Computing in Additive Networks with Bounded-Information Codes
aa r X i v : . [ c s . D C ] A ug Computing in Additive Networks with Bounded-Information Codes
Keren Censor-Hillel ∗ [email protected] Erez Kantor † [email protected] Nancy Lynch † [email protected] Merav Parter †‡ [email protected] July 31, 2018
Abstract
This paper studies the theory of the additive wireless network model, in which the receivedsignal is abstracted as an addition of the transmitted signals. Our central observation is thatthe crucial challenge for computing in this model is not high contention, as assumed previously,but rather guaranteeing a bounded amount of information in each neighborhood per round, aproperty that we show is achievable using a new random coding technique.Technically, we provide efficient algorithms for fundamental distributed tasks in additive net-works, such as solving various symmetry breaking problems, approximating network parameters,and solving an asymmetry revealing problem such as computing a maximal input.The key method used is a novel random coding technique that allows a node to successfullydecode the received information, as long as it does not contain too many distinct values. Wethen design our algorithms to produce a limited amount of information in each neighborhoodin order to leverage our enriched toolbox for computing in additive networks.
The main challenge in wireless communication is the possibility of collisions, occurring when twonearby stations transmit at the same time. In general, collisions provide no information on thedata, and in some cases may not even be distinguishable from the case of no transmission at all.Indeed, the ability to merely detect collisions (a.k.a., the collision detection model) gives additionalpower to wireless networks, and separation results are known (e.g., [30]).Traditional approaches for dealing with interference (e.g., FDMA, TDMA) treat collisions assomething that should be avoided or at least minimized [14, 25, 27]. However, modern codingtechniques suggest the ability to retrieve information from collisions. These techniques significantlychange the notion of collisions, which now depends on the model or coding technique used. Forexample, in interference cancellation [2], the receivers may decode interfering signals that are ∗ Department of Computer Science, Technion, Haifa 32000, Israel. Supported in part by the Israel Science Foun-dation (grant 1696/14). † CSAIL, Massachusetts Institute of Technology, MA 01239, USA. Supported in a part by NSF Award NumbersCCF-1217506, CCF-AF-0937274, 0939370-CCF, and AFOSR Contract Numbers FA9550-14-1-0403 and FA9550-13-1-0042. ‡ Merav Parter is also supported by Rothschild and Fulbright Fellowships. cancel them from the received signal in order to decode their intendedmessage. Hence, from this viewpoint, collision occurs only when neither the desired signal nor thethe interfering signal are relatively strong enough.In this paper, we consider the additive network model , in which colliding signals add up at thereceiver and are hence informative in some cases. It has been shown that such models approximatethe capacity of networks with high signal-to-noise ratio [3], and that they can be useful in thesesettings for various coding techniques, such as ZigZag decoding [12, 26], and bounded-contentioncoding [6]. While in practice there are limitations for implementing such networks to the full extentof the model, the above previous research shows the importance of understanding the fundamentalstrength of models that allow the possibility of extracting information out of collisions. In a recenttheoretical work [6], the problems of local and global broadcast have been addressed in additivenetworks, under the assumption that the contention in the system is bounded .The central observation of this paper is that in order to leverage the additive behavior of thesystem, what needs to be bounded is not necessarily the contention, but rather the total amountof information a node has to process at a given round. This observation allows us to extend thequantification of the computational power of the additive network model in solving distributedtasks way beyond local and global broadcast. Our key approach in this paper is not to assume abound on the initial number of pieces of information in the system, but rather guarantee a boundon the number of distinct pieces of information in a neighborhood of every vertex. We then usea new random coding technique, which we refer to as
Bounded-Information Codes (BIC) , in orderto extract the information out of the received signals. This allows us to efficiently solve variouscornerstone distributed tasks.
On the technical side, we provide efficient algorithms for fundamental symmetry breaking tasks,such as leader election, and computing a BFS tree and a maximal independent set (MIS), as well asalgorithms for revealing asymmetry in the inputs, such as computing the maximum. We also provideefficient algorithms for approximating network parameters by a constant factor. Our key methodsare based on enriching the toolbox for computing in additive networks with various primitives thatleverage the additive behavior of received information and our coding technique.
Main techniques:
The work in [6] introduced Bounded-Contention Codes (BCC) as the maintechnique. BCC allows the decoding of the XOR of any collection of at most a codewords, where a isthe bound on the contention. As mentioned, our key approach in this paper is not to assume a boundon the contention, but rather to make sure that the amount of distinct information colliding at anode at a given round is limited. Our main ingredient is augmenting the deterministic BCC codeswith randomization, resulting in Bounded-Information Codes. BIC allows successful decoding ofany transmission of n nodes sending at most O ( a ) distinct values altogether, with high probability.Randomization plays a key role in the presented scheme in two different aspects. First, thedrawback of the standard BCC code is that the transmission of the same message by an evennumber of neighbors is cancelled out. By increasing the message size by factor of O (log n ) andusing randomization, BIC codes add random “noise” to the original BCC codeword so that theprobability that two BIC messages cause cancellation becomes negligible.Another useful aspect of randomization is intimately related to the fact that our informationbounds are logarithmic in n . This allows for a win-win situation: if the number of distinct pieces2f information (in a given neighborhood) is small (i.e., O (log n )), the decoding is successful thanksto the BIC codes. On the other hand, if the number of distinct pieces of information is large (i.e.,Ω(log n )), there are sufficiently many transmitting vertices in the neighborhood which allows oneto obtain good concentration bounds by, e.g., using Chernoff bounds (for example, in estimatingvarious network parameters). It is noteworthy that our estimation technique bares some similarityto the well-known decay strategy [4] which is widely used in radio-networks. The key distinctionbetween the long line of works that apply this scheme and this paper is the dimension to whichthis strategy in applied. Whereas so-far, the strategy was applied to the time axis (e.g., in round i , vertex u transmits with probability 2 − i ), here it is applied to the information (or message) axis(e.g., vertex u writes the specific information in the i ’th block of its message with probability 2 − i ).This highly improves the time bounds compared to the basic radio model (i.e., the statistics arecollected over the multiple blocks of the message instead of over multiple slots).An immediate application of BIC is a simple logarithmic simulation of algorithms for networksthat employ full-duplex radios (where a node can transmit and receive concurrently) by nodes whohave only half-duplex radios (where a node either transmits or receives in a given round). Thisallows us to consider algorithms for the stronger model of full-duplex radios and obtain a translationto half-duplex radios, and also allows us to compare our algorithms to a message-passing setting.To make justice with such comparisons, we note that a message-passing setting not only does notsuffer from collisions, but also is in some sense similar to having full duplex, as a node receives andsends information in the same round.Note that in the standard radio model, collision detection is not an integral part of the modelbut rather an external capability that can be chosen to be added. In BIC, collision detectionis an integral part of the model, where collision now refers to the situation where the numberof distinct pieces of information exceeds the allowed bound. To avoid confusion, the collisiondetection in the context of BIC, is hereafter referred to as information-overflow detection . We showthat information-overflow can be detected while inspecting the received codeword, without the needfor any additional mechanisms. Symmetry breaking:
The first type of algorithms we devise are for various symmetry breakingtasks. The main tool in this context is the select-level function, SL , that outputs two randomvalues according to a predefined distribution. Every vertex v computes the SL function locally,without any communication. The power of this function lies in its ability to assign random levels tonodes, such that with high probability the maximal level contains at most a logarithmic numberof nodes (i.e., below the information bound of the BIC code), and the nodes in the maximal levelhave different values for their second random variable.The SL function allows us to elect a leader in O ( D ) rounds, w.h.p., where D is the diameterof the network. The elected leader is the node with the maximal pair of values chosen by the SL function. A by-product of this algorithm is a 2-approximation of the diameter, and the analysis isdone over a BFS tree rooted at the leader. We also show how to construct a BFS tree rooted atan arbitrary given node in O ( D ) rounds, w.h.p, by employing both the SL function and BIC.Apart from the above new algorithms, our framework allows relatively simple translations ofknown algorithms for solving various tasks in message passing systems into additive networks. Thisincludes Luby’s MIS algorithm [22], Schneider and Wattenhofer’s coloring algorithm [28], and ap-proximating the minimum dominating set of Wattenhofer and Kuhn [19], improving significantly We use the term with high probability (w.h.p.) to denote a probability of at least 1 − /n c for a constant c ≥ Approximations:
