Consensus and Mutual Exclusion in a Multiple Access Channel

Transcription

1 Consensus and Mutual Exclusion in a Multiple Access Channel Jurek Czyzowicz 1,, Leszek Gasieniec 2,, Dariusz R. Kowalski 2,, and Andrzej Pelc 1, 1 Département d informatique, Université duquébec en Outaouais, Gatineau, Québec J8X 3X7, Canada 2 Department of Computer Science, University of Liverpool, Liverpool L69 3BX, UK Abstract. We consider deterministic feasibility and time complexity of two fundamental tasks in distributed computing: consensus and mutual exclusion. Processes have different labels and communicate through a multiple access channel. The adversary wakes up some processes in possibly different rounds. In any round every awake process either listens or transmits. The message of a process i is heard by all other awake processes, if i is the only process to transmit in a given round. If more than one process transmits simultaneously, there is a collision and no message is heard. We consider three characteristics that may or may not exist in the channel: collision detection (listening processes can distinguish collision from silence), the availablity of a global clock showing the round number, and the knowledge of the number n of all processes. If none of the above three characteristics is available in the channel, we prove that consensus and mutual exclusion are infeasible; if at least one of them is available, both tasks are feasible and we study their time complexity. Collision detection is shown to cause an exponential gap in complexity: if it is available, both tasks can be performed in time logarithmic in n, which is optimal, and without collision detection both tasks require linear time. We then investigate both consensus and mutual exclusion in the absence of collision detection, but under alternative presence of the two other features. With global clock, we give an algorithm whose time complexity linearly depends on n and on the wake-up time, and an algorithm whose complexity does not depend on the wake-up time and differs from the linear lower bound only by a factor O(log 2 n). If n is known, we also show an algorithm whose complexity differs from the linear lower bound only by a factor O(log 2 n). Keywords: consensus, mutual exclusion, multiple access channel, collision detection. Partially supported by NSERC discovery grant. Partially funded by the Royal Society International Joint Project, IJP /R1. This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/G023018/1]. Partially supported by NSERC discovery grant and by the Research Chair in Distributed Computing at the Université du Québec en Outaouais. I. Keidar (Ed.): DISC 2009, LNCS 5805, pp , c Springer-Verlag Berlin Heidelberg 2009

2 1 Introduction Consensus and Mutual Exclusion in a Multiple Access Channel 513 The background and the problem. We consider deterministic feasibility and time complexity of two fundamental tasks in distributed computing: consensus and mutual exclusion. Processes have different integer labels from 1 to n, andeachof them knows its own label. In the sequel we identify processes with their labels. They communicate through a multiple access channel (MAC) which is a well known and thoroughly studied communication medium. In order to capture the notion of collisions, that are the main difficulty of communicating over a MAC, time is considered as slotted into rounds, similarly as in the literature on radio communication, cf., e.g., [1,3,7,12,14,16]. The adversary wakes up some processes in possibly different rounds. In each round every awake process either listens or transmits. Transmitting processes do not hear anything. The message of process i is heard by all other awake listening processes, if i is the only process to transmit in a given round. If more than one process transmits simultaneously, there is a collision and no message is heard. We consider three features that may or may not exist in the MAC: collision detection (CD), the availablity of a global clock showing the round number (GC), and the knowledge of the number n of all processes (KN). Collision detection is the capacity of listening processes to distiguish collision (when more than one process transmits in a given round) from silence (when no process transmits). Silence is in fact the background noise occurring in the MAC when no process transmits, and a collision slightly increases the level of this noise. Hence detecting this difference requires a more sensitive receiving device. Global clock permits awake processes to see the same round number. In the absence of it, individual clocks of awake processes tick at the same rate indicating rounds, but each clock starts at 0 when the process is woken up by the adversary. Finally, knowledge of the number n of all processes may or may not be available, but we never assume the knowledge of the number of processes ever woken up or the knowledge of their waking rounds. We focus on the problem of whether consensus and mutual exclusion are deterministically feasible, and if so, what is their deterministic time complexity, depending on which of the features CD, GC, KN are available in the MAC over which processes communicate. It should be stressed that the fact that the adversary wakes up an arbitrary unknown subset of processes and that these processes are woken up in arbitrary rounds, significantly increases the difficulty of the problem. The tasks and the power of the adversary. Since communication between processes is done over a MAC, we define a transmission schedule that is an infinite binary sequence π determining the communication actions of a process. For any non-negative integer i, π(i) = 1 means that the process transmits in round i after its wakeup, and π(i) = 0 means that the process listens in round i after its wakeup. Round 0 is the round in which the process is woken up. We now describe the two tasks under consideration, in the context of the communication over a MAC, and define the power of the adversary for each task. The adversary wakes up some of the processes in some, possibly different

