Radio Network Distributed Algorithms in the Unknown Neighborhood Model

Transcription

1 Radio Network Distributed Algorithms in the Unknown Neighborhood Model Bilel Derbel and El-Ghazali Talbi Laboratoire d Informatique Fondamentale de Lille LIFL Université des Sciences et Technologies de Lille France USTL Bilel.Derbel@lifl.fr, El-ghazali.talbi@lifl.fr Abstract. The paper deals with radio network distributed algorithms where initially no information about node degrees is available. We show that the lack of such an information affects the time complexity of existing fundamental algorithms by only a polylogarithmic factor. More precisely, given an n-node graph modeling a multi-hop radio network, we provide a Olog 2 n time distributed algorithm that computes w.h.p., a constant approximation value of the degree of each node. We also provide a O log n+log 2 n time distributed algorithm that computes w.h.p., a constant approximation value of the local maximum degree of each node, where the global maximum degree of the graph is not known. Using our algorithm as a plug-and-play procedure, we show that the local maximum degree can be used instead of to break the symmetry efficiently. We illustrate this claim by revisiting some fundamental algorithms that use as a key parameter. First, we investigate the generic problem of simulating any point-to-point interference-free message passing algorithm in the radio network model. Then, we study the fundamental coloring problem in unit disk graphs. The obtained results show that the local maximum degree allows nodes to self-organize in a local manner and to avoid the radio interferences from being a communication bottleneck. 1 Introduction Motivation and goals We consider a multi-hop network of radio devices modeled by a graph. We assume that the nodes of the graph are not yet organized in any specific manner, that is, we consider a newly deployed network where no specific communication structure is available. In such a network, it is crucial that initially the nodes self-organize in some accurate manner before any tasks could be conducted. For that purpose, nodes have to communicate with their neighbors and coordinate their computations. Compared to a classical pointto-point message passing network where any two neighboring nodes are related by a direct link allowing them to communicate safely, communications between neighbors in a radio network are subject to interferences, i.e., a message sent by a node may get lost due to interferences caused by other message sending. Coping A preliminary version of this paper appeared as an inria internal research report [6] The two authors are supported by the Equipe Projet INRIA DOLPHIN.

2 with these interferences is indeed one of the major obstacles to design efficient distributed algorithms for radio networks. In this paper, we focus on what knowledge about the network is required to avoid interferences while organizing the network nodes efficiently. More specifically, we study the impact of two parameters: the degree of each node and the maximum network degree, on designing time-efficient distributed algorithms for solving basic radio network tasks. These two parameters are in fact used as crucial ingredients to compute fundamental distributed structures in many past works, e.g., node/edge coloring [10, 11, 5], matching [2], single-hop emulation [3], broadcasting and gossiping [7, 1, 4], etc. Unfortunately, nodes in a newly deployed network could even not be aware of their own neighbors. Thus, efficient algorithms working under the assumption that no knowledge about node degrees is available are more likely to be applied in a real setting. The key challenge of this paper is to provide time-efficient algorithms working under an unknown neighborhood model, i.e., a model where no information about node degrees is available. Apart from being of a great importance in realistic settings, the unknown neighborhood model is also of a great theoretical interest. In fact, the time complexity of solving many basic tasks is tightly related to the degrees of nodes lower and upper bounds are often functions of those degrees. To illustrate this claim, let us assume a communication model where a node can hear a message from one neighbor in a given time slot if and only if no other neighbors are being sending in that time slot. It is obvious that if a given node v has to receive one message from each of its neighbors, then the degree of v is a time lower bound. Assuming that v s neighbors know v s degree, if each neighbor of v transmits with a probability inversely proportional to v s degree, then we can show that the latter lower bound is almost tight. Now if we consider the problem of broadcasting a message from every node to its neighbors, then it can be shown that tuning the sending probabilities according to the maximum degree of the network leads to almost time-optimal algorithms. These observations show that a knowledge about node degree can significantly affect the most primitive network tasks. More generally, tuning the sending probability of a node according to that knowledge appears to be very helpful to avoid interferences caused by simultaneous transmissions. It is in fact a key idea to coordinate node transmissions distributively and to break the symmetry efficiently. In this work, we show how to overcome the absence of such a knowledge when thinking and designing efficient distributed algorithms: i by giving a polylogarithmic time algorithm that estimates the degree of each node and ii by showing that the maximum degree of the network is not a bottleneck parameter for computing some fundamental structures such as coloring. The Model We consider a classical graph based radio network model. More precisely, the network is modeled by an n-node graph G = V, E. The nodes represent a set of radio devices. There is an edge between two nodes if the two nodes are in the transmission range of one another. Nodes can communicate by sending and receiving messages. For the sake of analysis, we assume that the nodes have access to a global clock generating discrete pulses slots. We