We design algorithms for approximating various network parameters. Weshow how to compute a constant approximation of the degree of a node, as well as a constantapproximation of the size and diameter of the network. (Our coding scheme only requires nodes toknow a polynomial bound N on the network size n .) Our algorithms naturally extend to solve themore general tasks of local-sum and global-sum approximations that have been recently consideredin [21]. Yet, the additive setting allows us to obtain much better bounds than those of [21]. Asymmetry revealing:
In addition to the above symmetry breaking algorithms, we show thatadditive networks also allow for fast solutions for tasks which do not require symmetry breaking,but rather already begin with inputs whose asymmetry needs to be revealed: we give an algorithmthat computes the exact maximal value of all inputs in the network in O ( D · log n/ log log n ) rounds,w.h.p. (in contrast, a 2-approximation for the maximal value can be computed within Θ( D ) rounds).We obtain this because our coding scheme allows us to perform a tournament at a high rate. Forexample, for single-hop networks, in each round only a O (log n ) fraction of the remaining competingvertices survive for the next round.In some sense, asymmetry revealing can be viewed as the counterpart of symmetry breaking.Clearly, if we compute the maximal input in the system then we can obtain a leader as a by-product.However, the opposite does not hold, and indeed in our leader-election algorithm mentioned abovewe significantly exploit the fact that the leader need not be predetermined, and use our new toolboxto obtain a leader within only O ( D ) rounds. First, we compare our results with previous theoretical work on the additive network model. Thework of [6] assumes a bound a on the contention in the system, i.e., there are at most a initialinputs in total in the network. The main method for obtaining global broadcast in the above workis random linear network coding, which can be shown to allow an efficient flow of information inthe system. However, this is what requires the bound on the contention. Our BIC coding methodbares some technical similarity to the approach of random linear network coding, but allows us torefrain from making assumptions on the total information present in the network.The aforementioned global broadcast algorithm requires O ( D + a + log n ) rounds. While thisalgorithm can be used to solve many of the problems that we address in this paper, such as electinga leader and computing the maximal input, it would require O ( n ) rounds, as for these problemsit holds that a can be as large as the total number of nodes in the network. In comparison, our O ( D )-round leader election algorithm is optimal, and our O ( D log n/ log log n )-round algorithmfor computing the maximal input is nearly-optimal, as O ( D ) is a natural lower bound for bothproblems, even in the message-passing model.It is important to mention that our algorithms use messages of size O (log n ). While a standardassumption might be that the message size is O (log n ) bits, this difference is far from renderingour results easy. In comparison, the global broadcast algorithm of [6] requires a message size of These are generalizations of degree-approximation and network-size approximation, respectively. ( a log n + ℓ ) bits for inputs of size ℓ and contention bounded by a . In our setting, we assume ℓ fits the message size (say, is logarithmic in n ), but since a can be as large as n , such a messagesize would be unacceptable. In addition, if we compare our results to algorithms for the muchless restricted message-passing setting, it is crucial to note that even unbounded message sizesdo not make distributed tasks trivial. For example, it is possible to compute an MIS in generalgraphs in O (log n ) rounds even with messages of size O (1) [23], but the best known lower boundis Ω(log ∆ + √ log n ) even with unbounded messages [17]. Recently, Barenboim at el. [5] showed arandomized MIS algorithm with O (log ∆ · √ log n ) rounds using unbounded messages.In appendix A, we overview results that address the same tasks as this paper in the standardradio network model and in the message-passing model. An additive network can be viewed aslying somewhere in between these two models, as it does suffer from collisions, but to a smallerextent. Nevertheless, while our coding methods assist us in overcoming collisions, the additivenetwork model is still subject to the broadcast nature of the transmissions, and therefore it ishighly non-trivial to translate algorithms for the message-passing setting that make use of theability to send different messages on different links concurrently. The related work overviewed inthe aforementioned appendix, include algorithms and lower bounds for various problems in radionetworks, such as the wake-up problem [9], MIS with and without collision detection [24, 30] orwith multiple channels [7], leader election [11], and approximation of local parameters [21], as wellas MIS algorithms for message passing systems [1, 22, 29] and lower bounds [17, 20]. The Additive Network Model: A radio network consists of stations that can transmit andreceive information. We address a synchronous system, in which in each round of communicationeach station can either transmit or listen to other transmissions. This is called the half-duplexmode of operation. Mainly due to theoretical interest, we also consider the full-duplex mode ofoperation which is considered harder to implement. We follow the standard abstraction in whichstations are modeled as nodes of a graph G = ( V, E ), with edges connecting nodes that can receiveeach other’s transmissions.In the standard radio network model, a node v ∈ V receives a message m in a given round if andonly if in that round exactly one of its neighbors transmits, and its transmitted message is m . Inthe half-duplex mode, it also needs to hold that v is listening in that round, and not transmitting.If none of v ’s neighbors transmit then v hears silence, and if at least two of v ’s neighbors transmitsimultaneously then a collision occurs at v . In both cases, v does not receive any message.Some networks allow for collision detection , where the effect at node v of a collision is differentfrom that of no message being transmitted, i.e., v can distinguish a collision from silence (despitereceiving no message in both). Other networks operate without a collision detection mechanism,i.e., a node cannot distinguish a collision from silence. It is known that the ability to detect collisionshas a significant impact on the computational power of the network [30].In contrast, in this paper, we study the additive network model , in which a collision of transmis-sions is not completely lost, but rather is modeled as receiving the XOR of the bit representation ofall transmissions. More specifically, we model a transmission of a message m by node v as a stringof bits. A node v that receives a collision of transmissions of messages { m u | u ∈ Γ( v ) } , receivestheir bitwise XOR, i.e., receives the message y = L u ∈ Γ( v ) m u . Here Γ( v ) is the set of neighbors of v . Note that the above notation does not distinguish between the case where a node u transmits5o that where it does not, because we model the string of a node that does not transmit as all-zero.The network topology is unknown, and only a polynomial upper bound N = n O (1) is known forthe number of nodes n . Throughout, we assume that each vertex v has a unique identifier id v inthe range [1 , . . . , n c ] for some constant c ≥
1. The bandwidth is O (poly log n ) bits per message. Bounded-Contention Coding (BCC):
Bounded-Contention Codes were introduced in [6] forthe purpose of obtaining fast local and global broadcast in additive networks. Given parameters M and a , a BCC code is a set of M codewords such that the XOR of any subset of size at most a is uniquely decodable. As such, BCC codes can leverage situations where the number of initialmessages is bounded by some number a , and can be used (along with additional mechanisms) forglobal broadcast in additive networks. Formally, Bounded-Contention Codes are defined as follows. Definition 2.1 An [ M, m, a ] -BCC-code is a set C ⊆ { , } m of size | C | = M such that for anytwo subsets S , S ⊆ C (with S = S ) of sizes | S | , | S | ≤ a it holds that L S = L S . Simple BCC codes can be constructed using the dual of linear codes. We refer the reader to [6] foradditional details and a construction of an [
M, a log
M, a ]-BCC code for given values of M and a . In this section we enrich the toolbox for computing in additive networks with the following threetechniques. The first is a method for encoding information such that it can be successfully decodednot when the number of transmitters in limited, but rather when the amount of distinct pieces ofinformation is limited (even if sent by multiple transmitters concurrently). The second techniqueis a general simulation of any algorithm for full-duplex radios in a setting of half-duplex radioswithin a logarithmic number of rounds. Finally, we show that we can detect whether the numberof distinct messages exceeds the given threshold.
Bounded-Information Codes (BIC).