3 514 J. Czyzowicz et al. rounds. Every process starts executing its protocol in its wake-up round. Actions of an awake process in a given round depend on its label, on its input value, on the previously heard messages (or noise, if collision detection is available), on the number of rounds since its wake-up, on the global round number, if there is a global clock, and on the number n of processes, if this number is known. Consensus Let {1,..., α}, forα 2, be the range of possible input values of processes. The adversary chooses a function v : {1,..., n} {1,..., α} which assigns an input value to every process. A consensus algorithm is distributedly run by all awake processes. Each action of a process can be either listening or transmitting some message and/or deciding a value from {1,..., α}. The following three conditions must be satisfied: Termination: each awake process eventually decides Validity: a decision is on one of the input values of awake processes Agreement: all awake processes decide the same value The time complexity of a consensus algorithm is the maximum number of rounds, over all awake processes, between the wake-up time and the decision time. Mutual Exclusion A mutual exclusion algorithm is distributedly run by all awake processes. Each process executes a protocol partitioned into the following sections: Entry (trying): the part of the protocol executed in preparation for entering the critical section Critical: the part of the protocol to be protected from concurrent execution Exit: the part of the protocol executed on leaving the critical section Remainder: the rest of the protocol Each process executes these sections cyclically in the order: remainder, entry, critical, and exit. In the traditional mutual exclusion problem, as defined in [2,31] in the context of the shared-memory model, the adversary controls the sections remainder and critical (in particular it controls their duration in each cycle, only subject to the obvious assumption that this duration in each cycle is finite), while an algorithm provides a protocol for the entry and exit sections of each process. In the model of communication over a MAC, each action of a process can be either listening or transmitting some message, as well as changing sections of the protocol: entering the critical section, if the process is currently in the entry section, and entering the remainder section, if the process is currently in the exit section. We assume that changing sections occurs momentarily between consecutive rounds, i.e., in each round a process is exactly in one section of the protocol. The following assumption is specific for mutual exclusion with communication over a MAC, replacing the traditional communication by shared variables: the MAC is not used by the adversary in the sections remainder and critical (otherwise the adversary would have an unlimited power of creating collisions in

4 Consensus and Mutual Exclusion in a Multiple Access Channel 515 the MAC, thus preventing communication if collision detection is not available). Instead, the protocol can use the MAC while a process is in the critical section by sending the message occupied. Any mutual exclusion algorithm has to satisfy the following two properties: Exclusion: in every round of any execution, at most one process is in the critical section. No deadlock: in every round r of any execution, if there is a process in the entry section at round r then some process will enter the critical section eventually after round r. Note that we do not require the no lockout property, stronger than no deadlock: in every round r of any execution, if there is a process in the entry section at round r, thenthis process will enter the critical section eventually after round r. The time complexity of a mutual exclusion algorithm, called the makespan, is the maximum number of rounds in any interval when there is some process in the entry section and there is no process in the critical section. Our results. If none of the three characteristics (collision detection, global clock, knowledge of the number n of all processes) is available in the channel, we prove that consensus and mutual exclusion are infeasible. If at least one of them is available, both tasks are feasible and we study their time complexity. Collision detection is shown to cause an exponential gap in complexity. If it is available, both tasks can be performed in logarithmic time, which is optimal. More precisely, consensus with values in the range {1,..., α} can be performed in time O(min(log n, log α)) and mutual exclusion in time O(log n), and both these orders of magnitude are tight. If collision detection is not available, we show that both consensus and mutual exclusion require time Ω(n). We then investigate both consensus and mutual exclusion in the absence of collision detection, but under alternative presence of the two other characteristics. With global clock, we give an algorithm for consensus and mutual exclusion whose time complexity linearly depends on n and on the wake-up time, and an algorithm whose complexity does not depend on the wake-up time and differs from the linear lower bound only by a factor O(log 2 n). If n is known, we also show an algorithm whose complexity differs from the linear lower bound only by a factor O(log 2 n). The paper is organized as follows. In Section 2 we show infeasibility of the considered tasks in the weakest model. In Section 3 we present a consensus algorithm with collision detection, show that it is optimal and prove a linear lower bound on the complexity of our tasks without collision detection. In Sections 4 and 5 we present consensus algorithms assuming the availability only of a global clock, resp. only of the knowledge of the number of processes, in the absence of the two other characteristics. Section 6 is devoted to presenting a scheme that transforms a consensus algorithm to a mutual exclusion algorithm. Thus we obtain corollaries for the complexity of mutual exclusion from the previous results. Section 7 contains conclusions and open problems. Due to lack of space, proofs of the results are omitted. They will appear in the full version of the paper.

