Improved lower bound for deterministic broadcasting in radio networks

Impro v ed lo w er b ound for deterministic broadcasting in radio net w orks Carlos Fisch Brito ∗ Shailesh V a y a † No v emb er 11, 2018 Abstract W e consider the problem of deterministic broadcasting in radio net w ork s wh en the n odes ha ve limited knowledge ab out the top ology of the netw ork. W e sho w that for every determin- istic broadcasting proto col there exists a net w ork, of radius 2, for whic h the p rotocol tak es at least Ω( n 1 2 ) round s for completing the broadcast. Ou r argument can b e extend ed to pr o v e a lo wer b oun d of Ω (( nD ) 1 2 ) roun d s for broadcasting in radio net w orks o f radius D . This resolv es one of the open problems p osed in [29], where in the authors prov ed a lo wer b ound of Ω( n 1 4 ) roun ds for broadcasting in constan t diameter n et w orks . W e pr o v e the n ew lo w er b oun d f or a sp ecial family of radius 2 n et w orks . Eac h net work of this family co nsists of O ( √ n ) comp onen ts whic h are conn ect ed to eac h other via only the source no de. A t the heart of the pr o of is a nov el sim ulation argument, whic h essent ially s ays that an y arbitrarily complicated strategy of the source no de can b e sim ulated by th e n odes of the netw orks, if the sour ce no de just transmits partial topological knowledge ab out some comp onen t ins tea d of arbitrary complicated messages. T o the b est of our kno w ledge this t yp e of sim ulation argumen t is no v el and ma y b e useful in furth er impro ving the lo w er b ound or ma y find use in other applications. Keyw ords: radio n et w orks, d ete rministic broadcast, lo w er b oun d , advice string, simulatio n, selectiv e families, limited top ological knowle dge. ∗ Departamento de Computa¸ c˜ ao, Universidade F eder al do Cear´ a , F o rtaleza, Brazil. E-mail: carlos@lia.ufc.br. Research in this w ork was co nducted while the author w as a gr aduate studen t at UCLA. † Department o f Co mputer Science and Engineering, Indian Institute of T echnology Madras, Chennai, India - 60003 6. E-mail: v ay a@ c se.iitm.ernet.in. Resear c h in this work w a s conducted while the a uthor w a s a g raduate student at UCLA. 1 1 In tro ductio n One of the most fu ndamen tal and well stu died p roblems in d istributed computing is th e problem of br oadca sting. In broadcasting there is a sour c e no de whic h p ossesses a message that should b e sent to all the remaining no des of the net work. If the source no de is d ir ect ly connected to all the no des of the netw ork then the message can b e sen t to the rest of the no des in a single transmission. What mak es the p roblem of broadcasting non-trivial is that the source no de m a y not b e directly connected to the rest of the net wo rk and the top ology of the net work m a y b e arbitrary . Th e minim u m requiremen t on the top ology of a net work so that broadcasting can b e completed is that the n et w ork is connected. There are sev eral metrics to measure the complexit y of a broadcasting pr otocol, like roun d complexit y (min- im u m num b er of r ounds n eeded to complete broadcasting), message complexit y (minimum n um b er of messages n eeded to b e sen t to complete b roadcasti ng) etc ., which can b e defined in terms of the num b er of no des in the net work n , the radiu s of th e net w ork D , etc. The most imp ortan t and p rev ale n t complexit y measure under wh ic h the broadcasting problem has b een studied is the r ound c omplexity of the broadcasting p rotocol. This wo rk fo cuses on the round complexit y of d eterministic broadcasting proto cols in sp ecial kinds of netw orks called r adio networks . Ad-ho c radio ne tw orks. Broadcasting proto cols find many applications in A d-ho c wir e- less netw orks. Ad-ho c wireless n et w orks are used in s cenarios lik e battlefields, emergency dis- aster reliefs an d other situations where there is n o infrastructure for communicatio n net w orks . Sensor net works are also an examp le of w ireless radio net works. Unlike the traditional w ir e- less net works these netw orks d o not h av e a base station to wh ic h the no des can comm unicate. No des which are within the range of their radio signals comm unicate via radio transmissions, while n odes that are far off rely on other no des of the n et w ork to exc hange messages. Comm unication in these net works is stru ctur ed using sync hr onous time-slots. In ev ery round eac h no de either acts as a transmitter or as a receiv er. A radio net wo rk can b e mo deled as an undir e cte d connected graph as follo ws: Eac h no de in the graph repr esen ts a pro cessor, and t w o n odes are connected by an edge if the corresp onding pro cessors lie within the transmission r ange of eac h other. A message tr ansmitted by a no de can p oten tially reac h all its neigh b ors . Ho w ev er, if more than one neigh b oring n ode send a message in the s ame round, then a collision o ccurs. This is the mo del that w as prop osed in the seminal pap er on radio net works [3], and has b een generally considered in the literature [29 ]. In [34], the authors adopt a more p essimistic mo del for radio transmission, according to wh ic h when tw o or more neigh b orin g no des transmit a message in the same round, the receiving no de receiv es the message fr om one of th em and the messages from others are lost. The m odel considered in this w ork is th e one traditionally considered in the literature on radio broadcasting, i.e. as stated in [3 ], [29], [30]. 