Faster Deterministic Communication in Radio Networks

LIPICS - Leibniz International Proceedings in Informatics, Aug 2016

In this paper we improve the deterministic complexity of two fundamental communication primitives in the classical model of ad-hoc radio networks with unknown topology: broadcasting and wake-up. We consider an unknown radio network, in which all nodes have no prior knowledge about network topology, and know only the size of the network n, the maximum in-degree of any node Delta, and the eccentricity of the network D. For such networks, we first give an algorithm for wake-up, in both directed and undirected networks, based on the existence of small universal synchronizers. This algorithm runs in O((min{n,D*Delta}*log(n)*log(Delta))/(log(log(Delta)))) time, improving over the previous best O(n*log^2(n))-time result across all ranges of parameters, but particularly when maximum in-degree is small. Next, we introduce a new combinatorial framework of block synchronizers and prove the existence of such objects of low size. Using this framework, we design a new deterministic algorithm for the fundamental problem of broadcasting, running in O(n*log(D)*log(log((D*Delta)/n))) time. This is the fastest known algorithm for this problems, improving upon the O(n*log(n)*log*log(n))-time algorithm of De Marco (2010) and the O(n*log^2(D))-time algorithm due to Czumaj and Rytter (2003), the previous fastest results for directed networks, and is the first to come within a log-logarithmic factor of the Omega(n*log(D)) lower bound due to Clementi et al. (2003). Our results have also direct implications on the fastest deterministic leader election and clock synchronization algorithms in both directed and undirected radio networks, tasks which are commonly used as building blocks for more complex procedures.

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Faster Deterministic Communication in Radio Networks

I C A L P Faster Deterministic Communication in Radio Networks?? Artur Czumaj 0 Peter Davies 0 0 Department of Computer Science and Centre for Discrete Mathematics and its Applications, University of Warwick, Warwick, United Kingdom Department of Computer Science and Centre for Discrete Mathematics and its Applications, University of Warwick , Warwick , United Kingdom In this paper we improve the deterministic complexity of two fundamental communication primitives in the classical model of ad-hoc radio networks with unknown topology: broadcasting and wake-up. We consider an unknown radio network, in which all nodes have no prior knowledge about network topology, and know only the size of the network n, the maximum in-degree of any node ?, and the eccentricity of the network D. For such networks, we first give an algorithm for wake-up, in both directed and undirected networks, based on the existence of small universal synchronizers. This algorithm runs in O( min{n,Dlog?l}oglo?g n log ? ) time, improving over the previous best O(n log2 n)-time result across all ranges of parameters, but particularly when maximum in-degree is small. Next, we introduce a new combinatorial framework of block synchronizers and prove the existence of such objects of low size. Using this framework, we design a new deterministic algorithm for the fundamental problem of broadcasting, running in O(n log D log log Dn? ) time. This is the fastest known algorithm for this problems, improving upon the O(n log n log log n)time algorithm of De Marco (2010) and the O(n log2 D)-time algorithm due to Czumaj and Rytter (2003), the previous fastest results for directed networks, and is the first to come within a log-logarithmic factor of the ?(n log D) lower bound due to Clementi et al. (2003). Our results have also direct implications on the fastest deterministic leader election and clock synchronization algorithms in both directed and undirected radio networks, tasks which are commonly used as building blocks for more complex procedures. and phrases Radio networks; Communication networks; Broadcasting; Wake-Up; Deterministic algorithms - 1.1 Introduction Model of communication networks We consider the classical model of directed ad-hoc radio networks with unknown structure. A radio network is modeled by a directed network N = (V, E), where the set of nodes 139:2 corresponds to the set of transmitter-receiver stations. The nodes of the network are assigned different identifiers (IDs), and throughout this paper we assume that all IDs are distinct numbers in {1, . . . , |V |}. A directed edge (v, u) ? E means that node v can send a message directly to node u. To make propagation of information feasible, we assume that every node in V is reachable in N from any other. In accordance with the standard model of unknown (ad-hoc) radio networks (for more elaborate discussion about the model, see, e.g., [1, 2, 5, 9, 10, 12, 17, 19, 22]), we make the assumption that a node does not have any prior knowledge about the topology of the network, its in-degree and out-degree, or the set of its neighbors. We assume that the only knowledge of each node is the size of the network n, the maximum in-degree of any node ?, and the eccentricity of the network D, which is the maximum distance from the source node to any node in N. For a discussion of these assumptions, see the full version of this paper. Nodes operate in discrete, synchronous time steps, but we do not need to assume knowledge of a global clock. When we