Using BCC and randomization allows one to controlthe number of distinct pieces of information in the neighborhood. Let G = ( V, E ) be an n -vertex network and assume that all the messages are integers in the range [0 , n ]. We show thatfor a bandwidth of size O (log n ), one can use randomization and BCC codes to guarantee thatevery vertex v , whose neighbors transmit O (log n ) distinct messages (i.e., hence bounded piecesof information) in a given round, can decode all messages correctly with high probability (i.e.,regardless of the number of transmitting neighbors). Let C be an [ n, log n, log n ]-BCC code and x ∈ [0 , n ]. By the definition of C , the codeword C ( x ) = [ b , . . . , b k ] ∈ { , } k contains k = O (log n )bits. Due to the XOR operation, co-transmissions of the same value even number of times arecancelled out. To prevent this, we use a randomized code, named hereafter as a BIC code (or BICfor short) as defined next.
Definition 3.1
Let C be an [ n, log n, log n ] -BCC code. An [ n, c log n, log n ]-BIC code for C is arandom code C I defined as follows. The codeword C I ( x ) consists of k ′ = ⌈ c · log n ⌉ blocks, for someconstant c ≥ , each block is of size k = O (log n ) (the maximal length of a BCC codeword), and the i ’th block contains C ( x ) with probability / and the zero word otherwise, for every i ∈ { , . . . , k ′ } . The definition of the BIC code can be given for any bound a on the number of distinct values. Since we care formessages of polylogarithmic size, we provide the definition for specific bound a = O (log n ).
6n other words, for vertex v with value x , let m ( v ) = C I ( x ) be the message containing the BICcodeword of x and let m i ( v ) denote the i ’th block of v ’s message. Then, m i ( v ) = C ( x ) withprobability 1 / m i ( v ) = 0 k otherwise. Let m ′ ( v ) = L u ∈ Γ( v ) m ( u ) be the received messageobtained by adding the BIC codewords of v ’s neighbors. Then the decoding is performed by usingBCC to decode each block m ′ i ( v ) separately for every i ∈ { , . . . , k ′ } , and taking a union over alldecoded blocks. Lemma 3.2
Let V ′ ⊆ V be a set of transmitting vertices with values X ′ = S v ∈ V ′ Val ( v ) where | X ′ | = O (log n ) . For every v ∈ V ′ , let C Iv be an [ n, c · log n, log n ] -BIC code, for constant c ≥ . Let m ( v ) be the C Iv codeword of Val ( v ) . Then, the decoding of L v ∈ V ′ m ( v ) is successful with probabilityat least − /n c − .Proof. For every x ∈ X ′ , let V x = { v ∈ V ′ | Val ( v ) = x } be the set of transmitting verticesin V ′ with the value x . For x ∈ X ′ and i ∈ { , . . . , k ′ } , let V ix = { v ∈ V x | m i ( v ) = C ( x ) } bethe set of vertices v whose i ’th block m i ( v ) contains the codeword C ( x ). We say that block i is successful for value x ∈ X ′ , if | V ix | is odd (hence, the messages of V x are not cancelled out in thisblock). Let M i ⊆ X ′ be the set of values for which the i ’th block is successful, and let V ′ i containone representative vertex with a value in M i . We first claim that with high probability, every value x ∈ X ′ has at least one successful block i x ∈ { , . . . , k ′ } . We then show that the decoding of this i x ’th block is successful. The probability that the i ’th block is successful for x is 1 / i ∈ { , . . . , k ′ } . By the independence between blocks, the probability that x has no successful blockis at most 1 /n c . By applying the union bound over all m ≤ n distinct messages, we get that withprobability at least 1 − /n c − , every value x ∈ X has at least one successful block i x in the message.Let m ′ = L v ∈ V ′ m ( v ) be the received message and let m ′ i be the i ’th block of the received message.It then holds that m ′ i = L v ∈ V ′ m i ( v ) = L v ∈ V ′ i m i ( v ) . To see this, observe that the values witheven parity in the i ’th block are cancelled out and the XOR of an odd number of messages withthe same value C ( x ) is simply C ( x ). Since m ′ i corresponds to the XOR of | V ′ i | = O (log n ) distinctmessages, the claim follows by the properties of the BCC code. (cid:3) In our algorithms, the messages may contain several fields (mostly a constant) each containing avalue in [0 , n c ] for some constant c ≥
1. To guarantee a proper decoding on each field, the messagesare required to be aligned correctly. For example, a message containing ℓ fields where the i ’th fieldcontains x i ∈ [0 , n ] is split evenly into ℓ blocks and all bits are initialized to zero. The BIC codewordof x i , denoted by C I ( x i ), is written at the beginning of the i ’th block. Hence, when the messages areadded up, all codewords of a given block are added up separately. To avoid cumbersome notation,a multiple-field message is denoted by concatenation of the BIC codewords of each field, e.g., thecontent of a two-field message containing x and x is referred as C I ( x ) ◦ C I ( x ), where formallythe message is divided into two equi-length blocks and C I ( x ) (resp., C I ( x )) is written at thebeginning of the first (resp., second) block. From full-duplex to half-duplex.
The algorithms provided in this paper are mostly concernedwith the full-duplex setting. However, in the additive network model, one can easily simulate a full-duplex protocol P f by half-duplex protocol P h with a multiplicative overhead of O (log n ) roundswith high probability, as explain below.Consider a full-duplex protocol P f in the additive network model. We will describe a half-duplexsimulation of P f , denoted by P h . A round t is said to be successful for vertex v , if v can decode7ll messages it receives in this round. With BIC codes, a round is successful if v ’s neighbors sent O (log n ) distinct pieces of information. Lemma 3.3
Each round of a full-duplex protocol P f can be simulated by half-duplex radios using O (log n ) rounds, w.h.p. That is, if t is a successful round for v in P f , then v receives all the piecesof information sent to it in this round in P f , within O (log n ) rounds in P h , w.h.p.Proof. Consider round t and let S t be the set of transmitting vertices in P f . Phase t in the half-duplex protocol P h consists of O (log n ) rounds. In each such round, every vertex v ∈ S t chooses tolistens or to transmits (if needed) with equal probability. We show that if round t is successful forvertex v in P f , then phase t is successful for vertex v in P h , with high probability.Let V t be the set of vertices for which t was a successful round in P f . Since each vertex v ∈ V t receives O (log n ) distinct messages in round t in P f , it implies that there are O ( n log n )communication links ( u, v ) ∈ E that need to be satisfied in round t . In each of the O (log n ) roundsin phase t , u transmits and v listens, with probability 1 /
4. Since the set of transmitting stations ineach round is a subset of S t (i.e., v ∈ V can successfully decode when all the vertices in S t transmit), v decodes u ’s message with probability 1 / t contains O (log n ) rounds,by a Chernoff bound the probability that v did not decode u ’s message in any of these rounds is atmost 1 /n c . The claim holds by applying the union bound over all O ( n log n ) required links. (cid:3) Information-Overflow Detection.
