), fn s, gs N ), so that for all N, th avrag sourc dcoding rror probability is boundd by: P s,(n) 2( log L (N) W I N ) Thus, whn log L (N) > W I N and log L(N) W I N N, squnc (L (N), fn s, gs N ) is achivabl. Th proof is in Appndix B. Th cor of th proof is to us random coding mthod to construct th codbook {z l } l=1 L(N), in particular, choos z l i.i.d. from U (Z K;N (n 1,..., n K )), and show th avrag of P s,(n) ovr all possibl codbooks satisfis th bound in Thorm 1, thus thr must xists at last on codbook satisfying this bound. Thn by assigning th sourc ncoding function f s N (s) = arg min 1 l L (N) d(s, z l), and th sourc dcoding function g s N (l) = z l, w will obtain th sourc coding squnc (L (N), fn s, gs N ) with th rror probability boundd by Thorm 1. From Thorm 1, w can s W I N can b usd to masur th amount of asymptotic partition information of th sourc. And it xplicitly shows th partition information, as wll as its diffrnc from th rquird information to rstor all stats in furthr rmarks.

14 14 Rmark 1: For group tsting, if w dfin WG,N I as that in (5), obviously w hav: ( ) N WG,N I = log (10) K Thus WN I = log ( ( N K ) ( K ) K) log k=1 n k of partition problm is smallr by a factor log k=1 n k than that of group tsting. W nxt rmark on th ffct of th ordr of K as compard with N on th achivd mutual information, as wll as th rror probability. Rmark 2: First, lt s show th xplicit xprssion of W I N. From rstriction of [n k] K k=1 it is asy to s without rquiring n k to b a intgr, thn th optimal valus of n k ar Thus n 1 = n 2 =... = n K = N K. W I N log in (8), ( ) ( ) K N N log (11) K K Th quality is achivd whn K divids N, and it is a good approximation whn N K. Also, w hav th inqualitis: ( N (a) K) ) K KK K! ( N K (b) K, (12) Equality of (a) will b approximatly achivd whn K N, and th quality of (b) rquirs 1 K N. Rmark 3: Whn K = O(N),.g. K = ηn and 0 < η < 1 is a constant, w hav: lim N lim N 1 N W N I = (1 η) log(1 η) (13) 1 N W G,N I =H(η) η log η (1 η) log(1 η) (14) Thy ar obtaind by a tight bound of ( N K) drivd by Wozncraft and Riffn, s in Sction 17.5 in [28]. Thus w can dfin an achivabl sourc information rat R s for th partition problm (not th unit of th rat dfind hr is bits/usr), so that for any R R s + ξ, whr ξ > 0 is any constant, thr xists an achivabl coding squnc (L (N) = 2 NR, fn s, gs N ), and P s,(n) 0, whn N (15)

15 1 By Thorm 1 and Eq. (13), w can s that R s = lim N N W N, I whn K = ηn, sinc w can always construct th achivabl coding squnc of L (N) = 2 NR that for all ξ > 0, and R R s + ξ, P s,(n) 2N(R Rs) 0 (16) Not that th rror is doubly xponntial. Whil for group tsting, if w dfin Rs g similarly to R s, w can s by (13) and (14) that Rs g 1 = lim N N W G,N I = R s + ( η log η) > R s. Thus, w nd highr rat to rprsnt th stats of usrs than to partition thm. Rmark 4: Whn K = o(n), lim N W I N 15 KK = log. A spcial xampl is that K is a constant, thn lim N W I N is also a constant. W can s th proposd achivabl rat R s = 0 by (12), i.., 1 N W I N K N log 0. By Thorm 1, for any L(N) = f(n), whr f(n) is a function satisfying f(n) N, w can always construct a sourc coding squnc with codbook siz L (N) = f(n), and P s,(n) 2 ( ) log f(n) log KK K! K! 0, whn N (17) It can b sn that w can choos L (N) to b of any ordr of N to guarant th convrgnc of P s,(n),.g., L (N) = log log N. Whil for group tsting, w should always nd L (N) = ( N K) to rprsnt th sourc, which can b much largr than that of partition problm. Howvr, diffrnt choics of f(n) will influnc th spd of convrgnc,.g., if an xponntial convrgnc spd is rquird, i.., P s,(n) N for som > 0, thr should b L (N) = O(N). V. THE BRUTE FORCE METHOD Givn th sourc codbooks randomly gnratd in th sourc coding problm, w propos a corrsponding channl coding schm. In this schm, th channl codbook X is cratd by first collcting all partitions (or codwords) in th sourc codbook; th dcodr thn chcks ach partition (or sourc codword) xhaustdly with th hlp of th Boolan channl. Mor spcially, if th partition st C is givn as a sourc codbook, and T 0 slots ar ndd to chck if a partition z C is a valid partition, thn at most T = T 0 C slots ar ndd to chck all partitions in C. This is th brut forc mthod. For a givn L (N), w can find a sourc codbook {z l } L(N) l=1 to rprsnt th sourc undr rror probability P s,(n) by Thorm 1. Thus if a matrix X is dsignd to chck whthr z l is th

