The Expressiveness of Silence: Tight Bounds for Synchronous Communication of. Information Using Bits and Silence. Una-May O'Reilly and Nicola Santoro

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1 The Expressiveness of Silence: Tight Bounds for Synchronous Communication of Information Using Bits and Silence Una-May O'Reilly and Nicola Santoro School of Comper Science, Carleton University Ottawa, K1S 5B6, Canada Abstract. We establish worst-case and average-case lower bounds on the trade-o between the time and bit complexity for two-party communication in synchronous networs. We prove that the bounds are tight by presenting a protocol which has bit-time complexity matching the ones expressed by the lower bounds. We actually show that the algorithm is everywhere optimal: at any point of the trade-o and for any universe of data to be communicated, no other solion has better complexity to communicate any element of that universe (within a xed relabelling). Similar results are derived when transmissions are subject to corruptions. 1 Introduction A distribed system is a collection of processing entities (each with a local cloc and local memory) which communicate by transmitting bounded sequences of bits called messages. A fully synchronous system is a distribed system where the following two conditions hold: 1. all local clocs 'tic' simultaneously (although they might not sign the same time) and that the interval s between two consecive cloc tics is constant; 2. there exists a nown upper bound! on the transmission and queueing delays. Since! is nown a priori to all entities and all local clocs tic simultaneously, the unit of time can be redened so that transmission delays are unitary; that is, a message sent at cloc tic t to a neighbor will arrive and be processed there at cloc tic t + 1. Therefore, we can just consider unitary transmission delays witho any loss of generality. To avoid paradoxical situations, it is assumed that at any cloc tic only one message can be send to the same neighbor. A great deal of theoretical research is based on this synchronous distribed compation model. For example, on the election problem in a synchronous ring, alone, there are numerous results [4, 5, 6, 7, 8, 9, 11, 12, 13, 14]. In a fully synchronous system, the process of eciently and accurately communicating information along a lin obeys a set of rules, called the two-party communication (TPC) protocol; its choice can greatly aect the overall performance of the higher-level protocols employed in the system. Associated with any TPC protocol are two related cost measures: the total number of bits transmitted and the total number of cloc tics elapsed during the communication; the study of the two-party communication problem in synchronous networs is really the study of the trade-o between time and bits. Most of the research activities on this subject have been carried o within coding theory and have focused on determining, for the information

2 to be communicated, encoding-decoding schemes which satisfy some specic constraints (e.g., error- detection, error correction, etc.) or optimize some performance parameters (e.g., average number of bits, etc.). Related investigations include the study of transmitting sequences of bits over unreliable channels in synchronous systems (e.g., [1, 2]). It is worthwhile noting that, unlie their asynchronous counterpart, in fully distribed systems transmission of bits is not the only way of communicating information; for example, in a fault-free system, if no bit is received at local cloc time t + 1, then none was transmitted at cloc time t. Hence, absence of transmission, or silence, is detectable and can be used to convey information. In other words, while the transmission alphabet is binary (i.e., T = f0; 1g), in a fully synchronous networ the communication alphabet is ternary (i.e., C = f0; 1; silenceg). Moreover, it is possible to communicate any positive integer x transmitting less than dlog 3 xe bits by using more time; for example, it is well nown that any positive integer x can be communicated transmitting only two bits in time linear in x. The tradeo between time and bits is, unfortunately, not linear. For example, using only a constant > 2 number of bits, the time complexity can be reduced to sublinear; e.g., [10, 12]. Several solions have been presented each oering dierent trade-os. Unfortunately, previous to this wor, no lower-bounds were nown on the time-bit complexity maing it dicult to evaluate the eciency of the proposed solions. In this paper, we investigate the two-party communication problem when a bit can be transmitted in a time unit. The problem is studied both in fault-free networs and in networs where transmissions are subject to corruption; in both cases, we establish tight bounds on the trade-o between time and bit complexity. Following is a summary of the results; unless otherwise specied, they are established for both types of networs. 1. We establish lower-bounds on the trade-o between time and bit complexity both for the average and the worst case. The lower-bounds hold even when the universe of discourse (from which the information to be communicated is drawn) is nite; thus, they are also a lower-bound for the case of a (countable) innite universe. 2. We present a solion protocol and show that its complexity matches both the worst-case and the average-case lower-bounds. The protocol wors even when the information to be communicated is drawn from an innite (b countable) universe. 3. We prove the stronger result that our protocol is everywhere optimal: at any point of the trade-o and for any universe, no other solion has better complexity to communicate any element of that universe (within a xed relabelling) 4. The protocol for networs where transmissions are subject to corruptions can tolerate any number of faults. Note that results 1 and 2 above imply that the existence and nowledge of a bound on the size of the universe does not aect the bit-time trade-o. The paper is organized as follows. In the next section, basic denitions are presented and some existing protocols are described to provide the reader with an insight into the nature of the problem. In Section 3, two distinct lower bounds are established on the trade-o between time and bits, depending on whether or not the

