Adhoc Affectanceselective Families for
Layer Dissemination
Abstract
Information dissemination protocols for adhoc wireless networks frequently use a minimal subset of the available communication links, defining a rooted “broadcast” tree. In this work, we focus on the core challenge of disseminating from one layer to the next one of such tree. We call this problem Layer Dissemination. We study Layer Dissemination under a generalized model of interference, called affectance. The affectance model subsumes previous models, such as Radio Network and Signal to InteferenceplusNoise Ratio. We present randomized and deterministic protocols for Layer Dissemination. These protocols are based on a combinatorial object that we call Affectanceselective Families. Our approach combines an engineering solution with theoretical guarantees. That is, we provide a method to characterize the network with a global measure of affectance based on measurements of interference in the specific deployment area. Then, our protocols distributedly produce an adhoc transmissions schedule for dissemination. In the randomized protocol only the network characterization is needed, whereas the deterministic protocol requires full knowledge of affectance. Our theoretical analysis provides guarantees on schedule length. We also present simulations of a real networkdeployment area contrasting the performance of our randomized protocol, which takes into account affectance, against previous work for interference models that ignore some physical constraints. The striking improvement in performance shown by our simulations show the importance of utilizing a more physicallyaccurate model of interference that takes into account other effects beyond distance to transmitters.
1 Introduction
The problem of disseminating information in adhoc wireless communication networks (for instance, embedded in the Internet of Things) has been studied in theory and practice. To reduce traffic, dissemination protocols often use a minimal subset of the available communication links, call it . Given that nodes communicate through radio broadcast, nodes may still receive through other links, but to provide performance guarantees only is assumed to be available, albeit taking into account the interference of the rest of the links.
When the dissemination task involves delivery to all nodes, defines a tree topology. (Since all nodes must be reachable but the set is minimal.) Either because there is a single source node (e.g. [11, 12]), or because packets are first aggregated at a single node for later dissemination (e.g. [10, 13]), the problem reduces to disseminate from a root to all other nodes through a Broadcast Tree. Moreover, as we observed in [11, 12], when packets are disseminated from layer to layer in a BFS fashion, the bottleneck for fast dissemination on broadcast trees occurs at layers with high interference. Indeed, we have shown in [11, 12] that in the long run throughput is a function of maximum layer interference, and it is independent of interference in paths where packets can be easily pipelined. This phenomenon has also been observed in other works, such as in the following fragment in [5].
In fact, if one has a fast way of transmitting one or more messages from one layer to the next, repeating this and using pipelining would yield a fast broadcast algorithm. Thus, the crux of the broadcast problem lies in how fast can this task be solved in bipartite graphs.
Thus, in this work, we focus on the core challenge of dissemination through one layer of a broadcast tree. We model such layer as a bipartite graph where (the transmitters) and (the receivers) are sets of nodes and is the set of links from to . We study the Layer Dissemination problem in assuming that initially all the transmitters have an identical piece of information, called message or packet indistinctively. To complete the task, all the nodes in have to receive the message.
We do not assume any underlying communication infrastructure. That is, transmitters attempt to deliver the message by radio broadcast but, if two or more nodes transmit at the same time, mutual interference may prevent reception of the message. To take into account this phenomenon, we study Layer Dissemination under a general model of interference called affectance. As in [11, 12] we parameterize affectance with a real value that represents the affectance of each transmitter on each link . An affectance model of interference from links on links was introduced by Kesselheim [8] in the context of link scheduling. Affectance is a general model of interference in the sense that comprises other particular models studied before (cf. [12]). Moreover, previous models do not accurately represent the physical constraints in realworld deployments. For instance, in the Radio Network model [2] interference from nonneighboring nodes is neglected, and Signal to InteferenceplusNoise Ratio (SINR) [6, 15] is a simplified model because other constraints, such as obstacles, are not taken into account.
Layer Dissemination is closely related to the combinatorial problem of computing selective families. The notion of selective families was introduced in [3] as a generalization of the dissemination problem in the Radio Network model to a combinatorial problem. Later in [4], Clementi et al. showed how to compute selective families adhoc, that is, for a given input family. The results are applicable to dissemination under the Radio Network model of interference when the topology is known.
In this work, we followup on [3] and [4] introducing the concept of Affectanceselective Families. That is, we generalize the dissemination problem in bipartite graphs also to a combinatorial problem, but taking into account the specific conditions to achieve a successful transmission under our generalized model of interference. Under certain conditions, we show the existence of families of subsets of that are affectanceselective for a given family of subsets of . We also present randomized and deterministic distributed protocols for Layer Dissemination based on those affectanceselective families, and we provide running time theoretical guarantees.