3 also assume that the nodes wake up synchronously, i.e., at pulse 0 all nodes are awaken. At each pulse, a node can transmit or stay silent. A node receives a message in a given pulse iff it is silent then and precisely one of its neighbors transmits. No collision detection mechanism is available to nodes in any way. We assume that nodes do not have any knowledge about the number of their neighbors. Nevertheless, we assume that an estimate ñ on the number of nodes n in the network is known. In practice, these assumptions can be interpreted as following: nodes know that they may have between 1 and ñ neighbors, but they do not know how many. As we will show, we remark that the running time of all our algorithms depends polylogarithmically on ñ, i.e., a rough estimate value ñ of n e.g., ñ = n O1 affects the time complexity by only a constant factor. Therefore, for a sake of clarity our algorithms are described for ñ = n. For every node v, we denote Nv the set of v s neighbors and d v = Nv the degree of v. We denote N + v = Nv {v}. The maximum degree of the network is denoted by = max {d v v V }. The local maximum degree of a node v is defined by v = max {d u u N + v}. We say that an event occurs with high probability w.h.p., for short if the probability that the event occurs is at least 1 1/n α for a positive constant α which can be made arbitrarily large by setting the corresponding parameters to large enough values. Contributions and Outline Our first contribution Section 2 is a Olog 2 n time distributed algorithm that computes w.h.p., a constant approximation value of the degree of each node. The idea of the algorithm is based on the claim that if the nodes transmit with the same probability p then the probability that a given node v hears a message from its neighbors is maximized for p = 1/d v. Thus, to decide on an estimate value of its degree, a node v changes its sending probability periodically and tries to detect the sending probability for which it hears the most often messages from its neighbors. Our second contribution Section 3 is a O log n+log 2 n time distributed algorithm that computes w.h.p., a constant approximation value v of the local maximum degree v of every node v. The idea of the algorithm is based on the claim that if all the neighbors of a given node v send with the same probability p = 1/d v then w.h.p., node v hears a message from each neighbor during a Od v log n time period. Thus, by combining this claim with our first algorithm, we are able to coordinate the nodes by periodically varying their sending probabilities so that every node can communicate its approximate degree to neighbors having a lower degree. The third result of the paper Section 4 is more general in the sense that it allows us to overcome the lack of information about the global maximum degree. Our analysis reveals that fundamental tasks using the knowledge of as a key assumption can still be scheduled efficiently when such a knowledge is not available. We illustrate this claim for three fundamental problems: i the SRS Single Round Simulation problem [1] which consists in simulating the rounds of a synchronous point-to-point message passing algorithm into the radio network model, ii the classical node coloring problem [10, 11] which consists in coloring the nodes such that every two neighbors have distinct colors iii a

