Radio networks are a long-studied model for distributed system of devices which communicate wirelessly. When these devices are mobile or have limited capabilities, the system is often best modeled by the ad-hoc variant, in which the devices do not know the structure of the network. A large body of work has been devoted to designing algorithms for the ad-hoc model, particularly for fundamental communications tasks such as broadcasting. Most of these algorithms, however, assume that devices have some network knowledge (usually bounds on the number of nodes in the network @math , and the diameter @math ), which may not always be realistic in systems with weak devices or gradual deployment. Very little is known about what can be done when this information is not available. This is the issue we address in this work, by presenting the first broadcasting algorithms for networks in which nodes have no prior knowledge whatsoever. We demonstrate that lack of parameter knowledge can be overcome at only a small increase in running time. Specifically, we show that in networks without collision detection, broadcast can be achieved in @math time, almost reaching the @math lower bound. We also give an algorithm for directed networks with collision detection, which requires only @math time.
A deterministic algorithm for broadcasting in radio networks without parameter knowledge is given in @cite_1 , with a running time of @math , where @math is the range of unique IDs with which nodes are equipped.
@cite_1
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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 @math , the maximum in-degree of any node @math , and the eccentricity of the network @math . For such networks, we first give an algorithm for wake-up, based on the existence of small universal synchronizers. This algorithm runs in @math time, the fastest known in both directed and undirected networks, improving over the previous best @math -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 ... Ad-hoc radio networks and multiple access channels are classical and well-studied models of distributed systems, with a large body of literature on deterministic algorithms for fundamental communications primitives such as broadcasting and wake-up. However, almost all of these algorithms assume knowledge of the number of participating nodes and the range of possible IDs, and often make the further assumption that the latter is linear in the former. These are very strong assumptions for models which were designed to capture networks of weak devices organized in an ad-hoc manner. It was believed that without this knowledge, deterministic algorithms must necessarily be much less efficient. In this paper we address this fundamental question and show that this is not the case. We present deterministic algorithms for blind networks (in which nodes know only their own IDs), which match or nearly match the running times of the fastest algorithms which assume network knowledge (and even surpass the previous fastest algorithms which assume parameter knowledge but not small labels). We consider deterministic broadcasting in geometric radio networks (GRN) whose nodes know only a limited part of the network. Nodes of a GRN are situated in the plane and each of them is equipped with a transmitter of some range r. A signal from this node can reach all nodes at distance at most r from it but if a node u is situated within the range of two nodes transmitting simultaneously, then a collision occurs at u and u cannot get any message. Each node knows the part of the network within knowledge radius s from it, i.e., it knows the positions, labels and ranges of all nodes at distance at most s. The aim of this paper is to study the impact of knowledge radius s on the time of deterministic broadcasting in a GRN with n nodes and eccentricity D of the source. Our results show sharp contrasts between the efficiency of broadcasting in geometric radio networks as compared to broadcasting in arbitrary graphs. They also show quantitatively the impact of various types of knowledge available to nodes on broadcasting time in GRN. Efficiency of broadcasting is influenced by knowledge radius, knowledge of individual positions when knowledge radius is zero, and awareness of collisions. We consider the problem of broadcasting in an unknown radio network modeled as a directed graph @math , where @math . In unknown networks, every node knows only its own label, while it is unaware of any other parameter of the network, including its neighborhood and even any upper bound on the number of nodes. We show an @math upper bound on the time complexity of deterministic broadcasting. This is an improvement over the currently best upper bound @math for arbitrary networks, thus shrinking exponentially the existing gap between the lower bound @math and the upper bound from @math to @math . We consider broadcasting in radio networks, modeled as undirected graphs, whose nodes know only their own label and labels of their neighbors. In every step every node acts either as a transmitter or as a receiver. A node acting as a transmitter sends a message which can potentially reach all of its neighbors. A node acting as a receiver in a given step gets a message if and only if exactly one of its neighbors transmits in this step. Bar-Yehuda, Goldreich, and Itai [J. Comput. System Sci., 45 (1992), pp. 104--126] considered broadcasting in this model. They claimed a linear lower bound on the time of deterministic