Efficient k -shot broadcasting in radio networks

Discrete Applied Mathematics (

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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam

Efficient k-shot broadcasting in radio networks✩ Erez Kantor a,∗ , David Peleg b a

CSAIL MIT, Cambridge, MA, United States

b

Weizmann Institute of Science, Rehovot, Israel

article

info

Article history: Received 2 September 2012 Received in revised form 15 August 2015 Accepted 20 August 2015 Available online xxxx Keywords: Wireless network Radio network Broadcasting

abstract The paper concerns time-efficient k-shot broadcasting in undirected radio networks for nnode graphs of diameter D. In a k-shot broadcasting algorithm, each node in the network is allowed to transmit at most k times. Both known and unknown topology models are considered. For the known topology model, the problem has been studied before by Ga¸sieniec et al. (2008). We improve both the upper and the lower bound of that paper providing a randomized algorithm for constructing a k-shot broadcasting schedule of length D + O(kn1/2k log2+1/k n) on undirected graphs, and a lower bound of D + Ω (k · (n − D)1/2k ), which almost closes the gap between these bounds. For the unknown topology model, we provide the first k-shot broadcasting algorithm. Assuming that each node knows only the network size n (or a linear upper bound on it), our randomized k-shot broadcasting algorithm completes broadcasting in O((D + min{D · k, log n})· n1/(k−1) log n) rounds with high probability for k ≥ 2, and in O(D · n2 log n) rounds with high probability for k = 1. Moreover, we present an Θ (log n)-shot broadcasting algorithm that completes broadcasting in at most O(D log n + log2 n) rounds with high probability. This algorithm matches the broadcasting time of the algorithm of Bar-Yehuda et al. (1992), which assumes no limitation on the maximum number of transmissions per node. © 2015 Elsevier B.V. All rights reserved.

1. Introduction In this paper we study the fundamental task of broadcastingin synchronous radio networks, in both the known and unknown topology models. A radio network consists of stations that can act, at any given time step (round), as either a transmitter or a receiver. The network is modeled as an undirected graph G(V , E ), where V represents the set of stations and E represents communication feasibility, i.e., two nodes u, v ∈ V can communicate directly with each other iff (u, v) ∈ E. Energy efficiency is a central issue in designing the operation of ad-hoc radio networks and sensor networks, as in many cases the only energy sources for the stations are limited lifetime batteries. This paper concerns the use of k-shot algorithms, where each node in the network is allowed to transmit at most k times, hence energy is preserved at each of the stations. Such a strategy for energy efficient radio communication was studied in the context of broadcasting and gossiping in radio networks of random topology, see [4,13].

✩ This work is partially supported by AFOSR FA9550-13-1-0042 and NSF grants Nos. CCF-0939370, CCF-1217506, and CCF-AF-0937274 and by a grant from the Israel Ministry of Science (36478). ∗ Corresponding author. E-mail addresses: [email protected] (E. Kantor), [email protected] (D. Peleg).

http://dx.doi.org/10.1016/j.dam.2015.08.021 0166-218X/© 2015 Elsevier B.V. All rights reserved.

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In the unknown topology model, we assume that each node knows only a linear upper bound on the number of nodes n, but does not know anything else concerning the topology. This model is often used to describe sensor networks and is particularly suitable for ad hoc networks. A sensor network is composed of a large number of sensor nodes, which can be densely deployed in the targeted environment, and communicate via an ad hoc wireless network. Sensor networks are expected to take part in many civil and military applications, for example, in earthquake and tsunami warnings, fire detection, etc. These networks possess some unique features, which introduce also new algorithmic challenges. The sensor devices are very cheap, they are prone to failures, and the number of sensors in a network is generally very large. The battery’s life time in each unit is limited, hence energy saving is an acute concern. In many cases, the deployment of sensor networks is such that the positions of the sensors are unknown. For example, in certain cases the sensors are spread out from the air by an airplane; in such cases the topology of the network is unknown. On the other hand, in many other settings the network topology is known, and the stations may either know the topology in advance or can learn the topology during some preprocessing procedure. We study this model as well, and present an algorithm that computes a fast broadcasting scheduler under this model. We consider a synchronous network, where communication is performed in rounds and is assumed to have the following property: (1) in each round a node either acts as a transmitter, or does not act as a transmitter (in which case we say that it acts as a receiver); (2) a message transmitted by a station reaches all its neighbors; and (3) a node u ∈ V receives a message M in a given round if and only if on that round it does not act as a transmitter and exactly one of its neighbors acts as a transmitter and transmits M. Otherwise (in case u does not act as a transmitter), there are two possibilities: if none of u’s neighbors transmits, then u hears silence, and if at least two of u’s neighbors transmit simultaneously, then a collision occurs at u. In both cases, u does not receive any message. We consider broadcasting, which is the following communication task. A distinguished node s, called the source, has a message M that has to be delivered to all other nodes in the network. A broadcasting schedule S in a radio network is a list (T1 , T2 , . . . , Tt ) of subsets of V that describes the order of transmissions: for each round i = 1, 2, . . . , t, the set Ti ⊆ V specifies the nodes that have to act as transmitters on round i. We assume that a node v scheduled to act as a transmitter on round t will transmit the source message M if it has already received it from one of its neighbors in some previous round. The length of the schedule S is the number of rounds, t, and S is said to complete broadcasting if by time t, all the network nodes have received M. Our contribution: We study the k-shot broadcasting in undirected radio networks. Both the known and unknown topology model are considered. For the known topology model, the problem has been studied before by Ga¸sieniec et al. [13]. √ That paper presented a deterministic 1-shot broadcasting protocol that completes the broadcasting task in D + O( n log n) rounds, and a randomized k-shot broadcasting protocol for k ≥ 3, that completes the broadcasting task in D + O(kn1/(k−2) log2 n) rounds with high probability. In addition, the authors proved a lower bound of D + Ω ((n − D)1/2k ) on the length of k-shot broadcasting schedules for n-node graphs of diameter D. We improve both the upper and the lower bound. Specifically, in Section 3 we present a randomized algorithm for constructing a k-shot broadcasting schedule of length D + O(kn1/2k log2+1/k n) on undirected graphs, which almost matches the lower bound. For the lower bound we show that on binomial bipartite graphs, presented in [13] (see Section 3.5), any broadcasting schedule requires at least Ω (k · n1/2k ) rounds, implying a lower bound of D + Ω (k · (n − D)1/2k ) rounds on arbitrary undirected graphs. For the unknown topology model, we present in Section 2 a first k-shot broadcasting algorithm. Assuming that each node knows only the network size n (or a linear upper bound on it), our randomized k-shot broadcasting algorithm completes broadcasting in O((D + min{D · k, log n})· n1/(k−1) log n) rounds with high probability for k ≥ 2, and in O(D · n2 log n) rounds with high probability for k = 1. Moreover, we present a Θ (log n)-shot broadcasting algorithm that completes broadcasting in at most O(D · log n + log2 n) rounds with high probability. This algorithm is competitive with the expected O(log2 n/ log(n/D))-shots randomized broadcasting algorithm with optimal broadcasting time of O(D log(n/D) + log2 n) presented by Berenbrink et al. [4], and matches the broadcasting time of the algorithm of Bar-Yehuda et al. [3], which assumes no limitation on the maximum number of transmissions per node (and is, in effect, an O(log2 n)-shot broadcasting algorithm using expected O(log n)-shots per node). (Note that the broadcasting algorithm of Berenbrink et al. [4] and the broadcasting algorithms of Czumaj and Rytter [10] and Kowalski and Pelc [18] all use O(log2 n) shots per node, i.e., all are O(log2 n)-shot broadcasting algorithms.) A comparative summary of these results is provided in Table 1. Related work: Deterministic centralized broadcasting in radio networks was first studied by Chlamtac and Kutten [5], who formulated the radio network model. A lower bound of Ω (log2 n) time for broadcasting, even in O(1)-diameter networks, was established in [1] by showing the existence of a family of radius-2 n-node networks for which any broadcast schedule requires at least Ω (log2 n) rounds. On the other hand, for the known topology model, a sequence of papers presented increasingly tighter upper bounds. In [6], Chlamtac and Weinstein presented an O(D log2 n)-time broadcasting algorithm for n-node radio networks of diameter D. In [12], Gaber and Mansour proposed an O(D + log5 n)-time broadcasting algorithm. Subsequently, Elkin and Kortsarz [11] presented a deterministic algorithm yielding schedules of length O(D + log4 n), Ga¸sieniec et al. [14] presented a deterministic algorithm for constructing schedules of length D + O(log3 n) and a randomized algorithm for computing schedules of length D + O(log2 n), and finally Kowalski and Pelc [20] gave an optimal deterministic algorithm yielding schedules of O(D + log2 n) rounds.