16 16 corrct output on by on, th avrag rror probability P (N) thus approachs zro whn log L (N) > W I N and log L(N) W I N is statd as follows: will bhav th sam as P s,(n), and. Th brut forc mthod 1) Sourc coding: For L (N), choos th codbook {z l } L(N) l=1, and th sourc coding squnc (L (N), fn s, gs N ) basd on Thorm 1. 2) Joint coding: Gnrat X by L (N) submatrics of dimnsion N K, X = [X 1,..., X L (N)]. Thus th dimnsion of X is N T, whr T = KL (N) (T 0 = K to chck ach possibl partition). Each X l is a N K matrix, so that 1 i N, 1 k K, th (i, k)-th lmnt of X l satisfis: S Fig. 4 for an xampl. 1, z l;i = k; x l;i,k =. 0, othrwis 3) Dcoding: Now th outputs ar sparatd into L (N) blocks: and y = [y 1 ;... ; y L (N)], y l = X l s is a K 1 column vctor. If thr xists y l = 1 K 1, whr 1 K 1 is a K 1 column vctor with all componnts qual to 1, thn th joint dcodr is g(y) = z l ; if thr xist mor than on, w can slct on of thm,.g., th first on; othrwis thr is dcoding rror. Not that if y l = 1 K 1, thn thr xists at last on activ usr in ach of k groups assignd by z l. And sinc w know thr ar xactly K activ usrs, only on activ usr is assignd in ach group. Thn dfinitly d(s, z l ) = 0, i.., it is basd on th following fact: i j G s, z i z j i G s {z i } = K.

17 17 usr dimnsion z X 1 X 2 #1 #2 #3 #1 #2 # z tim dimnsion Fig. 4. Exampl of th gnration of X in brut forc mthod, whr N = 6, K = 3, and sourc codbook of siz L (N) = 2 is chosn. Obviously in th brut forc mthod th numbr of channl uss T BF = KL (N). In addition, sinc in this mthod if thr xists z l in codbook {z l } L(N) l=1 so that d(s, z l ) = 0, thn dfinitly d(s, g(y)) = 0, th avrag rror of th brut forc mthod is th sam as cntralizd sourc coding. Thn basd on th analysis of cntralizd sourc coding as in Thorm 1, w hav Thorm 2 (Brut forc mthod): For th brut forc mthod, if th siz of cntralizd sourc codbook is L (N), thn and th avrag rror probability is P (N) ( T BF K T BF = KL (N), / ) 2 W N I = 2( log L (N) W I N ) Although th brut forc mthod is vry simpl and obviously not optimal, it highlights som faturs of th partition problm. First, if K is a fixd numbr, thn as statd in Rmark 4 in th last sction, only T BF = KK+1 f(n) is ndd for th convrgnc of P (N) K! (sinc P (N) f(n) ), whr KK+1 is a constant and f(n) is any function satisfying lim f(n) =. In this cas, th K! N thrshold ffct of th convrgnc dosn t xist as that in group tsting or comprssiv snsing [4], and th choic of f(n) is rlatd to th spd of convrgnc of P (N). Howvr, whn K is larg,.g. whn 1 K N, 2 W I N K, T BF should b largr than K K to guarant th convrgnc of P (N), which may b vn largr than th tim ndd for group tsting T G = O(K log N). It is xpctd sinc th brut forc mthod is not optimal. In particular,