broadcasting in such radio networks of diameter 3. This claim turns out to be incorrect in this model (although it is valid in a more pessimistic model [R. Bar-Yehuda, O. Goldreich, and A. Itai, Errata Regarding "On the time complexity of broadcast in radio networks: An exponential gap between determinism and randomization," http: www.wisdom.weizmann.ac.il mathusers oded p .html, 2002]). We construct an algorithm that broadcasts in logarithmic time on all graphs from the Bar-Yehuda, Goldreich, and Itai paper (BGI). Moreover, we show how to broadcast in sublinear time on all n-node graphs of diameter @math . On the other hand, we construct a class of graphs of diameter 4, such that every broadcasting algorithm requires time @math on these graphs. In view of the randomized algorithm from BGI, running in expected time @math on all @math -node graphs of diameter D (cf. also a recent @math -time algorithm from [D. Kowalski and A. Pelc, Proceedings of the 22nd Annual ACM Symposium on Principles of Distributed Computing, Boston, 2003, pp. 73--82; A. Czumaj and W. Rytter, Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, Cambridge, MA, 2003, pp. 492--501]), our lower bound gives the first correct proof of an exponential gap between determinism and randomization in the time of radio broadcasting, under the considered model of radio communication. We consider the problem of distributed deterministic broadcasting in radio networks whose nodes are located on a line. Nodes send messages in synchronous time-slots. Each node υ has a given transmission range. All nodes located within this range can receive messages from υ. However, a node situated in the range of two or more nodes that send messages simultaneously, cannot receive these messages and hears only noise. Each node knows only its own position and range, as well as the maximum of all ranges. Broadcasting is adaptive: nodes can decide on the action to take on the basis of previously received messages, silence or noise. We prove lower bounds on broadcasting time in this model and construct broadcasting protocols whose performance nearly matches these bounds for the simplest case when nodes are situated on a line. We also show that if nodes do not even know their own range, every broadcasting protocol must be hopelessly slow. While distributed randomized broadcasting algorithms, and, on the other hand, deterministic off-line broadcasting algorithms assuming full knowledge of the radio network, have been extensively studied in the literature, ours are the first results concerning broadcasting algorithms that are distributed and deterministic at the same time. We show that in this case, information available to nodes influences the efficiency of broadcasting in a significant way. We study deterministic gossiping in ad hoc radio networks with large node labels. The labels (identifiers) of the nodes come from a domain of size N which may be much larger than the size n of the network (the number of nodes). Most of the work on deterministic communication has been done for the model with small labels which assumes N = O(n). A notable exception is Peleg's paper, where the problem of deterministic communication in ad hoc radio networks with large labels is raised and a deterministic broadcasting algorithm is proposed, which runs in O(n2log n) time for N polynomially large in n. The O(nlog2n)-time deterministic broadcasting algorithm for networks with small labels given by implies deterministic O(n log N log n)-time broadcasting and O(n2log2N log n)-time gossiping in networks with large labels. We propose two new deterministic gossiping algorithms for ad hoc radio networks with large labels, which are the first such algorithms with subquadratic time for polynomially large N. More specifically, we propose: a deterministic O(n3 2log2N log n)-time gossiping algorithm for directed networks; and a deterministic O(n log2N log2n)-time gossiping algorithm for undirected networks. We study the broadcast version of the CONGEST CLIQUE model of distributed computing. In this model, in each round, any node in a network of size @math can send the same message (i.e. broadcast a message) of limited size to every other node in the network. Nanongkai presented in [STOC'14] a randomized @math -approximation algorithm to compute all pairs shortest paths (APSP) in time @math on weighted graphs, where we use the convention that @math is essentially @math polylog @math and @math is essentially @math polylog @math . We complement this result by proving that any randomized @math -approximation of APSP and @math -approximation of the diameter of a graph takes @math time in the worst case. This demonstrates that getting a negligible improvement in the approximation factor requires significantly more time. Furthermore this bound implies that already computing a @math -approximation of all pairs shortest paths is among the hardest graph-problems in the broadcast-version of the CONGEST CLIQUE model and contrasts a recent @math -approximation for APSP that runs in time @math in the unicast version of the CONGEST CLIQUE model. On the positive side we provide a deterministic version of Nanongkai's @math -approximation algorithm for APSP. To do so we present a fast deterministic construction of small hitting sets. We also show how to replace another randomized part within Nanongkai's algorithm with a deterministic source-detection algorithm designed for the CONGEST model presented by Lenzen and Peleg at PODC'13. We consider