4 variant of the maximum matching problem [5] which consists in computing a set of distance-2 pairwise disjoint edges. We revised these three problems under the assumption that no information about node degrees is available and we show that using an approximate value of the local maximum degree of each node is sufficient to obtain essentially the same time complexity. 2 From scratch approximation of node degrees 2.1 Description of the algorithm Input: An n-node graph G, an integer k = Θlog n, and a small constant β. e Output: A constant approximate value f d v of d v. for i = 1 to log n do c i := 0; p i := 1/2 i ; for j = 1 to k do v sends a ping message with probability p i; if v hears a ping message then c i := c i + 1 ; if c i e β k then f d v := 2 i ; Fig. 1: Algorithm InitNetwork; code for a node v. Algorithm InitNetwork Fig. 1 works in log n phases. At each phase i, every node transmits with a decreasing probability p i = p i 1 /2 for k = Θlog n rounds. At each phase, each node counts the number of times it hears a message variable c i. If that number exceeds a threshold β k then the node decides that its degree is order of 1/p i. Roughly speaking, we will show that if all the neighbors of a node v are sending with the same probability p i, then w.h.p., the number of times node v hears a message is roughly order of k when p i is roughly order of 1/d v. Thus, by varying geometrically the sending probability at each phase from 1/2 to 1/n, each node is likely to experience a phase where it hears order of k messages, and can decide on an approximate value of its degree. In order to obtain the w.h.p. correctness property of our algorithm, the precise value of β and k are to be fixed later in the analysis. 2.2 Analysis Fact 1 e.g., [10,11] x 1 and t x, e t 1 t2 x 1 + t x x e t. Fact 2 Chernoff and Hoeffding s Inequalities, e.g., [9] Let X 1,, X k be k independent Bernoulli trials, PX i = 1 = p = 1 PX i = 0. Let X = k i=1 X i, and EX = µ. For any ǫ and ǫ verifying 0 < ǫ < 1 and ǫ > 0, we have: PX < 1 ǫµ < e µǫ2 /2 and PX µ kε e 2kε 2.

5 Let c i v be the value of counter c i of node v at the end of a given phase i. Lemma 1. Given a node v V, we have Ec i v = k d v p i 1 p i dv. Proof. Due to lack of space, this proof is omitted see [6]. Lemma 2. There exist positive constants ν, β, ǫ > 0 such that: 1 For any node v and phase i such that i 1 log d v i, Ec i v β k, and 2 For any node v and phase i / [logd v 2, logd v + 4], Ec i v 1 ǫβ ν k. Proof. Due to lack of space, this proof is omitted see [6]. In the following, we set β = 1 ǫβ in algorithm InitNetwork where ǫ and β are the constants from the previous lemma. We also set k = α max { β/2ε 2, 1/2ν 2 } log n, where α 1 is any constant. We say that a node decides on the value of d v at phase i if the condition c i βk is true. Lemma 3. Given a node v and a phase i verifying i [logd v, logd v + 1], v decides on the value of d v at phase i with probability at least 1 1/n α. Proof. At phase i [logd v, logd v + 1], we have 1/p i 1 d v 1/p i. Thus, using the Chernoff s inequality, we have: P c i v < 1 εec i v Fact 2 exp ε2 2 Ec iv Lemma 2 exp ε2 2 β k 1 n α Now, using Lemma 2, for i [logd v, logd v + 1], we have Ec i v βk. Thus P c i v < 1 εβk = P c i v < βk 1/n α, and the lemma is proved. In the next lemma, we prove that w.h.p., a node v does not decide on the value of dv at a phase not in the range of log d v, i.e., w.h.p., the condition c i v βk does not hold for i not order of log d v. Lemma 4. Given a node v and a phase i verifying i / [logd v 2, logd v + 4], v does not decide on the value of d v at phase i with probability at least 1 1/n α. Proof. Take a phase i / [logd v 2, logd v + 4]. By Lemma 2, we have Ec i v β νk. Thus, using the Hoeffding s inequality, we have P c i v βk = P c i v Ec i v β Ec iv k k Fact 2 exp 2k β Ec 2 iv k Lemma 2 exp 2kν 2 1 n α