In the standard radio model, a collision corresponds to thescenario where multiple vertices transmit in the same round to a given mutual neighbor. In anadditive network, this may not be a problem, since with BIC codes, the decoding is successful aslong as there are O (log n ) distinct pieces of information in a given neighborhood. In this section, wedescribe a scheme for detecting an event of information-overflow. Our scheme is adapted from thecontention estimation scheme of [6], designed for the setting of detecting whether there are morethan a certain number of initial messages throughout the network. In our setting, the nodes generatevalues by themselves, and we will later wish to use the fact that we can detect whether too manydifferent values were generated. The key observation within this context, is that using a BIC codewith a doubled information-limit allows one to detect failings with high probability. To see this,assume an information bound K = c log n for constant c ≥ n, K log n, K ]-BCCcode C . The BIC code C I based on C supports 2 K distinct messages. Lemma 3.4
With high probability, either it is detected that the number of distinct values exceeds K , or each value w is decoded successfully.Proof. Fix a received codeword, and consider a fixed value w that is sent. For each block i , let X i be the values z = w whose parity in the i ’th block is odd. If X i is decodable into more than K values, or if its decoding is illegal then this is detected if the parity of w in that block is even. Thishappens with probability 1 /
2. Else, X i is decodable into a set Q of size less than K . We claim thatif the parity of w is odd, an event which holds with probability 1 /
2, then w is successfully decodedregardless of whether X i is correctly decoded. The reason is that the XOR of Q and w decodesuniquely because it contains at most K + 1 values and the BCC code supports 2 K distinct values.This holds even if Q is not the correct set of values included in X i . To summarize, for each value w and for each block i , with probability at least 1 / w is decoded or a failure is detected.Since there are c · log n blocks, the probability the none of these two events happen is at most 1 /n c .The correctness of the scheme holds by applying the union bound over all O ( n ) values. (cid:3) Symmetry Breaking Tasks
In this section we show how to solve symmetry breaking tasks efficiently in additive networks. Asa key example, we focus on the problem of leader election. In Appendix B we consider additionaltasks that involve symmetry breaking such as computing a BFS tree, computing an MIS and findinga proper vertex coloring. A key ingredient in many of our algorithms is having the vertices chooserandom variables according to some carefully chosen probabilities, which, at a high level, are usedto reduce the amount of information that is sent throughout the network. We refer to this as the SL (Select Level) function and describe it as follows.The SL function does not require communication, and only produces two local random values,an r -value and an z -value, that can be considered as primary and secondary values for breakingthe symmetry between the vertices. The r -value is defined by letting r = j with probability of 2 − j ,and the z -value, z , is sampled uniformly at random from the set { , ..., r } .Note that SL does not require the knowledge of the number of vertices n . We next show thatthe maximum value of r ( v ) is concentrated around O (log n ) and that not to many vertices collideon the maximum value. Let j SL max = max { r ( v ) | v ∈ V } and S SL max = { v ∈ V | r ( v ) = j SL max } . Lemma 4.1
With high probability, it holds that (a) j SL max ≤ n + 1 ; (b) | S SL max | ≤ n ; and(c) z ( v ) = z ( v ′ ) for every v, v ′ ∈ S SL max .Proof. Let P v = P ( r ( v ) ≥ n + 1). Then, by definition, P v = P ∞ i =3 log n +1 − i = 1 /n . Byapplying the union bound over all vertices in S , we get that with probability at least 1 − /n , r ( v ) ≤ n + 1, for every v ∈ S , as needed for Part (a).We now turn to bound the cardinality of S SL max . The random choice of r ( v ) can be viewed as arandom process in which each vertex flips a coin with probability 1 / r ( v ) corresponds to the first time when it gets a “tail”. We now claimthat the probability that | S SL max | > n is very small. This holds since the probability that all of2 log n coin flips are “tails” is exactly 2 − n which is less than the probability that | S SL max | > n and none of the vertices in S SL max succeeded in getting another head (and hence in having a larger r -value). Hence, the probability that | S SL max | ≤ n is at least 1 − − n = 1 − /n , as neededfor Part (b).Finally, consider Part (c). It is sufficient to show that the z -values (of vertices of S SL max ) aresampled from a sufficient large range. Note that, the size of this range is 2 · j SL max . We later showthat j SL max ≥ log n/ z -values) is atleast n with high probability. Assume that j SL max ≥ log n/
2, then the probability that z ( v ) = z ( v ′ ),for any pair v, v ′ ∈ S SL max is at most 1 /n . Applying the union bound over all pairs in S SL max givesthe claim, since | S SL max | ≤ n .In the remaining, we show that indeed, j SL max ≥ log n/ v ∈ V ,let x v be an indicator variable for the event that r ( v ) ≥ log n/
2, i.e., x v = 1, if r ( v ) ≥ log n/ x v = 0, otherwise. Let X = P v ∈ V x v . Note that, the probability that X ≥ j SL max ≥ log n/
2. In addition, Pr[ x v = 1] = 2 − (log n/ ≥ − log n/ and hence (bythe linearity of expectation) E [ X ] = P v ∈ V Pr[ x v = 1] = √ n . By Chernoff bound, the probabilitythat X = 0 is exponentially small. Hence, X ≥ j SL max ≥ log n/ (cid:3) .1 Leader Election A Leader-Election protocol is a distributed algorithm run by any vertex such that each nodeeventually decides whether it is a leader or not, subject to the constraint that there is exactly oneleader. Moreover, at the end of the algorithm all vertices know the SL function values of the leader. We first describe a two-round leader election protocol for single-hop networks. Let C I be an[ N, O (log N ) , O (log N )]-BIC code sampled uniformly at random from the distribution of all ran-dom codes that are based on a particular [ N, O (log N ) , O (log N )]-BCC code C (which is used byall vertices). First, the vertices apply the SL function to compute r ( v ) , z ( v ). To do that, in the firstcommunication round, every vertex v transmits C I ( r ( v )). Since with high probability, by Lemma4.1(a), j SL max ≤ n , the information is bounded and by Claim 3.1, each vertex can compute S SL max w.h.p. In the second communication round, every vertex v with r ( v ) = j SL max , transmits C I ( z ( v )).That is, in the second phase only the vertices of S SL max transmit the codeword of their z ′ s value.Since by Lemma 4.1(b), with high probability, | S SL max | = O (log n ), and by Claim 3.1 again, the z -values of all vertices in S SL max are known to every vertex in the network w.h.p. Finally, the leaderis the vertex v ∗ ∈ S SL max with the largest z -value, i.e., z ( v ∗ ) = max v ′ ∈ S SL max z ( v ′ ). In this section, we consider the general case of network G with diameter D . We present Algorithm LeaderElection that elects a leader within O ( D ) rounds w.h.p. To enable the termination of theprotocol, the vertices compute an approximation for D throughout the course of the leader electionprocess, thereby obtaining a 2-approximation of the diameter is a byproduct of this algorithm.Initially, every vertex v computes the random values ( r ( v ) , z ( v )) as defined for the single-hopcase. In the first two communication rounds, every vertex v transmits the codeword of the maximum r -value it has observed so far, and in the third round, if the maximum received r -value equals r ( v ),then it transmits C I ( r ( v )) ◦ C I ( z ( v )).From now on, the algorithm proceeds in stages , each stage t , consists of four communicationrounds, ( t, i ) for i ∈ { , , , } . The following notation is useful. For vertex v in stage t , let r t ( v ) be the maximum r -value that v has observed so-far (thus r ( v ) = r ( v )) and let z t ( v ) be thecorresponding z -value (if received without collisions). Let d t ( v ) be the distance to the vertex v ∗ t satisfying that r t ( v ) = r ( v ∗ t ) and z t ( v ) = z ( v ∗ t ). Hence, the vertex v ∗ t can be thought of as thelocal maximum in the t -neighborhood of v . Finally, let d ∗ t ( v ) be the maximum distance from v ∗ t ,observed by v . To avoid cumbersome notation, we override notation and write C I whenever a BICcode is in use. Yet, it is important to keep in mind that each application of BIC code, requires anindependent sampling of such an instance.In round ( t, r -value it has observedso-far, i.e., C I ( r t ( v )). In round ( t, C I ( r t ( v )) ◦ C I ( z t ( v )), if its r t ( v ) value is themaximal received. In round ( t, C I ( r t ( v )) ◦ C I ( z t ( v )) ◦ C I ( d t ( v )) if its d t ( v ) value isfinite. Finally, in round ( t, C I ( r t ( v )) ◦ C I ( z t ( v )) ◦ C I ( d ∗ t ( v )), if its d ∗ t ( v )value is finite.A vertex that has not receive an update value for none of the fields d ∗ t ( v ) or r t ( v ) for more than2 stages, terminates. This completes the description of the protocol. For a detailed pseudocode,see Figure 1. 