refer to the ?running time? of an algorithm, we mean the number of time steps which elapse before completion (i.e., we are not concerned with the number of calculations nodes perform within time steps). In each time step a node can either transmit a message to all of its out-neighbors at once or can remain silent and listen to the messages from its in-neighbors. We do not make any restriction on the size of messages. The distinguishing feature of radio networks is the interfering behavior of transmissions. In the most standard radio networks model, the model without collision detection (see, e.g., [1, 2, 10, 22]), which is studied in this paper, if a node v listens in a given round and precisely one of its in-neighbors transmits, then v receives the message. In all other cases v receives nothing; in particular, the lack of collision detection means that v is unable to distinguish between zero of its in-neighbors transmitting and more than one. The model without collision detection describes the most restrictive interfering behavior of transmissions; also considered in the literature is a less restrictive variant, the model with collision detection, where a node listening in a given round can distinguish between zero of its in-neighbors transmitting and more than one (see, e.g., [12, 22]). 1.2 Communications primitives: broadcasting and wake-up In this paper we consider two fundamental communications primitives, namely broadcasting and wake-up, and consider deterministic protocols for each of these tasks. 1.2.1 Broadcasting Broadcasting is one of the most fundamental problems in communication networks and has been extensively studied for many decades (see, e.g., [22] and the references therein). The premise of the broadcasting task is that one particular node, called the source, has a message which must become known to all other nodes. We assume that all other nodes start in a dormant state and do not participate until they are ?woken up? by receiving the source message (this is referred to in some works as the ?no spontaneous transmissions? rule). As a result, while the model does not assume knowledge of a global clock, we can make this assumption in practice, since the current time can be appended to the source message as it propagates, and therefore will be known be all active nodes. This is important since it allows us to synchronize node behavior into fixed-length blocks. 1.2.2 Wake-up The wake-up problem (see, e.g., [15]) is a related fundamental communication problem that arises in networks where there is no designated ?source? node, and no synchronized time-step at which all nodes begin communicating. The goal is for all nodes to become ?active? by receiving some transmission. Rather than a single source node which begins active, we instead assume that some subset of nodes spontaneously become active at arbitrary time-steps. The task can be seen as a variant of broadcast, with possibly multiple sources, and without the ability to assume a global clock. This last point is important, and results in wake-up protocols being slower than those for broadcast, since nodes cannot co-ordinate their behavior. 1.3 Related work As a fundamental communications primitive, the task of broadcasting has been extensively studied for various network models for many decades. For the model studied in this paper, directed radio networks with unknown structure and without collision detection, the first sub-quadratic deterministic broadcasting algorithm was proposed by Chlebus et al. [5], who gave an O(n11/6)-time broadcasting algorithm. After several small improvements (cf. [6, 21]), Chrobak et al. [9] designed an almost optimal algorithm that completes the task in O(n log2 n) time, the first to be only a poly-logarithmic factor away from linear dependency. Kowalski and Pelc [17] improved this bound to obtain an algorithm of complexity O(n log n log D) and Czumaj and Rytter [11] gave a broadcasting algorithm running in time O(n log2 D). Finally, De Marco [20] designed an algorithm that completes broadcasting in O(n log n log log n) time steps. Thus, in summary, the state of the art result for deterministic broadcasting in directed radio networks with unknown structure (without collision detection) is the complexity of O(n min{log n log log n, log2 D}) [11, 20]. The best known lower bound is ?(n log D) due to Clementi et al. [10]. Broadcasting has been also studied in various related models, including undirected networks, randomized broadcasting protocols, models with collision detection, and models in which the entire network structure is known. For example, if the underlying network is undirected, then Kowalski [16] gave a deterministic broadcasting algorithm running in time O(n log D). If spontaneous transmissions are allowed and a global clock available, then deterministic broadcast can be performed in O(n) time in undirected networks [5]. Randomized broadcasting has been also extensively studied, and in a seminal paper, BarYehuda et al. [2] designed an almost optimal broadcasting algorithm achieving the running time of O((D + log n) ? log n). This bound has been later improved by Czumaj and Rytter [11], and independently Kowalski and Pelc [18], who gave optimal randomized broadcasting algorithms that complete the task in O(D log Dn +log2 n) time with high probability, matching a known lower bound from [19]. In the model with collision detection, an O(D + log6 n)-time randomized