6 Theorem 1. In Olog 2 n time, algorithm InitNetwork outputs w.h.p., a constant approximation value of the degree of any node. Proof. The running time is straightforward. Using the two previous lemmas, a node v decides only at a phase i verifying i [logd v 2, logd v + 4] with probability at least 1 1/n α k. Thus all nodes decide on such a phase with probability at least : 1 1 k n α = 1 1 k n n α Fact n α 2 v V The last inequality holds for a sufficiently large constant α and k = Θlog n. 3 An algorithm for computing the local maximum degree Input: An n-node graph G and a parameter k = Θlog n. Output: A constant approximate value f v of v. Run algorithm InitNetwork and compute f d v; f v := f d v; i := 0; while i Θlog f d v do i := i + 1; p i := 1/2 i ; for j = 1 to k/p i do v sends message f d v with probability p i; if v hears a message d f w from node w and d f w > f v then f v := d f w; Fig. 2: Algorithm LocalMaxDegree; code for a node v. The idea of algorithm LocalMaxDegree Fig. 2 is to coordinate the sending probabilities so that each node can successfully send its approximate degree to neighbors having lower degrees. Given a node v, the crucial observation is that if all its neighbors are sending with probability 1/d v, then within Od v log n time, v will receive a message from each neighbor w.h.p.. The main difficulty is that a neighbor u of v does not know the degree of v. Thus, neighbor u cannot set its sending probability to the required value. Moreover, node u may have other neighbors with different degrees. Thus, node u should change its sending probability according to those neighbors. The key idea of our algorithm is to decrease geometrically the sending probability p i each Θ1/p i log n time period, from p i = 1/2 until reaching p i = Θ1/d v. Thus, at a phase i, nodes having a degree order of 1/p i can hear a message from neighbors having larger degrees. Note that nodes with low degree will finish their while loop early. Theorem 2. In O log n + log 2 n time, algorithm LocalMaxDegree outputs, w.h.p., a constant approximation of the local maximum degree of any node.

7 Proof. Let us consider a phase i and a node v verifying d v = Θ1/p i. Assuming that algorithm InitNetwork outputs the estimate value of the degree of each node, and from the initialization step line 2 of algorithm LocalMaxDegree, the condition of the while loop is still true for all nodes having degree at least order of 1/p i, i.e., neighbors of v with higher degree are still active at phase i. Let u be a neighbor of v with degree at least order of Θ1/p i. We denote u v j,i the event node v hears a message from neighbor u at a given iteration j of the for loop of phase i and u v,i the event node v hears a message from neighbor u at phase i. Thus, we have that: P u v j,i p i 1 p i dv 1 Thus, = P u v j,i Fact 1 p i e pidv 1 d v p 2 i = Θ 1 d v e Θ1 = Θ 1 d v 1 Θ 1 d v P u v,i k Θkdv 1 p 1 Θ i 1 1 Θ e Θk d v d v Thus, the probability that nodes having degree Θ1/p i do not receive a message from their neighbors with larger degree can be upper bounded as following: n Θ 1 e Θk 1 p i n Θ1 v V,d v=θ1/p i u Nv,d u>d v The last inequality holds for well chosen k = Θlog n. Thus, w.h.p., every node with degree order of 1/p i receives a message from each of its neighbors having a larger degree. We remark that we have implicitly assumed that for a node v with a degree order of 1/p i, there exist some neighbors having larger degree up to a constant. If this is not true, then from the initialization step in line 2, node v will decide that its local maximum degree is its own degree. Since the value d u sent by a node u is w.h.p. a constant approximation of u s degree, a simple verification shows that w.h.p., all nodes with a degree order of 1/p i succeeds computing their local maximum degree at phase i. This allows us to easily conclude that at the end of the algorithm, w.h.p., for every node v, v = Θ v. Since d v Θ for every node v, the time complexity is at most: Olog 2 n + Θlog i=1 Θlog k = Olog 2 n + k p i i=1 2 i = Olog 2 n + log n 2 log = Olog 2 n + log n