Our approach combines an engineering solution with theoretical guarantees. That is, we provide a method to characterize the network with a global measure of affectance based on measurements of interference in the specific deployment area. Then, our protocols distributedly produce an adhoc transmissions schedule for dissemination. The randomized protocol only requires knowledge of the network characterization (which could be hardwired), whereas the deterministic protocol requires full knowledge of the affectance values and it is computationally intensive. Similar approaches have been explored in practice, e.g. Conflict Maps (CMAP) [16], where nodes probe the network to build a map of conflicting transmissions.
In order to show the impact of a more accurate model of interference, we run simulations for a realworld deployment area. We compare the performance of our randomized protocol with previous protocols designed for the Radio Network and SINR models. Our experimental results expose a striking improvement in running time. Notably, this improvement does not come from algorithmic novelty, since all three protocols rely on transmitting with some probability, but from the careful choice of this probability as a function of the interference measured experimentally.
Roadmap
In Section 2 we overview previous related work. In Section 3 we specify the details of our models and the relation between affectanceselective families and dissemination in bipartite graphs. In Section 4 we specify the results obtained highlighting the novelty of our contribution. Section 5 contains our analysis and the protocols presented, and in Section 6 we present our simulation results.
2 Related Work
Before our work in [11, 12], the generalized affectance model was introduced and used only in the context of onehop communication, more specifically, to link scheduling by Kesselheim [8, 9]. He also showed how to use it for dynamic link scheduling in batches. This model was inspired by the affectance parameter introduced in the more restricted SINR setting [6]. They give a characteristic of a set of links, based on affectance, that influence the time of successful scheduling these links under the SINR model. In [11, 12], we generalized this characteristic, called the maximum average treelayer affectance, to be applicable to multihop communication tasks such as broadcast, together with another characteristic, called the maximum path affectance.
Layer Dissemination is closely related to the combinatorial problem of computing selective families adhoc for a given family of sets. The notion of selective families was introduced in [3] and it is defined as follows. Given any set of items , a family of subsets of is called selective for the set if and only if for any , such that , there is a set satisfying . Here, we introduce the concept of affectanceselective families, taking into account the specific conditions to achieve a successful transmission under affectance.
With respect to selective families, our work can be seen as an extension of [4] to affectance. Indeed, in [4], Clementi et al. showed how to compute selective families adhoc for a given family. That is, their algorithm can be used for dissemination under the Radio Network model. For instance, the input families can be seen as the different subsets of nodes that may be active at a given time, or as in Layer Dissemination as the subsets of transmitters connected to each receiver. Here, we revisit this problem under affectance, that is, we show the existence of affectanceselective families (the precise notion is defined in Section 3), we present randomized and deterministic protocols to solve Layer Dissemination based on the affectanceselective families, and we analyze their performance.
3 Model and Problem
We model the network topology as a bipartite graph , where is the set of transmitters, is the set of receivers, , and is the set of links from to . That is, for every , we have and . For each , we denote by the set of links incoming to receiver , and by the set of transmitters of those links.
Following [12], we model the interference among transmissions with an affectance matrix
where is a real number in quantifying the interference that the transmitter introduces to the communication through link . We denote as the total affectance of a set of transmitters on a link (i.e., the sum of affectances on over all nodes in ), and further, as the total affectance of a set of transmitters on a set of links (i.e., the sum of affectances of over all links in ). We do not restrict the affectance function, as long as its effect is additive; that is,
Without loss of generality we assume that time is slotted. Then, under the above affectance model, a successful transmission in a time slot is defined as follows. For any link , a transmission from is received at in time slot if and only if:

transmits in time slot , and

, where is the set of nodes transmitting in time slot .
The event of a nonsuccessful transmission, that is, when the affectance is at least , is called a collision. We assume that a node listening to the channel cannot distinguish between a collision and background noise present in the channel in absence of transmissions; in other words, the model is without collision detection.
Under the model above, the Layer Dissemination problem is defined as follows: for each node , must receive a successful transmission from some node in .