algorithm due to Ghaffari et al. [12] is the first to exploit collisions and surpass the algorithms (and lower bound) for broadcasting without collision detection. For more details, see e.g., [22] and the references therein. The wake-up problem (see, e.g., [15]) is a related communication problem that arises in networks where there is no designated ?source? node, and no synchronized time-step at which all nodes begin communicating. Before any more complex communication can take place, we must first require all nodes to be ?active,? i.e., aware that they should be communicating. This is the goal of wake-up, and it is a fundamental starting point for most other tasks in this setting, for example leader election and clock synchronization [8]. The first sub-quadratic deterministic wake-up protocol was given in by Chrobak et al. [8], who introduced the concept of radio synchronizers to abstract the essence of the problem. They give an O(n5/3 log n)-time protocol. Since then, there have been two improvements in running time, both making use of the radio synchronizer machinery: firstly to O(n3/2 log n) [4], and then to O(n log2 n) [3]. Unlike for the problem of broadcast, the fastest known protocol for directed networks is also the fastest for undirected networks. A recent survey of the current state of research on the wake-up problem is given in [15]. 1.4 Our results In this paper we present a new construction of universal radio synchronizers and introduce and analyze a new concept of block synchronizers to improve the deterministic complexity of two fundamental communication primitives in the model of ad-hoc radio networks with unknown topology: broadcasting and wake-up. By applying the analysis of block synchronizers, we present a new deterministic broadcasting algorithm (Algorithm 1) in directed ad-hoc radio networks with unknown structure, without collision detection, that for any directed network N with n nodes, with eccentricity D, and maximum in-degree ?, completes broadcasting in O(n log D log log Dn? ) time-steps. This result almost matches a lower bound of ?(n log D) due to Clementi et al. [10], and improves upon the previous fastest algorithms due to De Marco [20] and due to Czumaj and Rytter [11], which require O(n log n log log n) and O(n log2 D) time-steps, respectively. Our result reveals that a non-trivial speed-up can be achieved for a broad spectrum of network parameters. Since ? ? n, our algorithm has the complexity at most O(n log D log log D). Therefore, in particular, it significantly improves the complexity of broadcasting for shallow networks, where D nO(1). Furthermore, the dependency on ? reduces the complexity even further for networks where the product D? is near linear in n including sparse networks which can appear in many natural scenarios. Our broadcasting result has also direct implications on the fastest deterministic leader election algorithm in directed and undirected radio networks. It is known that leader election can be completed in O(log n) times broadcasting time (see, e.g., [9, 13]) (assuming the broadcast algorithm extends to multiple sources, which is the case here as long as we have a global clock), and so our result improves the bound to achieve a deterministic leader election algorithm running in O(n log n log D log log Dn? ) time. For undirected networks the best result is O(n log3/2 n?log log n) time [7] (we note that the O(n log D) broadcast protocol of [16] cannot be used at a log n slowdown for leader election, since it relies on token traversal and does not extend to multiple sources). Our result therefore favorably compares for shallow networks (for small D) even in undirected networks. We also present a deterministic algorithm (Algorithm 2) for the related task of wake-up. We show the existence of universal radio synchronizers of delay g(k) = O( n lloogg lnoglokg k ), and demonstrate that this yields a wake-up protocol taking time O( min{n,Dlog?l}oglo?g n log ? ). This improves over the previous best result, the O(n log2 n)-time protocol of [3]; the improvement is largest when ? is small, but even when it is polynomial in n, our algorithm is a log log n-factor faster. Our improved result for wake-up has direct applications to communication algorithms in networks that do not have access to a global clock, where wake-up is an essential starting point for most more complex communication tasks. For example, wake-up is used as a subroutine in the fastest known protocols for fundamental tasks of leader election and clock synchronization (cf. [8]). These are two fundamental tasks in networks without global clocks, since they allow initially unsynchronized networks to be brought to a state in which synchronization can be assumed, and results from the better-understood setting with a global clock can then be applied. Our wake-up protocol yields O( min{n,D?} log2 n log ? )-time leader log log ? election and clock synchronization algorithms, which are the fastest known in both directed and undirected networks. 