1.1 Related W ork The study of br oa dcasting in radio n et w ork s w as initiated by Bar-Y eh ud a, Goldreic h and Itai in [3]. In [29], Ko w alski and P elc establish a low er b ound of Ω ( n 1 / 4 ) r ounds for the deterministic broadcasting p roblem on radio net w orks of diameter 4 (and ( nD 3 ) 1 / 4 rounds for net works of d iamet er D ) in the mo del considered traditionally , i.e., whic h is the mo del stated in [3 ]. When the to p ology of the r adio n et w ork is kno wn to a ll the n od es of the net work, a deterministic broadcasting p rotocol of O ( D lg 2 2 n ) was give n in [11], for net works of r adius D . This cen tralized br oadca sting proto col h as recen tly b een imp ro v ed to O ( D lg 2 n + lg 2 2 n ) rounds b y [30]. A proto col r unning in O ( D + lg 5 2 n ) round s w as giv en in [21] and this has b een most recen tly impro ved to O ( D + lg 4 2 n ) in [20]. A low er b ound of Ω(lg 2 n ) rou n ds has b een pro v ed in [1], for the cen tralized setting. The setting wh ere the no des are giv en only their o wn lab els, but not the lab els of their neigh b orin g n od es has also b een studied quite extensiv ely . Researc h in this pr oblem has led to the introd uction of v ery in teresting com b inatorial concepts like sele ctive fam ilies . The use of selectiv e families in the d esign of d ete rministic p r otocols for u nkno wn netw orks was in tro duced b y Chlebus et. al. in [12]. Several recen t wo rks exploit this com bin ato rial tool, sp ecifically the use of p r obabilistic metho d, for obtaining goo d lo w er and u pp er b oun ds for the b roadcast ing pr oblem [16], [12], [14]. The problem of b roadcasti ng on d irecte d rad io n etw orks h as receiv ed muc h attent ion. The proto col give n by [12] requires O ( n 2 ) roun d s for completion. This upp er b ound was reduced to O ( n 3 2 ) round s by a b reakthrough result of [13]. In a further br eakt hrough, [14], the authors deplo y ed t he pr obabilistic metho d to b ring the upp er b ound fr om O ( n lg 2 2 n ) to within logarithmic factors of the b est known low er b oun d Ω( n lg 2 D ). In [29], a fu rther impro v ement has b een made for small diameter net w orks. They establish an upp er b oun d of O ( n lg 2 n lg 2 D ). In [9], the authors defin e and deploy new combinato rial stru ctures to reduce this gap b et ween lo wer and u pp er b ounds to O (lg 2 D ) f actor i.e., giv e a b roadcasti ng p r otocol runn in g in O ( n lg 2 2 D ) r ounds. Randomized proto cols f or the br oadca sting problem ha ve b een stu d ied in [3], [32], [27]. In these proto cols, the nod es do not kn o w the lab els of their n eigh b oring n odes and ma y in fact ha v e non-uniqu e lab els. In [3], a br oadcasting proto col run n ing in O ( D lg 2 n + lg 2 2 n ) rounds is give n. A low er b ou n d of Ω( D lg 2 ( N/D )) rounds w as prov ed f or this problem in [27 ]. The lo w er b ound of Ω (lg 2 2 n ) roun ds for b r oadca sting p rotocols also h olds for this prob lem. A broadcasting proto col running in O ( D lg 2 ( n/D ) + lg n 2 ), matching the lo wer b oun d, wa s prop osed in [32]. In the w ak e u p pr oblem, eac h no de in the net w ork either w ake s-up sp on taneously or is activ at ed by receiving a wa k e-up signal from another no de. Eac h activ e no de transmits the w ak e-up signal according to a giv en proto col. The r unning time of a w ak e-up proto col is the num b er of steps counted from t he fir st sp onta neous wak e-up, u n til all no des become activ at ed. T his p roblem has b een studied under differen t assumptions in [15], [8 ], [22], [25 ]. Async h ronous radio broadcasting has b een studied in [10]. In the n ext section, we shall present an outline of the pro of of the low er b ound. The rest of the p aper shall b e dev oted to form ali zing this outline. 2 Some early dev elop men ts and outline of the Pro of The s tu dy of br oa dcasting in radio net works was initiated b y Bar-Y ehuda, Goldreich an d Itai in [3], where they constructed a class of net w orks, referr ed to as simple BGI-net w orks (see Figure 1(a)), f or whic h they pr esented a low er b ou n d for deterministic broadcasting. In [29], Ko walski and Pelc observ ed that for simple BGI-net w ork s the role of the source n ode cannot b e ignored in ac hieving faster broadcast. They go on to d escrib e a deterministic proto col in whic h the source assists other no des in the n etw ork to co mplete broadcast on an y BGI-net work in O (lg 2 n ) r ounds. Figure 1: (a) BGI-net work and (b) C2 net w ork F or the result in [4], [5], w e in tro duced a more elab orate class of net w orks, (see Figure 1(b)) for which w e ask ed the questions: ”If the broadcast pr oto col run s for at most r rounds , how m u c h assistance can the source nod e pr ovide? Can this assista nce b e