10 Initially: ( r ( v ) , z ( v )) ← SL , TERMINATE=FLASE Send C I ( r ( v )) r ′ ← the maximum received value in this round If r ′ = r ( v ) send C I ( r ( v )) ◦ C I ( z ( v )) Else, r ( v ) ← r ′ t ← t + 1 While TERMINATE=FALSE: Round ( t, Send C I ( r t ( v )) r ′ t ← the maximum received value Round ( t, If r ′ t > r t ( v ) then r t ( v ) ← r ′ t , d t ( v ) ← ∞ , and d ∗ t ( v ) ← ∞ Else, send C I ( r t ( v )) ◦ C I ( z t ( v )) If the received r -value is r t ( v ) then z t ( v ) ← the maximum received z -value Round ( t, If d t ( v ) = ∞ , send C I ( r t ( v )) ◦ C I ( z t ( v )) ◦ C I ( d t ( v )) r ′ t , z ′ t , d ′ t ← the received values If r t ( v ) = r ′ t and z t ( v ) = z ′ t then d t ( v ) ← d ′ t + 1 and d ∗ t ( v ) ← d t ( v ) Round ( t, If d ∗ t ( v ) = ∞ then send C I ( r t ( v )) ◦ C I ( z t ( v )) ◦ C I ( d ∗ t ( v )) r ′ t , z ′ t , d ′′ t ← the received values If r t ( v ) = r ′ t , z t ( v ) = z ′ t and d ∗ t ( v ) < d ′′ t then d ∗ t ( v ) ← d ′′ t t ← t + 1 If ( z t ( v ) , r t ( v )) = ( z t − ( v ) , r t − ( v )) = ( z t − ( v ) , r t − ( v )) and d ∗ t ( v ) = d ∗ t − ( v ) = d ∗ t − ( v ) then TERMINATE=TRUE Algorithm 1:
LeaderElection protocol for vertex v .11 nalysis. Let v ∗ be the vertex with maximum r ( v ∗ ) and z ( v ∗ ) values. That is, v ∗ is the desig-nated leader in the network (global maximum). Throughout the analysis, we show that with highprobability every vertex terminates within O ( D ) rounds and that the final leader ℓ ( v ) of every ver-tex v is the leader v ∗ . It is convenient to analyze the process on the BFS tree rooted at the leader v ∗ . Let L i = { v ∈ V | dist( v, v ∗ , G ) = i } be the vertices at distance i from v ∗ , and e L i = S j ≤ i L i be the vertices up to distance i from v ∗ in G . For every vertex v , let D v = max u ∈ V dist( u, v, G )denote its local diameter. We begin by showing the following. Claim 4.2
With high probability, for every stage t ∈ { , . . . , D v ∗ } , it holds that:(a) after round ( t, , r t ( v ) = r ( v ∗ ) for every v ∈ e L t +2 ;(b) after round ( t, , z t ( v ) = z ( v ∗ ) for every v ∈ e L t +1 ;(c) after round ( t, , d t ( v ) = dist( v, v ∗ ) and d ∗ t ( v ) = max u ∈ e L t dist( u, v ∗ ) for every v ∈ e L t .Proof. We prove this by induction on t . For the base of the induction, consider t = 1. In thefirst communication round, the vertices transmit their r -value. Since there are O (log n ) distinct r -values, by Claim 3.1 there are no collisions when transmitting C I ( r ( v )). Hence, the verticesin L (neighbors of v ∗ ) know r ( v ∗ ). In the second communication round, all the vertices of L transmit C I ( r ( v ∗ )). Since there are no collisions on r -values, the vertices of L know r ( v ∗ ). Inthe third round, vertices v whose r -value is the maximum r -value they observed so far, transmit C I ( r ( v )) ◦ C I ( z ( v )). Hence, the only vertices in e L that transmit in the third round, are thosethat obtain the same r -value as v ∗ . By Lemma 4.1(b), there are O (log n ) such vertices and hencethere are no collisions at the vertices of L and they successfully decode the values of the leader r ( v ∗ ) , z ( v ∗ ). After round (1 , L transmit r ( v ∗ ) to L and since there are nocollisions the vertices of L know r ( v ∗ ). Part (a) of the induction base holds. In round (1 , L , L , L that transmit are those that have a z -value that corresponds to r ( v ∗ ). Hence, by Lemma 4.1(b), there are O (log n ) distinct z -values, implying that the vertices of L successfully receive z ( v ∗ ) from L . Part (b) of the induction base holds. In the beginning ofround (1 , L , L know ( r ( v ∗ ) , z ( v ∗ )) but they do not know their distance from v ∗ . Hence, the only transmitting vertex in e L is v ∗ , implying that the vertices in L successfullyreceive (( r ( v ∗ ) , z ( v ∗ ) ,
0) and hence can set d ( v ) = 1. Part (c) of the induction base holds.Assume the claim holds up to stage t − t . By the induction assumptionPart (a) for t −
1, it holds that the vertices in e L t +1 know r ( v ∗ ).In round ( t, L t +1 transmit r ( v ∗ ) and since there are no collisions on this value,the vertices of L t +2 know r ( v ∗ ) and Part (a) holds.In round ( t, L t , L t +1 , L t +2 transmit r ( v ∗ ) and some z -value that correspondsto it. Hence, by Lemma 4.1(b), as there are O (log n ) distinct z -values that correspond to r ( v ∗ ), itholds that there are no collisions for the vertices in L t +1 and they successfully receive ( r ( v ∗ ) , z ( v ∗ ))from the vertices of L t and Part (b) holds.In the beginning of round ( t, L ′ = L t − ∪ L t ∪ L t +1 know ( r ( v ∗ ) , z ( v ∗ )).Hence, the only vertices in L ′ that transmit are those that have finite distance from v ∗ , namely, L t − . Hence the vertices in L t successfully receive the message ( r ( v ∗ ) , z ( v ∗ ) , t −
1) and by increasingthe distance by one, they have the correct distance. Part (c) holds. (cid:3)
Claim 4.3
With high probability, the following hold:(a) The vertices of L D v ∗ terminate in stage D v ∗ + 2 with the correct values.(b) For every t ∈ { , . . . , D v ∗ } , every vertex v ∈ L D v ∗ − t knows D v ∗ after round ( D v ∗ − t + 1 , andterminates in stage D v ∗ − t + 3 . roof. Note that in every two stages, as long as the vertex does not obtain the correct values ofthe leader and the local radius D v ∗ , every vertex either receives an improved maximum distancefrom its current leader (local maximum in its neighborhood) or a notification of a new leader.First, observe that by Claim 4.2(c), after round ( D v ∗ , d D v ∗ ( v ) = d ∗ D v ∗ ( v ) = dist( v, v ∗ ) = D v ∗ forevery v ∈ L D v ∗ . Within two rounds, no update is received on either a new leader (since the globalmaximum has been found) or on an improved maximum distance, and hence the leaf vertices of L D v ∗ terminate in stage D v ∗ + 2. Consider Part (b). It is easy to see that for every stage t ≥
1, allvertices in L t ′ for t ′ ≤ t hold the same value of d ∗ t ( v ). That is for every t ′ ≤ t , there exists a value ℓ ′ such that d ∗ t ( v ) = ℓ ′ for every v ∈ L t ′ .We prove the claim by a reversed induction on the stage. In round ( D v ∗ , L D v ∗ transmit D v ∗ and since the vertices in L D v ∗ − receive a message from at most 3 layers, they cansuccessfully decode the value of D v ∗ . Since they get no update within 2 stages (i.e., they hold themaximum distance from the global maximum vertex), these vertices terminate in stage D v ∗ + 2.Assume that the claim holds up to stage t ′ = D v ∗ − ( t + 1) and consider t ′ + 1. By the inductionassumption for stage t ′ , the vertices of L t ′ receive D v ∗ in ( D v ∗ − t, D v ∗ − t + 1 , D v ∗ to L t ′ +1 . Since every vertex in this layer receives a message from at most threedistinct layers, they can decode successfully D v ∗ . As there are no future updates, they terminatein stage D v ∗ − t + 3, as desired. (cid:3) This completes the correctness of the leader-election protocol. Note that this protocol can alsobe used to compute a 2-approximation for the diameter of the network.