1.5 Previous approaches Almost all deterministic broadcasting protocols with sub-quadratic complexity (that is, since [5]) have made use of the concept of selective families (or some similar variant thereof, such as selectors). These are families of sets for which one can guarantee that any subset of [n] := {1, 2, . . . , n} below a certain size has an intersection of size exactly 1 with some member of the family. They are useful in the context of radio networks because if the members of the family are interpreted to be the set of nodes which are allowed to transmit in a particular time-step, then after going through each member, any node with an active in-neighbor and an in-neighborhood smaller than the size threshold will be informed. Most of the recent improvements in broadcasting time have been due to a combination of proving smaller selective families exist, and finding more efficient ways to apply them (i.e., choosing which size of family to apply at which time). One of the drawbacks of selective-family based algorithms is that applying them requires coordination between nodes. For the problem of broadcast, this means that some time may be wasted waiting for the current selective family to finish, and also that nodes cannot alter their behavior based on the time since they were informed, which might be desirable. For the problem of wake-up, this is even more of a difficulty; since we cannot assume a global clock, we cannot synchronize node behavior and hence cannot use selective families at all. To tackle this issue, Chrobak et al. [8] introduced the concept of radio synchronizers. These are a development of selective families which allow nodes to begin their behavior at different times. A further extension to universal synchronizers in [4] allowed effectiveness across all in-neighborhood sizes. However, the adaptability to different node start times comes at a cost of increased size, meaning that synchronizer-based wake-up algorithms were slightly slower than selective family-based broadcasting algorithms . The proofs of existence for selective families and synchronizers follow similar lines: a probabilistic candidate object is generated by deciding on each element independently at random with certain carefully chosen probabilities, and then it is proven that the candidate satisfies the desired properties with positive probability, and so such an object must exist. The proofs are all non-constructive (and therefore all resulting algorithms non-explicit; cf. Indyk [14] for an explicit construction of selective families). Returning to the problem of broadcastin g, a breakthrough came in 2010 with a paper by De Marco [20] which took a new approach. Rather than having all nodes synchronize their behavior, it instead had them begin their own unique pattern, starting immediately upon being informed. These behavior patterns were collated into a transmission matrix. The existence of a transition matrix with appropriate selective properties was then proven probabilistically. The ability for a node to transmit with a frequency which decayed over time allowed De Marco?s method to inform nodes with a very large in-neighborhood faster, and this in turn reduced total broadcasting time from O(n log2 D) [11] to O(n log n log log n). A downside of this new approach is that having nodes begin immediately, rather than wait until the beginning of the next selector, gives rise to a far greater number of possible starting-time scenarios that have be accounted for during the probabilistic proof. This caused the logarithmic factor in running time to be log n rather than log D. Further, the method was comparatively slow to inform nodes of low in-degree, compared to a selective family of appropriate size. These are the difficulties that our approach will have to overcome. 1.6 Overview of our approach Our wake-up result follows a similar line to the previous works; we prove the existence of smaller universal synchronizers than previously known, using the probabilistic method. Our improvement stems from new techniques in analysis rather than method, which allow us to gain a log-logarithmic factor by choosing what we believe are the optimal probabilities by which to construct a randomized candidate. Our broadcasting result takes a new direction, some elements of which are new and some of which can be seen as a compromise between selective family-type objects and the transmission schedules of De Marco [20]. We first note that nodes of small in-degree can be quickly dealt with by repeatedly applying (n, Dn )-selective families ?in the background? of the algorithm. This allows us to tailor the more novel part of the approach to nodes of large in-degree. We have nodes perform their own behavior patterns with decaying transmission frequency over time, but they are semi-synchronized to ?blocks? of length roughly Dn , in order to cut down the number of circumstances we must consider. This idea is formalized by the concept of block synchronizers, combinatorial objects which can be seen as an extension of the radio synchronizers used for wake-up. An important new concept used in our analysis of block synchronizers (and also in our proof of small universal synchronizers) is that of cores. Cores reduce a set of nodes and starting times to a (usually smaller) set of nodes which are active during a critical period. In this way we can combine many different circumstances into a single case, and demonstrate that for our purposes they all behave in the same way. The most technically involved part of both of the proofs is the selection of the probabilities with which we generate a randomized candidate object (universal synchronizer or block synchronizer). Intuitively, when thinking about radio networks, a node in our network is aiming to inform its out-neighbors, and it should assume that as time goes on, only those with large