quan tified in terms of the n um b er of rou n ds, the topology of the net work and argued to b e insufficient to significan tly exp edite b roadcasting?” . The fir st attempt to formalize this id ea w as an in formatio n theoretic argument presented in [4 ]. Basically it is argued in [4] that even if the role of the source no de is sub stituted b y an advice string enco ding some partial knowle dge ab out the top ology of the un derlying net w ork in consideration, the assistance is found to b e insufficien t, in formation theoretically sp eaking, to complete broadcast in √ n roun ds on at least some of th e net works. Th is argumen t emplo ys a straigh tforw ard generalization of a lo wer b oun d on the size of selectiv e families, [16]. In this generalization w e found a tec hnical gap. The generaliza tion is p erhaps true for families of subsets with sp ecial p rop erties, but we b eliev e that form ulating an argument f or the lo wer b ound along the original lines wo uld mak e the pro of u n necessarily complicated. Along the same lines an oracle argumen t, [28], wa s also sk etc hed in [5]. This argumen t h as the conceptual problem to ameliorate which the original inf ormatio n theoretic argument was pr oposed in [4]. Namely , if an oracle provides the iden tities of the t w o n odes w ith least IDs which transmitted in a giv en round when a collision is d etec ted, then dep ending on the sp ecification of the proto col the information p ro vided by the oracle can also leak partial kno w ledge ab out the top olog y of the net w ork. F or example, it may b e the case that the pr otocol sp ecifies that some no des in la ye r L 1 , with IDs smaller than the ones rev ealed by the oracle also transmit in this round if connected to the appropriate n o de in la yer L 2 (see Figur e 2). Then, an interpretation of the oracle message is that these n odes are not connected to the corresp onding no des in L 2 in the give n netw ork. Th is kno wledge ab out the top ology of the net w ork ma y b e exploited in ac hieving faster br oadca st in future an d cann ot just b e ignored. In this work w e pr esen t a more d irect pro of of th e lo wer b oun d, along lines similar to the original in formatio n theoretic argument. The p roof consists of three main comp onen ts: In the first comp onen t it is shown thr ough a series of sim u latio n b ased reductions that, if an arbitrary deterministic b roadcast proto col π 0 completes br oadca st in r rounds on ev ery C 2 net work, then there must exist another deterministic b roadcast proto col in whic h the sour ce is giv en an advice string at the b eginning of the pr otocol π ′ whic h the source tr ansmits along with the broadcast message at the b eginning of the pr otocol an d remains silen t for the rest of the protocol. This advice string consists of r blo c ks of binary strings eac h of whic h enco des one of the follo w ing tw o t yp es of messages: φ (wh ic h denotes a collision or empt y m essag e), or a tuplet < i, C i > where C i denotes the configuration of the i th BGI-comp onen t in the net work. The proto col π ′ m u st complete broadcast in at most 3 ∗ r r ounds on every netw ork of the C 2 family . Th ese redu ctions are presented in Section 5. The essence of the second comp onen t is to u tiliz e meaningful connections b et w een the messages encoded in th e advice string and the to p ology of the co rresp onding underlying net work. F or this we consider the set of all th e unique advice s tr ings pro vided to the source no de for d ifferen t n et w orks from C 2 family . W e select one suc h advice string whic h m us t b e pro vided to the source no de for a large subset of C 2 net works. This subset of net wo rks has the prop ert y that all the d ifferen t p ossible configurations of at least one BGI-comp onen t are present in some n et w ork of the sub s et. These arguments are presented in Section 6. The ab o ve t w o comp onen ts are combined with a kno wn lo wer b ound on the size of selec tiv e families, or alternative ly a hitting game argument fr om [3], to complete the pro of of the lo wer b ound, Theorem 4.2. 2.1 Organization of the re st of the pap er The rest of the pap er is d ev oted to form aliz ing the ab o ve argument. S ect ion 3 describ es the mo del in detail and introdu ces notations and definitions used in th e text. Section 4 states some auxiliary results and prov es the main theorem. The statemen t of the redu ctio ns and their pro ofs are p resen ted in Section 5. Section 6 describ es the pr ocedur e PRU NE() that c ho oses an app r opriate advice string and a corresp onding subset of C 2 net works as menti oned ab o ve . 3 Descrip t ion of the mo del and general definitio ns Definition [3],[29]: A broadcast proto col π for a radio netw ork is a sync hronous multipro- cessor proto col which pro ceeds in r ounds as follo w s: 1. No des ha v e distinct lab els from the set { 0 , 1 , . . . , m } , wh er e m is a p olynomial on the n um b er of n odes in the netw ork. A distinguish ed no de with lab el 0 is called the sour c e no de. 2. All no des execute iden tical copies of the same proto col π . 