distributed broadcasting in radio networks, modeled as undirected graphs, whose nodes have no information on the topology of the network, nor even on their immediate neighborhood. For randomized broadcasting, we give an algorithm working in expected time O(D log(n D) + log2 n) in n-node radio networks of diameter D, which is optimal, as it matches the lower bounds of [1] and Kushilevitz and Mansour [14]. Our algorithm improves the best previously known randomized broadcasting algorithm of Bar-Yehuda, Goldreich and Itai [3], running in expected time O(D log n + log2 n). For deterministic broadcasting, we show the lower bound Ω(n(log n) (log (n D)))) on broadcasting time in n-node radio networks of diameter D. This implies previously known lower bounds of Bar-Yehuda, Goldreich and Itai [3] and Bruschi and Del Pinto [5], and is sharper than any of them in many cases. We also give an algorithm working in time O(n log n), thus shrinking -- for the first time -- the gap between the upper and the lower bound on deterministic broadcasting time to a logarithmic factor. We study the problem of waking up a collection of @math processors connected by a multihop ad hoc ratio network with unknown topology, no access to a global clock, and no collision detection mechanism available. Each node in the network either wakes up spontaneously or gets activated by receiving a wake-up signal from another node. All active nodes transmit the wake-up signals according to a given protocol @math . The running time of @math is the number of steps counted from the first spontaneous wake-up until all nodes become activated. We provide two protocols for this problem. The first one is a deterministic protocol with running time @math . Our protocol is based on a novel concept of a shift-tolerant selector to which we refer as a (radio) synchronizer. The second protocol is randomized, and its expected running time is @math , where @math is the diameter of the network. Subsequently we show how to employ our wake-up protocols to solve two other communication primitives: leader election and clock synchronization. We consider broadcasting in radio networks: one node of the network knows a message that needs to be learned by all the remaining nodes. We seek distributed deterministic algorithms to perform this task. Radio networks are modeled as directed graphs. They are unknown, in the sense that nodes are not assumed to know their neighbors, nor the size of the network, they are aware only of their individual identifying numbers. If more than one message is delivered to a node in a step then the node cannot hear any of them. Nodes cannot distinguish between such collisions and the case when no messages have been delivered in a step. The fastest previously known deterministic algorithm for deterministic distributed broadcasting in unknown radio networks was presented in [6], it worked in time O(n11 6). We develop three new deterministic distributed algorithms. Algorithm A develops further the ideas of [6] and operates in time O(n1:77291) = O(n9 5), for general networks, and in time O(n1+a+H(a)+o(1)) for sparse networks with in-degrees O(na) fora < 1=2; here H is the entropy function. Algorithm B uses a new approach and works in time O(n3 2 log1 2 n) for general networks or O(n1+a+o(1)) for sparse networks. Algorithm C further improves the performance for general networks running in time O(n3 2). In this paper we present new randomized and deterministic algorithms for the classical problem of broadcasting in radio networks with unknown topology. We consider directed n-node radio networks with specified eccentricity D (maximum distance from the source node to any other node). Bar- presented an algorithm that for any n-node radio network with eccentricity D completes the broadcasting in O(D log n + log2 n) time, with high probability. This result is almost optimal, since as it has been shown by Kushilevitz and Mansour and , every randomized algorithm requires Ω (D log(n D) + log2 n) expected time to complete broadcasting.Our first main result closes the gap between the lower and upper bound: we describe an optimal randomized broadcasting algorithm whose running time complexity is O(D log (n D + log2 n), with high probability. In particular, we obtain a randomized algorithm that completes broadcasting in any n-node radio network in time O(n), with high probability.The main source of our improvement is a better "selecting sequence" used by the algorithm that brings some stronger property and improves the broadcasting time. Two types of "selecting sequences" are considered: randomized and deterministic ones. The algorithm with a randomized sequence is easier (more intuitive) to analyze but both randomized and deterministic sequences give algorithms of the same asymptotic complexity. Energy is often the most constrained resource in networks of battery-powered devices, and as devices become smaller, they spend a larger fraction of their energy on communication (transceiver usage) not computation. As an imperfect proxy for true energy usage, we define energy complexity to be the number of time slots a device transmits listens; idle time and computation are free. In this paper we investigate the energy complexity of fundamental communication primitives such as broadcast in multi-hop radio networks. We consider models with collision detection (CD) and without (No-CD), as well as both randomized and deterministic algorithms. Some take-away messages from this work include: 1. The energy complexity of broadcast in a multi-hop network is intimately connected to the time complexity of leader election in a single-hop (clique) network. Many existing lower bounds on time complexity immediately transfer to energy complexity. For example, in the CD and No-CD models, we need @math and @math energy, respectively. 