in-neighborhoods will remain uninformed (because these nodes are harder to inform quickly). Therefore a node should transmit with ever-decreasing frequency, roughly inversely proportional to how large it estimates remaining uninformed neighbors? in-neighborhoods must be. However, these in-neighborhoods cannot be estimated exactly, and so we must tweak the probabilities slightly to cover the possible range. In block synchronizers we do this using phases of length O(log log Dn? ) during which nodes halve their transmission probability every step, but since behavior must be synchronized to achieve this we cannot do the same for radio synchronizers. Instead, we allow our estimate to be further from the true value, and require more time-steps around the same value to compensate. As with previous results based on selective families, synchronizers, or similar combinatorial structures, the proofs of the structures we give are non-constructive, and therefore the algorithms are non-explicit. 2 Combinatorial tools Our communications protocols rely upon the existence of objects with certain combinatorial properties, and we will separate these more abstract results from their applications to radio networks. In this section, we will define the combinatorial objects we will need to make use of, and prove their existence. Next, in Sections 3?4, we will demonstrate in details how these combinatorial objects can be used to obtain fast algorithms for broadcasting and wake-up. 2.1 Selective families We begin with a brief discussion about selective families, whose importance in the context of broadcasting was first observed by Chlebus et al. [5]. A selective family is a family of subsets of [n] := {1, . . . , n} such that every subset of [n] below a certain size has intersection of size exactly 1 with a member of the family. For the sake of consistency with successive definitions, rather than defining the family of subsets Si, we will instead use the equivalent definition of a set of binary sequences Sv (that is, Siv = 1 if and only if v ? Si). For some m ? N, let each v ? [n] have its own length-m binary sequence Sv = S0vS1vS2v . . . Smv?1. I Definition 1. S = {Sv}v?[n] is an (n, k)-selective family if for any X ? [n] with 1 ? |X| ? k, there exists j ? [0, m) such that Pv?X Sjv = 1. (We say that such j hits X.) 2.1.1 Application to radio networks During the course of radio network protocols we can ?apply? a selective family S on an n-node network by having each node v transmit in time-step j if and only if v has a message it wishes to transmit and Sjv = 1 (see, e.g., [5, 10]). Some previous protocols involved nodes starting to transmit immediately if they were informed of a message during the application of a selective family (or a variant called a selector designed for such a purpose), but here we will require nodes to wait until the current selective family is completed before they start participating. That is, nodes only attempt to transmit their message if they knew it at the beginning of the current application. The result of applying an (n, k)-selective family is that any node u which has between 1 and k active neighbors before the application will be informed of a message upon its conclusion. This is because there must be some time-step j which hits the set of u?s active neighbors, and therefore exactly one transmits in that time-step, so u receives a message. This method of selective family application in radio networks was first used in [5]. 2.1.2 Existence of small selective families The following standard lemma (see, e.g., [10]) posits the existence of (n, k)-selective families of size O(k log nk ). This has been shown to be asymptotically optimal [10]. I Lemma 2 (Small selective families). For some constant c and for any 1 ? k ? n there exists an (n, k)-selective family of size at most m = ck log n . k 2.2 Radio synchronizers Radio synchronizers are an extension of selective families designed to account for nodes in a radio network starting their behavior patterns at different times, and without access to a global clock. They were first introduced in [8] and used in an algorithm for performing wake-up, and this is also the purpose for which we will apply them. To define radio synchronizers, we first define the concept of activation schedule. I Definition 3. An n-activation schedule is a function ? : [n] ? N. We will extend the definition to subsets X ? [n] by setting ?(X) = minv?X ?(v). As for selective families, let each v ? [n] have its own length-m binary sequence Sv = S0vS1vS2v . . . Smv?1. We then define radio synchronizers as follows: I Definition 4. S = {Sv}v?[n] is an (n, k, m)-radio synchronizer if for any activation schedule ? and for any X ? [n] with 1 ? |X| ? k, there exists j ? [?(X), ?(X) + m) such that Pv?X Sjv??(v) = 1. One can see that the definition is very similar to that of selective families (Definition 1), except that now each v?s sequence is offset by the value ?(v). To keep track of this shift in expressions such as the sum in the definition, we will call such values j columns. As with selective families, we say that any column j satisfying the condition in Definition 4 hits X. In [4], this concept was extended to universal radio synchronizers which cover the whole range of k from 1 to n. Let g : [n] ? N be a non-decreasing function, which we will call the delay function. I Definition 5. S = {Sv}v?