5 516 J. Czyzowicz et al. Related work. The multiple access channel (MAC) is a well-studied communication medium. Research concerning the MAC can be divided into two parts: one assuming that some communicating processes are woken up by the adversary in possibly different rounds (this is the model used in this paper), and the other assuming that all processes are awake from the beginning. In the first model two tasks were mainly studied in the literature: the wake-up problem in which one process has to transmit alone in some round, thus waking up all other processes [9,17,24,26] and the continuous broadcast, in which processes start to broadcast possibly multiple messages in different rounds, the broadcast being successful when the process transmits alone in some round. The latter problem is subject to dynamic packet arrival, either modeled by an adversarial queuing framework (see, e.g., [4,11]), or by queue-free framework (see, e.g., [28]), or by stochastic distributions (see, e.g., [21]). One of the fundamental problems investigated assuming that all processes communicating over a MAC are awake from the beginning is the leader election problem. For deterministic leader election without collision detection and with a known number n of processes, matching bounds on time Ω(n log n) and O(n log n) follow from [14], with the upper bound being non-constructive. A constructive upper bound O(n polylog (n)) follows from [24]. For the time of deterministic algorithms with collision detection, matching bounds are also known: Ω(log n) follows from [22], and O(log n) follows from [5,23,35]. For the expected time of randomized algorithms without collision detection, the same matching bounds are known: Ω(log n) follows from [30] and O(log n) from [3]. Randomized leader election with collision detection can be done faster: matching bounds Ω(log log n) (forfairprotocols)ando(log log n) were proved in [36]. Further references can be found in [25,32]. Communication with possible failures (e.g., crash or Byzantine) has been investigated in the above model, e.g., in [15,18,20]. It should be noted that the MAC is equivalent to a special case of the popular radio network model, namely when the underlying graph is complete. General radio networks were intensely studied in the context of the broadcasting problem, starting with the seminal paper [10]. Most researchers worked in the model without collision detection: deterministic broadcasting in this model was studied, e.g., in [8,14,16] and randomized broadcasting in [1,16,29,30]. Fewer papers were devoted to broadcasting with collision detection, cf. [8,19]. Communication with possible failures (such as crash, Byzantine, probabilistic) has also been studied in multi-hop radio network models, see, e.g., [15,27,34] Consensus and mutual exclusion are two classic problems in distributed computing, mostly studied assuming that processes communicate by shared variables or through message passing networks [2,31]. In [6], feasibility and complexity of consensus in a multiple access channel with synchronized starting points and crash failures were studied in the context of different collision detectors the tools introduced in that work by the analogy to classic failure detectors. To the best of our knowledge, consensus and mutual exclusion were never studied in the context of a multiple access channel with non-synchronized wake-up times.