2. The energy lower bounds above can almost be achieved, given sufficient ( @math ) time. In the CD and No-CD models we can solve broadcast using @math energy and @math energy, respectively. 3. The complexity measures of Energy and Time are in conflict, and it is an open problem whether both can be minimized simultaneously. We give a tradeoff showing it is possible to be nearly optimal in both measures simultaneously. For any constant @math , broadcast can be solved in @math time with @math energy, where @math is the diameter of the network. We study broadcasting time in radio networks, modeled as unit disk graphs (UDG). showed that broadcasting time depends on two parameters of the UDG network, namely, its diameter D(in hops) and its granularityg. The latter is the inverse of the densitydof the network which is the minimum Euclidean distance between any two stations. They proved that the minimum broadcasting time is @math , assuming that each node knows the density of the network and knows exactly its own position in the plane. In many situations these assumptions are unrealistic. Does removing them influence broadcasting time? The aim of this paper is to answer this question, hence we assume that density is unknown and nodes perceive their position with some unknown error margin i¾?. It turns out that this combination of missing and inaccurate information substantially changes the problem: the main new challenge becomes fast broadcasting in sparse networks (with constant density), when optimal time is O(D). Nevertheless, under our very weak scenario, we construct a broadcasting algorithm that maintains optimal time @math for all networks with at least 2 nodes, of diameter Dand granularity g, if each node perceives its position with error margin i¾?= i¾?d, for any (unknown) constant i¾?< 1 2. Rather surprisingly, the minimum time of an algorithm stopping if the source is alone, turns out to be i¾?(D+ g2). Thus, the mere stopping requirement for the special case of the lonely source causes an exponential increase in broadcasting time, for networks of any density and any small diameter. Finally, broadcasting is impossible if i¾?i¾? d 2. We study deterministic broadcasting in radio networks in the recently introduced framework of network algorithms with advice. We concentrate on the problem of trade-offs between the number of bits of information (size of advice) available to nodes and the time in which broadcasting can be accomplished. In particular, we ask what is the minimum number of bits of information that must be available to nodes of the network, in order to broadcast very fast. For networks in which constant time broadcast is possible under a complete knowledge of the network we give a tight answer to the above question: O(n) bits of advice are sufficient but o(n) bits are not, in order to achieve constant broadcasting time in all these networks. This is in sharp contrast with geometric radio networks of constant broadcasting time: we show that in these networks a constant number of bits suffices to broadcast in constant time. For arbitrary radio networks we present a broadcasting algorithm whose time is inverse-proportional to the size of the advice. For every fixed constant @math , we design an algorithm for computing the @math -sparse Walsh-Hadamard transform of an @math -dimensional vector @math in time @math . Specifically, the algorithm is given query access to @math and computes a @math -sparse @math satisfying @math , for an absolute constant @math , where @math is the transform of @math and @math is its best @math -sparse approximation. Our algorithm is fully deterministic and only uses non-adaptive queries to @math (i.e., all queries are determined and performed in parallel when the algorithm starts). An important technical tool that we use is a construction of nearly optimal and linear lossless condensers which is a careful instantiation of the GUV condenser (Guruswami, Umans, Vadhan, JACM 2009). Moreover, we design a deterministic and non-adaptive @math compressed sensing scheme based on general lossless condensers that is equipped with a fast reconstruction algorithm running in time @math (for the GUV-based condenser) and is of independent interest. Our scheme significantly simplifies and improves an earlier expander-based construction due to Berinde, Gilbert, Indyk, Karloff, Strauss (Allerton 2008). Our methods use linear lossless condensers in a black box fashion; therefore, any future improvement on explicit constructions of such condensers would immediately translate to improved parameters in our framework (potentially leading to @math reconstruction time with a reduced exponent in the poly-logarithmic factor, and eliminating the extra parameter @math ). Finally, by allowing the algorithm to use randomness, while still using non-adaptive queries, the running time of the algorithm can be improved to @math . In a seminal paper, Bar- (1992) considered broadcasting in radio networks whose nodes know only their own label and labels of their neighbors. They claimed a linear lower bound on the time of deterministic broadcasting in such radio networks, by constructing a class of graphs of diameter 