3 transmission is reliable. In Section 4, for both cases, we present protocols for the two-party communication problem and prove that they are everywhere optimal. 2 Denitions And Previous Results The purpose of this section is to provide the basic denitions and introduce the terminology used in the paper. In addition, we also describe some of the existing protocols to provide the reader with an insight into the nature of the problem. 2.1 Basic Denitions Consider a synchronous system composed of two entities, called the sender and the receiver, connected by a direct lin; at each time unit, the sender can either transmit a bit or remain silent; a bit transmitted by the sender at time t will be received and processed by the receiver at time t + 1 (sender's time). A quantum of silence (or, simply, quantum) is the number of cloc tics between the transmission of two successive bit transmissions; the quantum is zero if the bits are sent at two consecive cloc tics. Given a countable (and possibly innite) universe U, the two-party communication problem for U (denoted by TPC(U)), is the problem of the sender communicating witho ambiguity to the receiver arbitrary elements of U using any combination of bit transmissions and silence. Since U is countable, we will assume witho loss of generality that U is a set of consecive integers starting from 0. Given a solion protocol A for TPC(U), let t(x) and b(x) denote the time and the number of bits requires by A to communicate x 2 U; the couple ht(x); b(x)i will be called the time-bits complexity of A for x. Let A(x; b) denote the time required by A to communicate x 2 U using at most b bits; that is, A(x; b) = t(x) if b b(x), A(x; b) = 1 otherwise. We can now dene time-bits worst-case optimality: Denition 1 Time-Bits Worst-Case Optimality. A solion protocol A is timebits worst-case optimal for U if for every solion protocol B and 8b 2 maxfa(x; b) : x 2 Ug maxfb(x; b) : x 2 Ug That is, a worst-case optimal algorithm is worst-case optimal at any point of the time-bit trade-o. Similarly, we can dene time-bits average-case optimality: Denition 2 Time-Bits Average-Case Optimality. A solion algorithm A is average-case optimal for U if for every solion protocol B and 8b 2 AvefA(x; b) : x 2 Ug AvefB(x; b) : x 2 Ug A much stronger notion of optimality is everywhere-optimality. Denition 3 Everywhere Optimality. A solion protocol A is everywhere optimal for U ior every solion protocol B and 8b 2, there exists a permation of the elements of U such that 8x 2 U : A(x; b) B((x); b)