In this section we consider approximation tasks. As a key example, we focus on the task ofapproximating the degree, i.e., each vertex v is required to compute an approximation for itsdegree in the graph G . We describe Algorithm
AppDegree that computes with high probability a constant approximationfor the degree of the vertices within O (1) rounds. For vertex v and graph G , let deg( v, G ) = | Γ( v, G ) | be the degree of v in G . When the graph G is clear from the context, we may omit it and simplywrite deg( v ). Recall that we assume that each vertex v has a unique identifier id v in the range of[1 , . . . , n c ] for some constant c ≥ v with degreedeg( v ) ≤ c · log n . The second round computes a constant approximation for high-degree vertices v with deg( v ) > c · log n . In the first communication round, every vertex v uses a random instance C Iv of an [ N, c · log N, c · log N ]-BIC code to encode its ID and transmits C Iv ( id v ) as part of m ( v ).In addition, the vertices use the Information-Overflow Detection scheme of Section 3 to verify iftheir BIC decoding is successful (that is, the message m ( v ) consists of two fields, the first encodesthe ID and the second is devoted for overflow detection). Upon receiving m ′ ( v ) = L u ∈ Γ( v ) m ( u ),the vertex applies BIC decoding to the first field of the message and applies Information-OverflowDetection to the second field to verify the correctness of the decoding. Note that by the propertiesof the BIC code, in this round, the low-degree vertices compute their exact degree in G .13he second round aims at computing a constant factor approximation for the remaining verticeswith high-degree. Set a = 40 · log N and b = 2 log N . Every vertex v sends an ( a · b )-bit message m ( v ) defined by a collection of a random numbers in the range of { , . . . , b } sampled independentlyby each vertex v . Specifically, for every v and i ∈ { , . . . , a } , r i ( v ) is sampled according to thegeometric distribution, letting r i ( v ) = j for j ∈ { , . . . , b − } with probability 2 − j , and r i ( v ) = b with probability 2 − b +1 (the remaining probability). For every i ∈ { , . . . , a } and every j ∈{ , . . . , b } , let x i,j ( v ) = 1 if j < r i ( v ) and x i,j ( v ) = 0 otherwise. Let X i ( v ) = x i,b ( v ) · · · x i, ( v ) · x i, ( v )and let m ( v ) = X ( v ) = X a ( v ) · · · X ( v ) · X ( v ) be the transmitted message of v . Let Y ( v ) = L u ∈ Γ( v ) X ( u ) be the received message of v . The decoding is applied to each of the a blocks of Y ( v )separately, i.e., treating Y ( v ) as Y ( v ) = Y a ( v ) · · · Y ( v ) · Y ( v ), where Y i ( v ) = y i,b ( v ) · · · y i, ( v ) · y i, ( v ),such that y i,j ( v ) = L u ∈ Γ( v ) x i,j ( u ). For every j ∈ { , ..., b } and every v ∈ V , define SUM ( j, v ) = P ai =1 y i,j ( v ). Finally, define j ∗ ( v ) = min { j | SUM ( j, v ) ≤ . · a } , if there exists an index j such that SUM ( j, v ) ≤ . · a (we later show that such index do exists with high probability) and j ∗ ( v ) = 0,otherwise as a default value. The approximation δ ( v ) is then given by 2 j ∗ ( v ) − . This completes thedescription of the algorithm.As mentioned earlier, the correctness for low-degree vertices follows immediately by the proper-ties of the BIC code and the information-overflow detection (Lemma 3.2 and Lemma 3.4). Hence, itremains to show that in the second round, for high-degree vertices v , we have δ ( v ) / deg( v ) = O (1). Lemma 5.1
With high probability, if v has high-degree, then δ ( v ) ∈ [deg( v ) / , v )] .Proof. First, we would like to show that j ∗ > SUM ( j ∗ − , v ) ≥ . · a and SUM ( j ∗ , v ) < . · a .Note that the complementary event occurs, if eitherCase (1) SUM (1 , v ) ≤ . · a and then j ∗ = 1 and SUM ( j ∗ − , v ) is not defined; orCase (2) SUM ( j, v ) > . · a , for every j ∈ { , ..., b } and then j ∗ = 0 and SUM ( j ∗ = 0 , v ) is notdefined. We show that with high probability SUM ( b, v ) ≤ . · a ; (1)which, necessarily, implies that case (2) above does not occur, and that SUM (1 , v ) > . · a , (2)which implies that case (10 does not hold as well. Before proving that inequalities (1) and (2)hold with high probability, we analyze the expectation of SUM ( j, v ). For every i ∈ { , . . . , a } and j ∈ { , , . . . , b } , define V i,j = { u | r i ( u ) > j } (note that, by this definition, V i, = Γ( v )). Forevery j ∈ { , ..., b } , it then holds that E [ SUM ( j, v )] = a X i =1 P [ y i,j = 1] = a X i =1 P [ | V i,j ∩ Γ( v ) | is odd]= a X i =1 P (cid:2) | V i,j − ∩ Γ( v ) | ≥ (cid:3) / a · (1 − (1 − − j +1 ) deg( v ) ) / . (3)Now, we prove that Inequality (1) holds with high probability. Recall that b = 2 log N and deg( v ) ≤ n . Combining this with Inequality (3), we get that E [ SUM ( b, v )] ≤ a · (1 − (1 − /n ) n ) / ≤ . a, n >
10. Thus, by Chernoff bound P [ SUM ( b, v ) ≥ . · a ] ≤ exp( − . a/ ≤ /n , as needed for Inequality (1), where the last inequality holds, since a = 40 log n . We now show thatwith high probability Inequality (2) holds too. Recall that deg( v ) >
1. By Inequality (3), we have E [ SUM ( b, a/ . Thus, by Chernoff bound P [ SUM (1 , v ) < . · a ] ≤ exp( − . · a/ ≤ /n , as needed for Inequality (2), where the last inequality holds, since a = 40 log n .As mentions above, Inequalities (1) and (2) implies that j ∗ > SUM ( j ∗ − , v ) ≥ . · a and SUM ( j ∗ , v ) < . · a . Intuitively, we use the last two inequalities together with the fact that SUM ( j ∗ − , v ) and SUM ( j ∗ , v ) concentrating around theirs expectations to bound from below andbound from above the expectations E [ SUM ( j ∗ − , v )] and E [ SUM ( j ∗ , v )], respectively. Next, weuse the fact that the expectations E [ SUM ( j ∗ − , v )] and E [ SUM ( j ∗ , v )] are functions of deg( v )and hence 2 j ∗ approximates deg( v ). Formally, by Chernoff bound, we get, on the one hand, that P [ SUM ( j ∗ , v ) < . · a | E [ SUM ( j ∗ , v )] ≥ . · a ] ≤ exp( − . · a/ ≤ /n , where the last inequality holds, since a ≥
40 log n . Therefore, with high probability, E [ SUM ( j ∗ , v )] ≤ . · a . (4)On the other hand, by Chernoff bound, we have P [ SUM ( j ∗ − , v ) > . · a | E [ SUM ( j ∗ − , v )] ≤ . · a ] ≤ exp( − . · a/ ≤ /n , where the last inequality holds, since a ≥
30 log n . Thus, with high probability, E [ SUM ( j ∗ − , v )] ≥ . · a . (5)Combining Inequality (4) and Inequality (3), we get that(1 − − j ∗ +1 ) deg( v ) ≥ . , (6)and similarly by combining Inequality (5) and Inequality (3), we get that(1 − − j ∗ ) deg( v ) ≤ . . (7)Recall that (1 − /x ) x < exp( − x >
0. Thus, by combining it with Inequality (6), wehave exp( − deg( v ) · − j ∗ +1 ) ≥ (1 − − j ∗ +1 ) deg( v ) ≥ . , hencedeg( v ) · − j ∗ ≤ . . (8)On the other hand, (1 − /x ) x > exp( − . x ≥
64. By combining with Inequality (7),exp( − .
01 deg( v ) · − j ∗ +1 ) ≤ (1 − − j ∗ +1 ) deg( v ) ≤ . , v ) · − j ∗ +1 > . . (9)Overall, by Inequalities (8) and (9), we have thatdeg( v )3 . ≤ j ∗ − ≤ deg( v )0 . , as required, and the lemma follows. (cid:3) We thus have the following.