6 Consensus and Mutual Exclusion in a Multiple Access Channel Infeasibility in the Weakest Model We start with a negative result that neither consensus nor mutual exclusion are feasible in the weakest of all models considered in this paper, the model in which none of the assumptions CD, GC, KN holds. Theorem 1. Consensus and mutual exclusion are infeasible without collision detection, without a global clock and with an unknown number n of processes. The above impossibility result should be contrasted with the positive solution of the wake-up problem in the same model. Indeed, it was shown in [17] that wake-up in a MAC can be achieved in polynomial time without a global clock, collision detection or the knowledge of the number of processes. Hence wake-up in the weakest of our models is strictly easier than consensus. This difference can be also viewed as follows. Consider the special case of the consensus problem in which the input value of each process is equal to its label. This is called label consensus and it is clearly equivalent to leader election. While wake-up in the weakest model is feasible and all other awake processes can elect as the leader the first process to speak alone, this process itself cannot become aware that it is the leader. 3 Impact of Collision Detection Collision detection permits a listening process to distinguish between silence and collision noise, which occurs when at least two messages are sent. Hence any listening process hears either the silence, or collision noise, or the content of the message transmitted. We say that a listening process hears signal μ, if one or more processes transmit in the given round. A round is called silent for a listening process i, ifi hears silence in this round, and it is noisy for i, ifi hears signal μ. 3.1 Availability of Collision Detection We first prove a lower bound on the time of consensus and on the makespan of mutual exclusion even in the strongest of our models. Theorem 2. Any consensus algorithm, even with collision detection, global clock and known number n of processes, requires time Ω(min(log n, log α)). Anymutual exclusion algorithm, even with collision detection, global clock and known number n of processes, has makespan Ω(log n). We now present a consensus algorithm matching the above lower bound Ω(min(log n, log α)), if collision detection is available, even without the global clock or the knowledge of n. We first design a consensus algorithm working in time O(log α). Let B v(i) = b 1 b 2...b l denote the string of bits in the binary representation of the input value v(i) of process i, written in reverse order, i.e. starting from the least significant

7 518 J. Czyzowicz et al. bit. Only meaningful bits of this representation are considered. Hence the last bit b l of B v(i) always equals 1 and B v(i) = log v(i). Consider the first round r when some process is woken up. In the case when there is only one process wokenupinroundr the consensus value is the input value of this process. Otherwise, the consensus value v(i ) is the input value of process i woken up in round r, such that for any process i woken up in round r, B v(i) B v(i ), where denotes the natural lexicographic order of bit-strings (we also write B v(i) B v(j),ifb v(i) B v(j) and B v(i) B v(j) ). The transmission schedule π i,v(i) begins with 0001 followed by the infinite sequence of repetitions of bit-string b 1 1b b l 1 1b l 001, that we will call the value transmission pattern. The only message ever transmitted by a process is the contact message consisting of bit 1. During the algorithm execution each process i may be either active, when it follows its transmission schedule π i,v(i), or passive, when it listens forever. At least one process remains active forever. This process will follow its value transmission pattern periodically and its input value becomes the consensus value. Passive processes decode this value from a sequence of silent and noisy rounds. Algorithm ConsensusCD1 (integer v) 1 active: Listen for three rounds; if silence is always heard then transmit in round 4 else goto passive; 2 Follow value transmission pattern for v; if μ is heard at some listening round then goto passive; 3 Decide on v; 4 forever follow value transmission pattern for v periodically. 5 passive: Wait until silence is heard for two consecutive rounds ρ, ρ +1; 6 Start counting rounds r 1,r 2,...,withr 1 = ρ +2; Listen until silence is heard in the first odd-numbered round r 2k+1 ; For each even-numbered round r 2j,for0< 2j <2k +1, store bit 0 if silence was heard and bit 1 otherwise; Interpret the reverse of the stored bit-string as a binary representation of value x; Decide on x. Each process i runs Algorithm ConsensusCD1 with its input value v(i) asthe parameter v. Lemma 1. Algorithm ConsensusCD1 reaches consensus with collision detection in time O(log α). We now present our main consensus algorithm with collision detection, working in time O(min(log n, log α)). It essentially consists in running in parallel ConsensusCD1(v(i)) and ConsensusCD1(i). This may be done by reserving oddnumbered rounds for one algorithm and even-numbered rounds for the other one.