We define affectanceselective families as a purely combinatorial problem on a family of subsets of integers and a matrix of real numbers. (Refer to Section 2 for a definition of classic selective families.) The relation with Layer Dissemination is the following. For each receiver , consider the set of transmitters connected to . These sets of transmitters define a family of subsets of nodes in . On the other hand, for a given Layer Dissemination protocol, the schedule of transmissions from nodes in can also be viewed as a family of subsets of nodes. Specifically, for each time slot , the subset of nodes in transmitting in is a member of . In the Radio Network model, the family is called selective on the family if and only if for any there is some such that . This is because successfully receives a message if and only if exactly one node in transmits. Given an integer , consider a family of subsets of integers in . Let be a matrix of real numbers in defined on in such a way that for each there is a corresponding row, and for each and each there is a column in corresponding to the pair . Then, we say that a family of subsets of is affectanceselective on the family if for each there exists such that:

, and

for some it is .
We say that the family has length , and that each is affectanceselected, or simply selected for short.
In terms of Layer Dissemination, labeling the transmitters as well as the receivers with consecutive integers in , each is the subset of transmitters connected to receiver , is the affectance matrix, and each value in corresponds to the affectance of node on link . Then, the family is a solution for Layer Dissemination setting each node in set to transmit in time slot , for each .
4 Our Results
In this work, for a given family of subsets of integers in and a given affectance matrix , we first show the existence of a family of subsets of that is affectanceselective on . Under certain conditions on the relation between and , the family is proved to have a number of sets that is in . That is, at most logarithmic on and logarithmic on the maximum average affectance . The latter is a characterization based on and . Specifically,
The proof of that bound is existential because it is based on the probabilistic method (as in [4]). Nevertheless, it provides a method to derive algorithms for Layer Dissemination. We present two Layer Dissemination distributed protocols, one randomized and one deterministic. We show that both protocols have the same running time guarantee, which is asymptotically the same as the size of the affectanceselective family shown. That is, . The randomized protocol is Monte Carlo, it is very simple (a version of Decay [1]), and only requires knowledge of , , and two constants. The deterministic protocol (inspired on [4]) provides worstcase guarantees, but nodes need to know the topology and the affectance matrix , and its computational complexity is exponential.
We also include simulations to evaluate the impact of using a more accurate model of interference. We compare our randomized protocol with previous work for the Radio Network and SINR models. Our experimental results show a striking improvement in performance because the Radio Network protocol neglects interference from nonneighboring nodes, whereas SINR protocols do not take advantage of low interference from nodes that, although located at a short distance, are blocked by obstacles. Our results also show that for the particular inputs tested our randomized protocol performs better than predicted by our theoretical analysis.
5 Analysis
5.1 Existence of an Affectanceselective Family of Polylogarithmic Size
Theorem 1
For any , consider a family of subsets of integers in and any affectance matrix defined on . For each , let be the maximum average affectance on . If there exists a constant such that for all , then, there exists a family that is affectanceselective on , and its size satisfies
where is the maximum average affectance.
Proof.
We prove the claim using the probabilistic method. That is, we show a randomly generated family such that the probability that does not select some set in is strictly less than one.
Let be a subset of defined as follows. For each , independently include in with some probability (we will discuss the best choice for later). Let be a random variable indicating whether is in or not. Let be a random variable indicating whether is selected or not. The probability that is not selected given that some is in is
The above inequality is true because, for not to be selected, the affectance in all pairs such that and has to be at least one. The inequality is due to the righthand side possibly including events where some pairs have affectance less than one, but others have affectance larger than one so that the overall sum is still larger than . This righthand side can be bounded as follows using a Markovtype inequality that can be proved as in [14].
Replacing , for any , we get the following bound.
Then, we have that
(1)  
Consider now a family of subsets of where is obtained including each independently with probability for and . If , replacing in Equation 1 we have that for , which is strictly smaller than . Otherwise, if , we know that, for some , it is . Replacing,
Using that for some constant , we obtain
Replacing we get
To show that there is a positive probability that is selected, we show that for each constant there is a constant such that the latter is strictly smaller than as follows.
The left hand side is equal to for and monotonically decreasing for any such that , which is the range of for any .
Having proved that that there is a positive probability that is selected, we add a multiplicity on the sets to show that the probability that some is not selected is small, as follows.
We redefine as the family of subsets of where the set is obtained including each in independently with probability , for each and each .
Then, the probability that a given is not selected is , where is some constant as shown above. Using the union bound, the probability that some is not selected is , which is smaller than for some , showing the existence of an affectanceselective family of size .