4 In other words, for every choice of the number of transmitted bits, A requires no more time to communicate any element of U (within a relabelling) than any other solion algorithm. Obviously, everywhere optimality implies both worst-case and average-case time-bits optimality. Among all solion protocols, of particular interest are the ones which solve the two-party communication problem for every universe U (and are, thus, independent of U). Denition4 Universal Communicator. A protocol A is a universal communicator if it solves TPC(U) for every U. We consider two dierent versions of TPC(U) depending on whether transmissions are reliable or subject to corruption, and we denote them by R(U) and C(U), respectively. In R(U), transmissions are reliable and, thus, the value of the bits can be used to convey information. In C(U), transmissions are subject to corruption and, thus, the value of the received bits can not be relied upon by the protocol. A solion protocol for C(U) is therefore resilient to corruptions during transmission. Given a positive integer b 2 Z +, let R b (U) and C b (U) denote the restriction of R(U) and C(U), respectively, when exactly b bits are transmitted. In the following, b i will denote a bit and q i will denote a quantum; the subscript will specify the order in the communication. 2.2 Existing Protocols There exists a well nown solion protocol for C 2 (U): to communicate any x 2 U, the sender waits x time units between the transmission of the rst bit b 0 and second bit b 1 ; the receiver decodes the quantum of silence as x. The time-bits complexity is exactly hx; 2i. If the bits can be used to carry information (problem R 2 (U)) the quantum of silence can be reduced by a factor of four: the rst bit, b 0, is used to indicate whether x is odd; the second bit, b 1, is used to indicate whether w = b xc 2 is odd; the quantum waited is z = b w c. To obtain x the receiver simply compes 2 4z + 2b 1 + b 0 (here both bits are treated as integer values). This protocol has timebit complexity h x ; 2i and has been successfully employed to obtain a bit-optimal 4 election algorithm for ring networs of unnown size [3]. For the case b = 3, consider the following protocol for C 3 (U): to communicate x 2 U, the sender transmits b 0, waits q 1 = b p xc transmits b 1, waits q 2 = x? q1, 2 transmits b 2, where the value of the b i 's is arbitrary. To obtain x the receiver simply compes q1+q 2 2. Since x?b p xc 2 2b p xc, this protocol has time-bits complexity at most h3b p xc+3; 3i. In the case of R 3 (U), the rst bit, b 0, is used to indicate whether y = b p xc is odd; the second bit, b 2, is used to indicate whether z = x? b p xc 2 is odd; the third bit, b 3, is used to indicate whether w = b z c is odd. The two quanta 2 waited are q 0 = b y c and q 2 1 = b w c. To obtain x the receiver simply compes 2 (2q 0 + b 0 ) 2 + (4q 1 + 2b 1 + b 2 ) where the bits are treated as integer values. For example, if x = 7387, we have y = 85, z = 162 and w = 81; thus, the two quanta are q 0 = 42 and q 1 = 40 while the bits are b 0 = 1, b 1 = 0 and b 2 = 1. The reader can easily verify that the quantity (2q 0 + b 0 ) 2 + (4q 1 + 2b 1 + b 2 ) comped by the receiver is indeed x. Notice that q 0 = b y c = b bp xc c and, since z 2bp xc, then 2 2

5 = b w c = b b 2 z c c b p xc ; thus, this protocol has time-bits complexity at most hb p xc + 3; 3i (e.g., see [10]). A solion protocol for R b (U) can be obtained by extending the protocol for R 3 (U). For simplicity, let = b?1 be a power of two. Given x 2 U, the encoding E(x) E(x) = b 0 E(X 1 )b q 1 of x is dened recursively as follows: E(X i ) = E(X2i )b i E(X 2i+1 ) if 1 < i < quantum of length X i if i 2 where X i b b x 2 c c, X 2 2i = b p X i c, X 2i+1 = b X i?x 2 2i c, b 2 i = X 2i+1 mod 2 b = b xc mod 2 2. The encoded information to be communicated is then the sequence of bits and quanta dened by the recurrence relation dened above. For example, let x = 927 and = 4; the encoding is h1; 3; 0; 3; 0; 1; 0; 1; 1i where the quanta are denoted with a bar. To obtain x, the receiver will recursively compe X i = X2i 2 + (2X 2i+1 + b i ) until X 1 is determined; then, x = 4X 1 +2b +b 1. The protocol is provably correct. Exactly quanta will be used, and b = +1 bits will be transmitted. It is easy to verify that X 2i+1 b p X i c ; since X 2i = b p X i c by denition, it follows that each quantum is at most ( x) 1. Hence, the time complexity is at most ( x) 1 + b. 4 4 For the general case of b = + 1 bits ( quanta), a solion protocol for C b (U) has been recently proposed by Schmeltz, and it requires time x 1?2 + l.o.t. in the worst case [12]. 3 Lower Bounds In this section we establish lower-bounds on the time-bits trade-o for the two-party communication problem both in the worst and in the average case. The bounds are established when the universe U is nite; as will be shown in section 4, these bounds are tight even when U is countable b innite. We consider both versions of the problem, C and R; that is, when bit transmissions are subject to corruption and when transmission are fault-free, respectively. The bounds apply to any solion protocol, regardless of the schemes employed for encoding, transmitting and decoding. 3.1 Communication in Presence of Corruption If transmissions are subject to corruptions, the value of the received bits cannot be relied upon. Hence, the only meaningful information is whether or not a transmission occurs at a given time; obviously, the time before the rst transmission and after the last transmission cannot be used to convey information. Consider C b (U); i.e., the two-party communication problem for U using exactly b bit transmissions which are subject to corruption. Observe that b time units will be required by any solion algorithm for C b (U) to transmit the b bits; hence, the concern is on the amount of additional time required by the protocol. In the following, unless otherwise specied, \time" refers to \additional" time. Given a nite universe U, let c(u; b) denote the number of time units needed in the worst case to solve C b (U). To derive a bound on c(u; b), we will consider the dual problem of determining the size!(t; b) of the largest set U for which c( U ; b) t; that is, U is the largest set for which the two-party communication problem can