8 Consensus and Mutual Exclusion in a Multiple Access Channel 519 Since global clock is not available, some synchronization is necessary in order for each process to recognize at some point the round parity. Similarly as before we consider the string B i =b 1b 2...b m - which is the inverse of the binary representation of label i. Wecallthelabel transmission pattern of process i the sequence of bits b 1 1b b m 1 1b m001. Unless specified otherwise, the message transmitted in each transmission round is the contact message. Algorithm ConsensusCD2 (integer i, integer v) 1 active: Listen for six rounds; if silenceisalwaysheardthen transmit in round 7 else goto passive; 2 Starting from the 7th round after wake-up consider the numbering of rounds r 1,r 2,... Follow b 1 1b b l 1 1b l the value transmission pattern of v in the odd-numbered rounds Follow b 11b 21...b m 11b m001 - the label transmission pattern for i in the even-numbered rounds if μ is heard at some listening round in the above patterns then goto passive; 3 if the end of the value transmission pattern is achieved then Decide on v forever follow the pattern b 1 1b b l 1 1b l periodically; 4 if the end of the label transmission pattern is achieved then Decide on v forever follow the pattern periodically, where v is transmitted in each transmission round. 5 passive: Listen until silence is heard for five consecutive rounds; Listen until silence is heard again for five consecutive rounds ρ +1,...,ρ+ 5 and signal μ is heard in round ρ +6; 6 if silence is heard in round ρ +7then Decide on x (inthiscaseamessagewasheardinroundρ +6); 7 else Start counting rounds r 1,r 2,...,withr 1 = ρ +7 Listen until silence is heard in the first odd-numbered round r 2k+1 For each even-numbered round r 2j, for 2j <2k +1, store bit 0 if silence was heard and bit 1 otherwise Interpret the reverse of the stored bit-string as a binary representation of integer x Decide on x. Each process i runs Algorithm ConsensusCD2 with its label i and its input value v(i) as the parameters of the algorithm. Theorem 3. Algorithm ConsensusCD2 reaches consensus with collision detection in time O(min(log n, log α)).

9 520 J. Czyzowicz et al. 3.2 Absence of Collision Detection The following lower bound on the complexity of consensus and mutual exclusion in the absence of collision detection shows an exponential gap caused by the lack of this characteristic of the MAC. Theorem 4. Any consensus (resp. mutual exclusion) algorithm in the model without collision detection, even with global clock and known number n of processes, requires time (resp. makespan) at least n/2. 4 Global Clock In this section we assume that a global clock is available to all awake processes, but we do not assume collision detection or the knowledge of the number n of processes. We present algorithms for consensus based on the following scheme. Algorithmic scheme GlobalClock The set of natural numbers (corresponding to the round numbers given by the global clock) is partitioned into an infinite family A 1,A 2,... of pairwise disjoint infinite sets. The set A i is the set of rounds reserved for process i, i.e., no other process transmits in these rounds. Process i that was woken up in (global) round t listens in rounds t, t +1,..., t,wheret + 1 is the first integer larger than t belonging to A i. If silence was heard in all these rounds, then process i decides on its value v(i) and in all rounds larger than t transmits value v(i). If some value w was heard in one of the rounds t, t +1,..., t, then process i decides on value w and remains silent forever. Lemma 2. The algorithmic scheme GlobalClock reaches consensus with global clock, for any family A 1,A 2,... of pairwise disjoint infinite sets of natural numbers. Depending on the particular family of sets A 1,A 2,..., the algorithmic scheme GlobalClock can produce various consensus algorithms. We show two such algorithms with incomparable complexities. The first one, Algorithm GlobalClock1, has complexity O(n + t), where t is the largest wake-up round of any process. It matches the lower bound Ω(n) from Theorem 4, for small values of t. Algorithm GlobalClock1 It is enough to define the family of sets A 1,A 2,... First partition the set N of natural numbers into consecutive segments C 0,C 1,... called blocks. BlockC i has length 2 i.forafixedi, letr 1,..., r 2 i be elements of C i. We define the function f i : C i N by f i (r j )=j. This gives the function f : N N defined as f i on block C i. The function f corresponds to the sequence (1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8,...). Now the set A i is defined as f 1 ({i}).