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Table 1 Summary of results on n -node networks of diameter D. Topology Known Unknown k=1 k≥2 k = Θ (log n) k = Θ (log2 n)

Our result

Previous 1/2k

2+1/k

D + O(k · n log D + Ω (k · n1/2k )

D + O(kn1/(k−2) log2 n) [13] D + Ω (n1/2k ) [13]

n)

O (D + min{D, log n}) · n2 log n





  O (D + min{D · k, log n}) · n1/(k−1) log n 2 O(D log n + log n) –

– – – O(D log(n/D) + log2 n) [4,10,18]

For the unknown topology model, Bar-Yehuda et al. [3] were the first to study distributed broadcasting, and presented a randomized protocol that achieves successful broadcast in O(D log n + log2 n) rounds with high probability. (The paper assumes that every node knows its neighborhood, but the result also holds for a model where each node knows only its own label.) Kushilevitz and Mansour [21] proved a lower bound of Ω (D log(n/D)) on the problem, and Czumaj and Rytter [10] and Kowalski and Pelc [18] later showed that this bound and the lower bound Ω (log2 n) of [1] are tight by presenting a randomized broadcasting algorithm whose time complexity is O(D log(n/D) + log2 n), with high probability. In the deterministic case, for directed n-node networks of diameter D, Chrobak et al. [7] showed that there exists a deterministic broadcast algorithm with time O(n log2 n). Later, Kowalski and Pelc [19] improved this result and established an O(n log n log D) bound, and recently, De-Marco [22] established an O(n log n log log n) bound. All proofs are nonconstructive. For undirected n-node networks of diameter D, Kowalski and Pelc [18] presented an algorithm working in time O(n log n) and later Kowalski [17] improved this result and established a broadcast algorithm working in time O(n log D). On the other hand, a lower bound of Ω (n log D) was given in [9] for directed n-node networks of diameter D, and a lower bound of Ω (n log n/ log(n/D)) was given in [18] for undirected n-node networks of diameter D. Energy efficient radio broadcasting was studied in the context of geometric networks, where the network nodes are embedded on the Euclidean plane. One of the main problems studied in this context is the energy efficient broadcast tree problem, where the goal is to find a transmission schedule that minimizes the total power consumption, based on a directed spanning tree rooted at the source node s. When the stations are spread in d-dimensional Euclidean space (d > 1), the minimum spanning tree (MST) based algorithm achieves constant approximation ratio for the problem (see [2,8,16,24]). On the other hand, the problem is known to be NP-hard [8] and if the distance function is arbitrary, then the problem has no logarithmic factor approximation unless P = NP [15]. 2. A k-shot broadcast algorithm in unknown topology This section considers k-shot broadcasting in an unknown topology, where the knowledge available to each node is limited to n, the number of nodes in the network (or a linear upper bound on it). The section is organized as follows. Section 2.1 presents a k-shot broadcast algorithm that is time efficient (namely, that completes broadcasting on G in log n O((D + min{D · k, log n}) · n1/(k−1) log n) rounds with high probability) for k ≤ 6 . In Section 2.2 we show how to extend this algorithm to a k-shot broadcast algorithm for large k (i.e., k ≈ (log n)/δ for constant δ ≥ 1), which completes broadcasting almost as fast as the best broadcast algorithm without any constraints on the allowed number of transmissions per node. 2.1. A k-shot broadcast algorithm for small k Consider a network G(V , E ) of diameter D, where |V | = n. We present a randomized 1-shot broadcast algorithm that completes broadcasting on G in O(D · n2 log n) rounds with high probability, and a randomized k-shot broadcast algorithm for k ≥ 2 that completes broadcasting on G in O((D + min{D · k, log n}) · n1/(k−1) log n) rounds with high probability. (We say that an event holds with high probability if its probability is at least 1 − n−1/(k−1) for k ≥ 2 and 1 − 2n−1 for k = 1.) Let ΓG (w) ⊆ V be the set of neighbors of w in G and let degG (w) = |ΓG (w)| be w ’s degree in G. (We may omit the subscript G when it is clear from the context.) In our broadcasting algorithm, time is divided into epochs, with each epoch divided into Ψ phases (for integer Ψ ≥ 1), and with each phase consisting of O(log n) rounds (time slots). In particular, each node u stays silent and waits to receive the message M. When u receives M during some round, it waits (silent) until the end of the current epoch. Then, the broadcasting algorithm performs k invocations of Procedure Epoch at u, and in each invocation it randomly selects one of the Ψ phases of the epoch (with uniform distribution), and assigns u to that phase. In that phase, u randomly selects exactly one of the O(log n) rounds of the phase (with some specific distribution) and transmits in that round. The operation of a broadcast algorithm can be viewed from two different angles. One is the viewpoint of a node v ∈ V that has already received the message M, and whose goal is to deliver the message M to its neighbors Γ (v). The other viewpoint is that of a node w ∈ V that has not received M yet; the goal of the algorithm is to ensure that w does receive M from one of its neighbors in Γ (w). Our analysis takes the second viewpoint.

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Fig. 1. Procedure Phase.

For proving that our broadcast algorithm completes broadcasting within the stated time with the claimed probability, we bound the probability of the complementary event, namely, the event that not all the nodes receive the message M by that time. The probability that there exists a node that did not receive M is the same as the probability that there exists a node w that did not receive M, but at least one of its neighbors did receive M. In this event, there exist at least k executions of Procedure Epoch in which at least one of w ’s neighbors participated. The key component of the proof (outlined below and formally proved in Lemma 2.3) involves showing that the probability that w did not receive M in an execution of Procedure Epoch in which at least one of w ’s neighbors participated is at most 1/Ψ . Thus in turn, the probability that w did not receive M, is the same as the probability that for at least k times, at least one of w ’s neighbors participated and executed Procedure Epoch, and w did not receive the message in any of those executions, which is at most (1/Ψ )k . By the union bound, the probability that there exists a node that did not receive M is at most n · (1/Ψ )k . We select Ψ so that the complementary event, in which all the nodes receive the message, occurs with high probability. Regarding the time complexity of the algorithm, we have a trivial upper bound of O(D · k · Ψ log n) rounds. For D ≥ log n we intuitively use the fact (formally proved in Lemma 2.4) that the expected number of executions in which at least one of w ’s neighbors participated and executed Procedure Epoch, and w did not receive the message in any of those executions, is constant (in fact less than 2). This yields the desire broadcasting time of O((D + min{D · k, log n}) · Ψ log n) rounds. We next give a high-level explanation for why w receives M with probability at least 1 − 1/Ψ in an epoch in which at least one of w ’s neighbors participated. Consider a node w ∈ V that has not received M so far, and let Λ(w) ⊆ Γ (w) be a subset of w ’s neighbors that have already received M and have participated in a given epoch. Assume that Λ(w) ̸= ∅. All the nodes of Λ(w) execute Procedure Epoch in parallel, where this procedure assigns each node to a single transmission round, and its goal (w.r.t. w ) is to have a round in which exactly one node of Λ(w) will transmit, which will ensure that w receives M on that round. The goal is to assign the set Λ(w) to a large number of phases among all considered phases. We say that a phase is active with respect to w if a nonempty subset of w ’s neighbors is assigned to this phase. For a nonempty subset Λ′ (w) ⊆ Λ(w) assigned to the same phase, the algorithm invokes Procedure Phase in parallel on those nodes. It is obvious that if |Λ′ (w)| = 1, then there will be a round in that phase with exactly one node of Λ′ (w) transmitting. We show in Lemma 2.1 that if |Λ′ (w)| ≥ 2, then there will be a round in that phase with exactly one node of Λ′ (w) transmitting, with probability at least 31/48. Thus intuitively, when the broadcast algorithm executes an epoch in parallel among Λ(w)’s nodes, it ensures that with probability least 1 − 1/Ψ for ‘‘small’’ Λ(w), there will be more than |Λ(w)|/2 active phases (w.r.t. w ), which in turn implies that w will receive the message. For ‘‘large’’ Λ(w), the algorithm ensures that with probability at least q (formally fixed in the proof of Lemma 2.3) there will be more than Ψ /16 active phases (w.r.t. w ). Conditioned on this event occurring, w will receive the message with probability at least 1 − (31/48)Ψ /16 . We then conclude that (for ‘‘large’’ Λ(w)) w will receive the message with probability at least of 1 − 1/Ψ , by showing that 1 − 1/Ψ ≥ q · (1 − (31/48)Ψ /16 ). We now turn to describing the broadcasting algorithm in detail. We first describe a basic 1-shot transmission procedure named Phase. Let T = ⌈log n⌉ + 5. A node that participates in Procedure Phase randomly selects a number r to be i ∈ {1, . . . , T − 1} with probability 2−i and otherwise selects r = T (with probability 2−T +1 ). Next, it transmits the message M to all its neighbors on round t ≡ r mod T . The formal code is presented in Fig. 1. Procedure Phase is a simple T -round 1-shot procedure, where each node transmits in exactly one round. Recall that by our definition of rounds, Procedure Phase always starts at a round corresponding to t ≡ 1 mod T and ends at a round corresponding to t ≡ 0 mod T . Lemma 2.1. Consider a node w ∈ V and a non-negative integer z. If a nonempty subset of w ’s neighbors executes Procedure Phase during the time interval [z T + 1, (z + 1)T ], and they all start at time t = z T + 1, then the probability that w receives the message M during this time interval is at least 31/48. Proof. Assume, without loss of generality, that z = 0. Let Λ(w) be the subset of w ’s neighbors that execute Procedure Phase during t = 1, 2, . . . , T . Let r ′ (v) and r (v) be the random selection made by the node v ∈ Λ(w) in Procedure Phase (see Fig. 1). Let Λi (w) = {v | v ∈ Λ(w), r (v) ≥ i}, for i = 1, . . . , T , and let Λ′i (w) = {v | v ∈ Λ(w), r ′ (v) ≥ i} for i = 1, 2, . . .. It is clear that

Λi (w) = Λ′i (w), for every i = 1, . . . , T .