[n] is an (n, g)-universal radio synchronizer if for any activation schedule ?, and for any X ? [n], there exists column j ? [?(X), ?(X) + g(|X|)) such that Pv?X Sjv??(v) = 1. 2.2.1 Application of universal radio synchronizers to radio networks One can apply universal radio synchronizers to the problem of wake-up in radio networks by having ?(v) represent the time-step in which node v becomes active during the course of a protocol (either spontaneously or by receiving a transmission). Subsequently, v interprets Sv as the pattern in which it should transmit, starting immediately from time-step ?(v). That is, in each time-step j after activation, v checks the next value in Sv (i.e., Sv j??(v)), transmits if it is 1 and stays silent otherwise. Then, the selective property specified by the definition guarantees that any node u with an in-neighborhood of size q hears a transmission within at most g(q) steps of its first in-neighbor becoming active. 2.2.2 Existence of small universal radio synchronizers We will prove existence of small universal synchronizers. I Theorem 6. For some constant c and for any n ? N there exists an (n, g)-universal radio synchronizer with g(q) = cqlloogglqogloqg n . The proof of Theorem 6 is deferred to the full version, but here we present its main ideas. Our universal synchronizer must hit any set X within g(X) columns of the set?s start column ?(X). Our first step is to narrow down the amount of sets and activation schedules we must consider by defining the core of a set. The core removes elements of X which only become active after we must already have hit X (i.e. after column ?(X) + g(X)), and then shifts all values of ? so that ?(X) = 0. Multiple different sets X under multiple different activation schedules can have the same core, and therefore behave the same way during the critical period. So, by considering cores instead of the original sets and activation schedules, we can reduce the amount of cases we must account for. We then probabilistically generate a candidate universal synchronizer, and prove that it does indeed satisfy the required property with positive probability. The crucial part of this method is our choice of probabilities with which we include each element in the candidate. We expect to be able to hit a core of size q within cq log q log n columns. Therefore, for log log q a particular element v and for columns greater than ?(v) + cq log q log n , all remaining cores log log q containing v that we have left to hit should be larger than q. If we want to hit a set of size roughly q by randomly selecting each element of [n], then we optimize our probability of doing so by selecting each element with probability roughly 1q . This translates to setting Svcq log q log n = 1 with probability roughly 1q . Substituting j = cqlloogglqogloqg n , this works out to log log q give the probability of Sjv = 1 to be roughly ljolgojglloogg nj . However, as it transpires, this is not quite the optimal choice. To see what we should choose instead, we must first examine how we analyze our candidate. We define the load fC (j) of a column j with respect to a core C; this is the expected number of elements of the core selected in that column. We show that the probability of hitting the core on j is at least fC (j)2?fC(j). Ideally, we want fC (j) to be constant for cores of roughly the size we should currently be hitting, and then this hitting probability is also constant. However, if we use the probabilities P[Sjv = 1] = ljolgojglloogg nj , we in fact find that fC (j) can be between ?(1) and O(log j), depending upon the times at which its elements became active. This log j-sized gap cannot be closed, since we can easily find cores which do indeed have loads at either end of the range. What we can do, however, is tweak the probabilities to shift this range down and maximize hitting probability over it. Specifically, if we shift probabilities down by a factor of lolgolgojg j , i.e. to P[Sjv = 1] ? logj n , we shift the range of possible loads to be between ?( lolgolgojg j ) and O(log log j), ensuring that the hitting probability is ?( lolgolgojg j ). This is as close to constant as we can get; the gap is log q what necessitates the log log q factor in our delay function g. With this bound on the probability of hitting a particular core at a particular column, we can use independence to obtain an lower bound for hitting a core over the whole range of valid columns, and then use a union bound to show that we can hit all cores with positive probability. 2.3 Block synchronizers Next, we introduce block synchronizers, which are a new type of combinatorial object designed for use in a fast broadcasting algorithm. They can be seen as an extension of both radio synchronizers and the transmission matrix formulation of De Marco [20]. Let each v ? [n] have its own length-m binary sequence Sv = S0vS1vS2v . . . Smv?1. Define a function ?B : N ? N, for some fixed B, which rounds its input up to the next multiple of B, that is, ?B(x) = min{pB : p ? Bx , p ? N}; we will call s(v) := ?B(?(v)) the start column of v. We extend s to subsets of [n] in the obvious way, s(X) = ?B(?(X)). I Definition 7. S = {Sv}v?