10 Consensus and Mutual Exclusion in a Multiple Access Channel 521 Lemma 3. Algorithm GlobalClock1 reaches consensus with global clock in time O(n + t), wheret is the largest wake-up round of any process. While the complexity of Algorithm GlobalClock1 matches the lower bound Ω(n) for small values of t, it may be arbitrarily large (as compared to the number of processes), if processes are woken up late in global history. Hence it is natural to seek a consensus algorithm whose complexity does not depend on the times of wake-up of processes by the adversary. Our next algorithm, also based on the algorithmic scheme GlobalClock, satisfies this requirement. Algorithm GlobalClock2 Again it suffices to define the family of sets A 1,A 2,... First partition the set N of natural numbers into consecutive blocks B 0,B 1,... BlockB i has length 8 2 i i 2 and it is formed of positions enumerated from 1 to 8 2 i i 2. We subdivide B i into i + 1 pairwise disjoint lists S i (1),..., S i (i) andr i. Consider a list S i (l), where 2 k 1 l 2 k 1forsome1 k log i. The consecutive elements of such S i (l)arepositions2 k x+2 k 1 in B i for all integers x satisfying x l (mod 2 k 1 ). Note that consecutive elements in such S i (l) are at distance 2 k 2 k 1 =2 2k 1 in B i. We now show that for any 1 l 1 <l 2 i the lists S i (l 1 )ands i (l 2 )donot intersect. First, if 2 k 1 l 1 <l 2 2 k 1forsome1 k log i, then values x 1 l 1 (mod 2 k 1 )andx 2 l 2 (mod 2 k 1 ) can only be equal when l 1 l 2 > 2 k 1, which is impossible. Assume now that 2 k1 1 l 1 2 k1 1and 2 k2 1 l 2 2 k2 1, for some 1 k 1 <k 2 log i. Note that in this case integers in S i (l 1 ) are certain multiples of 2 k1 1 but not multiples of 2 k1. Since integers in S i (l 2 ) are certain multiples of 2 k2 1 which are multiples of 2 k1 when k 2 >k 1, the intersection of S i (l 1 )ands i (l 2 ) is also empty. Finally, the list R i contains all remaining positions in B i, i.e., R i = B i \ (S i (1) S i (l)). For each i and for l = 1,..., i, we now define functions g l : S i (l) {2 l 1,..., 2 l 1} as follows. The function g l assigns elements from the set {2 l 1,..., 2 l 1} to consecutive elements from the list S i (l) in a round-robin fashion, i.e., forming the sequence of values (2 l 1,..., 2 l 1, 2 l 1,..., 2 l 1, 2 l 1,..., 2 l 1,...). We additionally define the function h i as the function constantly equal 1 on the domain R i. Since the lists S i (1),..., S i (i),r i form a disjoint partition of the block B i,the above functions define, for each i, a function φ i : B i {1,..., 2 i 1}. Thisin turn gives the function φ : N N defined as φ i on block B i.nowtheseta i is defined as φ 1 ({i}). Lemma 4. Algorithm GlobalClock2 reaches consensus with global clock in time O(n log 2 n). By interleaving algorithms GlobalClock1 and GlobalClock2 on even and odd rounds, respectively, listening in the first two rounds after wake-up and keeping silence on both threads as soon as a process hears some value in one of the threads, we get the following result.

11 522 J. Czyzowicz et al. Theorem 5. There exists an algorithm reaching consensus with global clock in time O(min(n+t, n log 2 n)), wheret is the largest wake-up round of any process. 5 Known Number of Processes In this section, we assume that the number n of all processes is known to every process, but we do not assume global clock or collision detection. Our consensus algorithm uses the notion of a fixed transmission schedule, introduced in [17]. A fixed transmission schedule of process i is a finite binary sequence π i depending only on the label i of the process and on the parameter n. The interpretation of π i is the following. If process i is woken up in round t, then i transmits in round t + u 1ifπ i (u) =1andi listens in round t + u 1 if π i (u) = 0. It was proved in [17] that, for every n, there exists a set of fixed transmission schedules {π i : i =1,..., n} of length s O(n log 2 n), such that regardless of the (non-empty) set of processes woken up by the adversary and regardless of the wake-up rounds of these processes, there exists a process and a round t + s,wheret is the earliest wake-up round of any process and s s, in which this process transmits alone. Thus the easier problem of wake-up can be solved in time O(n log 2 n). Our aim is to give a consensus algorithm with the same time complexity. Algorithm KnownNumber Starting in its wake-up round t, process i listens for s rounds. If it hears some input value in one of these rounds, it decides on this value and remains silent forever. If it hears silence in all these s rounds, it starts transmitting its input value according to the schedule π i. If it hears some input value in one of the following s rounds, it decides on this value and remains silent forever. If it does not hear any message in all the 2s rounds, it decides on its own input value and transmits it according to the schedule π i repeated periodically forever. Theorem 6. Algorithm KnownNumber reaches consensus in time O(n log 2 n), for any known number n of processes. 6 From Consensus to Mutual Exclusion In this section we propose a generic mutual exclusion algorithm, called MacMEx, which uses a consensus algorithm as a subroutine and solves the problem of mutual exclusion, preserving the complexity of the consensus solution. Using the consensus algorithms developped in the previous sections, we obtain mutual exclusion algorithms in the respective models. Consider our consensus algorithms in the case when the input value of every process is equal to the label of the process (label consensus). All our algorithms have the following two properties.