(1)

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Note that the existence of a round I ∈ {1, . . . , T } in which |ΛI (w)| = 1 means that in the Ith round of the phase, exactly one of w ’s neighbors transmits, which implies that w will receive M during that round. We prove a lower bound on the probability of that event. We begin with an inductive proof that shows that the event ε ′ (w) in which there exists an index I ′ ∈ {1, 2, . . .} such that |Λ′I ′ (w)| = 1 occurs with probability at least 2/3 (a similar proof appears in [3]). For two integers ℓ ≥ 1 and i ≥ 1, let f (ℓ, i) = Pr[ε ′ (w) | |Λ′i (w)| = ℓ]. Note that 2−r −k+1 = 2−k . Pr[r ′ (v) ≥ r + k | r ′ (v) ≥ r ] = 2−r +1

(2)

The above probability depends on k, but does not depend on r. This implies that, for any integrals r ≥ r ′ ≥ 0, i′ ≥ i ≥ 0 and j ≥ 0 we have

    Pr |Λ′i′ (w)| = r ′ | |Λ′i (w)| = r = Pr |Λ′i′ +j (w)| = r ′ | |Λ′i+j (w)| = r , which implies, in particular, that f (ℓ, i) = f (ℓ, j),

for every ℓ, i, j ∈ {1, 2, . . .}.

(3)

So, let f (ℓ) = f (ℓ, i)

for every i, ℓ ∈ {1, 2, . . .}.

(4)

Equality (2) implies a basic property of the sequence sets Λ1 (w), Λ2 (w), . . . , namely, if the node v belongs to Λi (w), then with probability 1/2 it belongs to Λ′i+1 (w) as well. Thus, it is clear that f (1) = 1 and that ′

′

′

f (2) = 1 − (1/4 + 1/42 + 1/43 + · · · ) ≥ 2/3, which is the base of the induction. For the induction step, let ℓ > 2, and assume that f (z ) ≥ 2/3 for any z < ℓ. Let κi = |Λ′i (w)|, for i = 1, 2, . . .. We have f (ℓ) = f (ℓ, i)

= Pr[κi+1 = 1 | κi = ℓ] +

ℓ−1  

 Pr[κi+1 = ℓ′ | κi = ℓ] · f (ℓ′ , i + 1) + Pr[κi+1 = ℓ | κi = ℓ] · f (ℓ, i + 1)

ℓ′ =2

≥ Pr[κi+1 = 1 | κi = ℓ] +

2 3

Pr[1 < κi+1 < ℓ | κi = ℓ] + Pr[κi+1 = ℓ | κi = ℓ] · f (ℓ, i + 1),

(5)

where the inequality holds since by the inductive assumption and Equality (4), f (ℓ′ ) = (ℓ′ , i + 1) ≥ 2/3 for every ℓ′ = 2, . . . , ℓ − 1. Note that for every i = 1, 2, . . ., we have Pr[κi+1 = 1 | κi = ℓ] = ℓ−1  

  ℓ 1

· 2−ℓ = ℓ · 2−ℓ ,

 Pr[κi+1 = ℓ′ | κi = ℓ] = 1 −

ℓ′ =2

  ℓ 0

2−ℓ −

(6)

    ℓ −ℓ ℓ −ℓ 2 − 2 = 1 − (ℓ + 2) · 2−ℓ ℓ 1

(7)

and Pr[κi+1 = ℓ | κi = ℓ] =

  ℓ · 2−ℓ = 2−ℓ . ℓ

By pulling together inequalities (5)–(8), we have, f (ℓ) ≥

= = = ≥

  2 ℓ · 2−ℓ + (1 − (ℓ + 2) · 2−ℓ ) + 2−ℓ f (ℓ) 3      1  2  3 2 ℓ · 2−ℓ + (1 − (ℓ + 2) · 2−ℓ ) · 1 + 2−ℓ + 2−ℓ + 2−ℓ + · · · 3   2 1 −ℓ −ℓ ℓ · 2 + (1 − (ℓ + 2) · 2 ) · 3 1 − 2−ℓ    2 2 1 · + 0.5ℓ − 2 · 2−ℓ 3 3 1 − 2−ℓ 2/3,

(8)

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Fig. 2. Procedure Epoch.

where the second equality holds by a simple algebraic calculation of recursion and the third equality by an algebraic calculation of a sum of geometric sequence. The last inequality holds for any ℓ > 2. We have shown that the probability that there exists an index I ′ in which |Λ′I ′ (w)| = 1 is at least 2/3. The probability that I ′ ≤ T is trivially at least 1 − |Λ1 (w)| · 2−T ≥ 1 − n · 2−(log n+5) = 1 − 1/32. Combining these probabilities, we get that the probability that there exists such an index in which |ΛI (w)| = 1 is at least (1 − 1/32) · 2/3 = 31/48.  Next we describe another 1-shot procedure named Epoch, which consists of Ψ phases (i.e., Ψ T rounds). A node executing Procedure Epoch selects uniformly at random exactly one of Ψ potential phases in which it participates and executes Procedure Phase. The formal code is presented in Fig. 2. Procedure Epoch takes at most Ψ T rounds. We next prove that for any given node w , if a nonempty subset of w ’s neighbors executes Procedure Epoch simultaneously, then w will receive the message M with probability at least 1 − 1/Ψ during the Ψ T rounds of the procedure. We start with a lemma stating three technical observations referring to placing balls in bins. Given x balls and y bins, consider a process in which each ball is placed uniformly and independently in a random bin (with repetitions, i.e., multiple balls can be placed in the same bin). For x ≤ y/8, we say that the process ended successfully (referring to it as success of the first type) when there exists a bin occupied with exactly one ball. In order to give a lower bound for the probability of successes of the first type, we bound from below the probability that more than x/2 bins are occupied, which necessarily yields a bin with exactly one ball. For x > y/8, we say that the process ended successfully (referring to it as success of the second type) when more than y/16 bins are occupied. Let P (x, y) be the probability of success of the first type, i.e., of having more than x/2 occupied bins, and let P ′ (x, y) be the probability of success of the second type, i.e., of having more than y/16 occupied bins. Lemma 2.2. Let y ≥ 64. The success probabilities of the first and the second type satisfy the following. (C1) P (x, y) ≥ 1 − 1/y, for x = 2, 3, (C2) P (x, y) ≥ 1 − (2e/y)2 , for 3 < x ≤ y/8, and (C3) P ′ (x, y) > 1 − (2e/y)2 , for x > y/8. Proof. For x < y, let f (x, y) =





y

⌊x/2⌋

⌊x/2⌋

y

x

.

Note that f (x, y) is an upper bound on the probability that at most ⌊x/2⌋ bins are occupied (the probability of failure), i.e., f (x, y) ≥ 1 − P (x, y), and trivially, f (2x, y) > f (2x + 1, y),

for every x < y/2.

(9)

For x = 2, 3 we simply have f (3, y) < f (2, y) = 1/y, hence claim (C1) of the lemma is established. Next, we prove claim (C2) by bounding the desired probability for 4 ≤ x ≤ y/8. Let g (x, y) =



e·x

x

y

.

(10)

Note that f (2x, y) =

   2x y

x

x

y

≤

 e · y x  x 2x x

y

 =

e·x y

x

= g (x, y).

(11)

This function is monotonically decreasing with respect to x in the range [1, y/e2 ] and monotonically increasing with respect to x in the range (y/e2 , ∞), since d dx

g (x, y) =



e·x y

x

· (ln(x/y) + 2) ,

and the sign of this expression depends only on the sign of ln(x/y) + 2, hence d g dx

(x, y) > 0 for any x ∈ (y/e2 , ∞). This implies that max{g (x, y) | x ∈ [2, z ]} = max{g (2, y), g (z , y)},

for any z ∈ [2, ∞].

d g dx

(x, y) ≤ 0 for any x ∈ [1, y/e2 ] and (12)

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We next show that for any y ≥ 64, g (x, y) < (2e/y)2 ,

for every x ∈ [2, y/4].

(13)

By Ineq. (12) and Eq. (10), for every x ∈ [2, y/4], we have g (x, y) ≤ max{g (2, y), g (y/4, y)} = max{(2e/y)2 , ey/4 (1/4)y/4 }. Thus, to show that Inequality (13) holds, it remain to show that for any y ≥ 64, (e/4)y/4 ≤ (2e/y)2 , or equivalently,

(e/4)y/4 < 1, (2e/y)2

for any y ≥ 64.

(14)

Let

 e 4 (e/4)y/4 2 = y /(2e)2 (2e/y)2 4  d ′ g (y) = yc y (2 + y ln c )/(2e)2 which is negative for any y > 21 and let c = 4 4e . Note that, g ′ (y) = y2 · c y /(2e)2 and dy ′ and thus implies that g (y) is monotonically decreasing in (21, ∞). In addition, one can verify, by plotting the function, that g ′ (64) < 1, thus by the monotonicity of g ′ in [21, ∞) we get that g ′ (y) ≤ g ′ (64) < 1 for any y ≥ 64. This shows that y

g ′ (y) =

Inequality (14) holds and hence Inequality (13) holds. Combining Inequality (13) with inequalities (9) and (11), we get that f (x, y) <

 2 2e y

for every x ∈ [4, y/8] and y ≥ 64 as needed for claim (C2).