6 always be solved using b transmissions and at most t additional time units. Notice that, with b bit transmissions, it is only possible to distinguish = b? 1 quanta; hence, the dual problem can be rephrased as follows: Determine the largest positive integer n =!(t; b) such that every x 2 Z n = f0; 1; : : :; ng can be communicated using = b? 1 distinguished quanta whose total sum is at most t. This problem has an exact solion which will enable us to establish the desired bounds. Theorem 5.!(t; b) = t + Proof. Let n =!(t; b); by denition, it must be possible to communicate any element in Z n = f0; 1; : : :; ng using = b? 1 distinguished quanta requiring at most time t. In other words,!(t; + 1) is equal to the number of distinct -tuples ht 1 ; t 2 ; : : : ; t i of positive integers such that P 1i t i t. Given a positive integer x, let T [x] denote the number of compositions of x of size ; i.e.,. Since T [x] = x + X T [x] = jfhx 1 ; x 2 ; : : : ; xi : x j = x; x j 2 Z + gj, it follows that!(t; + 1) = X i T [i] = X i i + = t + which proves the theorem. Given two positive integers x and, let f(x; ) be the smallest integer t such that x!(t; + 1). Theorem 6 Worst Case Lower Bound. c(u; b) = f(juj; ); that is, any solion protocol for C +1 (U) requires f(juj; ) time units in the worst case. Proof. from Theorem 5. Theorem 7. Let f(juj; ) = t. For any solion protocol P for C +1 (U), there exists a partition of U into t + 1 disjoint subsets U 0 ; U 1 ; : : : ; U t such that 1. ju i j = i + ; 0 i < t; ju tj t + 2. the time P (x) required by P to communicate x 2 U i is P (x) i. Proof. Since f(juj; ) = t; by Theorem 5, U is the largest set for which the twoparty communication problem can always be solved using b = + 1 transmissions and at most t additional time units. Given a protocol P for C +1 (U), order the elements x 2 U according to the time P (x) required by P to communicate them; let U be the corresponding ordered set. Dene U i to be the subset composed of the elements of U whose raning, with respect to the ordering dened above, is in the range P 0j<i j + ; P 0ji j +. Since f(juj; ) = t, it follows

7 that j Ui j = i + for 0 i < t and j Ut t + j which proves part 1 of the Theorem. We will now show that, for every x 2 Ui ; P (x) i. By contradiction, let this not be the case. Let j t be the smallest index for which there exists an x 2 Ui such that P (x) < j. This implies that there exists a j0 < t such that jfx 2 U : P (x) = j0gj > j0 +. In other words, in protocol P, the number of elements which are uniquely identied using quanta for a total of j0 time is greater than the j0 + number T [j0] = compositions of j0 of size ; a clear contradiction. Hence, for every x 2 Ui ; P (x) i, proving part 2 of the theorem. Theorem8 Average Case Lower Bound. Any solion protocol for C +1 (U) requires tm + P 0i<t i i + time on the average where t = f(juj; ) and m = t(juj? P 0i<t i i + juj ). Proof. From Theorem Reliable Transmission If bit transmissions are error-free, the value of a received bit can be trusted. Consider R b (U); i.e., the two-party communication problem for U using only b bit transmissions in a fault-free system. As in C b (U), the time before the rst transmission and after the last transmission cannot be used to convey information, and the concern is on the amount of time in addition to the one needed for bit transmissions. Given a nite universe U, let r(u; b) denote the number of time units needed in the worst case to solve R b (U). To derive a bound on r(u; b), we will consider the dual problem of determining the size (t; b) of the largest set U for which r(u ; b) t; that is, U is the largest set for which the two-party communication problem can always be solved transmitting b bits and at most t additional time units. Since with b bit transmissions it is only possible to distinguish = b? 1 quanta, the dual problem can be rephrased as follows: Determine the largest positive integer n = (t; b) such that every x 2 Z n = f0; 1; : : :; ng can be communicated using b bits and = b? 1 distinguished quanta whose total sum is at most t. This problem has an exact solion which will enable us to establish the desired bounds. Theorem9. (t; b) = 2 +1 t +