3. In eac h round, every no de either acts as a trans mitter or as a receiv er (or is inactiv e). 4. A no de receiv es a message in a sp ecific r ound if and only if it acts as a r ece iv er and exactly one of its n eig h b ors transmits in that roun d. Otherwise, it receiv es φ . W e assume that the messages are authentica ted, that is, when a no de r ece iv es a message it kno w s the lab el of the transmitting no de. 5. The action of a n od e in a sp ecific roun d is d ete rmined by (a) Initial inpu t, whic h conta ins its o wn lab el an d th e lab els of its neigh b ors . (b) Messages receiv ed b y the no de in previous rounds . 6. In round 0, only the source no de transmits th e broadcast message. 7. Only no des that hav e receiv ed a message are allo w ed to transmit. That is, th e only sp ontane ous transmission is the one b y the source in round 0. 8. Broadcast is completed in r round s if all the no des receiv e the source message in one of the rounds 0 , 1 , . . . , r − 1. 3.1 C 2 net w orks Figure 2: C2 net w ork W e pro v e the low er b ound f or a family of n et w orks, called C 2 net works, with radiu s 2 and a simp le comm unication structure. Intuitiv ely , a C 2 net work is formed b y connecting a n um b er of BGI-net works via a common source n o de (see Figure 2). More p recisely , the no des of a C 2 net work are d ivided into three la ye rs: L 0 , L 1 and L 2 , where: • la yer L 0 consists only of the source no de; • la yer L 1 consists of √ n groups of √ n no des eac h, all of whic h are connected to the source no de. • la yer L 2 consists of √ n n odes, eac h one of whic h is asso ciated w ith a distinct group of no des in L 1 and connected to an arbitrary subset of no des of th e group. W e call eac h group of no des in L 1 together with the asso ciat ed no de in L 2 a BGI- comp onen t, or alternativ ely a comp onen t, of the netw ork. The lab els of th e no des in a net w ork are arbitrary b ut fixed for all the C 2 net works. Th us, w hat distin gu ish es t w o net works N and M of C 2 family is only the top ology of one or more BGI-co mp onen ts. Ob serving th at there exists 2 √ n distinct p ossible top ologi es for a BGI-comp onen t, we can giv e a complete description of a comp onent by a tuplet < i, τ > , where i is an ind ex for th e comp onent w h ic h is b eing describ ed and τ is an in teger in th e range [0 . . . 2 √ n − 1]. 3.2 Advice string υ ( π , N , t ) An advice string υ ( π , N , t ) is an arr a y of t ele men ts, eac h of whic h could p ote n tially enco de partial top ologic al information ab out the netw ork N . More p recisel y , eac h element consists of either φ or a tup let < i, τ i > whic h d escribes the top ology of a BGI-comp onen t of the net work N. The sp ecific con ten ts of the advice strin g υ ( π , N , t ) d ep end s on the first t r ounds of the execution of proto col π on netw ork N . 4 Main theor e m In this section we present the main theorem 4.2 which prov es the low er b oun d. Theorem 4.2 com bin es the three parts of the argument Lemm a 4.1 (simulati on based reductions), Lemma 4.2 (selection of an advice string and an appropriate sub set of netw orks) an d Theorem 4.1 (a w ell kn o wn lo wer b ound on the size of selectiv e families). Lemma 4.1 Assume that ther e exists a pr oto c ol π that c omp letes br o adc ast in at most r r ounds on ev ery network of the C 2 family. Then, ther e exists a deterministic br o adc ast pr oto c ol π ′ such tha t • the no des in layer L i tr ansmit only in r ounds t ≡ i (mo d 3), for i = 0 , 1 , 2 . • for e ach network N of the C 2 family ther e is an advic e string υ ( π 3 , N , r ) which is pr ovide d to the sour c e no de at the set up phase along with the br o adc ast message. • the sour c e tr ansmits the br o adc ast message and the string υ ( π 3 , N , 3 r ) in r ound 0 . • the sour c e r emains silent in ev ery r ound i > 0 . • the pr oto c ol π ′ c ompletes br o adc ast on network N i n at most 3 ∗ r r ounds Lemma 4.2 L et π ′ b e the pr oto c ol given by lemma 4.1, with r < √ n 2 . Then, ther e exists an advic e string υ and a subset of networks S ⊂ C 2 such tha t 1. Pr oto c ol π ′ c ompletes br o ad c ast in at most 3 ∗ r r ounds on every network of subset S , when the sour c e no de is pr ovide d with the advic e string υ in the set u p phase. 2. Ther e exists an index i such that, for e ach p ossible top olo g y of a BGI-c omp onent, ther e is a network N ∈ S whose i th c omp onent has e xactly this top olo gy. Definition 4.1 ([16 ]) L et [ n ] = { 1 , . . . , n } and let k ≤ n . A family F of subsets of [ n ] is a ( n, k ) -sele ctive if for every non empty subset Z of [ n ] such that | Z | ≤ k , ther e is a set F in F su ch that | Z T F | = 1 . Theorem 4.1 ([16]) L et F b e a ( n, k ) -sele ctive family, with n > 2 and 2 ≤ k ≤ n/ 64 . Then it holds that, |F | ≥ k 24 log 2 n k . Theorem 4.2 F or every deterministic pr oto c ol π that runs f or o ( √ n ) r ound s, ther e exists a C 2 network N such that π do es not c omplete br o adc ast when exe cu te d on N . Pro of: Assume, to the cont rary , that th er e exists a broadcast p rotocol π that completes broadcast in less than √ n/ 1536 rounds on ev ery C 2 net work. T hen by