[n] is an (n, ?, r, B)-block synchronizer if for any activation schedule ? and any set X ? [n] with |X| ? ?, there exists column j ? [s(X), s(X) + Bd |Xr | e) such that Pv?X Sjv?s(v) = 1. Block synchronizers differ from radio synchronizers in two ways: Firstly, on top of the offsetting effect of the activation schedule, there is also the function ?B that effectively ?snaps? behavior patterns to blocks of size B, hence the name block synchronizer. Secondly, the size of the range in which we must hit X is linearly dependent on |X| rather than being fixed. The parameter r is the increment by which each block increases the size of sets we can hit. 2.3.1 Application of block synchronizers to radio networks The idea of our broadcasting algorithm will be that any node v waits until the start of the first block after its activation time ?(v), and then begins its transmission pattern Sv. The definition of block synchronizer aims to model this scenario. The hitting condition ensures that any node with an in-neighborhood of size q ? ? will be informed within Bd qr e time-steps of the start of the block in which its first in-neighbor begins transmitting. 2.3.2 Existence of small block synchronizers We prove the following theorem. I Theorem 8. For some constant c and for any n, D, ? ? N with D, ? ? n < D?, there exists an (n, ?, Dn , c Dn log D log log Dn? )-block synchronizer. The proof of this theorem is deferred to the full version, but we outline its main ideas. We begin in a similar way to that of Theorem 6, in that we define the concept of a core to narrow down the range of possibilities we must consider. The difference is that now elements v have their behavior patterns snapped to blocks, so the core need only store the number of the first block v becomes active, rather than the exact column. This significantly reduces the number of possible cores, and is essential for obtaining the log D factor in size, rather than log n. As before, cores also shift the activation schedule to begin at 0. We wish to proceed in a similar manner as in the proof of small universal synchronizers, generating a candidate probabilistically and then proving that it satisfies the desired property with positive probability. While we could do this directly for block synchronizers using the same method, we would obtain a size of ?(n log D log log ?). Our improved bound of O(n log D log log Dn? ) results from noting that we can afford to hit cores smaller than Dn using selective families, leaving a narrower range of cores to be hit by our randomized candidate object. Therefore, we define an upper block synchronizer which need only hit cores above a certain size threshold (in our case, Dn ), and this is the object we prove the existence of with the probabilistic method. Once again, the most technical aspect of the proof lies in the probabilities with which we choose elements in our candidate object. Again, we define the load of a column fC(j) to be the expected number of elements in a core C which are selected in column j, and show that the probability of hitting the core at j is at least fC(j)2?fC(j). By choosing our probabilities to be roughly inversely proportional to the size of cores we expect to remain un-hit, we can guarantee that on a constant fraction of columns we have fC(j) = ?(1) and fC(j) = O(log Dnj ) = O(log Dn? ). However, the way in which we deal with this range differs from the case of universal synchronizers. Since our elements? behavior is snapped to blocks, we can synchronize changes in probability of selection between all elements. Specifically, we have O(log log Dn? )-length phases in which the selection probability of all elements halves every consecutive column (in addition to the slight decrease that occurs between columns naturally). This means that fC(j) also roughly halves every column within phases, and so there is at least one column within each phase in which it is within some constant range (we use the interval ( 13 , 1)). Then, the probability of hitting the core at that column is also at least some constant. With this bound, we can again use independence of columns and a union bound to show that our candidate upper block synchronizer hits all cores larger than Dn with positive probability. Then, it remains only to insert a (n, Dn )-selective family at the start of every block in order to hit smaller cores, and we meet the conditions of a block synchronizer. 3 Algorithms for broadcasting and wake-up In this section we present our algorithms for broadcasting and wake-up in radio networks. 3.1 Broadcasting We will assume that D? > n, otherwise an earlier O(D? log ?n )-time protocol from [10] can be used to achieve O(D? log ?n ) = O(n log D) time. Let S be an (n, ?, Dn , B)-block synchronizer, with B = c Dn log D log log Dn? , and recall that ?B(x) = min{pB : p ? Bx , p ? N}. We will say that the source node becomes active at time-step 0, and any other node v becomes active in a time-step i if it received its first transmission at time-step i ? 1. Our broadcasting algorithm is the following (Algorithm 1): Algorithm 1 Broadcast at a node v Let i be the time-step in which v becomes active for j from 0 to DB ? 1, in time-step ?B(i) + j do v transmits source message iff Sjv = 1 end for Let S be an (n, g)-universal radio synchronizer with g(q) = cqlloogglqogloqg n . We will say that a node v becomes active in a time-step i if it either spontaneous wakes up at i, or received its first transmission at time-step i ? 1. Our wake-up algorithm is the following (Algorithm 2): Algorithm 2 Wake-up at a node v Let i be the time-step in which v becomes active for j from 0 to g(n) ? 