Theorem 5.2
There exists an O (1) -round algorithm that computes w.h.p. the exact degree deg( v ) for vertices with deg( v ) = O (log n ) and a constant approximation if deg( v ) = Ω(log n ) . In this section, we present Algorithm
ApproxNetSize that approximates the network size by aconstant factor within O ( D ) rounds w.h.p. At a high level, the nodes choose random variablesaccording to a probability distribution for which we can obtain an approximation for the numberof vertices that choose each value. By aggregating the amount of numbers chosen in intervals thatgrow in size by a constant factor, the nodes try to estimate the size of the network. While intervalswith too many values cannot be decoded, they indicate that the number of nodes in the graph islarger. Hence, the nodes look for the interval of largest values in which there are enough valuesso that with high probability indeed it estimates well the total number of nodes. We obtain thefollowing. Theorem 5.3
For every network G = ( V, E ) of diameter D , a constant approximation for the sizeof the network n = | V | can be computed with high probability within O ( D ) communication rounds. Recall that it is assumed the vertices know a polynomial bound N on the size of the network(this bound corresponds for example to the message size which is O (log n )). For simplicity, let N be a power of 2. The algorithm uses message size of O (log N ) bits and consists of O ( D )communication rounds. To ease the communication scheme, in the first phase, the vertices applyAlgorithm LeaderElection of Section 4.1.2 to elect a leader v ∗ and construct a BFS tree T rooted at v ∗ of depth D v ∗ .The second phase of the algorithm consists of two parts. The first part is by convergecast ofinformation to the leader v ∗ , which is the root of T . In the second part, the leader down casts theresult of the computation through T to all the nodes. At a high level, the nodes choose randomvariables according to a probability distribution for which we can obtain an approximation for thenumber of vertices that chose each value. By aggregating the amount of numbers chosen in intervalsthat grow in size by a constant factor, the nodes try to estimate the size of the network. Intervalswith too many values chosen cannot be decoded, but will also indicate that the number of nodesin the graph is larger. Hence, the nodes look for the interval of largest values in which there are enough values so that with high probability indeed it estimates well the total number of nodes.Formally, for every τ ∈ { , . . . , D v ∗ } , let L τ = { v | dist( v, v ∗ , G ) = τ } be the τ ′ th level of T . The second phase uses the following codes: an [ N, log N, log N ]-BCC code C that is used toencode the IDs of the vertices and an [ N, log N, log N ]-BIC code C I based on C that supports log N N blocks, where the i ’thblock is used to examine the possibility that the number of vertices in the network is Θ(2 i ). At thebeginning of this phase (before communication starts), every vertex v , selects a value b r ( v ) accordingto the geometric distribution such that b r ( v ) = i with probability p i = i/ i for i ∈ { , . . . , log N } .In the next D v ∗ communication rounds, τ ∈ { , . . . , D v ∗ } , the vertices v of L D v ∗ − τ +1 transmit amessage m ( v ), defined as follows. The message consists of log N blocks, each of size log N (i.e.,the maximal size of an BIC codeword). For every i ∈ { , . . . , log N } , let b V i ( v ) be the IDs of vertices u with b r ( u ) = i that are known to v . Initially, b V i ′ ( v ) = { v } for i ′ = b r ( v ) and b V i ( v ) = ∅ for every i = i ′ .These sets are updated by v by decoding the message m ′ ( v ) received in round τ − v is not a leaf). If v is not a leaf, each of the log N blocks of the received message m ′ ( v ) is decodedseparately by applying the BIC decoding. If the i ’th block cannot be decoded successfully, it isadded to the list F B ( v ) of failed blocks (blocks that couldn’t be decoded). Note that we do notassume that the vertices know if the decoding of a given block is decoded (this can be obtained byusing the information-overflow detection scheme of Section 3, but it is not needed in our algorithm,as will be shown in the analysis). For every block i / ∈ F B ( v ) (i.e., a block that could be decoded),the decoded values (corresponding to vertex IDs) are added to b V i ( v ). Using the updated b V i ( v )sets, the i ’th block of m ( v ), denoted by m i ( v ), is defined as follows. If i ∈ F B ( v ), then the i ’block contains the BIC codeword of a special word that indicates failure. Otherwise, (i.e., the i ’th block of the received message was decoded successfully), m i ( v ) contains the XOR of the BICcodewords of the IDs in b V i ( v ). Formally, let C Iv,j be random instances of [ N, log N, log N ]-BICcodes for j ∈ { , . . . , | b V i ( v ) |} , then m i ( v ) = L id j ∈ b V i ( v ) C Iv,j ( id j ). This completes the description ofthe convergecast communication on T .Let m ′ ( v ∗ ) be the message received by the root vertex v ∗ and let i ∗ ∈ { , . . . , log N } be the last index i satisfying that i / ∈ BF and | b V i ( v ∗ ) | ≥ i/
4. The estimate n alg for the number of vertices isthen n alg = 2 i ∗ . Finally, the root vertex v ∗ downcasts n alg on T . This completes the description ofthe algorithm. We now turn to the analysis. Clearly, the protocol consists of O ( D ) communicationrounds, so it remains to show correctness. Lemma 5.4
With high probability, n alg = Θ( n ) .Proof. For every i ∈ { , . . . , log N } , let V i = { v | b r ( v ) = i } be the vertices whose b r ( v ) value is i .Let j ∗ be the index satisfying that n ∈ [2 j ∗ , j ∗ +1 ].We first claim that with high probability j / ∈ S v ∈ V F B ( v ) for every j ∈ { j ∗ , . . . , log N } . In otherwords, we show that w.h.p. each of the last log N − j ∗ + 1 blocks are successfully decoded at each vertex. By the properties of the BIC code (see Lemma 3.2), it is sufficient to show that with highprobability | V j | ≤ n for every j ∈ { j ∗ , . . . , log N } , which indeed holds by a Chernoff bound.As a corollary, we get that b V j ( v ∗ ) = { ID ( v ) | v ∈ V j } = V j for every j ∈ { j ∗ , . . . , log N } . Thatis, after v ∗ decodes of m ′ ( v ∗ ), it knows the IDs of the vertices in V j for every j ∈ { j ∗ , . . . , log N } ,since the decoding of these blocks was successful all along with high probability.For j ≥ j ∗ + 6, the expected number of vertices in V j is n · j/ j ≤ j/
8. Since j ≥ log n , by aChernoff bound, we get that with high probability | V j | > j/ j > j ∗ + 6. Since the j ∗ ’thblock satisfies that | V j ∗ | ≥ j ∗ / i ∗ that satisfiesthis, it holds that i ∗ ∈ { j ∗ , . . . , j ∗ + 6 } . We therefore have that n alg = 2 i ∗ = Θ(2 j ∗ ) = Θ( n ). Theclaim follows. (cid:3) O ( D ) rounds, givesTheorem 5.3. Γ r ( v ) . Using BIC codes, one can compute, w.h.p, a constant approximation of any value x ( v ) ∈ { , . . . , n c } in the r -neighborhood within O ( r ) rounds. This is done by letting v transmit the BIC codewordof the value j v satisfying that x ( v ) ∈ [2 j v − , j v ]. Since there are logarithmic such distinct values,the communication simulates the message passing scheme. Theorem 5.5
Computing a constant approximation for the minimum or maximum value in the r -neighborhood can be done within O ( r ) rounds with high probability. Consider the setting where every vertex is given an input value (corresponding to its rank, for exam-ple) and the goal is to find the vertex with the maximum value. We will show that BCC codes withmessage size of O (log n ) allow one to perform many simultaneous competitions between Ω(log n )candidates, which result in a tournament process of O ( D · log n/ log log n ) rounds for a network ofdiameter D . Specifically, the fact that the BCC code provides successful decoding when there are O (log n ) concurrent transmitting neighbors, allows us to reduce the number of competitors by afactor of Ω(log n ) in every round, and hence the winner is found within O ( D · log n/ log log n ) rounds.We begin by describing the protocol for single-hop networks and in Section 6.1.2, we generalize itfor any network of diameter D >
1, which requires some subtle modifications.