applying lemma 4.1 w e obtain a pr otocol π ′ . Let υ , S, i b e the advice string, subset of net w orks and index giv en b y Lemma 4.2 for proto col π ′ . The pro of is based on the f oll o win g observ ation. Supp ose that x is a no de in la yer L 1 and y is a n ode in la y er L 2 of the i th BGI-comp onen t of the net w ork. According to th e defi nition of a broadcast proto col, the b eha vior of no de x in a giv en round dep ends on its o wn lab el, wh ic h is fixed, the lab els of its neighbors and the sequ en ce of m essag es r eceiv ed in previous roun ds. The source is alwa ys a n eigh b or of x , b ut x ma y or ma y not b e connected to y . Ho w ev er, if x is n ot connected to y , then x cannot transmit the message to y an d complete broadcast on this BGI-comp onent , so we fo cus our atten tion on the situation w hen x is connected to y . No w, from lemma 4.2, w e get that the source transm its the same message in r ound 0 when p rotocol π ′ is executed on every net work of s ubset S , n amely the br oadca st message and advice string υ . A t th is p oin t, we wo uld lik e to conclude th at, wh en connected to y , the no de x has exactly the same b ehavio r in the exec utions of protocol π ′ on different netw orks of sub set S . Ho wev er, w e still hav e to consider the messages r ece iv ed by x fr om the no de y , whic h can b e transmitted in d ifferen t rounds on the differen t netw orks b elonging to subset S . But then, we recall that the only sp on taneous transmission is made by the source in round 0, and so if y makes a transmission the task of broadcast has already b een completed on this BGI-comp onen t. Thus, we ma y co nclude that, when connected to y , no de x has alw a ys the same b eha vior wh en π ′ executes on ev ery net work of S , up to the round in whic h broadcast is completed on comp onen t i . Next, w e present the pro of of the lo w er b ound formally . Consider the family F = { F 0 , F 1 , . . . , F r − 1 } of subsets of no des fr om the BGI-comp onen t i , defin ed as follo ws. A no de x b elongs to subset F j if and only if there exists a net work N ∈ S s uc h that 1. x is connected to y in the n et w ork N 2. x transmits in round 3 ∗ j + 1 wh en π ′ is executed on net w ork N 3. y do es n ot receiv e an y message up till round 3 ∗ j + 1 wh en π ′ is executed on N . No w, w e claim that F corresp ond s to a ( √ n, √ n )-selec tiv e family . T o s ee this, n ote that for an arbitrary n et w ork N , the subs et of no des in la yer L 1 of the i th BGI-comp onen t of net work N that are connected to the nod e y , corr esp ond to an arbitrary subset of [ √ n ]. Let this sub set b e called Z . Since π ′ completes broadcasting on net w ork N in at most 3 ∗ r rounds, there must exist some j < r su c h that in round 3 j + 1, for the fi rst time, exactly one no de fr om subset Z transmits. But this is equiv alen t to sa ying that there exists a su bset F j in F suc h that | F j ∩ Z | = 1. Finally , to appy the theorem 4.1, w e restrict ourselve s to ( n, k )-selectiv e families, w here k ≤ n 64 . Ho w ever, it is easy to see that if F is ( √ n, √ n )-selectiv e, then it is also ( √ n, √ n/ 64) -selectiv e. Now, r = |F | ≥ √ n 64 · 24 ∗ log n ≥ √ n 1536 . But this con tradicts our initial assumption that there exists a d ete rministic broadcast proto col π that completes b roadcast in less than √ n/ 1536 round s on every n etw ork of C 2 family . W e note here th at b etter constants could b e d eriv ed by using the hitting game argumen t from [3 ]. How ev er, we b eliev e that the current lo wer b ound is not asymp toti cally close to the optimal and h ence hav e c hosen n ot to optimize the constan ts. Belo w, w e n ote a coroll ary to the ab o v e theorem f or net w orks of arbitrary d iameter D . Corollary 4.1 Every deterministic br o adc ast pr oto c ol must take Ω( √ n ∗ D ) r ounds to c om- plete b r o adc ast on r adio networks of diam eter D . 5 Reduction s ( Pro of of the Lemma 4.1) In this section we pr esen t a series of sim ulation based reductions to p ro v e Lemma 4.1. 5.1 Reduction 1 The fir st reduction sho ws that w e ma y consider proto cols with a simp lified communicat ion structure, at the cost of a constant facto r (= 3) increase in the num b er of rounds. I n these proto cols, the no des co ordinate their transmissions so that collisions inv olving no des f r om differen t la ye rs do not o ccur. Lemma 5.1 Assume that ther e exists a pr oto c ol π 0 that c ompletes br o adc ast in at most r r ounds on every C 2 network. Then, ther e exists a pr oto c ol π 1 that c ompletes br o adc ast in at most 3 ∗ r r ounds on every network of C 2 family, su ch that the no des in layer L i tr ansmit in r ound t only if t ≡ i (mo d 3), for i = 0 , 1 , 2 . Pro of: Proto col π 1 sim u late s eac h round t of pr otocol π 0 in the s equ ence of r ounds 3 t , 3 t + 1 and 3 t + 2. The idea is that, in roun d 3 t + i , eac h no de in la y er L i tak es th e action that it w ould take in roun d t under proto col π 0 . Assuming that π 1 includes the description of pr otocol π 0 , then it is sufficien t to sho w that eac h no de w in the n et w ork can compu te the list of messages