8 4 Applications 4.1 On simulating the message passing model The first application of the previous algorithms is on simulating the standard synchronous point-to-point message passing model on the graph-based radio model. This problem was first studied in [1]. Paraphrasing the introduction in [1], that study was motivated by the fact that whenever a type of communication mode emerges, new algorithms have to be developed for it for all standard network operations. Thus, simulation procedures could help to convert algorithms designed for networks with the same topology but different means of communication to algorithms for the new communication mode. In particular, since designing algorithms for radio networks from scratch turns to be a hard task, the simulation of algorithms for standard message-passing systems may prove to be a plausible approach. This motivation is still of interest since, compared to the advances gained by the research community in the classical message passing model, solving efficiently distributed tasks in a radio network model is still open for many fundamental problems. The work in [1] concentrated on round-by-round simulations where a separate phase of radio transmission is dedicated to simulate each single round of the original algorithm: in a single round of the original algorithm, it is assumed that each node can send a message to each neighbor. A general primitive called single-round simulation SRS as a building block is provided. The goal of this primitive is to ensure that every message passed by the original algorithm will be effectively transmitted and received during the simulation. We refer to this as the SRS problem. For the general message passing model where a node can send a different message to its neighbors at each pulse, a randomized distributed SRS algorithm with running time O 2 log n was given. For the so-called uniform message passing model where a node sends the same message to its neighbors at each pulse, the authors gave a randomized distributed SRS algorithm with O log n running time. The SRS algorithms in [1] use as a key parameter to schedule the sending probability of each node. Thus, these algorithms are impracticable in the unknown neighborhood model. By using our algorithms to compute an approximation of the local maximum degree of each node, and by locally using the computed value instead of, we can prove the following: Theorem 3. In the unknown neighborhood model, the nodes of any n-node graph can be initialized in O log n + log 2 n time so that the SRS problem can be solved w.h.p., for the uniform resp. general message passing model in O log n resp. O 2 log n time. Proof. First we consider a uniform point-to-point message passing algorithm A. For now let us assume that d v and v are known for each node v. Consider a round of A. To solve the SRS problem: each node v must send a message M v to all its neighbors during that round the message must be received by each neighbor. Consider the following SRS algorithm: each node v transmits message M v with probability 1/ v for r v rounds r v is to be fixed later.

9 Given a node v, let u v denote the event v hears the message M u in a single transmission round. Let u v denote the event v hears the message M u during the r v transmission rounds. Thus, we have: P u v = 1 u w N + v\{u} Using the fact that w N + v, w d v, we obtain 1 1 w P u v dv 1 Θ u d v u = P u v ru 1 ru 1 exp Θ Θ u u Hence, the probability that node u fails sending M u to at least one neighbor v satisfies P u v, v Nu P u v ru d u exp Θ u v Nu Thus by choosing r u = Θ u logd u n, node u succeeds sending M u to all its neighbors with probability at least 1 1/n Θ1. Thus, the SRS algorithm succeeds for all nodes with probability at least 1 1/n Θ1 n 1 1/n Θ1. Note that the constant hidden in the Θ notation when fixing r u is to be tuned carefully to obtain the w.h.p., property. In particular, a constant approximation value of u and d u affects r u by only a constant. Since algorithm LocalMaxDegree succeeds for all nodes w.h.p., it is not difficult to conclude that w.h.p., the SRS algorithm is successful for all nodes. Here, let us remark that the SRS algorithm has the nice property that sparse nodes are not penalized by dense regions, i.e., a node u terminates a SRS within r u = O u log n time. Now, let us consider the general point-to-point message passing model, i.e., at each round of the original algorithm, a node v can send a different message to each neighbor at most d v messages each round. It is easy to see that the previous SRS algorithm can be easily extended to this general case: each node v applies the previous SRS procedure each O v log n time period for each of the d v messages to be delivered. It is not difficult to prove that w.h.p., the messages are received within Od v v log n. However, this is not sufficient to guarantee the correctness of the SRS algorithm since a message sent from node u to node v can be received by another neighbor w of u. By concatenating nodes identifiers to messages, a node receiving a message can easily know if the message was for him. If unique identifiers are not available, then each node selects randomly an identifier from the { 1,, n O1}. Hence, it is easy to show that each node picks a unique identifier w.h.p. Then, each node sends its identifier to its neighbors using the uniform SRS procedure. Thus, w.h.p., in O log n time, each node can learn the identifiers of its neighbors.