Finally, the probability that more than y/16 bins are occupied after placing x balls where x > y/8, is greater than the probability that more than y/16 bins are occupied after placing exactly y/8 balls into y bins, i.e., P ′ (x, y) ≥ P (y/8, y) for any x > y/8. Hence the lower bound on P (y/8, y) shown previously (in claim (C2)) yields claim (C3).  Consider a node w ∈ V . The process of placing balls in bins is used to model a random selection of transmission phases made by w ’s neighbors. We say that a phase is active with respect to w if at least one of its neighbors participates in that phase. Note that success of the first type (where more than x/2 bins have been occupied) implies the existence of a phase selected by exactly one neighbor of w among all considered neighbors, which necessarily yields a round where exactly one neighbor of w transmits, and thus implies that w will receive M on that round. Success of the second type (where more than y/16 bins have been occupied) implies the existence of more than y/16 active phases among all considered subsets of phases, which implies that w will receive M during these phases with high probability. Using Lemmas 2.1 and 2.2, we prove the following. Lemma 2.3. Let Ψ ≥ 64. Consider a node w ∈ V and a nonnegative integer z. If an nonempty subset of w ’s neighbors executes Procedure Epoch during the time interval [z Ψ T + 1, (z + 1)Ψ T ] and they all start at t = z Ψ T + 1, then the probability that w receives the message M during this time interval is at least 1 − 1/Ψ . Proof. Assume, without loss of generality, that z = 0. Let Λ(w) be the set of neighbors of w that executes Procedure Epoch during t = 1, 2, . . . , Ψ T . We consider three cases. The first is when exactly one of w ’s neighbors executes Procedure Epoch, (i.e., |Λ(w)| = 1), and in that case w will receive M with probability 1. The second case is when 1 < |Λ(w)| ≤ Ψ /8. The probability that there exists a phase with exactly one of w ’s neighbors active is the same as the probability that after placing |Λ(w)| balls uniformly at random into Ψ bins, there exists a bin with exactly one ball, which is at least 1 − 1/Ψ by claims (C1) and (C2) of Lemma 2.2, since Ψ ≥ 64. The third case is when |Λ(w)| > Ψ /8. The probability that more than Ψ /16 phases are active is the same as the probability that after placing uniformly at random at least Ψ /8 balls into Ψ bins, at least ⌊Ψ /16⌋ + 1 bins are

 2

by claim (C3) of Lemma 2.2. In each active phase the probability that w receives occupied, which is at least 1 − 2e Ψ the message is at least 31 / 48, by Lemma  2.1. Combining both probabilities, we get that w receives M with probability 

 2e 2

− (17/48)⌊Ψ /16⌋+1 ≥ 1 − 1/Ψ , when Ψ ≥ 64. (The last inequality holds since 1/Ψ ≥ 1/75 ≥  2  2  2 +(17/48)5 > 2e +(17/48)⌊Ψ /16⌋+1 when 64 ≤ Ψ ≤ 75, and 1/Ψ ≥ 2e +(17/48)Ψ /16 ≥ 2e +(17/48)⌊Ψ /16⌋+1 64 Ψ Ψ Ψ when Ψ > 75.)  at least

1−

Ψ

 2e 2

Now we present our k-shot broadcast algorithm, named Broadcast. In this algorithm, once a node receives the source message M, it executes Procedure Epoch(M , Ψ , T ) k times, starting on rounds corresponding to t ≡ 1 mod Ψ T . The formal code of Algorithm Broadcast is presented in Fig. 3. Relying on Lemma 2.3, we get the following. Lemma 2.4. Let Ψ ≥ 64 and fix the parameter k. Algorithm Broadcast(M , k, Ψ ) is a k-shot algorithm, and it completes broadcasting on an n-node network of diameter D in O((D + min{D · k, log n}) · Ψ T ) rounds with probability at least (1 − 1/n)(1 − n(1/Ψ )k ).

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Fig. 3. Algorithm Broadcast.

Proof. Clearly, each node transmits at most k times (in fact, a node receiving the message transmits exactly k times), hence Algorithm Broadcast(M , k, Ψ ) is a k-shot algorithm. We now prove that Algorithm Broadcast completes broadcasting within the stated time with the claimed probability, by bounding the probability of the complementary event, namely, that not all the nodes receive the message M. The probability that there exists a node that did not receive M is the same as the probability that there exists a node w that did not receive M, but at least one of its neighbors did receive M. This probability, in turn is the same as probability that for at least k times, at least one of w ’s neighbors participated and executed Procedure Epoch, and in all those executions w did not receive the message, which is at most (1/Ψ )k . (Recall that by Lemma 2.3, the probability that w does not receive M while at least one of its neighbors executes Procedure Epoch is at most 1/Ψ .) Let E (w) be the event that w did receive M after at most k executions of Procedure Epoch by at least one of its neighbors which has already received the message M. By the union bound, the probability that the event E (w) did not occur for some node w is at most n · (1/Ψ )k . Regarding the time complexity of the algorithm, under the event E (w) for every w ∈ V (except the source), we have a trivial upper bound of O(D · k · Ψ T ) rounds. For D ≥ log n we have the following. Consider a node w ∈ V . Let di be a random variable measuring the length of the shortest path between w and the set of nodes which have received the source message M at the end of epoch i. At the beginning, the source has the message M, hence d0 ≤ D. By Lemma 2.3, we have

 di+1 =

di − 1 di ,

with probability 1 − 1/Ψ , if di > 0, otherwise.

Let R = 12 · max{D, log n} and let d′0 = D and d′i+1 =



d′i − 1 di , ′

with probability 1/2, otherwise.

The probability that w receives M after R rounds is at least the probability that dR = 0, which is at least the probability that d′R ≤ 0. Therefore, it is sufficient to bound from below the probability that d′R ≤ 0. Define a random variable Xi = d′i − d′i−1 , for every 1 ≤ i ≤ R, let X =

R

i =1

Xi and let µ = E [X ] = R/2 be the expectation on X . Hence,

Pr[dR ≤ 0] = 1 − Pr[X < D], ′

where Pr[X < D] ≤ Pr[X < µ/6], recalling that µ/6 ≥ D. Thus by applying Chernoff bound (cf. [23]), we have 2 Pr[X < µ/6] < e(−(5/6) µ/2) ≤ e(−2 log n) = 1/n2 ,

where the last inequality holds since µ = R/2 ≥ 6 log n. Taking the union bound on all nodes, we get that with probability at least 1 − 1/n all the nodes did receive M after O(RΨ T ) rounds. Thus with probability at least (1 − 1/n)(1 − n(1/Ψ )k ) all the nodes of the network did receive the message M after O((D + min{D · k, log n}) · Ψ T ) rounds.  Letting T = log n + 5 and

Ψ =



max{64, n2 }, max{64, ⌈2n1/(k−1) ⌉},

k = 1, k ≥ 2,

for a fixed parameter k, Lemma 2.4 yields the following. Theorem 2.1. Algorithm Broadcast(M , k = 1, max{64, n2 }) is a 1-shot algorithm and it completes broadcasting in O((D + min{D, log n}) · n2 log n) rounds with probability at least 1 − 2n2 . For k ≥ 2, algorithm Broadcast(M , k, max{64, ⌈2n1/(k−1) ⌉}) is a k-shot algorithm and it completes broadcasting in O((D + min{D · k, log n}) · n1/(k−1) log n) rounds with probability at least 1 − n−1/(k−1) . 2.2. A fast k-shot broadcasting protocol We now show that for any constant integer parameter δ ≥ 1, setting k = ⌈(log n)/δ⌉ and Ψ = 4δ , Algorithm Broadcast(M , k, Ψ ) is an O(log n)-shot algorithm that completes broadcasting in O(D log n + log2 n) · Ψ = O((D log n) + log2 n) rounds with probability at least 1 − 2/n. For δ > 2, this follows immediately by Lemma 2.4. That lemma does not apply for δ = 1, 2, since Ψ must be at least 64 for Lemma 2.3 to apply. Thus, for δ = 1, 2, we analyze the success probability of Procedure Epoch directly (using Lemma 2.2).

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9

Remark 2.1. Consider a node w ∈ V and some integer z ≥ 0. If a nonempty subset of w ’s neighbors executes Procedure Epoch(M , 4δ ) during the time interval [z · 4δ T + 1, (z + 1) · 4δ T ] and they all start at time t = z · 4δ T + 1, then the probability that w receives the message M during this time interval is at least 1 − 1/4δ , for δ = 1, 2. Proof. Assume, without loss of generality, that z = 0. Let Λ(w) be the subset of w ’s neighbors that execute Procedure Epoch during t = 1, 2, . . . , 4δ T . Recall that (P1) if there exist more than |Λ(w)|/2 phases that are active with respect to w , then w will receive M with probability 1; and (P2) if exactly z phases are active with respect to w , then by Lemma 2.1, w will receive M with probability at least 1 − (17/48)z . We consider the following cases. If |Λ(w)| = 1, then w will receive M with probability 1. If |Λ(w)| = 2, 3, then trivially, with probability at least 1 − 1/4δ , for δ = 1, 2, there will be at least two active phases with respect to w . This implies by property (P1) that w will receive M with probability at least 1 − 1/4δ , for δ = 1, 2 as well. For |Λ(w)| ≥ 4 and δ = 1, the probability that at most two phases are active with respect to w is Pr[there are at most two active phases w.r.t. w] =

   |Λ(w)| 4

1

2

2

=

6 2|Λ(w)|

.

