Brief Announcement: Randomized Blind Radio Networks
Brief Announcement: Randomized Blind Radio Networks
Peter Davies 0 1
0 University of Warwick , Coventry , UK
1 Artur Czumaj University of Warwick , Coventry , UK
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 best modeled by the ad-hoc variant, in which the devices do not know the structure of the network. Much 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 n, and the diameter D), which may not be realistic in systems with weak devices or gradual deployment. Little is known about what can be done without this information. This is the issue we address in this work, by presenting the first randomized broadcasting algorithms for blind 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 O(D log Dn log2 log Dn + log2 n) time, almost reaching the Ω(D log Dn + log2 n) lower bound. We also give an even faster algorithm for directed networks with collision detection.
and phrases Broadcasting; Randomized Algorithms; Radio Networks
Model and problem
In the ad-hoc multi-hop radio network model, a communications network is represented as a
graph, with nodes corresponding to devices with wireless capability. A directed edge (u, v)
in the graph means that device u can reach device v via direct transmission. Efficiency of
algorithms is measured in terms of number of nodes n in the network, and eccentricity D
(maximum distance between any pair of nodes). The defining feature of radio networks is
the rule for how nodes can communicate: time is divided into discrete synchronous steps,
in which each node can choose whether to transmit a message or listen for messages. A
listening node in a given time-step then hears a message iff exactly one of its in-neighbors
transmits. In the model with collision detection, a listening node can distinguish between the
cases of having 0 in-neighbors transmit and having more than one, but in the model without
collision detection these scenarios are indistinguishable.
Randomized Blind Radio Networks
While, in the ad-hoc model, the underlying graph is unknown to the nodes, it is usual to
assume that nodes do know the values of n and D. We do not make this assumption, and
thus are dealing with a more restrictive model, which we call blind radio networks, in which
nodes have no prior network knowledge whatsoever.
We design randomized algorithms for the task of broadcasting, in which a single designated
source node starts with a message, and must inform all nodes in the network via transmissions.
We assume that all nodes except the source begin in an inactive state, and become active
when they receive a transmission. Our algorithms are Monte-Carlo algorithms succeeding
with high probability (i.e., their failure probability is at most n−c for some c > 0).
Broadcasting is possibly the most studied problem in radio networks, and has a wealth of
literature in various settings. In networks without collision detection, optimal broadcasting
was achieved by Czumaj and Rytter , and Kowalski and Pelc , who gave randomized
algorithms that complete the task in O(D log Dn + log2 n) time with high probability. This
matched a known Ω(D log Dn + log2 n) lower bound for the task [1, 6]. However, their
algorithms intrinsically require parameter knowledge, and algorithms that do not require
such knowledge have been little studied. The closest analogue in the literature is the work of
Jurdzinski and Stachowiak , who give algorithms for wake-up in single-hop radio networks
under a wide range of node knowledge assumptions. Their Use-Factorial-Representation
algorithm is the most relevant; the running time is given as O((log n log log n)3) for single-hop
networks, but a similar analysis as we present would demonstrate that the algorithm also
performs broadcasting in multi-hop networks in O((D + log n) log2 Dn log3 log Dn ) time.
We present a randomized algorithm for broadcasting in blind (directed or undirected)
networks without collision detection which succeeds with high probability within time
O(D log Dn log2 log Dn + log2 n). This improves over the running time of  and comes
within a poly-log log factor of the Ω(D log Dn + log2 n) lower bound [1, 6]. We also present an
O(D log Dn log log log Dn +log2 n)-time algorithm for directed networks with collision detection.
The main idea of our randomized algorithms in blind radio networks is as follows: when
considering a particular node v we wish to inform, all of its active in-neighbors will be
transmitting with some probability. We wish to make the sum of these probabilities
approximately constant (say 12 ), since then we can show that v will be informed with good
probability. However, we do not know the size of v’s active in-neighborhood, so choosing
appropriate probabilities is difficult. To do so, we have the source node generate a global
random variable from some distribution Y for each time-step, which will function as a ‘guess’
of in-neighborhood size. By appending these variables to the source message, we can ensure
that all active nodes are aware of them. Then, based on these global variables and upon
local randomness, the active nodes decide whether to transmit.
By choosing and analyzing the distribution Y we can obtain some bound on the probability
that a node with active neighbors is informed, in each time-step. We then show a recipe for
converting these probabilities to a running time for broadcasting.
A. Czumaj and P. Davies
Algorithm 1 Broadcast Framework.
Blind radio networks without collision detection
In networks without collision detection, we take Y to be the sum of two components, which
account for different network conditions. Under most circumstances, we use
GeneralBroadcast; where the source “guesses” a neighborhood size from 1 to ∞ in each time-step,
with a probability that decreases in neighborhood size in order to converge. In low diameter
networks, we improve upon this with Shallow-Broadcast component, which quickly informs
networks of low diameter using T to approximate in-neighborhood size.
I Theorem 1. Broadcasting can be performed in networks without collision detection in
O(D log Dn log2 log Dn + log2 n) time, succeeding with high probability.
Directed blind radio networks with collision detection
When collision detection (and a global clock) is available, nodes can learn their exact distance
dv from the source node within O(D) time, via a process known as beep waves . The
local transmission probabilities that nodes use during our broadcasting algorithm can then
depend on dv, as well as T and the global randomness provided by the source. We add two
new components to the two already defined which exploit this: Deep-Broadcast, which
quickly informs nodes far from the source, and Semi-Shallow-Broadcast, which removes
a running-time bottleneck when D is small.
I Theorem 2. Broadcasting can be performed in networks with collision detection in
O(D log Dn log log log Dn + log2 n) time, succeeding with high probability.
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