10 Remark 1. In the correctness proof of algorithm LocalMaxDegree Theorem 2, we used the fact that nodes having a degree smaller than order of p i at a phase i have already finished computing their local maximum degree and so they are silent, i.e., only nodes having a degree larger than order of p i remain active and can transmit. This assumption is no longer true if the nodes begin running the simulation process described in Theorem 3 after terminating algorithm LocalMaxDegree. However, in our SRS procedure, each node v transmits with probability order of v. Thus, Equation 1 in the proof of Theorem 2 still holds when replacing p i by Θp i. The other inequalities given in Theorem 2 are then affected by only a constant and Theorem 2 still holds w.h.p. Corollary 1. Given a uniform resp. general point-to-point message passing algorithm A running on an n-node graph G in τ time τ = n O1, there exists a distributed algorithm that simulates w.h.p., A in the unknown neighborhood radio model in O log n τ + log 2 n time resp. O 2 log n τ + log 2 n time. Corollary 2. Consider a point-to-point message passing algorithm A running on an n-node graph G in τ time τ = n O1. Assume that at each round of algorithm A, each node sends the same message to a bounded number of neighbors and another message to the remaining neighbors. Then, there exists a distributed algorithm that simulates w.h.p., algorithm A in the unknown neighborhood radio model in O log n τ + log 2 n time. Consider for instance the local broadcasting problem where each node has to broadcast a message to its neighbors. Using the previous results it is straightforward that this problem can be solved w.h.p., in O log n + log 2 n time. This could be for instance compared with a randomized O log 3 n time protocol for the same problem in the SINR Signal-to-Interference-plus-Noise-Ratio radio model not the graph based model given recently in [8]. 4.2 On computing a coloring in unit disk graphs Node and edge coloring are one of the most important and fundamental tasks in distributed radio networks. In fact, coloring can be considered as a basic tool to initially organize unstructured wireless ad hoc and sensor networks. This is wellmotivated for instance by associating different colors with different time-slots in a time-division multiple access TDMA scheme. In the following paragraphs, we revised some coloring related problems in the unknown neighborhood model. Node coloring A correct node coloring is an assignment of a color to each node in the graph, such that any two adjacent nodes have different colors. In [10, 11], the authors provide an algorithm that produces w.h.p., a correct coloring with O colors in O log n time for unit disk graphs. That algorithm requires that is known to all nodes. In the conclusion of [10, 11], it was asked whether we can get rid of. Using our algorithms, we answer that question in the positive:

11 Proposition 1. In the unknown neighborhood model, there exists a distributed algorithm that produces w.h.p., a O node coloring within O log n + log 2 n time on any n-node unit disk graph. Proof Sketch. Due to lack of space, we only give the very general guidelines to prove the proposition. More specifically, we run the coloring algorithm of [10, 11] by using the approximated local maximum degree for each node instead of parameter. By carefully redefining the constants used in [10,11] according to our approximation factor, the reasoning used in [10, 11] can still be proved to hold w.h.p. In particular, we use some arguments from Remark 1 to ensure that the coloring process from [10, 11] does not interfere with the initialization step. One should however note that, at the same time and independently of our result [6], Schneider and Wattenhofer [12] came up with a new O + log 2 n time algorithm for -coloring without knowledge of. Partial strong edge coloring A correct edge coloring is an assignment of a color to each edge in the graph, such that any two adjacent edges have a different color. One can find many variants of the edge coloring problem. In particular, in [5], the authors considered the problem of computing a strong or a distance-2 edge coloring, that is the problem of assigning distinct colors to any pair of edges between which there is a path of length at most two. The algorithms described there are based on computing an approximate maximum strong matching by an algorithm originally described in [2]. Computing a strong maximum matching consists in computing a set, with maximum cardinality, of edges mutually at distance at least 2. In [5], it is proved that a O1-approximate solution to the strong maximum matching can be computed in Oρ log n time for unit disk graphs 1, where ρ denotes the time it takes to compute the active degree of each node. Following the same terminology than in [5], a node is said to be active in a given round if it decides to transmit. Then, the active degree of a node is the number of its neighbors that are active in the current round. Using our algorithms, we can show that ρ = Olog 2 n. Thus, we obtain the following: Proposition 2. In the unknown neighborhood model, there exists a distributed algorithm that produces w.h.p., a O1-approximate strong matching in Olog 3 n time on any n-node unit disk graph. 5 Conclusion In this paper, we have shown that computing in the unknown neighborhood model is up to a polylogarithmic factor similar to computing in the known neighborhood model. We have also shown that, for some distributed tasks, computing the local maximum degree of each node is sufficient to overcome the need to 1 In [5], it is also assumed that a node can hear collisions which is actually stronger than the assumptions of our model.