The bound shown matches the bound for the Radio Network model in [4], because in our setting the number of subsets to select is , and in the Radio Network model it is .
5.2 Randomized Layer Dissemination Protocol
The proof of Theorem 1, showing the existence of an affectanceselective family, yields a Monte Carlo distributed randomized protocol for Layer Dissemination applicable to settings where the conditions of the theorem hold. I.e., there exists a constant bounding for each receiver . The protocol requires that all transmitters have knowledge of the maximum average affectance , the constant , the number of transmitters , and the constant computed in the proof of Theorem 1. The protocol, detailed in Algorithm 1, is a version of the Decay protocol [1] extended to the affectance model. Its correctness and running time are established in the following theorem.
Theorem 2
Consider a layer of a Radio Network with affectance matrix and topology , where , where for each receiver there is at least one transmitter such that . Then, if there exists a constant such that for all , where is the maximum average affectance on , Algorithm 1 solves the Layer Dissemination problem with high probability ^{1}^{1}1We say that an event occurs with high probability if it occurs with probability at least , for some constant ., and the running time is in , where is the maximum average affectance.
5.3 Deterministic Layer Dissemination Protocol
Algorithm 1 is simple and it is easily distributed because only requires knowledge of a few global parameters (namely , , and ), and also does not require intensive computations at each node. However, the running time guarantee is only stochastic. In this section we present a deterministic algorithm that provides the same running time guarantee but worstcase, although to implement it distributedly knowledge of the graph and the affectance matrix is required.
The ideas of algorithm [4] can be reused here to compute a transmission schedule that solves Layer Dissemination, but cannot be used asis because it does not cope with affectance or families of sets with different sizes. So, building upon the ideas of , we present in this section an algorithm for Layer Dissemination under the affectance model. That is, the transmission schedule is computed to cope with affectance, and without assuming anything about the number of neighbors of each receiver. We specify such protocol in Algorithm 2 and an explanation of the details follow.
The receivers pending to be selected (initially all) are partitioned in subsets so that, for each receiver , it is
The expectations in Lines 2 and 2 of the protocol correspond to the following. Recall that we assume the transmitters to be labeled by consecutive integers. That is, the set of transmitters is . Then, in Algorithm 2, for each time slot , we keep track of whether each node in transmits or not in an array of booleans , where index of the array is true if transmits in and false otherwise. The array is filled incrementally for as follows. For each index , let if , or otherwise. Likewise, let if , or otherwise.
Then, for each value of , taking into account the action of transmitters in that was already decided, we decide whether transmitter transmits or not in computing the expected number of receivers from a given subset that will be affectanceselected, if transmits and the actions of transmitters in is chosen at random with probability (Line 2). We do the same for the case that transmitter does not transmit (Line 2). The expectations are taken over the random choice of transmitters in . Such computation is feasible given that every transmitter is assumed to know and the affectance matrix . The specific computation of expectations is the following.
The calculation corresponds to the th iteration of the inner loop (Line 2) and probability for some . Let be an indicator variable defined as follows. The variable is random if , and deterministic otherwise. For each , if and only if . For each , with probability or with probability . Finally, it is to compute the expectation (Line 2) or to compute the expectation (Line 2). Also, let be a random variable indicating whether receiver is selected or not.
Then, it is
Where,
In the following theorem, we prove that each time the probability is updated to (Line 2), at least a constant fraction of receivers is selected, solving Layer Dissemination in a logarithmic number of steps.
Theorem 3
Consider a layer of a Radio Network with affectance matrix and topology , where , where for each receiver there is at least one transmitter such that . Then, if there exists a constant such that for all , where is the maximum average affectance on , Algorithm 2 solves the Layer Dissemination problem, and the running time is in , where is the maximum average affectance.
Proof. Algorithm 2 is correct as long as it terminates, as it does not stop until (Line 2). Then, to prove the claim, it is enough to prove the upper bound on the running time, which we do as follows.
Consider the execution divided in stages, where a new stage starts each time that is set to (Line 2 and Line 2). Moreover, consider each stage divided in rounds according to the value of . That is, starting from round when , a new round starts each time that and are updated in Lines 2 and 2. Thus, each stage is composed by rounds when respectively, and when becomes smaller or equal than , a new stage begins and is reset to in Line 2.
We show now that, in any given round , a constant fraction of receivers in is selected. Thus, a constant fraction of receivers is selected in each stage, which yields stages, each of rounds, proving the claimed running time.