18 18 whn K incrass, th siz of cntralizd sourc codbook incrass fast, and it bcoms so infficint to chck thm on by on. VI. RANDOM CODING AND REFORMULATION AS HYPERGRAPH Th brut forc mthod was inspird by a cntralizd sourc coding and it works wll only for small K. To find th achivabl bound of T for gnral cas, w dsign th cod from anothr way by randomly gnrating X first and thn mploying MAP dcoding. Howvr, to hav a mor amiabl approach to driv an achivabl rat, and to provid mor insights on th intrnal structur of th problm in dpth, a nw angl from graph thory is proposd in this sction, which transforms th ffct of channl to a sris of oprations on hyprgraphs. It is shown that sking an accptabl partition is quivalnt to obtaining a common strong colorabl hyprgraph by all usrs, and thn coloring this hyprgraph. Bcaus w ar only concrnd about an achivabl rat, th computational cost associatd with th coloring is not countd in our framwork. A. Random coding and MAP dcoding Random coding is frquntly usd in th proof of achivability in information thory, and has bn provn usful for group tsting [26]. Th binary matrix X is gnratd randomly, whr ach lmnt x i,t B(p) follows th i.i.d Brnoulli distribution with p paramtr (othr distributions of X can also b considrd, but that is out of scop of this papr). Th probability of X is dnotd by Q(X). Thn th avrag probability of rror ovr th ralization of X is givn by: P (N) = X = X (a) = X Q(X)P (N) (X) Q(X) Q(X) y s S K;N p(s)p y s;x (y s)1(d(s, g(y)) 0) y p y s;x (y s 0 )1(d(s 0, g(y)) 0) (18) Sinc w don t considr obsrvation nois in this papr, p y s;x (y s) = 1 (y = X s),

19 19 and quality of (a) abov follows from th symmtry of th gnration of X, so w can choos any particular s 0 as input to analyz. W will choos G s0 = {1, 2} in th rst of th papr. Sinc th drivd achivabl T (N) c S c, so that for any T satisfying log(n) P (N) N for random coding is of ordr log N, w dfin an achivabl rat T 0. Which also implis thr xists a X such that P (N) w will driv such a S c. S c ξ, whr ξ > 0 is an arbitrary constant, w hav (X ) N 0. In this sction, Th optimal dcoding mthod is MAP dcoding, i.., givn fdback y, choos z = g(y) so that z z Z K;N, th following holds which is quivalnt to p z y;x (z y) p z y;x (z y), s S K;N 1 (d(s, z ) = 0) p y s;x (y s) s S K;N 1 (d(s, z) = 0) p y s;x (y s) (19) If thr is mor than on z with th maximum valu, choos any on. Not that hr w sarch all possibl z in all possibl z Z K;N ; howvr, considring th sourc coding rsults, w can just sarch z Z K;N (n 1,..., n K ) without loss of gnrality. As sn in th dfinition of MAP dcoding, to find MAP of z, w should count all s S(z) satisfying y = X s. Whil many s S(z) has common activ usrs, so 1(y = X s) ar corrlatd for diffrnt s sharing parts of common activ usrs. Thus, it is xtrmly difficult to compar th postrior probability of diffrnt z. Th obstacl ariss bcaus in MAP dcoding, fw inhrnt structurs of th problm ar found and utilizd. To furthr rval this inhrnt problm structur, a novl formulation from th prspctiv of hyprgraph is proposd in th nxt sction, which provs to b hlpful in rducing complxity of prformanc analysis. B. Rformulation as Hyprgraph Th procss of random coding can b illustratd in th uppr part of Fig. 5. For an input s 0, th channl output y = i G s0 x i is obsrvd, and thn a candidat subst of S K;N that is capabl of gnrating y can b infrrd: S y = {s S K;N : y = X s}