12 know the global parameter. Many open issues can however be investigated. For instance, although the correctness of our algorithms was proved analytically, it would be nice to give an experimental analysis and/or to study the impact of changing the values the constants used in our algorithms through simulation. Another interesting field of research is to derive new algorithms which are both time and energy efficient and to extend our algorithms to non graph based models and/or to multi-channel radio networks. References 1. Noga Alon, Amotz Bar-Noy, Nathan Linial, and David Peleg. On the complexity of radio communication. In 21 st annual ACM Symposium on Theory of computing STOC 89, pages , New York, NY, USA, ACM. 2. Hari Balakrishnan, Christopher L. Barrett, V. S. Anil Kumar, Madhav V. Marathe, and Shripad Thite. The distance-2 matching problem and its relationship to the mac-layer capacity of ad hoc wireless networks. IEEE Journal on Selected Areas in Communications, 226: , Reuven Bar-Yehuda, Oded Goldreich, and Alon Itai. Efficient emulation of singlehop radio network with collision detection on multi-hop radio network with no collision detection. In 3 rd International Workshop on Distributed Algorithms WDAG 89, DISC nowadays, pages 24 32, London, UK, Springer-Verlag. 4. Reuven Bar-Yehuda, Amos Israeli, and Alon Itai. Multiple communication in multihop radio networks. In 8 th ACM Symposium on Principles of distributed computing PODC 89, pages , Christopher L. Barrett, V. S. Anil Kumar, Madhav V. Marathe, Shripad Thite, and Gabriel Istrate. Strong edge coloring for channel assignment in wireless radio networks. In 4 th annual IEEE international conference on Pervasive Computing and Communications Workshops PERCOMW 06, page 106, Washington, DC, USA, IEEE Computer Society. 6. Bilel Derbel and El-Ghazali Talbi. Radio network distributed algorithms in the unknown neighborhood model. In Technical Report HAL, inria , RR- 6581, June Leszek Gasieniec, David Peleg, and Qin Xin. Faster communication in known topology radio networks. In 24 th symposium on principles of distributed computing PODC 05, pages ACM, Olga Goussevskaia, Thomas Moscibroda, and Roger Wattenhofer. Local Broadcasting in the Physical Interference Model. In International Workshop on Foundations of Mobile Computing DialM-POMC, Toronto, Canada, pages 35 44, August Michael Mitzenmacher and Eli Upfal. Probability and Computing, Randomized Algorithms and probabilistic analysis. Cambridge University Press, Thomas Moscibroda and Roger Wattenhofer. Coloring Unstructured Radio Networks. In 17 th ACM Symposium on Parallelism in Algorithms and Architectures SPAA 05, pages 39 48, July Thomas Moscibroda and Roger Wattenhofer. Coloring unstructured radio networks. Distributed Computing, 214: , Johannes Schneider and Roger Wattenhofer. Coloring unstructured wireless multi-hop networks. In 28 th Symposium on Principles of Distributed Computing PODC 09, pages , 2009.