Fix any given round when . We focus then on showing that a constant fraction of receivers in is selected, knowing that, for each receiver , if it is , and if it is .
We showed in the proof of Theorem 1 that, for any , if a subset is chosen including each with a probability , for such that , the probability of selecting with is a positive constant . The specific bound on is dependent on whether or not, but still a constant for both cases. This bound applies to round for any receiver and a subset of transmitters, each chosen with probability . Thus, the expected number of receivers selected by from would be , that is, a constant fraction . Let this expectation be denoted as , where each indicates whether .
Then, to complete the proof, now we show that the expected number of receivers selected from by the set of transmitters defined by the array after completing the loop in Lines 22 (which indeed is the actual number because no random choice is made in the last iteration) is at least . Indeed, we prove the stronger claim that for each iteration of the loop, which we show by induction on the iteration index . For clarity, we denote as . For , we have that
Given that , the claim is true. Now, assuming that the claim is true for iteration , we want to prove that for iteration , where
By inductive hypothesis we know that
(2) 
Call the expected number of receivers selected after we fix the value of in Line 2. That is,
Replacing in Equation 2, we have that
(3) 
We also have that
(4) 
6 Simulations
In this section we present our simulations, developed to evaluate the impact of a more accurate model of interference on Layer Dissemination. For that purpose, we run simulations for a realworld deployment area, comparing the performance of our randomized protocol with previous protocols designed for the Radio Network and SINR models. The details follow.
We used as a model of a network deployment area the floor plan of the Seidenberg School of Computer Science and Information Systems at Pace University, considering nodes installed in the intersections of each square of four ceiling panels (see Figure (a)a). To evaluate Layer Dissemination, we focused on one layer of this network going across various offices (see Figure (b)b). For simplicity, to evaluate performance as grows, we replicated the same office multiple times in a layer.
The walls of these offices have a metallic structure. Hence, each office behaves as a Faraday cage blocking radio transmissions (specially millimeter wave). Consequently, most of the radio waves propagate through doors (which are not metallic). We fixed the radio transmission power to be large enough to reach five grid cells, so that transmissions from layer to layer are possible. So, given the offices dimensions, transmitters within an office are connected to all receivers. On the other hand, the interference to other offices in the same layer is approximated by adding ten grid cells for each office of distance. The resulting topology can be seen in Figure (c)c, whereas the reason why affectance is more accurate than interference based on Euclidean distance is illustrated in Figure (d)d. For instance, it can be seen that transmitters that are close to a wall in one office have low affectance on links that are close to other side of that wall in the contiguous office, even though they are separated by only one gridcell in Euclidean distance.
Using the network topology and the resulting affectance matrix described above as input, and for , we simulated our randomized protocol in Algorithm 1, which requires knowledge of only global variables , , and . (Refer to Algorithm 1 for further details.) For comparison, we also simulated protocols designed for the Radio Network and SINR models of interference on the same inputs, but considering a transmission successful under the affectance model constraints, as defined in Section 3. We did not simulate our deterministic protocol in Algorithm 2 because the schedule computation has exponential complexity.
For the Radio Network model of interference, we simulated the classic Decay [1] protocol, whereas for SINR we simulated the Broadcast protocol in Algorithm 1 in [7]. (Most of the work for SINR is oriented to link scheduling, which cannot be accurately mapped to Layer Dissemination or Broadcast.) The former requires knowledge of global variable , which is the maximum indegree in the network, whereas the latter requires knowledge of global variables and , as defined in [7]. All three protocols provide guarantees on the number of rounds of communication needed to complete Broadcast, but running them for that fixed time would not provide any performance comparison. Instead, for each of the protocols we measured the number of rounds of communication passed until all receivers have received the message. In Algorithms 3 and 4 we specify how we adapted the Radio Network and SINR protocols respectively for our simulations. The results of the simulations are plotted in Figure 2 and analyzed in the following section.
7 Conclusions
As seen in the plot of Figure 2, our experimental results show a striking improvement in performance of our protocol with respect to Algorithms 3 and 4. Indeed, the running times of Algorithms 3 and 4 grow exponentially with (the scale of the y axis is logarithmic), whereas our algorithm’s running time grows exponentiallly slower. Moreover, the plot shows also the theoretical upper bound proved in Theorem 2. It can be seen that in these simulations our protocol performs better than the theoretical guarantees. This difference in performance could be due to an algorithmic improvement. However, at their core, all three algorithms are based on iteratively choosing to transmit with some probability. Thus, we conclude that the improvement is due to a careful choice of such transmission probability, making it a function of the network characteristic derived from the interference measured experimentally, rather than due to algorithmic novelty. This conclusion should not come as a surprise, given that Algorithm 3 was designed neglecting interference from nonneighboring nodes, whereas Algorithm 4 does not take advantage of low interference from nodes that, although located at a short distance, are blocked by obstacles. Therefore, the results of our experimental evaluation show the importance of studying information dissemination under more accurate models of interference.