8 Proof. The number of distinct assignment of values to +1 distinguished bits P is The number of distinct -tuples ht 1 ; t 2 ; ; t i of positive integers such that j t j t t + is!(t; b) (from 5). Therefore (t; b) = 2 +1!(t; b) = Given two positive integers x and, let (x; ) denote the smallest t such that x (t; + 1). Theorem 10 Worst Case Lower Bound. Any protocol for R +1 (U) requires (juj; ) time units in the worst case; that is, r(u; b) = (juj; ). Proof. By Theorem 9. Theorem 11. Let (juj; ) = t. For any solion protocol P for R +1 (U), there exists a partition of U into t + 1 disjoint subsets U 0 ; U 1 ; : : : ; U t, such that i + t + 1. ju i j = 2 +1 ; 0 i < t, and ju t j = the time P (x) required by P to communicate x 2 U i is P (x) i. Proof. Similar to Theorem 7. P Theorem 12 Average Case lower Bound. Any protocol for R +1 (U) requires on i i + the average time where t = (juj; ) t + Proof. By Theorem Upper Bounds In this section we present solion algorithms for the two versions of the two-party communication problem, C and R. These algorithms solve the problem even when the the universe U of information is innite. As will be shown later, their complexity matches the global lower-bounds for the problems. 4.1 P 1: An Everywhere Optimal Solion for C Given two -tuples q = hq 1 ; q 2 ; : : : ; q i and q0 = hq0 1 ; q0 2 ; : : : ; q0 i of positive integers, we say that q < q0 if q j = q0 j for 1 j < l, and q l < q0 l for some index l, 1 l +1. For a given, let V t be the ordered set of -tuples q = hq 1 ; q 2 ; : : :; q i where q P i 2 Z + and i q i t; that is V t [i] < V t [i + 1]. Obviously, the size of V t is t + t +. Any two integers t and P i, 1 i, uniquely identies a - tuple V t [i] = hq 1 ; q 2 ; : : : ; q i where i q P i t; conversely, any -tuple hq 1 ; q 2 ; : : : ; q i t + uniquely identies the integers t = i q i and i; 1 i, such that

9 V t [i] = hq 1 ; q 2 ; : : : ; q i. The solion algorithm, P 1, is described below; it comprises of an encoding scheme, a decoding scheme, and a communication protocol. The value X to be communicated will be encoded as a (2 + 1)-tuple hp 0 ; p 1 ; : : :; p 2 i, where the even elements p 0 ; p 2 ; : : : ; p 2 are arbitrary bits and the odd elements p 1 ; p 3 ; : : : ; p 2?1 form the -tuple corresponding to the X-th element of the set V f(x;) ; i.e., hp 1 ; p 3 ; : : :; p 2?1 i = V f(x;) [X]. Encoding Scheme: Given X and, 1. Let t be the smallest integer such that X t + ; i.e., t = f(x; ). 2. Determine V t [X] = hq 1 ; q 2 ; : : : ; q i 3. Set encoding(x) = hp 0 ; p 1 ; : : : ; p 2 i, where p 2i = b 2 f0; 1g and p 2i+1 = q i ; (0 i < ). Once the (2 + 1)-tuple hp 0 ; p 1 ; : : : ; p 2 i corresponding to the encoding of X has been determined, the actual communication can start. The encoded information is communicated as follows: the element p 2i = b 2 f0; 1g is transmitted and the element p 2i+1 = q i is communicated by waiting a quantum of silence of length q i. Once the last bit p 2 has been received, the receiving entity will reconstruct the (2 +1)-tuple hp 0 ; p 1 ; : : : ; p 2 i and apply to it the decoding scheme. Communication Protocol SEND(X): Compe encoding(x) = hp 0 ; p 1 ; : : : ; p 2 i for 0 i 2 if even(i) then transmit p i else wait p i time units endif endfor RECEIVE(Z): i := 0 receive(b) p 0 = b Repeat wait q until receive(b) p 2i+1 = q i := i + 1 p 2i = b Until i = Z = hp 0 ; p 1 ; : : : ; p 2 i Compe decoding(z) To decode hp 0 ; p 1 ; : : :; p 2 i, the receiver will extract the (+1)-tuple hq 1 ; q 2 ; : : : ; q i formed by the odd elements q i = p 2i+1 ; (0 i < ) and compe t = P i q i ; at this