20 20 s X S MAP z* H H H*, z* 6 1 G s G s G s Fig. 5. Rformulation from hyprgraph MAP dcodr tris to find z such that thr is th largst numbr of s S y satisfying d(z, s) = 0. This procss can b illustratd from th prspctiv of hyprgraphs, as shown in Fig. 5. 1) Sourc: Sinc all ral sourcs s 0 S K;N ar quiprobabl, a complt K-uniform hyprgraph H(V (H), E(H)) can b usd to xprss th knowldg of th sourc bfor obsrvation, whr th st of nods V (H) = N rprsnts N usrs, and th st of hyprdgs E(H) = { V (H) : = K} rprsnts all possibl inputs [30, 31]. It mans vry hypr-dg in H could b s 0. Actually th ral input is just an dg G s0 E(H), th objctiv of group tsting is to find xactly this dg to obtain vry usr s stat; whil for partition rsrvation systm, th objctiv is to sparat ach vrtx of G s0. 2) Transmission and obsrvation: th transmission and corrsponding obsrvation can b sn as a sris of dg dlting oprations on th hyprgraphs. Bcaus aftr obsrving ach fdback y t, 1 t T, som s could b dtrmind to b not possibl, and th candidat st S y could shrink. A sub-hyprgraph H (V (H ), E(H )) H(V (H), E(H)) is usd to dnot th candidat st S y aftr obsrving th fdback y. Not that w considr th nod st V (H ) = V (H) to b invariant, but actually thr will b many isolatd nods in V (H ) with zro dgr. Th dtails of th oprations will b shown in nxt subsction. Not that for th considrd noislss cas, w always hav s 0 S y, so G s0 H. 3) Partition: Finally, th partition z should b dcidd by obsrving H. First, w introduc th concpt of strong coloring. A strong coloring of a hyprgraph H is a map Ψ : V (H) N +, such that for any vrtics u, v for som E(H), Ψ(u) Ψ(v). Th valu of Ψ(u) is calld th color of nod u. In othr words, all vrtics of any dg should hav diffrnt colors.th corrsponding strong chromatic numbr χ s (H) is th last numbr of

21 21 colors so that H has a propr strong coloring [32]. Obviously for a K-uniform hyprgraph, χ s (H) K. W calld a strong coloring with K colors to b K-strong coloring. If zi is viwd as a color of nod i, actually z Z K;N givs a coloring mapping of V (H) with K colors. For MAP dcoding in (19), th mthod of finding z from S y is quivalnt to finding a hyprgraph H (V (H ), E(H )) H (V (H), E(H )), such that χ s (H ) = K, i.., H is K-strong colorabl, and th numbr of dltd dgs E(H ) \ E(H ) is minimum. Thn th output z can b any strong coloring of H. From th prospctiv of hyprgraph, th procss can b rprsntd as H H (H, z ), corrsponding to th xprssion from vctors s 0 S y z. Th procss is shown in Fig. 5 through an xampl of N = 6, K = 2. Not that th hyprgraph bcoms a graph whn K = 2. Compard with group tsting, whos objctiv is to obtain H = H with only on dg G s0 by dlting dgs through transmissions and obsrvations, our partition problm allows H and H to hav mor dgs, so lss ffort is ndd to dlt dgs, which is translatd to highr achivabl rat than that of th group tsting problm. W can s z is corrct iff G s0 E(H ) and H is K-strong colorabl, w will us this quivalnt condition to judg if th dcoding is corrct in th analysis. From th viw of th algorithm, first all usrs obtain a common good H which will lad to a corrct partition. Scond, w obtain H and choos a common consistnt z. Th scond stp, to obtain H by dlting th minimum numbr of dgs from H and find th K-strong coloring [32], dos not influnc th transmission tim ndd to arriv at a good H. This is bcaus onc all usrs hav th sam copy of H, th rmaining computation, including rmoval of dgs and coloring, can b don locally without furthr xpnding communication rsourcs. A furthr xplanation of oprations of th dlting dgs is introducd in th nxt subsction. C. Rduction stp: obtaining H from H Th ffct of transmissions and obsrvation using matrix X can b summarizd in two oprations: dlting vrtx and dlting cliqu. Assum at tim t, th st of activ usrs transmitting 1 is: G X (t) = {i N : x i,t = 1}. (20)