References
 [1] Reuven BarYehuda, Oded Goldreich, and Alon Itai. On the timecomplexity of broadcast in multihop radio networks: An exponential gap between determinism and randomization. Journal of Computer and System Sciences, 45(1):104–126, 1992.
 [2] Imrich Chlamtac and Shay Kutten. Treebased broadcasting in multihop radio networks. IEEE Trans. Computers, 36(10):1209–1223, 1987.
 [3] Bogdan S. Chlebus, Leszek Gasieniec, Alan Gibbons, Andrzej Pelc, and Wojciech Rytter. Deterministic broadcasting in ad hoc radio networks. Distributed Computing, 15(1):27–38, 2002.
 [4] Andrea EF Clementi, Pilu Crescenzi, Angelo Monti, Paolo Penna, and Riccardo Silvestri. On computing adhoc selective families. In Proc. of the 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, volume 2129 of Lecture Notes in Computer Science, pages 211–222, 2001.
 [5] Mohsen Ghaffari, Bernhard Haeupler, and Majid Khabbazian. The complexity of multimessage broadcast in radio networks with known topology. CoRR, abs/1205.7014, 2012.
 [6] Magnús M. Halldórsson and Roger Wattenhofer. Wireless communication is in apx. In Proc. of the 36th International Colloquium on Automata, Languages and Programming, Part I, pages 525–536, 2009.
 [7] Tomasz Jurdzinski, Dariusz R. Kowalski, Michal Rozanski, and Grzegorz Stachowiak. Distributed randomized broadcasting in wireless networks under the sinr model. In Yehuda Afek, editor, DISC, volume 8205 of Lecture Notes in Computer Science, pages 373–387. Springer, 2013.
 [8] Thomas Kesselheim. Dynamic packet scheduling in wireless networks. In Proc. of the 31st Annual ACM SIGACTSIGOPS Symposium on Principles of Distributed Computing, pages 281–290, 2012.
 [9] Thomas Kesselheim and Berthold Vöcking. Distributed contention resolution in wireless networks. In Proc. of the 24th International Symposium on Distributed Computing, volume 6343 of Lecture Notes in Computer Science, pages 163–178. SpringerVerlag, Berlin, 2010.
 [10] Majid Khabbazian and Dariusz R. Kowalski. Timeefficient randomized multiplemessage broadcast in radio networks. In Cyril Gavoille and Pierre Fraigniaud, editors, PODC, pages 373–380. ACM, 2011.
 [11] Dariusz R. Kowalski, Miguel A. Mosteiro, and Tevin Rouse. Dynamic multiplemessage broadcast: bounding throughput in the affectance model. In 10th ACM International Workshop on Foundations of Mobile Computing, FOMC 2014, Philadelphia, PA, USA, August 11, 2014, pages 39–46, 2014.
 [12] Dariusz R. Kowalski, Miguel A. Mosteiro, and Kevin Zaki. Dynamic multiplemessage broadcast: Bounding throughput in the affectance model. CoRR, abs/1512.00540, 2015.
 [13] Fredrik Manne and Qin Xin. Optimal gossiping with unit size messages in known topology radio networks. In Workshop on Combinatorial and Algorithmic Aspects of Networking, pages 125–134. Springer, 2006.
 [14] Michael Mitzenmacher and Eli Upfal. Probability and Computing. Cambridge University Press, 2005.
 [15] Christian Scheideler, Andréa W. Richa, and Paolo Santi. An o(log n) dominating set protocol for wireless adhoc networks under the physical interference model. In Proceedings of the 9th ACM International Symposium on Mobile Ad Hoc Networking and Computing, pages 91–100. ACM, 2008.
 [16] Mythili Vutukuru, Kyle Jamieson, and Hari Balakrishnan. Harnessing exposed terminals in wireless networks. In Proceedings of the 5th USENIX Symposium on Networked Systems Design and Implementation, pages 59–72, 2008.