(15)

For |Λ(w)| = 4 (respectively, |Λ(w)| = 5), the probability of a failure (that w will not receive M) is bound from above by the probability that there exists at most two active phases with respect to w and in the first active phase (among, either, one or two such active phases) there was a failure. By Ineq. (15), for |Λ(w)| = 4 (resp., |Λ(w)| = 5), with probability 6/16 (resp., 6/32) at most two phases are active. By property (P2), the probability that w will not  receive M in the first active 6 17 17 6 phase is at most 17/48. Therefore, the probability that w will receive M is at least 1 − 16 (resp., 1 − ), which is 48 32 48 greater than 1 − 1/4. For δ = 2 and |Λ(w)| ≥ 4, the probability that at most two phases are active with respect to w is Pr[there is at most two active phases w.r.t. w] =



  |Λ(w)|

16

1

2

8

=

120 8|Λ(w)|

.

(16)

We first consider |Λ(w)| = 4 (resp., |Λ(w)| = 5). By Inequality (16) for |Λ(w)| = 4 (resp., |Λ(w)| = 5), with probability 120/84 ≤ 1/16 (resp., 120/85 ≤ 1/16) at most two phases are active. Thus, in both cases, the probability that at least three phases are active (with respect to w ) is at least 1 − 1/16. Hence, by property (P1), w will receive M with probability at least 1 − 1/16 as well.

16  6

Finally, for |Λ(w)| ≥ 6, the probability that at least three phases are active is at least 1 − 2 · 81 and by property (P2), the probability that w receives M under that event  is at least 1 − (17/48)3 . Combining these probabilities, we get that w  receives M with probability at least 1 −

16  1 6   · 1 − (17/48)3 > 1 − 1/16. · 8 2



Theorem 2.2. Consider an n-node network. For constant δ , let k = ⌈(log n)/δ⌉ and Ψ = 4δ . Algorithm Broadcast(M , k, Ψ ) is an O(log n)-shot algorithm that completes broadcasting in O(D log n + log2 n)· 4δ = O((D log n)+ log2 n) rounds with probability at least 1 − 2/n. This result should be compared with the BGI algorithm [3], the fundamental broadcast algorithm for wireless networks of unknown topology. The BGI algorithm consists of 2 log n phases. In each phase, each node transmits on average twice, hence the expected number of transmissions per node is 4 log n and each node transmits at most 4 log2 n times. Note that for δ = 1, Algorithm Broadcast is fast as the BGI algorithm, and in addition it is energy efficient, as each node transmits at most log n times. 3. Near-optimal k-shot broadcasting in known topology Consider a radio network modeled as an n-node directed graph G(V , E ). In this section we present a (centralized) scheduling algorithm generating k-shot broadcast schedules of length O(kn1/2k log2+1/k n), which is almost optimal given the aforementioned lower bound of Ω (k · n1/2k ). Specifically, we first present a randomized algorithm named RandSchedule that produces broadcast schedules of length at most O(kn1/2k log1+1/k n) on bipartite graphs, and then use the technique of [13] to extend the algorithm to one applying to arbitrary graphs (paying an additional logarithmic factor in schedule length). The randomized algorithm RandSchedule applies to a bipartite graph B = (U , L, E ). Assume that all the nodes of U log n know the source message M and the goal is to deliver the message to all nodes of L. Assume that 2 ≤ k < 2 log log n . Note that

√

for k = 1 one can use the efficient deterministic 1-shot algorithm described in [13], obtaining schedules of length O( n). log n Moreover, for k ≥ 2 log log n one can execute the broadcast algorithm for unknown topology presented in Section 2.

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Our randomized scheduling algorithm improves upon the broadcast length of O(k · n1/(k−2) log n) achieved by Algorithm RandBroadcast (k) [13]. The bottleneck of that broadcast algorithm is nodes of degree at most 3. Our algorithm makes use of a technique of partitioning the schedule into sets in base k (see Section 3.1) that overcomes this bottleneck, by ensuring that all the nodes of degree at most 3 will receive the source message with probability 1. The section is organized as follows. In Section 3.1, we begin with a technical observation referring to partitions of a set into subsets Next, in Section 3.2, we present a scheduling algorithm that produces a broadcast schedule of  in base k. √ ˜ (k · k max{|U |, n}), only slightly improving on the previous algorithm of [13]. Later, in Section 3.3, we present length O a composition procedure, and in Section 3.4 we present our final algorithm, which constructs a broadcast schedule of the desired length, O(kn1/2k log1+1/k n). 3.1. Partitions of a set to subsets in base k In this subsection we describe a method for partitioning a set into subsets in base k, which will assist us in designing the scheduling algorithm. Let x be a positive integer of the form ψ k for integer ψ . For simplicity, denote [y] = {0, 1, . . . , y − 1} and [y]+ = {1, . . . , y} for any positive integer y, and denote X = [x] = {0, 1, . . . , x − 1}. Consider the representation in base ψ of an integer z ∈ X . Denote the ℓ’th digit of z in base ψ by iℓ (z ), i.e., iℓ (z ) = iℓ = ⌊ ψzℓ ⌋ mod ψ . Hence, z = ik−1 (z ) · ψ k−1 + ik−2 (z ) · ψ k−2 + · · · + i1 (z ) · ψ + i0 (z ). For a given function f : X → X , define a sequence of projection functions f0 , . . . , fk−1 , where fℓ : X → [ψ] yields the ℓ’th digit of the f value of the element of X . Namely, fℓ (z ) = iℓ (f (z )), for any z ∈ X and ℓ ∈ [k]. In addition, define a partition of X into a sequence of sets in base k with respect to f as a follows: j

X (f )ℓ = {z ∈ X | fℓ (z ) = j}. j

j

Let X (f , Z ) = {X (f )ℓ | X (f )ℓ ∩ Z ̸= ∅, ℓ ∈ [k] and j ∈ [ψ]} and f (Z ) = {f (z ) | z ∈ Z } for any subset Z ⊆ X and let j

fℓ (Z ) = {fℓ (z ) | z ∈ Z } for any subset Z ⊆ X and any ℓ ∈ [k]. Note that X (f , X ) = {X (f )ℓ | j ∈ [ψ], ℓ ∈ [k]} is a collection j

of k · ψ subset of X , and {X (f )ℓ | j ∈ [ψ]}, for any ℓ ∈ [k], is a partition of X into ψ disjoint subsets. We say that f : X → X (respectively, π : X → X ) is a random function (resp., permutation) if it is selected with uniform distribution from the set {f : X → X | f is a function } (resp., {π : X → X | π is a permutation}) of functions (resp., permutations) over X . Note that when f is a random function, so is fℓ for every ℓ ∈ [k], and conversely, if k−1 ℓ f0 , . . . , fk−1 : X → [ψ] are random functions, then their combined function f : X → X (such that f (z ) = ℓ=0 fℓ (z ) · ψ ) is random as well. Using Lemma 2.2 again, we have the following. Lemma 3.1. For integers ψ ≥ 64 and k ≥ 1, let x = ψ k and X = [x]. For a subset Z ⊆ X , we have: j

(C1) If |Z | ≤ 3, then there exist indices ℓ ∈ [k] and j ∈ [ψ] such that |X (π )ℓ ∩ Z | = 1.

j

(C2) If 4 ≤ |Z | ≤ ψ/8, then the probability that |X (π , Z )| > |Z |/2 (hence there exists a subset X (π )ℓ ∈ X (π , Z ) such that j

|X (π )ℓ ∩ Z | = 1) is at least 1 − (4e ) · x . (C3) If |Z | > ψ/8, then the probability that |X (π , Z )| > ψ/16 is at least 1 − (4e2 )k · x−2 . 2 k

−2

Proof. We begin by proving claim (C1). If |Z | = 1 then the assertion holds trivially. So suppose 2 ≤ |Z | ≤ 3, and let {z0 , z1 } ⊆ Z and zi′ = π (zi ) for i = 0, 1. There exists an index ℓ ∈ [k] such that iℓ (z0′ ) ̸= iℓ (z1′ ) (or equivalently, i (z ′ )

i (z ′ )

πℓ (z0 ) ̸= πℓ (z1 )), which implies, letting A0 = X (π )ℓℓ 0 and A1 = X (π )ℓℓ 1 , that z0 ∈ A0 \A1 and z1 ∈ A1 \A0 . Thus for |Z | = 2, both |A0 ∩ Z | = 1 and |A1 ∩ Z | = 1. Recall that A0 and A1 are disjoint, hence for |Z | = 3, the size of at least one of these sets is exactly one, yielding the claim. Now we prove claim (C2) for 4 ≤ |Z | ≤ ψ/8. Consider a random function f : X → X . The sets f (Z ) and π (Z ) are both random sets, and |f (Z )| ≤ |π (Z )| = |Z |. Therefore, it is clear that Pr[|πℓ (Z )| > p] ≥ Pr[|fℓ (Z )| > p],

for any ℓ ∈ [k] and 1 ≤ p < |Z |.

(17)

Consider an independent process of placing x balls with ID’s from the set X into x bins with ID’s from the set X , in which each ball is placed uniformly and independently with repetitions in a random bin. Let f (i) be the ID of the bin that contains ball i after placing all x balls in the above random process. The function f : X → X is randomly selected with uniform distribution from all functions over the set X . Let ℓ ∈ [k]. The size of fℓ (Z ) can be viewed as the number of occupied bins after placing |Z | ≤ ψ/8 balls into ψ bins, hence by claim (C2) of Lemma 2.2, Pr[|fℓ (Z )| > |Z |/2] > 1 −



2e

ψ

2

.

(18)

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11

Fig. 4. Algorithm RandSchedule− (B(U , L, E )).