10 point X, the communicated value, is the unique integer such that 1 X t + and V t [X] = hq 1 ; q 2 ; : : : ; q i. Decoding Scheme: Given Z = hp 0 ; p 1 ; : : : ; p 2 i and, 1. Let Y = hq 1 ; q 2 ; : : : ; q i where q i = p 2i+1 ; (0 i < ); let t = P i q i. 2. Find X such that V t [X] = Y. 3. Set decoding(z) = X. For a xed, let P 1(X) denote the amount of time required by algorithm P 1 to communicate integer X using bit transmissions. Recall (from Section 3) that f(x; ) is the smallest integer t such that x!(t; + 1). Lemma 13. For a xed, P 1(X) = f(x; ) for every integer X. Proof. By construction. Theorem 14. P 1 is worst-case optimal for every Z n = f0; 1; : : :; ng. Proof. By Lemma 13 and Theorem 6. Theorem 15. For a xed, P 1 is everywhere optimal for every Z n = f0; 1; : : :; ng. Proof. Given Z n, let t = h(n; ) be the smallest integer such that n!(t; + 1). Assume for simplicity that n = t +. Let S i = fx 2 Z n : P 1(x) = ig. By Lemma i + 13, for every x 2 Z n ; P 1(x) = h(x; ) t; hence, js i j = ; 0 i t. Recall that, by Theorem 6, for any solion algorithm A, there exists a partition i + of Z n into t + 1 disjoint subsets A 0 ; A 1 ; : : : ; A t such that ja i j = and A(x) i for every x 2 A i. Therefore, there exists a permation of Z n such that P 1(x) A((x)) for all x 2 Z n, proving the theorem. 4.2 P 2: An Everywhere Optimal Protocol for R t + Given, let W t be an ordered set of size 2 +1 where each element W t [i] is a distinct (2 + 1)-tuple p = hp 0 ; p 1 ; : : : ; p 2 i where p 2i 2 f0; 1g; p 2i+1 2 Z +, and Pi p 2i+1 t. The set is lexicographically ordered; that is, W t [i] < W t [i + 1] where hp 0 ; p 1 ; : : : ; p 2 i < hp0 0 ; p0 1 ; : : : ; p0 2 i if p j = p0 j for 1 j l and p l+1 < p0 l+1. A solion algorithm, P 2, is described below; it comprises of an encoding scheme, a decoding scheme, and uses the same communication protocol as P 1. Encoding Scheme: Given X and, 1. Let t be the smallest integer such that X 2 +1 t + ; i.e.,t = (X; ).

11 2. Find W t [X] = hp 0 ; p 1 ; : : : ; p 2 i 3. Set encoding(x) = W t [X]. Decoding Scheme: Given Z = hp 0 ; p 1 ; : : : ; p 2 i and, 1. Let Y = hq 1 ; q 2 ; : : : ; q i where q i = p 2i+1 (0 i < ); let t = P i q i. 2. Find X such that W t [X] = Y. 3. Set decoding(z) = X. Communication Protocol: same as in P 1 For a xed, let P 2(X) denote the amount of time required by algorithm P 1 to communicate integer X using bit transmissions. Recall (from Section 3) that (X; ) is the smallest integer t such that x (t; + 1). Lemma 16. For a xed, P 2(X) = (X; ) for every integer X. Corollary17. For a xed, P 2 is worst-case optimal for every Z n = f0; 1; : : :; ng Theorem18. For a xed, P 2 is everywhere optimal for every Z n = f0; 1; : : :; ng. The proofs of the above lemma and theorem are analogous to those presented in Section 4.1 and therefore are omitted here for brevity. 5 Conclusion We have presented an algorithm for two-party communication in synchronous networs which has bit-time complexity matching the ones expressed by the lower bounds. This algorithm (and the similar one derived when tranmissions are subject to corruption) has been shown to be everywhere optimal: at any point of the trade-o and for any universe of data to be communicated, no other solion has better complexity to communicate any element of that universe (within a xed relabelling). An extension of this algorithm, to extend its adequacy to a larger class of fault tolerant networs (e.g., noise and bit ommision), is desirable. Furthermore, an implication of this result is that ecient asynchronous to synchronous transforms may now be designed. This will lead to improvements on the complexity of existing synchronous solions and the establishment of bounds for problems for which no previous synchronous solion has existed. 6 Acnowledgements This wor has been supported in part by the Natural Sciences and Engineering Research Council of Canada. Ms. O'Reilly also acnowledges the support of Bell Northern Research Ltd.

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