1, in time-step i + j do v transmits source message iff Sjv = 1 end for 4 Analysis of broadcasting and wake-up algorithms In this section we show that our algorithms for broadcasting and wake-up have the claimed running times. We begin with the analysis of the broadcasting algorithm. I Theorem 9. Algorithm 1 performs broadcast in O(n log D log log Dn? ) time-steps. To begin the analysis, fix some arbitrary node v and let P be a shortest path from the source (or first informed node) x to v. Number the nodes in this path consecutively, e.g., P0 = x and Pdist(x,v) = v. Classify all other nodes into layers dependent upon the furthest node along the path P to which they are an in-neighbor (some nodes may not be an in-neighbor to any node in P ; these can be discounted from the analysis). That is, layer L` = {u ? V : maxu in-neighbour to Pi i = `} for ` ? dist(x, v). We separately define layer Ldist(x,v)+1 to be {v}. At any time step, we call a layer leading if it is the foremost layer containing an active node, and our goal is to progress through the network until the final layer is leading, i.e., v is active. The use of layers allows us to restrict to the set of nodes of our main interest: if we focus on the path node whose in-neighborhood contains the leading layer, we cannot have interference from earlier layers since they contain no in-neighbors of this path node, and we cannot have interference from later layers since they are not yet active. I Lemma 10. Let h : [?] ? N be a non-decreasing function, and define T (n, D, ?, h) to be the supremum of the function PiD=1 h(qi), where integers 1 ? qi ? ? satisfy the additional constraint PD i=1 qi ? n. If a broadcast or wake-up protocol ensures that any layer (under any choice of v) of size q remains leading for no more than h(q) time-steps, then all nodes become active within T (n, D, ?, h) time-steps. Proof. Let qi = |Li|. Layer Ldist(x,v)+1 must be leading (and thus node v active) once no other layers are leading, and so this occurs Pid=is1t(x,v) h(qi) time-steps after layer L1 becomes leading. Since Pid=is1t(x,v) h(qi) ? PiD=1 h(qi) and PiD=1 qi ? n, this is no more than T (n, D, ?, h) time-steps. Since v was chosen arbitrarily, all nodes must be active within T (n, D, ?, h) time-steps of x becoming active. J We make use of Lemma 10 to give bounds on the running times of our algorithms: I Lemma 11. Algorithm 1 ensures that any layer of size q remains leading for fewer than Bd q +rr e time-steps. Proof. For all nodes w, let ?(w) be the time-step that w becomes active during the course of the algorithm. By definition of a block selector, for any layer Li of size qi there is a time-step j < s(Li) + Bd qri e in which exactly one element of Li transmits. Then, either path node Pi hears the transmission (and so layer Li is no longer leading in time-step j + 1), or Pi has active in-neighbors not in Li, in which case these must be in a later layer so Li is not qi qi+r leading. Thus, Li can remain leading for no more than s(Li) + Bd r e ? ?(Li) < Bd r e time-steps. J With these tools, we are now ready to complete the proof of Theorem 9. Proof. Proof of Theorem 9 By Lemma 10, Algorithm 1 ensures that all nodes are active (and have therefore heard the source message) within T (n, D, ?, h) time-steps, where h(q) = Bd q +rr e. We will use an upper bound T (n, D, ?, h0), where h0(q) = B q+r2r . Since h0 is linear and increasing, PiD=1 h0(qi) subject to PiD=1 qi ? n is maximized whenever PiD=1 qi = n, for example at qi = Dn for all i ? [D]. So, the algorithm completes broadcast within i=1 D D X h0( Dn ) = X B i=1 Dn + 2r r time-steps. In a similar way, we can analyze Algorithm 2. = 3BD = 3c0n log D log log D? n J I Theorem 12. Algorithm 2 performs wake-up in O( min(n,D?) log n log ? ) time-steps. log log ? 5 Conclusions The task of broadcasting in radio networks is a longstanding, fundamental problem in communication networks. Our result for deterministic broadcasting in directed networks combines elements from several of the previous works with some new techniques, and, in doing so, makes a significant improvement to the fastest known running time. Our algorithm for wake-up also improves over the previous best running time, and relies on a proof of smaller universal synchronizers, a combinatorial object first defined in [4]. Neither of these algorithms are known to be optimal. The best known lower bound for both broadcasting and wake-up is ?(min(n log D, D? log ?n )) [10]; our broadcasting algorithm therefore comes within a log-logarithmic factor, but our wake-up algorithm remains a logarithmic factor away. As well as the obvious problems of closing these gaps, there are several other open questions regarding deterministic broadcasting in radio networks. Firstly, the lower bound for undirected networks is weaker than that for directed networks [18], and so one avenue of research would be to find an O(n log D) lower bound in undirected networks, matching the broadcasting time of [16]. Secondly, the algorithms given here, along with almost all previous work, are non-explicit, and therefore it remains an important challenge to develop explicit algorithms that can come close to the existential upper bound. The best constructive algorithm known to date is by Indyk [14], but it is a long way from optimality. 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Artur Czumaj, Peter Davies. Faster Deterministic Communication in Radio Networks, LIPICS - Leibniz International Proceedings in Informatics, 2016, 139:1-139:14, DOI: 10.4230/LIPIcs.ICALP.2016.139