As f is random (hence so are f0 , . . . , fk−1 ), the values |f0 (Z )|, |f1 (Z )|, . . . , |fk−1 (Z )| are independent, thus Pr[∃ℓ s. t. |fℓ (Z )| > |Z |/2] > 1 −



2e

2k

ψ

= 1 − (4e2 )k · x−2 ,

where the left inequality holds by inequality (18) and by the union bound and the right equality holds since ψ k = x. Combining this with inequality (17), with p = |Z |/2, we get that Pr[∃ℓ s. t. |πℓ (Z )| > |Z |/2] > 1 − (4e2 )k · x−2 .

(19) j

Note that |πℓ (Z )| > |Z |/2 implies the existence of an indices j ∈ [ψ] and ℓ ∈ [k] in which |X (π )ℓ ∩ Z | = 1, yielding claim (C2). Finally, we prove claim (C3), for |Z | > ψ/8. Let Z ′ ⊆ Z , where |Z ′ | = ψ/8. It is clear that |X (π , Z )| ≥ |X (π , Z ′ )|. Combining this with inequality (19), we get that Pr[|X (π , Z )| ≥ ψ/16] ≥ Pr[|X (π , Z ′ )| ≥ ψ/16] > 1 − (4e2 )k · x−2 , yielding claim (C3) and the lemma.



˜ (k · 3.2. A k-shot broadcast schedule of length O

 k

max{|U |,

√

n}) on bipartite graphs

Consider a bipartite graph B(U , L, E ), where U = {u0 , . . . , u|U |−1 }. In this subsection we present an algorithm named √ RandSchedule− , for finding a k-shot broadcast schedule whose length depends on max{|U |, n}. The algorithm consists of two stages. The first stage is centralized, and the second stage is local. The algorithm operates as follows. In the first (centralized) stage, initially set

ψ ← max

 k

  √  |U | , 64 2k n ,

x ← ψ k , X ← [x] and T ← 5 + log n. j

Next, select uniformity at random a permutation π : X → X . Define a collection of kψ subsets X (π )ℓ of X for every j ∈ [ψ] ψ−1 X (π )0ℓ , . . . , X (π )ℓ ,

and ℓ ∈ [k], where for every ℓ ∈ [k], is a partition of X into ψ disjoint subsets. In the second (local) stage, each node ui ∈ U selects k random numbers r0 (i), . . . , rk−1 (i) (similarly to the random selection in Procedure Phase of Section 2.1), where rℓ (i) is selected to be b ∈ {1, . . . , T − 1} with probability 2−b and j rℓ (i) = T otherwise (with probability 2−T +1 ), for every ℓ ∈ [k]. Finally, each subset X (π )ℓ is partitioned into T disjoint subsets X (π , r )ℓ,b according to the random number rℓ (i), i.e., X (π , r )ℓ,b ← {i | i ∈ X (π )ℓj and rℓ (i) = b}. j

j

j

The schedule is now defined as follows. Order the kψ T sets X (π , r )ℓ,b arbitrarily, getting the sequence X1 , . . . , Xkψ T . Now, for s = 1, . . . , kψ T , let all the vertices ui such that i ∈ Xs transmit simultaneously at round s. Hence the overall time required for this broadcasting schedule is kψ T rounds. Note that each vertex belongs to exactly k such subsets. The formal code of Algorithm RandSchedule− is described in Fig. 4. Using Lemma 3.1, we get: Lemma 3.2. The schedule S returned by Algorithm RandSchedule− maintains the following properties. (C1) S is a k-shot schedule.

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j

j

(C2) For every w ∈ L, there exists a transmitting set X (π , r )ℓ,b ∈ S such that |Γ (w) ∩ X (π , r )ℓ,b | = 1 (hence w receives the message), with probability at least 1 − 1/2n. (C3) Schedule S is broadcasting schedule with probability at least 1/2 . Proof. By Algorithm RandSchedule− , each node belongs to exactly k transmitting sets, hence the first claim holds. Hereafter, in the proof of this lemma, we simplify notations by identifying the vertex ui with its index i, using interchangeably as either the index or the corresponding vertex ui . Consider a node w ∈ L. Recall that Γ (w) ⊆ X . Now we prove the second claim. We consider two cases. Case 1: |Γ (w)| ≤ 8 log n. j For |Γ (w)| ≤ 3, by claim (C1) of Lemma 3.1, there exist indices j ∈ [ψ] and ℓ ∈ [k] such that |X (π )ℓ ∩ Γ (w)| = 1.

It remains to prove the claim for 4 ≤ |Γ (w)| ≤ 8 log n. Note that |Γ (w)| ≤ ψ/8 (since k <

log n 2 log log n

and x ≥ 64k

√

n,

which implies that 8 log n ≤ x1/k /8 = ψ/8). Hence by claim (C2) of Lemma 3.1, the probability that there exists a subset j j X (π)ℓ ∈ X (π , Γ (w)) such that |X (π )ℓ ∩ Γ (w)| = 1 is at least 1 − (4e2 )k · x−2 ≥ 1 − 1/4n (where the inequality holds √ k since x ≥ 64 n), yielding this case. Case 2: |Γ (w)| > 8 log n. By claims (C2) and (C3) of Lemma 3.1, Pr [|X (π , Γ (w))| > 4 log n] ≥ 1 − (4e2 )k · x−2 ≥ 1 − 1/4n.

(20)

Consider a subset Z ⊆ X such that Z ∩ Γ (w) ̸= ∅ and a random function r : X → [T ] , where r (i) is randomly set to be b ∈ {1, . . . , T − 1} with probability 2b and r (u) = T otherwise. Denote Zb = {i ∈ Z | r (i) = b} for any b ∈ [T ]+ . Let Λ(w) = Z ∩ Γ (w) and let Λb (w) = {i | i ∈ Λ(w), r (i) ≥ b}, for b = 1, . . . , T . Let I ∈ [T ]+ be the maximal index b for which Λb (w) ̸= ∅, i.e., I = max{b | Λb (w) ̸= ∅}. Similarly to the proof of Lemma 2.1, the probability that |ΛI (w)| = 1 is at least 31/48. Note that |ΛI (w)| = 1 means that |ZI ∩ Γ (w)| = 1. Thus, +

Pr ∃b∈[T ]+ s. t. |Zb ∩ Γ (w)| = 1 | Z ∩ Γ (w) ̸= ∅ ≥ 31/48.





j

j

(21) j

Let X (π )ℓ ∈ X (π , Γ (w)). By inequality (21), with Z = X (π )ℓ , Zb = X (π , r )ℓ,b and r = rℓ , we get that with probability j

at least 31/48 there exists an index b such that |X (π , r )ℓ,b ∩ Γ (w)| = 1. Combining these with the probability bound

of inequality (20), we get that with probability at least (1 − 1/4n) · (1 − (17/48)4 log n ) > 1 − 1/2n, there exist indices j b ∈ [T ]+ , ℓ ∈ [k] and j ∈ [ψ] such that |Γ (w) ∩ X (π , r )ℓ,b | = 1, yielding claim (C2) of the lemma. Finally, the schedule S is a broadcasting schedule if for every node w ∈ L there exists a transmitting set with exactly one of w ’s neighbors (which admits the delivery of the message to w ). There are at most n nodes in L, therefore, by claim (C2) of this lemma and by the union bound, the probability that all nodes of L have a transmitting set with exactly one neighbor (which implies that S is a broadcasting schedule) is at least 1 − |L|/2n > 1/2. 

√

The length of the schedule produced by Algorithm RandSchedule− depends on max{|U |, n}, but the size of U may ˜ (k · n1/k ). In order to reduce the schedule length to be linear in n. Thus the schedule length is bounded from above by O ˜O(k · n1/2k ), we next develop a composition procedure, presented in the next section, that reduces the number of nodes in U by merging some of them into larger ‘‘composed nodes’’, while maintaining some restrictions designed to ensure correctness and efficiency. Later, in Section 3.4, we show how to combine the ideas of Algorithm RandSchedule− and the composition procedure to design our final algorithm, named RandSchedule, that produces a broadcasting schedule of shorter length, ˜ (k · n1/2k ). O 3.3. The composition procedure Consider a bipartite graph B = (U , L, E ). In this subsection, we describe a composition procedure named Comp that transforms a bipartite graph B(U , L, E ) into another bipartite graph C = (U, L, E ), where L remains the same and U is a set of composed nodes. Each composed node consists of a subset of U, and together they partition U into disjoint subsets, i.e., V ∈U V = U and V ′ ∩ V ′′ = ∅ for every pair of composed nodes V ′ , V ′′ ∈ U. Consider a node w ∈ L. We say that there is an overlap in C with respect to w , if there exist two neighbors of w in U that were composed into the same node in U, i.e., there exists a V ∈ U such that |V ∩ ΓB (w)| ≥ 2. We use an integral shrinkage parameter ∆ > 1. The composition procedure maintains the following four properties. First, √ it ensures that |U| ≤ ∆ · |L|. Second, it ensures that if degB (w) ≤ ∆, then ensure that there is no overlap with respect to w in C , i.e., if degB (w) ≤ ∆, then |V ∩ ΓB (w)| ≤ 1 for every V ∈ U. Third, if degB (w) ≤ ∆, then Comp preserves the degree of w in C , (i.e., degC (w) = degB (w)) and if degB (w) > ∆, then an overlap with respect to w may occur, but the procedure ensures that degC (w) ≥ ∆. Fourth, it ensures that there exists an edge (V , w) ∈ E iff there exists a node u ∈ V such that (u, w) ∈ E, i.e., E = {(V , w) | ΓB (w) ∩ V ̸= ∅}. The composition procedure works as follows. First, initialize the set of composed nodes U to be a subset with exactly one node for each node in U, i.e., V0 ← {u0 }, . . . , V|U |−1 ← {u|U |−1 }. Then, update the edge set to satisfy the fourth property,

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Fig. 5. Procedure Comp .

i.e., E = {(Vi , w) | (ui , w) ∈ E }. Next, in a greedy manner, as long as there exist two composed nodes V ′ , V ′′ such that V ′ ∪ V ′′ does not create an overlap for any w of degree at most ∆ in C (i.e., degC (w) ≤ ∆), merge these nodes into a single composed node (deleting the other from U), and then update the set of edges, i.e., V ′ ← V ′ ∪ V ′′ , U ← U\{V ′′ } and E ← {(V , w) | ΓB (w) ∩ V ̸= ∅}. The formal code is described in Fig. 5. Lemma 3.3. The resulting graph C (U, L, E ) of Procedure Comp (B(U , L, E ), ∆) maintains the following properties.