22 22 Original complt graph v3 y = 0 Dlt vrtxs: v7 v3 y = 1 Dlt cliqus: (v1,v3,v4,v6,v8) v3 v4 v2 v4 v2 v4 v2 v5 v1 v5 v1 v5 v1 v6 v8 v6 v8 v6 v8 v X v7 v7 t = 0 t = 1 t = 2 y = 0 y = 1 Dlt vrtxs: (v4,v5,v6) Dlt cliqus: (v1,v3,v4,v5,v7) v3 v3 v4 v2 v4 v2 v5 v1 v5 v1 v6 v8 v6 v8 v7 t = 4 t = 3 v7 Fig. 6. Exampl of X ffcts on th oprations of a graph. (Hr N = 8, K = 2, T = 4, and G s0 = {1, 2}) Th opration at tim t can b classifid basd on th fdback y t : 1) If y t = 0, which mans any usrs in G X (t) should not b activ usrs, so ths vrtics should b dltd, i.., all dgs containing ths vrtics should b dltd. 2) If y t = 1, which mans at last on activ usr is transmitting 1 at tim t, so any dg compltly gnratd by vrtics in N \ G X (t) should b dltd. Othrwis if ths dgs ar actually th G s0, thr will b no activ usrs in G X (t) and y t = 0. In fact, it is quivalnt to dlting all K uniform hypr-cliqus gnratd by N \ G X (t). Th two ffcts ar illustratd by an xampl in Fig. 6 using a graph with squnc of nods rmovd and cliqu rmoving oprations. Thr ar 8 usrs and 4 slots ar usd for transmission. W can s th dgs rmoving procss starting from a complt hyprgraph at t = 0, to a graph of only 3 dgs at tim t = 4. At t = 1, 4, y t = 0, th corrsponding vrtics ar rmovd, whil at tim t = 2, 3 cliqus ar rmovd. Now it is clar that our problm can b viwd as a K-strong hyprgraph coloring problm, and th objctiv is to schdul a sris of dg rmoving oprations fficintly to construct such a hyprgraph so that all K activ usrs could b assignd a uniqu color (or transmission ordr). In nxt sction, a spcial cas of K = 2 is solvd; vn in this simpl cas, th problm is nontrivial.

23 23 VII. RANDOM CODING OF K = 2 For K = 2, two sub-optimal dcoding mthods inspird by MAP dcoding ar proposd to furthr simplify th calculation. A. Two simplifid dcoding mthods In MAP dcoding, th dcodr will find a K-strong colorabl graph H from H by dlting th minimum numbr of dgs, and th dcoding rsult is corrct if G s0 H. For K = 2, hyprgraph H bing 2-strong colorabl is quivalnt to H bing bipartit, or quivalntly, having no odd cycls. Furthr, assum G s0 = {1, 2}, odd cycls can b classifid into thr kinds: 1) Containing a cycl with vrtics 1 and 2, th cycl may or may not containing dg (1,2); 2) Containing on of th vrtics 1 and 2; 3) Containing nithr vrtx 1 or 2. Dnot th odd cycls containing dg (1, 2) by 1-odd cycls. Sinc (1, 2) always xists in H du to th noislss channl, it is asy to s H contains no first kind of cycls iff H contains no 1-odd cycls. Thus in th rst of papr, w just considr th xistnc of 1-odd cycls and th othr two kinds of odd cycls (somtims for simplification of notation, w also call 1-odd cycl th typ-1 odd cycl). W can assrt that if thr is no 1-odd cycls in H, th dcoding rsult will b surly corrct. Th rason is that for MAP dcoding, it braks all odd cycls in H to gt H by dlting last dgs. If thr is no 1-odd cycl in H, st G s0 will not b dltd during this procss. Thus, G s0 H, which implis th corrct dcoding. Thus, w hav P (N) P 1 odd X Q(X)Pr(H contains 1-odd cycls X, s 0 ) (21) P odd X Q(X)Pr(H contains odd cycls X, s 0 ) (22) In th following, P odd and P 1 odd ar both upprboundd by thir rspctiv union bounds, and it is shown thir upprbounds ar narly th sam whn N, which points to th possibility of using a suboptimal dcoding to advantag: whn H is 2-colorabl, find any z consistnt with it; othrwis announc an rror. Th rason is if MAP coding is usd, it is ncssary to obtain H by dlting minimum dgs of H, which is a NP hard problm[33];