3, with the property that every broadcasting algorithm requires linear time on one of these graphs. Due to a subtle error in the argument, this result is incorrect. We construct an algorithm that broadcasts in logarithmic time on all graphs from the work of Bar- Moreover, we show how to broadcast in sublinear time on all n-node graphs of diameter o(log log n). On the other hand, we construct a class of graphs of diameter 4, such that every broadcasting algorithm requires time spl Omega (4 spl radic n) on one of these graphs. In view of the randomized algorithm, running in expected time O(D log n + log sup 2 n) on all n-node graphs of diameter D, our lower bound gives the first correct proof of an exponential gap between determinism and randomization in the time of radio broadcasting. This paper considers the issue of broadcasting protocols in multihop radio networks. The objective of a broadcasting protocol is to deliver the broadcasted message to all network nodes. To efficiently achieve this objective the broad- casting protocol in this paper utilizes two basic properties of the multihop radio network. One is the broadcast nature of the radio which allows every single trasmission to reach all nodes that are in line of sight and within range of the transmitting node. The other, spatial reuse of the radio channel, which due to the multihop nature of the network allows multiple simultaneous transmissions to be received correctly. The proposed protocol incorporates these properties to obtain a collision free forwarding of the broadcasted message on a tree. Centralized and distributed algorithms for the tree construction are presented. The obtained trees are unique in incorporating radio oriented time ordering as part of their definition. In this way multiple copies of one or more broadcasted messages can be transmitted simultaneously without collision, requiring only a small number of message transmissions. Consequently, the protocol not only guarantees that the broadcasted message reaches all network nodes in bounded time, but also ensures that the broadcasting activity will use only limited channel bandwidth and node memory. The proposed broadcast protocol thus possesses the advantages of TDM solutions while allowing the channel bandwidth to be shared, concurrently with the broadcast, with other transmission activities as dictated, for instance, by data link protocols. Some NP-completeness proofs are also given. We consider distributed plurality consensus in a complete graph of size @math with @math initial opinions. We design an efficient and simple protocol in the asynchronous communication model that ensures that all nodes eventually agree on the initially most frequent opinion. In this model, each node is equipped with a random Poisson clock with parameter @math . Whenever a node's clock ticks, it samples some neighbors, uniformly at random and with replacement, and adjusts its opinion according to the sample. A prominent example is the so-called two-choices algorithm in the synchronous model, where in each round, every node chooses two neighbors uniformly at random, and if the two sampled opinions coincide, then that opinion is adopted. This protocol is very efficient and well-studied when @math . If @math for some small @math , we show that it converges to the initial plurality opinion within @math rounds, w.h.p., as long as the initial difference between the largest and second largest opinion is @math . On the other side, we show that there are cases in which @math rounds are needed, w.h.p. One can beat this lower bound in the synchronous model by combining the two-choices protocol with randomized broadcasting. Our main contribution is a non-trivial adaptation of this approach to the asynchronous model. If the support of the most frequent opinion is at least @math times that of the second-most frequent one and @math , then our protocol achieves the best possible run time of @math , w.h.p. We relax full synchronicity by allowing @math nodes to be poorly synchronized, and the well synchronized nodes are only required to be within a certain time difference from one another. We enforce this synchronicity by introducing a novel gadget into the protocol. We analyze the maximum gain that can be achieved through cooperative broadcast with energy accumulation and memory. We consider a linear network, where a known number of nodes are placed on a line, and derive an upper bound on @math , the gain of cooperative broadcast over noncooperative broadcast with respect to total power consumption. Specifically, we prove that, in linear networks with path loss exponent @math , @math , irrespective of the number of devices, the size of the network, the node placement strategy, and the cooperative broadcast strategy. We extend this result to any path loss exponent @math . We also show that the cooperation gain in short-range transmissions, wherein the circuit energy consumption is nonnegligible, is smaller than that in long-range transmissions. We further study the cooperative broadcast gain when the objective is to reduce the maximum transmission power used by any node in the network. In this case, we show that, when nodes are distributed uniformly at random, the maximum cooperation gain will be @math (log @math ), with high probability, where @math is the number of nodes in the network. These are important observations that should be considered in designing power-efficient broadcast algorithms in future network-wide cooperative broadcast.