In this paper, we address the problem of broadcasting in a wireless network under a novel communication model: the swamping communication model. In this model, nodes communicate only with those nodes at geometric distance greater than @math and at most @math from them. Communication between nearby nodes under this model can be very time consuming, as the length of the path between two nodes within distance @math is only bounded above by the diameter @math , in many cases. For the @math -node lattice networks, we present algorithms of optimal time complexity, respectively @math for the lattice line and @math for the two-dimensional lattice. We also consider networks of unknown topology of diameter @math and of a parameter @math ( granularity ). More specifically, we consider networks with @math the minimum distance between any two nodes and @math . We present broadcast algorithms for networks of nodes placed on the line and on the plane with respective time complexities @math and @math , where @math .
In @cite_21 , under the conditional wakeup model, @math was shown to be the tight lower bound on broadcasting time. However, for networks where nodes locations are restricted to the vertices of a grid of squares of size @math , the authors proposed an @math -time broadcasting algorithm, thus showing that the broadcast time is not always linearly dependent on @math .
@cite_21
2088895311 1984466682 1984355228 1973843421
The paper considers broadcasting in radio networks, modeled as unit disk graphs (UDG). Such networks occur in wireless communication between sites (e.g., stations or sensors) situated in a terrain. Network stations are represented by points in the Euclidean plane, where a station is connected to all stations at distance at most 1 from it. A message transmitted by a station reaches all its neighbors, but a station hears a message (receives the message correctly) only if exactly one of its neighbors transmits at a given time step. One station of the network, called the source, has a message which has to be disseminated to all other stations. Stations are unaware of the network topology. Two broadcasting models are considered. In the conditional wake up model, the stations other than the source are initially idle and cannot transmit until they hear a message for the first time. In the spontaneous wake up model, all stations are awake (and may transmit messages) from the beginning. It turns out that broadcasting time depends on two parameters of the UDG network, namely, its diameter D and its granularity g, which is the inverse of the minimum distance between any two stations. We present a deterministic broadcasting algorithm which works in time O (D g) under the conditional wake up model and prove that broadcasting in this model cannot be accomplished by any deterministic algorithm in time better than ( (D g ) ) . For the spontaneous wake up model, we design two deterministic broadcasting algorithms: the first works in time O (D + g 2) and the second in time O (D log g). While neither of these algorithms alone is optimal for all parameter values, we prove that the algorithm obtained by interleaving their steps, and thus working in time ( O ( D + g^2, D g ) ), turns out to be optimal by establishing a matching lower bound. The paper studies broadcasting in radio networks whose stations are represented by points in the Euclidean plane. In any given time step, a station can either receive or transmit. A message transmitted from station (v) is delivered to every station (u) at distance at most (1) from (v), but (u) successfully hears the message if and only if (v) is the only station at distance at most (1) from (u) that transmitted in this time step. A designated source station has a message that should be disseminated throughout the network. All stations other than the source are initially idle and wake up upon the first time they hear the source message. It is shown in [11] that the time complexity of broadcasting depends on two parameters of the network, namely, its diameter (in hops) (D) and a lower bound (d) on the Euclidean distance between any two stations. The inverse of (d) is called the granularity of the network, denoted by (g). Specifically, the authors of [11] present a broadcasting algorithm that works in time (O (D g)) and prove that every broadcasting algorithm requires Ω (D √g) time. In this paper, we distinguish between the arbitrary deployment setting, originally studied in [11], in which stations can be placed everywhere in the plane, and the new grid deployment setting, in which stations are only allowed to be placed on a (d)-spaced grid. Does the latter (more restricted) setting provides any speedup in broadcasting time complexity? Although the (O (D g)) broadcasting algorithm of [11] works under the (original) arbitrary deployment setting, it turns out that the Ω (D √g) lower bound remains valid under the grid deployment setting. Still, the above question is left unanswered. The current paper answers this question affirmatively by presenting a provable separation between the two deployment settings. We establish a tight lower bound on the time complexity of broadcasting under the arbitrary deployment setting proving that broadcasting cannot be completed in less than Ω ( D √g) time. For the grid deployment setting, we develop a broadcasting algorithm that runs in time O ( D g5 6 log g), thus breaking the linear dependency on (g). 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 the time of broadcasting in ad hoc radio networks modeled as undirected graphs. In such networks, every node knows only its own label and a linear bound on the number of nodes but is unaware of the topology of the network, or even of its own neighborhood. Our aim is to study to what extent the availability of two important characteristics of a broadcasting algorithm influences optimal broadcasting time. These characteristics are adaptiveness and randomization. Our contribution is establishing upper and lower bounds on optimal broadcasting time for three classes of algorithms: adaptive deterministic, oblivious randomized and oblivious deterministic. In two cases we present tight bounds, and in one case a small gap remains. We show that for deterministic adaptive algorithms time Ω(n) is required even for n-node networks of constant diameter. This lower bound is strongest possible, since linear time algorithms are known, and hence establishes optimal time Θ(n) for this class. For oblivious randomized algorithms we show an upper bound O(n min D, log n ) and a lower bound Ω(n) on optimal expected broadcasting time in n-node networks of diameter D. Finally, for oblivious deterministic algorithms we show matching upper and lower bounds Θ(n min D, √n ) on optimal broadcasting time. Our results imply that enforcing obliviousness has at least as strong negative impact on broadcasting time as enforcing determinism, and that algorithms having both these features are strictly less efficient than those having only one of them.