√

(P1) |U| ≤ ∆ · |L|. (P2) There is no overlap in C with respect to nodes of degree at most ∆ − 1, i.e., |ΓC (w) ∩ V | ≤ 1 for any V ∈ U and any w ∈ L such that degC (w) < ∆. (P3) degC (w) ≥ min{degB (w), ∆}, with equality when degB (w) ≤ ∆. (P4) There is an edge (V , w) ∈ E if and only if ΓB (w) ∩ V ̸= ∅. Proof. First we prove (P1). Let C ′ (U, L′ , E ′ ) be a subgraph of C (U, L, E ) containing only nodes of degree at most ∆ in L with respect to C , i.e., L′ = {w ∈ L | degC (w) ≤ ∆√ } and E ′ = {(V , w) ∈ E | w ∈ L′ }. ′ ′ Assume to the contrary that |U| > ∆ · |L|. The total number of edges in E ′ is √|E | ≤ ∆ · |L | ≤ ∆ · |L|. These two inequalities imply that there exists a composed node V ∈ U such that degC ′ (V ) < |L|. Consider some node w ∈ ΓC ′ (V ). Then degC ′ (w) ≤ ∆, hence |ΓC ′ (w)\{V }| ≤ ∆ − 1, which implies that

         ≤1+  ′ |ΓC ′ (w)\{V }| ≤ (∆ − 1) |L| + 1 < |U|. Γ (w) C    w∈ΓC ′ (V ) w∈ΓC ′ (V ) This implies that there exists a composed node V ′ ∈ U such that ΓC ′ √ (V ) ∩ ΓC ′ (V ′ ) = ∅, which contradicts the halting condition of the loop in Procedure Comp (see step 3). Hence |U| ≤ ∆ · |L| and property (P1) holds. (P2) immediately holds by the stopping condition of the loop (see step 3). (P3) trivially holds at the initialization part (where degC = degB (w)), and subsequently, in each iteration, the degree of w may decrease by 1 whenever the degree of w is greater than ∆, which implies that degC (w) ≥ ∆ for every w ∈ L such that degB (w) > ∆, and degC (w) = degB (w) for every w ∈ L such that degB (w) ≤ ∆. Finally, (P4) holds by the way we update the set of edges (see step 3c).  3.4. A k-shot broadcasting schedule on bipartite graphs of length O(kn1/2k log1+1/k n) Now we are ready to design our final algorithm, named RandSchedule, that produces a k-shot broadcasting schedule of length O(kn1/2k log1+1/k n) on bipartite graphs, which is almost optimal. Consider a bipartite graph B(U , L, E ), where U = {u0 , . . . , u|U |−1 }. Let y = |U | and Y = [y]. Algorithm RandSchedule operates as follows. First we execute Procedure Comp on B(U , L, E ) with parameter ∆ = 8 log n and get a bipartite graph C (U, L, E ). Subsequently, Algorithm RandSchedule is similar to Algorithm RandSchedule− . In particular, we apply the centralized stage of Algorithm RandSchedule− on U by setting

ψ = max

 k

  √  |U| , 64 2k n ,

x = ψ k and X = [x].

Then we select (uniformly at random) a permutation π : X → X over the set X and define a collection of kψ subsets √ j X (π)ℓ of X for every j ∈ [ψ] and ℓ ∈ [k]. Note that y may be linear in n, but x is bounded by O((64k + log n) n), since √ |U| = O( n log n) by property (P1) of Lemma 3.3. Thus ψ = O(n1/2k log1/k n). In the second stage, similarly to the local stage of Algorithm RandSchedule− , each node ui ∈ U selects k random numbers, where rℓ (i) is randomly set to be b ∈ {1, . . . , T − 1} with probability 2−b and otherwise set to rℓ (u) = T , for every ℓ ∈ [k]. j Then, we define a collection of kψ subsets Y (π , r , B, C )ℓ ⊆ Y , where j

j

Y (π , r , B, C )ℓ = {i | ui ∈ Vl and l ∈ X (π )ℓ }, j

j

ψ−1

for every j ∈ [ψ] and ℓ ∈ [k]. For simplicity, denote Yℓ = Y (π , r , B, C )ℓ . Note that X (π )0ℓ , . . . , X (π )ℓ

is a partition of X into ψ disjoint subsets and

ψ−1 Yℓ , . . . , Yℓ , 0

, for any ℓ ∈ [k],

for any ℓ ∈ [k], is a partition of Y into ψ disjoint subsets.

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Fig. 6. Algorithm RandSchedule(B(U , L, E )). j

j

j

Finally, each subset Yℓ is partitioned into T disjoint subsets Yℓ,b = Y (π , r , B, C )ℓ,b , according to the random numbers rℓ (i), j

j

i.e., Yℓ,b ← {i | i ∈ Yℓ and rℓ (i) = b}. j The schedule is now defined as follows. Order the kψ T sets Yℓ,b arbitrarily, getting the sequence Y1 , . . . , Ykψ T . Now for s = 1, . . . , kψ T , let all the vertices ui such that i ∈ Ys transmit simultaneously at round s. Hence the over all time required for this broadcasting schedule is kψ T rounds. Note that each vertex belongs to exactly k such subsets. The formal code of Algorithm RandSchedule described in Fig. 6. Using Lemmas 3.1–3.3, we get: Lemma 3.4. Algorithm RandSchedule returns a k-shot broadcasting schedule of length O(kn1/2k log1+1/k n) with probability at least 1/2. Proof. It is clear that each node ui ∈ U belongs to exactly k transmitting sets, hence S is a k-shot schedule. The schedule length is bounded by kψ T = O(kn1/2k log1+1/k n), since ψ = O(n1/2k log1/k n) and T = O(log n). Hereafter, in the proof of this lemma, we simplify notation by identifying the vertex ui ∈ U (resp., Vl ∈ U) with its index i, (resp., l), using interchangeably either the index i (resp. l) or the corresponding vertex ui (resp., Vl ). Consider a node w ∈ L. Recall that ΓB (w) ⊆ Y and ΓC (w) ⊆ X . We prove that with probability at least 1 − 1/2n, there j exist indices ℓ ∈ [k], j ∈ [ψ] and b ∈ [T ]+ such that |Yℓ,b ∩ ΓB (w)| = 1, hence w receives the message. We consider two cases. Case 1: |ΓC (w)| < 8 log n. By claims (C1) and (C2) of Lemma 3.1, the probability that there exist indices ℓ ∈ [k] and j ∈ [ψ] such that |X (π)ℓj ∩ ΓC (w)| = 1 is at least 1 − (4e2 )k · x−2 ≥ 1 − 1/4n. Assume that this event occurred, i.e., X (π )ℓj ∩ ΓC (w) = {l}, for some indices ℓ ∈ [k] and j ∈ [ψ]. Thus, property (P2) of Lemma 3.3 implies that |Vl ∩ ΓB (w)| = 1 and property (P4) j of Lemma 3.3 implies that V ∩ ΓB (w) = ∅ for every V ∈ {Vl′ | l′ ∈ X (π )ℓj }, which implies that |Yℓ ∩ ΓB (w)| = 1, which j

implies the existence of a transmitting set Yℓ,b containing exactly one neighbor of w in B, proving the claim for this case. Case 2: |ΓC (w)| ≥ 8 log n. By claims (C2) and (C3) of Lemma 3.1, the probability that |X (π , ΓC (w))| ≥ 4 log n is at least 1 − (4e2 )k · x−2 ≥ 1 − 1/4n. I.e., Pr [|X (π , ΓC (w))| > 4 log n] ≥ 1 − 1/4n. j

(22)

j

j

Let X (π )ℓ ∈ X (π , ΓC (w)). Recall that X (π )ℓ ∩ ΓC (w) ̸= ∅ implies that Yℓ ∩ ΓB (w) ̸= ∅, by property (P4) of Lemma 3.3.