receiv ed by itself up to roun d t under proto col π 0 , durin g the execution of π 1 . The claim is certainly tr u e for t = 0. Now, sup p ose that, at the b eginning of roun d 3 t , ev ery n o de in the net work has the correct list of messages receiv ed up to round t − 1 un der proto col π 0 . Then, eac h n o de can tak e the appropriate action during rounds 3 t , 3 t + 1 and 3 t + 2, and up d ate th eir list of receiv ed message s as follo ws . If w is a no de in la ye r L i , then a) if w transmits a message in r ound 3 t + i , it app ends φ to its list of receiv ed messages, since it cannot receiv e a message in a round in whic h it acts as a trans mitter. b) otherwise, if exactly one neigh b or of w transmits durin g round s 3 t , 3 t + 1 and 3 t + 2, it app ends this message to the list; else it app end s φ . Condition (a) is obvio us. The only issue on condition (b) is th e abilit y of the n o de w to detect that a single neighbor tr an s mitte d in roun ds 3 t , 3 t + 1 and 3 t + 2. If w b elongs to la y er L 0 or L 2 , then all its neigh b ors are in la ye r L 1 and, according to the definition of a br oadca st p rotocol, it receiv es a message in round 3 t + 1 if and only if exactly one of its neigh b ors transmit. If w b elongs to la yer L 1 , then it h as the source and p ossibly a no de f rom la ye r L 2 as neighbors. Bu t since in this case no collision is p ossible, w can d etec t exactly whic h of its neigh b ors transm itte d. Hence, the conditions ab ov e guarantee that, at the b eginning of round 3( t + 1), eac h no de in the net work h as the correct list of m essag es receiv ed un der proto co l π 0 up to round t . W e denote by Π 1 the class of proto cols satisfying th e conditions of lemma 5.1. 5.2 Reduction 2 The second reduction shows that, instead of transmitting arbitrary messages, the source can just retransmit whatev er message it r ecei v ed in the previous L 1 transmission roun d. The idea is that the no des of la ye r L 1 can s imulate the b eh avior of the source, once they h a v e the sequence of messages it has receiv ed so far, and in this w a y to reco v er the message that the source sh ould hav e transmitted dur ing the execution of some other proto col. Lemma 5.2 Assume tha t ther e e xists a pr oto c ol π 1 of the class Π 1 , that c ompletes br o adc ast in at most 3 r r ounds on every C 2 network. Then, ther e exists a pr oto c ol π 2 , also of the class Π 1 , that c ompletes br o adc ast in at most 3 r r ounds on every C 2 network su ch that, in r ound 3 t , the sour c e just r etr ansmits whatever message it r e c eive d i n r ound 3 t − 2 . Pro of: In proto col π 2 the n odes of the net w ork b eha v e as follo ws. The source n ode b eha ves according to th e statement of the lemma. The no des in la yer L 2 just sim ulate pr otocol π 1 . The n od es in lay er L 1 decide wh at actio n to p erform in round 3 t + 1 in t w o steps. First, they u se the list of messages receiv ed from the source u p to round 3 t to simulate the b eha vior of the source un der proto col π 1 , and obtain the sequence of messages that th e source sh ould hav e transmitted up to roun d 3 t in the execution of proto col π 1 . This second list, together with th e list of messages receiv ed fr om a p ossible neighbor in la ye r L 2 , allo ws them to sim ulate their o wn b eha vior under proto col π 1 and compute the appropriate action to p erform in roun d 3 t + 1. W e denote by Π 2 the class of proto cols satisfying th e conditions of lemma 5.2. 5.3 Reduction 3 This redu ctio n sho ws that, instead of transmitting arbitrary m essag es, the source can just send descrip tions of BGI-comp onen ts of the net work on whic h the pr otocol is b eing executed. The idea is that, with the descrip tion of a BGI-comp onen t and the list of messages transmitted b y the sour ce, an y no de of the net work can reco ver the messages transmitted by the no des of this comp onen t to the source, and this is all that is requ ired to simulate a p rotocol of th e class Π 2 . Lemma 5.3 Assume that ther e exists a pr oto c ol π 2 of the class Π 2 that c om pletes br o adc ast in at most 3 r r ounds on ev ery C 2 network. Then, ther e exi sts a br o adc ast pr oto c ol π 3 , of the class Π 1 , that c ompletes br o adc ast in at most 3 r r ounds on every C 2 network such that • the sour c e is pr ovide d, as input, with a c omplete description of the network on which the pr oto c ol is b eing exe cute d. • in r ound 3 t the sour c e tr ansmits: a) the br o ad c ast message µ , if t = 0 . b) φ , if no message is r e c eive d in r ound 3 t − 2 . c) < i, τ > , if a message is r e c eive d in r ound 3 t − 2 ; in this c ase, the tuplet < i, τ > is a description of the BGI-c omp onent of the tr ansmitting no de in the network on which the pr oto c ol is b eing exe cute d. Pro of: In proto col π 3 the n odes of the net w ork b eha v e as follo ws. The s ou r ce no de b eh a v es according to the statement of the lemma. The no des in la ye r L 2 just sim u lat e proto col π 2 . The no des in la yer L 1 need, for the sim ulation of protocol π 2 , to compute, in eac h round 3 t , the message r ece iv ed b y the sour ce in round 3 t − 2. W e consider t wo cases: 1. If the source transm its φ in roun d 3 t , this means that it has receiv ed no message in round 3 t − 2, either b ecause n o neigh b or transm itted or b ecause a collision o ccurred. 