24 24 howvr, it is asy to judg whthr H is a bipartit graph in linar stps of N. So whil th suboptimal dcoding mthod nds mor channl us, it is asir to comput. B. Main rsult: Achivabl bound of T for K = 2 cas To upprbound P (N) by P 1 odd, dnot: C(p) = (1 (1 p) 2 ) log ϕ(p) (1 p) 2 log(1 p) (23) whr W hav th following lmma: ϕ(p) = p + 4p 3p 2. (24) 2 Lmma 2: For K = 2 cas, if for any constant ξ > 0 such that log N (N) max C(p), w hav P P 1 odd 0 p 1 P (N) And similarly, by bounding P (N) N 0. by P odd, w hav th following thorm: Thorm 3: For K = 2 cas, for any constant ξ > 0 that P odd N 0. log N T T S c ξ, and S c S c ξ, w will hav Th proofs of Lmma 2 and Thorm 3 ar givn in Appndix C. Actually if th lmnts of X ar gnratd i.i.d. by Brnoulli distribution of paramtr p, w will hav P 1 odd approachs 0 if log N T both S c for two mthods to mak P (N) and P odd C(p) ξ, thus S c = max C(p). W can s th achivabl bounds ar p 0. Th main ida in th proof is to calculat th probability of xistnc of a particular odd cycl in H, and th calculation is similar for all thr kinds of odd cycls. As a ky factor in th rsult, ϕ(p) in (24) is actually a factor of th solution of th xtndd Fibonacci numbrs, which rvals som intrsting structur in th partition problm. A sktch of th proof of Lmma 2 is givn as blow, it is th sam for Thorm 3: 1) Considr th problm conditioning on [x 1, x 2 ] in a strong typical st A (T ɛ ) ; this will mak th algbra asir. Assum th probability of xistnc of a particular 1-odd cycl of M vrtics in H to b P ;M ; thr ar ( N 2 M 2) (M 2)! N M 2 such odd cycls and all of thm ar quiprobabl. Thus, P 1 odd M 3,M is odd 2 (M 2) log N P ;M + Pr([x 1, x 2 ] A (T ) ɛ ) (25)

25 25 Sinc Pr([x 1, x 2 ] A (T ɛ ) q(p,ɛ)t T ) 2 0, whr q(p, ɛ) is som constant, according to th proprtis of strong typical st[34]. W will show that P ;M 2 (M 2)C(p)T. Thus whn log N < (C(p) ξ)t, 2 (M 2) log N P ;M 2 (M 2)ξ, which mans th P 1 odd 0. Not that w can also s th P 1 odd gos to 0 with an xponntial spd with T, and thus, polynomial spd with N, i.., P 1 odd 2 1T = 1 N 2, whr 1 and 2 ar constant. 2) Divid T slots into four parts T u,v = {t : (x 1,t, x 2,t ) = (u, v)}, for four diffrnt (u, v) {0, 1} 2, according to th codwords of th ral input [x 1, x 2 ]. In th strong typical st, w just nd to considr whn T u,v p x (u)p x (v)t, whr p x (u) p1(u = 1) + (1 p)1(u = 0) is th probability distribution of Brnoulli variabl. And du to symmtry and indpndnc of th gnration of X, for any (u, v), w just nd to considr any slot t T u,v. 3) At t T u,v, dnot µ u,v;m to b th probability that th considrd 1-odd cycl of lngth M won t b dltd by th oprations, thn P ;M = Π u,v (µ u,v;m ) Tu,v Π u,v (µ u,v;m ) px(u)px(v)t (26) 4) W hav shown that for all t whr y t = 0, i.., t and th xponnt p x (0)p x (0) = (1 p) 2, thus (u,v) (0,0) T u,v, µ u,v;m = (1 p) M 2, (µ 0,0;M ) px(0)px(0)t = (1 p) (M 2)(1 p)2t = 2 (M 2)T (1 p)2 log(1 p) (27) Whil for y t = 1, i.., t T u,v, (u, v) (0, 0), w hav shown µ u,v;m = ϕ M 2 (p), and (u,v) (0,0) p x(u)p x (v) = 1 (1 p) 2. Thus, Π (u,v) 0 (µ u,v;m ) px(u)px(v)t = (1 p) (M 2)(1 (1 p)2 )T = 2 (M 2)T (1 (1 p))2 log ϕ(p) (28) Thn, combining (27) and (28), w obtain P ;M 2 (M 2)C(p)T, which complts th proof. W can provid an intuitiv xplanation. Th rsult in Lmma 2 can b xprssd as p, T > M 2 log N (M 2)C(p) (29) Intuitivly th problm can b statd as that w hav at most N M 2 1-odd cycls of lngth M, and aftr dtrmining(or liminating) all of thm, th rror probability bcoms 0. Thus, log N M 2 can b sn as an upprbound on sourc information, which dscribs th uncrtainty