Therefore, by inequality (21) of Lemma 3.2, with Z = Yjℓ , Zb = Yℓ,b and r = rℓ , we get that with probability at least 31/48 j there exists an index b such that |Yℓ,b ∩ ΓB (w)| = 1. Combining these with the probability bound of inequality (22), we get that with probability at least (1 − 1/4n) · (1 − (17/48)4 log n ) > 1 − 1/2n, there exist indices b ∈ [T ]+ , ℓ ∈ [k] and j ∈ [ψ] such that |ΓB (w) ∩ Yℓ,j b | = 1, hence w receives the message. Thus the claim holds in case 2 as well. There are |L| ≤ n − 1 nodes in L, therefore by the union bound we get that with probability at least 1 − |L|/2n > 1/2 there exists a transmitting set containing exactly one neighbor for all nodes of L, which yields the lemma.  j

We can execute Algorithm RandSchedule repeatedly until getting a schedule that completes broadcast (failing with negligible probability after n attempts). Therefore, we have the following. Theorem 3.1. Algorithm RandSchedule is a randomized k-shot broadcasting algorithm that constructs (with high probability, in polynomial time) broadcast schedules of length at most O(k · n1/2k log1+1/k n) on bipartite graphs.

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Fig. 7. Example of a binomial graph B(x) for x = 4.

Moreover, the conclusion of [13] that any O(f (n))-time k-shot broadcasting scheme for bipartite graphs admits D + O(f (n) log n) time broadcast in arbitrary graphs of diameter D (see the relation between Theorem 4.9 and Corollary 4.10 therein) implies the following. Corollary 3.1. There exist a randomized algorithm for generating (with high probability, in polynomial time) k-shot broadcast schedules of length at most D + O(k · n1/2k log2+1/k n), in every radio network of size n and diameter D. 3.5. Lower bound We establish a lower bound of D + Ω (k · (n − D)1/2k ) for broadcasting in known topologies, improving on the lower bound of D + Ω ((n − D)1/2k ) presented in [13]. Namely, we show that there exist radio networks in which every k-shot broadcasting schedule requires to be of length at least D + Ω (k ·(n − D)1/2k ). To prove this, we first show that for any positive integer n there exists an n-node bipartite graph on which any k-shot broadcasting schedule requires to be of length at least Ω (k · n1/2k ). x Consider the binomial graph B(x) = ({s} ∪ U ∪ L, E ) presented in [13] (see Fig. 7). This graph contains n = x + 2 + 1 nodes, where U = {u1 , . . . , ux } and L = {wij | 1 ≤ i < j ≤ x}. The node s is connected to all the nodes in U, and each node wij ∈ L (for 1 ≤ i < j ≤ x) is connected to exactly two nodes ui and uj in U, i.e., E = {(s, u) | u ∈ U } {(ui , wij ), (uj , wij ) | 1 ≤ i < j ≤ x}. In the first step, the message is transmitted by s to reach all the nodes in U. Our analysis concerns the process by which the message is disseminated from the nodes of U to the nodes of L. Lemma 3.5. Consider a binomial bipartite graph B(x), where x positive integer. Then any k-shot broadcasting schedule for B(x) 1/k requires at least k·xe − e transmission rounds.

Proof. Let S = ⟨T0 , T1 , . . . , Tt ⟩ be a broadcasting schedule of length t + 1 for B(x), where T0 = {s} and ⟨T1 , . . . , Tt ⟩ are rounds in which the message is disseminated from the nodes of U to the nodes of L. Let for every 1 ≤ i ≤ x,

Ai = {ℓ | ui ∈ Tℓ },

be the set of rounds on which the node ui transmits. It is clear that |Ai | ≤ k for every 1 ≤ i ≤ x. Moreover, Ai ̸= Aj for any 1 ≤ i < j ≤ x (since equality between any two such sets implies that node wij will not receive the message, hence S will fail to complete broadcasting). Therefore, S is required to be long enough so that a construction of at least x different subsets of [t ]+ , each of size at most k, be feasible. Let A′i = Ai ∪ {t + j | 1 ≤ j and j ≤ k − |Ai |},

for every 1 ≤ i ≤ x.

Note that A′i ⊆ [t + k]+ , |A′i | = k and if Ai ̸= Aj , then clearly A′i ̸= A′j , for every 1 ≤ i < j ≤ x. Thus, number of different subset of size at most k of [t ]+ , which implies that

 e

t +k k

k

 ≥

t +k

1/k and thus t ≥ k·xe − e.

k



t +k k

is greater than the

≥ x,



Recall that the number of nodes in B(x) is n = lower bounds for bipartite graphs.

x 2

+ x + 1 ≥ x2 /2, thus t must be greater than

√

2k·n1/(2k) e

− e, yields our

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Theorem 3.2. There exist bipartite graphs of size n in which any k-shot broadcasting schedule requires Ω (k · n1/2k ) transmission rounds. Using the same argument as in the proof of Theorem 2.1 in [13], we come to the following conclusion. Corollary 3.2. There exist bipartite graphs of size n and diameter D in which any k-shot broadcasting schedule requires D + Ω (k · (n − D)1/(2k) ) rounds. References [1] A. Alon, A. Bar-Noy, N. Linial, D. Peleg, A lower bound for radio broadcast, J. Comput. System Sci. 43 (1991) 290–298. [2] C. Ambühl, An optimal bound for the mst algorithm to compute energy efficient broadcast trees in wireless networks, in: Proc. 32nd Int. Colloq. on Automata, Languages and Programming, ICALP, 2005, pp. 1139–1150. [3] R. Bar-Yehuda, O. Goldreich, A. Itai, On the time complexity of broadcast in radio networks: an exponential gap between determinism and randomization, J. Comput. System Sci. 45 (1992) 104–126. [4] P. Berenbrink, C. Cooper, Z. Hu, Energy efficient randomised communication in unknown adhoc networks, in: Proc. 19th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA, 2007, pp. 250–259. [5] I. Chlamtac, S. Kutten, On broadcasting in radio networks—problem analysis and protocol design, IEEE Trans. Commun. 33 (1985) 1240–1246. [6] I. Chlamtac, O. Weinstein, The wave expansion approach to broadcasting in multihop radio networks, IEEE Trans. Commun. 39 (1991) 426–433. [7] M. Chrobak, L. Ga¸sieniec, W. Rytter, Fast broadcasting and gossiping in radio networks, in: Proc. 41st Symp. on Foundations of Computer Science, FOCS, 2000, pp. 575–581. [8] A.E.F. Clementi, P. Crescenzi, P. Penna, R. Rossi, P. Vocca, On the complexity of computing minimum energy consumption broadcast subgraphs, in: Proc. 18th Symp. on Theoretical Aspects of Computer Science, STACS, 2001, pp. 12–131. [9] A.E.F. Clementi, A. Monti, R. Silvestri, Selective families, superimposed codes, and broadcasting on unknown radio networks, in: Proc. 22nd ACM-SIAM Symp. on Discrete Algorithms, SODA, 2001, pp. 709–718. [10] A. Czumaj, W. Rytter, Broadcasting algorithms in radio networks with unkown topology, in: Proc. 44rd IEEE Symp. on Foundations of Computer Science, FOCS, 2003, pp. 492–501. [11] M. Elkin, G. Kortsarz, Improved schedule for radio broadcast, in: Proc. 26th ACM-SIAM Symp. on Discrete Algorithms, SODA, 2005, pp. 222–231. [12] I. Gaber, Y. Mansour, Centralized broadcast in multihop radio networks, J. Algorithms 46 (1) (2003) 1–20. [13] L. Ga¸sieniec, E. Kantor, D.R. Kowalski, D. Peleg, C. Su, Time efficient k-shot broadcasting in known topology radio networks, Distrib. Comput. 21 (2) (2008) 117–127. [14] L. Ga¸sieniec, D. Peleg, Q. Xin, Faster communication in known topology radio networks, in: Proc. 24th ACM Symp. on Principles of Distributed Computing, PODC, 2005, pp. 129–137. [15] S. Guha, S. Khuller, Improved methods for approximating node-weighted steiner trees and connected dominating sets, Inform. Comput. 150 (1999) 57–74. [16] R. Klasing, A. Navarra, A. Papadopoulos, S. Perennes, Adaptive broadcast consumption (abc), a new heuristic and new bounds for the minimum energy broadcast routing problem, Networking (2004) 866–877. [17] D.R. Kowalski, On selection problem in radio networks, in: Proc. 24th ACM Symp. on Principles of Distributed Computing, PODC, 2005, pp. 158–166. [18] D.R. Kowalski, A. Pelc, Broadcasting in undirected ad hoc radio networks, in: Proc. 22nd ACM Symp. on Principles of Distributed Computing, PODC, 2003, pp. 73–82. [19] D.R. Kowalski, A. Pelc, Faster deterministic broadcasting in ad hoc radio networks, in: Proc. 20th Symp. on Theoretical Aspects of Computer Science, STACS, 2003, pp. 109–120. [20] D.R. Kowalski, A. Pelc, Optimal deterministic broadcasting in known topology radio networks, Distrib. Comput. 19 (2007) 185–195. [21] E. Kushilevitz, Y. Mansour, An ω(d log(n/d)) lower bound for broadcast in radio networks, SIAM J. Comput. 27 (1998) 702–712. [22] G. De Marco, Distributed broadcast in unknown radio networks, in: Proc. 29th ACM-SIAM Symp. on Discrete Algorithms, SODA, 2008, pp. 208–217. [23] M. Mitzenmacher, E. Upfal, Probability and Computing, Cambridge University Press, 2005. [24] P.J. Wan, G. Calinescu, X.Y. Li, O. Frieder, Minimum-energy broadcast routing in static ad hoc wireless networks, in: Proc. 20th Joint Conf. of the IEEE Computer and Communications Societies, INFOCOM, 2001, pp. 1162–1171.