2. If the source transmits < i, τ > in round 3 t , this means that a single neigh b or, from the BGI-comp onen t i , tr an s mitte d in round 3 t − 2. T o reco ver this message , eac h no de in L 1 can sim ulate the b ehavio r of th e no des in comp onent i up to r ound 3 t − 2, using the description τ and the list of messages transmitted by the source u p to round 3( t − 1) under proto col π 2 . W e denote by Π 3 the class of proto cols that satisfy th e conditions of lemma 5.3 5.4 Reduction 4 This reduction sh o ws that, instead of pro viding the source with a complete d escrip tion of the top olog y of the net w ork, it is suffi cie n t to provide it with just a p artia l top ologic al information in the form of an advice string. Moreo v er, the b eha vior of the sour ce is further restricted so that no w it just transmits the adv ice string together with the broadcast m essage in round 0, and remains silent thereafter. The idea is that the advice s tring gives the sequ ence of messages transmitted b y the source u nder pr otocol π 3 , and with this inform atio n the no d es in la ye r L 1 can carry out a complete sim ulation of proto col π 3 on th e netw ork. Lemma 5.4 Assume that ther e exists a pr oto c ol π 3 fr om class Π 3 , that c ompletes br o adc ast in at most 3 r r ounds on every network of family C 2 . Then, ther e exists a br o adc ast pr oto c ol π 4 , of the class Π 1 , such that • for e ach network N in C 2 ther e is an advic e string υ ( π 3 , N , 3 r ) such that pr oto c ol π 4 c ompletes br o adc ast on network N in at most 3 r r ounds when the sour c e is pr ovide d with this advic e string in the set up phase. • the sour c e tr ansmits the br o adc ast message and υ ( π 3 , N , 3 r ) in r ound 0 . • the sour c e r emains silent in ev ery r ound i > 0 . Pro of: The con tent of the advice string υ ( π 3 , N , 3 r ) is just the sequence of messages trans- mitted b y the source no de in the first 3 r r ounds of the execution of proto col π 3 on net wo rk N . In proto col π 4 the no des of the net work b eha v e a s follo ws. The source no de b eha v es according to the statemen t of the lemma. The no des in lay er L 2 just sim ulate proto col π 3 . In eac h round 3 i + 1, the n odes in la y er L 1 read the i th en tr y of the advice strin g υ ( π 3 , N , 3 r ), to obtain the message that the source would transmit in round 3 i und er p rotocol π 3 , and then sim u late their o wn b eha vior under proto col π 3 to compute the appr opriate action to p erform in this r ound. The pr otocol π 4 giv en by Lemma 5.4 is the p rotocol π ′ men tioned in the Lemma 4.1 of Section 4. 6 Pro cedu r e Prune (Pro of of th e Lemma 4 .2) In this section we describ e a pr ocedur e to obtain an advice string υ and a corresp onding subset of netw orks S ⊂ C 2 for a proto col π ′ giv en b y Lemma 4.1, su c h that the follo wing tw o conditions hold true: 1. Proto col π ′ completes broadcast in at most 3 ∗ r r ou n ds on ev ery net w ork of subset S , when the source no de is pr o vided with the advice string υ in the set up phase. 2. There exists an index i suc h that, for eac h p ossible top ology of a BGI-comp onen t, there is a net work N ∈ S w hose i th comp onen t h as exactly this top olog y . Recall that protocol π ′ is obtained through a series of reductions, and let π 3 b e the proto col from w hic h π ′ is obtained in the last r eduction. Under proto col π 3 , the source transmits only the br oa dcast m essag e µ , the message φ , an d top ological information of BGI-comp onent s < i, τ > . The sequence of messages transmitted b y the source no de under protocol π 3 (disregarding message µ ) giv es the advice string υ ( π 3 , N , 3 r ) pro vided to the source in the execution of protocol π ′ . The subset of net w orks S is constructed by the execution of pro cedure Pr une(), and considers the execution of pr otocol π 3 on ev er y net wo rk in C 2 . In the d escription of Pr une() and in the argumen ts that f ollo w, w e us e the notation ε ( π 3 , M , 3 t ) to indicate whic h ev ent o ccurs in roun d 3 t when π 3 is executed on net work N . There are three p ossibilities: • ε ( π 3 , M , 3 t ) = φ in dicate s that the source transmits φ in round 3 t b ecause none of its neigh b ors transm itte d in round 3 t − 2. • ε ( π 3 , M , 3 t ) = ρ indicates that the source transmits φ in round 3 t b ecause more than one of its n eigh b ors transmitted in round 3 t − 2. • ε ( π 3 , M , 3 t ) = < i, τ > giv es the message tr an s mitte d by the source in round 3 t when only one no de from la y er L 1 transmits in r ound 3 t − 2. Procedur e Prune( π 3 ,r) { S := set with all networks of C 2 If r = 1 Then Return S For t := 0 To r − 1 Do { If there exists N ∈ S with ε ( π 3 , N , t ) = ρ Delete from S every network M with ε ( π 3 , M , t ) 6 = ρ Else, if there exists N ∈ S with ε ( π 3 , N , t ) = < i, τ > , for some i Fix an arbitrar y such network N , and Delete from S every network M with ε ( π 3 , M , t ) 6 = ε ( π 3 , N , t ) } Return( S ) } No w, observe that all the net w orks in the set S returned by Prun e() are asso ciated with the same s equence of ev en ts ε ( π 3 , M , 3) , . . . , ε ( π 3 , M , 3( r − 1)). F rom this, it easily follo w s that the source transmits the same sequence of messages when pr otocol π 3 is executed on ev ery net work of the subset S . Let υ denote the advice string corresp onding to this sequence of messages, th en the ab o ve condition (1) h olds. T o chec k th at cond itio n (2) also holds, we fi x an arbitrary net work N from S , execute proto col π 3 , on N and mark its comp onen ts as follo w s: 1. If ε ( π 3 , M , 3 t ) = φ , then no comp onen t is marked. 