12 Consensus and Mutual Exclusion in a Multiple Access Channel 523 P1. Every process listens in the round in which it is woken up. P2. If the decision is on value i, no process other than i transmits in the round when i makes its decision. Hence all our consensus algorithms, considered in the case of label consensus, can be transformed by having the winning process i transmit a special message my label i won in the round r when process i decides on its value and in all subsequent rounds. Indeed, all processes awake in round r will hear this message in round r, decide on i and remain silent forever, and all processes woken up in some round r >rwill hear this message in round r, decide on i and remain silent forever. The complexities of the transformed algorithms remain the same. Hence we may assume that the label consensus subroutine used by Algorithm MacMEx has the following two properties. P 1. Every process listens in the round in which it is woken up. P 2. Starting from the round in which process i decides on its own value, process i transmits the message my label i won forever, and all other processes listen forever. Algorithm MacMEx Entry section. Process i executes a consensus subroutine satifying properties P 1 and P 2, with its label as the input value, until one of the following events occurs: process i decides on its own label; in this case process i enters the critical section process i hears either the message occupied or the message my label j won ; in this case process i stops the execution of the consensus subroutine and listens on the MAC in the next round process i hears the message released ; in this case process i starts a new execution of the consensus subroutine with its label as the input value. Critical section. Process i transmits the message occupied on the MAC in each round when it is in the critical section. The rest of the behavior of the process in this section is controlled by the adversary. Exit section. Process i transmits the message released on the MAC and leaves the section. The proof of correctness of Algorithm MacMEx is based on the following invariant. Lemma 5. Exactly one of the following properties holds in any round r: Q1 the message occupied is heard in round r and its sender is the only process in the critical section in this round; additionally, no process is in the exit section and no process executes the consensus subroutine in round r; or

13 524 J. Czyzowicz et al. Q2 the message released is heard in round r and its sender is the only process in the exit section in this round; additionally, no process is in the critical section and no process executes the consensus subroutine in round r; or Q3 there is at least one process executing the consensus subroutine in round r; additionally, all such processes are exactly those in the entry section and no process is in the critical or exit sections in round r; or Q4 all processes are in the remainder section in round r. Using Lemma 5 we can prove the following theorem. Theorem 7. Algorithm MacMEx with a consensus subroutine satisfying properties P 1 and P 2, is a mutual exclusion algorithm with no deadlock. Moreover, the makespan of the MacMEx algorithm is the same as the time complexity of the consensus subroutine. Combining Theorem 7 with Theorems 3, 5 and 6 for the label-consensus version of the problem, we derive the following conclusions for mutual exclusion. Theorem 8. Algorithm MacMEx is a mutual exclusion algorithm with no deadlock in a multiple access channel having at least one of the following characteristics: collision detection, global clock, or the knowledge of the number n of processes. The makespan of algorithm MacMEx is: (i) O(log n), if collision detection is assumed; (ii) O(min(n + t, n log 2 n)), ifglobalclockisassumedandt is the largest round of the wake-up of any process; (iii) O(n log 2 n), ifknowledgeofn is assumed. Moreover, the first bound is tight, while the two others differ from the lower bound Ω(n) without collision detection at most by a factor of O(log 2 n). 7 Conclusion and Open Problems We provided almost optimal algorithms for consensus and mutual exclusion with processes communicating over a MAC. It would be interesting to close the O(log 2 n) factor gaps in the models without collision detection but with global clock or with a known number of processes. (In the model with collision detection our algorithms have optimal complexity.) It also remains open how randomization influences the complexity of these problems with MAC communication. Another set of open problems concerns energy consumption. We may assume that processes can not only transmit or listen, but can switch off. Then a natural measure of efficiency is the maximum or average number of rounds in which a process is active (listens or transmits). Finally, in the case of mutual exclusion, we guaranteed no deadlock, but not the stronger no lockout property. It remains open if mutual exclusion with no lockout is feasible in all models except the weakest one, and if so, what is its complexity.

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