26 26 of 1-odd cycls; (M 2)C(p) can b sn as th information transmitting rat of th channl, which rprsnts th spd of liminating th uncrtainty of odd cycls with M vrtics. To furthr xplain th maning of (M 2)C(p), w should us th ffct of X on hyprgraph statd in Sction VI-C. If a givn 1-odd cycl H ;M of M vrtics xists in H, in 1 t T, non of th M vrtics, or th cliqus containing th dgs of H ;M can b dltd. S an xampl in Fig. 7, whr H ;M is th outr boundary. It won t b rmovd if all th nods ar maintaind; and th cliqus to b dltd should not contain th conscutiv vrtics on th outr boundary. Sinc at any tst t, vrtics ar dltd if y t = 0; th probability of this happning is (1 p) 2. For a particular t whn y t = 0, th vrtx of an inactiv vrtx i is dltd only if x i,t = 1, so th probability that all M vrtics ar maintaind at tim t is µ 0,0;M = (1 p) M 2. On th othr hand, all th dgs of th odd cycl H ;M can t b dltd by th cliqu dlting opration. At any slot t so that y t = 1, whos probability is 1 (1 p) 2, thr ar 3 diffrnt cass (x 1,t, x 2,t ) = (u, v), (u, v) (0, 0), thir analysis is similar, lt us just considr (x 1,t, x 2,t ) = (1, 1). Assum H ;M = (1, 2, i 1,..., i M 2 ), so at any slot t, th probability that H ;M is not rmovd by dlting cliqus can b drivd: µ 1,1;M =1 Pr(H ;M is rmovd at slot t t T 1,1 ) =1 Pr( w {1,..., M 3}, (x iw (t), x iw+1 (t)) = (0, 0)) (a) = 1 p F (M, p) ϕ(p)m 2 (30) Th drivation of (a) is sn in Appndix C, and F (k, p) = k 1 2 j=0 ( k 1 j whr ϕ(p) = p + 4p 3p 2 j ) p k 1 j (1 p) j (31) = ϕ(p)k ψ(p) k ϕ(p) ψ(p), (32) 2 ; ψ(p) = p 4p 3p 2 2 is th solution of a gnralizd Fibonacci squnc [35]. Actually 1 F (k +2, p) is th probability p that thr ar no conscutiv 0s in a p-brnoulli squnc of lngth k. This fatur of Fibonacci squncs has also bn usd in gnrating cods without conscutiv 1s, known as Fibonacci (33)

27 27 = 3 v7 v1 v2 = 2 v3 v6 v5 v4 Fig. 7. Random cliqu dlting whil kping an particular odd cycl at a particular t such that y(t) = 1. (Hr th siz of th odd cycl is M = 7, K = 2, only th cliqus of siz 2 (dg) or 3 (triangl) consistd with non conscutiv vrtics can b dltd, as shown by (2, 4) and (2, 5, 7) for xampl.) coding. Th othr µ 1,0;M and µ 0,1;M can b drivd similarly. Thus, w can s (M 2)C(p) is wll xplaind as th rat of dlting vrtics or cliqus for an 1-odd cycl with M vrtics from abov. Now it is clar that Lmma 1 and Thorm 2 abov hav rvald th intrnal structur of th partition problm. Th partition information is rlatd to odd cycls, and X is constructd to dstroy th odd cycls by dlting vrtics or cliqus. Th Fibonacci structur mrgs sinc it is rlatd to considring conscutiv 0s in Brnoulli squncs, which may b a ky factor in partition problm and could b xtndd to mor gnral cass with K > 2. In th nxt sction, th fficincy is compard with random coding basd group tsting approach. VIII. COMPARISON As statd in th introduction, our partition rsrvation has clos rlation to dirct transmission and group tsting. Sinc th avrag rror considrd in dirct transmission systm is not th sam as th dfinition usd in this papr, w just compar with th group tsting. Atia and Saligrama hav provd th achivabl rat in Thorm III. 1 in [26] for group tsting with random coding, which shows that if for any ξ > 0 and log N T whr { C g (p) = min (1 p)h(p), 1 } 2 H((1 p)2 ) S cg ξ, S cg max C g (p), p th avrag rror probability P (N) 0 (Also, S cg is shown to b a capacity in Thorm IV. 1 in [26]). From Fig. 8, w can s C g (p) < C(p) for any 0 < p < 1. In particular, S cg =