2. If ε ( π 3 , M , 3 t ) = < i, τ > , then mark comp onen t i . 3. If ε ( π 3 , M , 3 t ) = ρ , then c h oose tw o of the no des that transmit in round 3 t − 2 arb itrarily , and mark the resp ectiv e comp onent s. Let B b e the set of comp onen ts marked in this pro cedure. Lemma 6.1 L et M b e a C 2 network whose c omp onents liste d in B have the same top olo gy as in network N . Then, M b elongs to the set S r eturne d b y Prune(). Pro of: W e prov e the lemma b y induction on r . The case of r = 1 is trivial. F or the ge neral case, we fi rst observ e that ev ery net work N ′ whic h satisfies ε ( π 3 , N ′ , t ) = ε ( π 3 , N , t ), f or t = 0 , 1 , . . . , r − 1, b elongs to the subset S . So, w e w ill pro v e that the net w ork M satisfies all those equations. Supp ose that the lemma h olds for r = k . Now let us consider the case of r = k + 1. By the ind uctiv e hyp othesis, w e ha ve that ε ( π 3 , M , t ) = ε ( π 3 , N , t ), for t = 0 , 1 , . . . , k − 1, so w e only n eed to show that ε ( π 3 , M , 3 k ) = ε ( π 3 , N , 3 k ). There are three cases to consider: Case 1: ε ( π 3 , N , 3 k ) = ρ In this case, more than one no de fr om L 1 transmit in roun d 3 k − 2 when π 3 is execute d on n et w ork N . Let x and y b e the no des chosen in step (3) of the marking pro cedure ab o v e, and supp ose that x and y b elong to the BGI-comp onen ts i and j , r esp ectiv ely . W e claim that x and y also transmit in round 3 k − 2 w hen π 3 is executed on netw ork M , whic h imp lies th at ε ( π 3 , M , 3 k ) = ρ . T o c hec k the claim, observe that th e BGI-comp onen t i and j h a v e the s ame top ologies in net w orks N and M . S ince the n o des in these comp onen ts receiv e the same sequence of messages from the source u p to round 3( k − 1), they ha ve iden tical b eha viors in rounds 3 k − 2 and 3 k − 1 when π 3 is executed on net w orks N and M . This follo ws from the definition of a d eterministic broadcast proto col. In particular, no des x and y transmit in roun d 3 k − 2 wh en π 3 is executed on net w ork M . Case 2: ε ( π 3 , N , 3 k ) = < i, τ > In this case, a single no de f r om L 1 transmits in round 3 k − 2 when π 3 is executed on netw ork N . Let x b e such no de, which, according to the message, b elongs to the i th BGI-comp onen t. No w, w e claim that x is also th e single no de to transmit in roun d 3 k − 2 w h en π 3 is executed on net work M . Before pro v in g the cla im, w e observ e that the BGI-comp onen t i is mark ed in the pro cedure ab o v e, and so it has exactly the same top ology on netw orks N and M . Thus, if the claim holds, we ha ve ε ( π 3 , M , 3 k ) = < i, τ > , as r equired. The fact that no de x transmits in round 3 ∗ k − 2 when proto col π 3 is executed on net work M follo ws b y the same argumen t giv en in case 1. No w, supp ose that some other no de in L 1 also transmits in the same r ound. Then, a collision would o ccur and the source w ould transmit ρ in its next transmission r ound 3 ∗ k . Ho wev er, b y the in ductiv e h yp othesis, the net work M survives up to the ( k + 1) th iteration of Prune(), an d so in this iteration th e net work N would b e purged f rom set S , whic h is a con tradiction. Hence, x is the only nod e to tr ansmit in round 3 k − 2 wh en π 3 is executed on net w ork M . Case 3: ε ( π 3 , N , 3 k ) = φ In this case, there is no trans mission in round 3 k − 2 when proto col π 3 is executed on N . A similar argument to the one giv en in case 2 allo ws to conclude that there is also no tr ansmission in round 3 k − 2 when proto col π 3 is executed on net w ork M . Hence, ε ( π 3 , M , 3 k ) = φ . Finally , condition 2 easily follo ws from lemma 6.1 by observing that, when r < √ n/ 2, the set B has less than √ n comp onen ts. Thus, if i is a BGI-comp onen t wh ic h do es not b elong to B , then every net w ork M whic h has exactly the same top ology as that of net work N excepting the i th comp onen t, b elongs to the subset S . Ac knowledgemen t(s): W e w ould lik e to thank Dariusz Ko w alski a nd And rzej P elc for useful discussions and encouraging u s to w ork on the lo wer b ound. W e thank E li Gafni for some initial co llab oratio n regardin g this wo rk, though he has dignifiedly c hosen not to b e a co-author. W e also thank Gunes Er cal and Ch en Avin for useful comments and suggestions ab out the pr esen tation of an earlier d raft of this wo rk. References [1] N. 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