Let V = { v , . . . , v n } be the vertices of the network and let X = { x , . . . , x n } , where x i ∈ { , . . . n } for all i , be the set of integral inputs such that vertex v i holds the input x i . Let max( X ) = max ni =1 x i be the maximum value in X . Note that by Section 5, a 2-approximation for the maximum canbe computed within a single round, w.h.p. The main contribution of this section is the exact computation of the maximum value. Theorem 6.1
The maximum value max( X ) can be computed within O (cid:16) log n log log n (cid:17) rounds, with highprobability. Algorithm
CompMaxSH consists of O (log n/ log log n ) communication rounds. For simplicity, as-sume that the input values are distinct. This can be obtained by appending to every input value ⌈ log n ⌉ least significant bits corresponding to the ID of the vertex. Let c ≥ ApproxNetSize and set τ = ⌈ c · log n/ log log n ⌉ . Initially, allvertices are active. In round t = { , . . . , τ } , let n t be a constant approximation for the number ofactive vertices at the beginning of round t , and let C be an [ n , c · log n , c · log n ]-BCC code . This approximation for the size of the network can be obtained by applying Algorithm
ApproxNetSize or sim-ply Algorithm
AppDegree in the case of single-hop networks (where only the active vertices participate in thesealgorithms). n t , every active vertex v j transmits C ( x j ) with probability p t = 4 c · log n /n t . Ifa vertex v i receives an input x j > x i in round t , it becomes inactive. The final result max( v i ) ofevery vertex v i corresponds to the maximum input value x j it received throughout the algorithm.This completes the description of the algorithm.We now analyze the algorithm and begin with correctness. Let A t be the active vertex set atthe beginning of round t . Note that A τ ⊆ . . . ⊆ A = V . Let v m be a vertex with maximum input,i.e., x m = max( X ). Lemma 6.2
For each round t ∈ { , . . . , τ } , with high probability it holds that | A t | = O ( n / log t − n ) and x m ∈ A t .Proof. The claim is shown by induction. For the base of the induction t = 1, we have that A = V ,and n ≤ c · n since by the properties of Algorithm ApproxNetSize it holds that with high probability n ∈ [ n/ , c · n ] for some constant c ≥
2. Assume that the claim holds up to step t − ≥ t . Order the values of the vertices in A t − in increasing order of their inputs andconsider the subset H t − ⊂ A t − of the ⌈| A t − | / log n ⌉ vertices with the highest input values in A t − . We first claim that with high probability, at least one of the vertices in H t − transmits inround t −
1. Since every vertex in A t − transmits with probability of p t − = 4 c log n /n t − and n t − ≤ c · | A t − | , in expectation there are at least 4 log n transmitting vertices in H t − and hence,by a Chernoff bound, w.h.p there is at least one transmitter in H t − .We proceed by showing that the number of transmitting vertices in round t − O (log n ).In expectation, the number of transmitting vertices in A t − is at most 8 c · log n , and hence byChernoff bound, with high probability there are less than 32 c log n transmitters. By the propertiesof the BCC code, all messages received in round t − H t − and as a result all vertices in V \ H t − become inactive.In other words, A t ⊆ H t − and hence n t ≤ | H t − | = | A t − | / log n = O ( n / log t n ), where the lastequality holds w.h.p by the induction assumption. Finally, by the induction assumption for t − v m ∈ A t − , since all messages were decoded successfully in round t − v m remains in A t as well. The claim follows. (cid:3) We thus have the following, which proves Theorem 6.1.
Lemma 6.3
With high probability max( v i ) = max( X ) for every vertex v i ∈ V .Proof. By Lemma 6.2, after τ rounds there are O (log n ) active transmitters w.h.p. Since thevertex v m with the maximum input max( X ) remains active in each round, it transmits in the lastround, and as its message can be successfully decoded by all vertices, the claim follows. (cid:3) The single-hop network protocol can be extended for general networks by paying an extra multi-plicative factor of O ( D ). Notice that this is not immediate from the fact that the diameter is D ,since it is not clear that concurrent instances of the algorithm for single-hop networks can be runin parallel without incurring an overhead in the number of rounds. Instead, we obtain the resultby simulating each round of the algorithm for single-hop networks in a general network in O ( D )rounds using a leader and a BFS tree rooted at it.19 heorem 1 For any network G of diameter D , the maximum value can be computed within O (cid:16) D · log n log log n (cid:17) rounds, with high probability.Proof. Algorithm
CompMaxMH operates in a very similar manner to the single-hop case with somemodifications. First, the vertices use an [ n , log n , log n ]-BIC code (instead of the deterministicBCC code) where n is an upper bound on the number of vertices in the network. In addition,the algorithm computes a BFS tree T rooted at some leader v ∗ using Algorithm LeaderElection andAlgorithm
ConstructBFS .Next, the algorithm proceeds by τ = O (log n/ log log n ) phases corresponding to the τ com-munication rounds in the single-hop case, each phase consists of O ( D ) communication rounds.Each phase t ∈ { , . . . , τ } begins by applying Algorithm ApproxNetSize to compute the value n t , a constant approximation for the number of active vertices | A t | . Then, every active vertex v i ∈ A t transmits C Ii,t ( x i ) with probability p t = log n /n t where C Ii,t is a random instance of the[ n , log n , log n ]-BIC code. These values are then upcast to the root v ∗ within O ( D ) rounds.The root v ∗ uses BIC decoding and downcasts the maximal value among all the values of the trans-mitting vertices in A t . If a vertex v i receives an input x j > x i in phase t then it becomes inactive.The final result max( v i ) of every vertex v i corresponds to the maximum input value x j it receivedthroughout the algorithm. This completes the description of the algorithm, and the correctnessfollows the same line of argumentation as in the single-hop case. (cid:3) It is clear that computing in the additive network model should be doable faster than in the standardradio network model. In this paper we quantify this intuition, by providing efficient algorithms forvarious cornerstone distributed tasks. Our work leaves open several important open questions forfurther research. First, it is natural to ask whether our algorithms can be improved. Specifically,most of our algorithms apply for the full-duplex model and translate into half-duplex by payingan extra factor of O (log n ). It would be interesting to obtain better bounds for half-duplex radioswithout using the full-duplex protocol as a black box. An additional axis that requires investigationis the multiple channels model. It would be interesting to study the tradeoff between running time,message size and the number of channels. Note, that whereas most of our algorithms are optimalfor full-duplex radios (up to constant factors), some leave room for improvements. For example, inthe problem of computing the maximum input, we believe that some pipelining of the simulation ofphases should be able to give a round complexity of O ( D + log n/ log log n ), instead of the current O ( D · log n/ log log n ). However, this is not immediate. Designing lower bounds for this modelis another important future goal. It seems that the problem of computing the maximum inputin a single-hop network, should be a good starting point, as we believe that this task requiresΩ(log n/ log log n ) rounds. Another interesting future direction involves the implementation of anabstract MAC layer over additive radio network model. Such an implementation was providedrecently [15] for the standard radio network model. Finally, we note that all our algorithms arerandomized, as opposed to the original definition of BCC codes. Is randomization necessary? Whatis the computational power of the additive network model without randomization?20 eferences [1] N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for themaximal independent set problem. J. Algorithms , 7(4):567–583, 1986.[2] J. Andrews. Interference cancellation for cellular systems: a contemporary.
SIAM Journal onComputing , 12(1):19 – 2, 2005.[3] A. S. Avestimehr, S. N. Diggavi, and D. Tse. Wireless network information flow: A determin-istic approach.
IEEE Trans. on Info. Theory , 57(4):1872–1905, 2011.[4] R. Bar-Yehuda, O. Goldreichh, and A. Itai. On the time-complexity of broadcast in multi-hopradio networks: An exponential gap between determinism and randomization.
J. of Compt.Syst. Sciences , 45:104 – 126, 1992.[5] L. Barenboim, M. Elkin, S. Pettie, and J. Schneider. The locality of distributed symmetrybreaking. In
FOCS , pages 321–330, 2012.[6] K. Censor-Hillel, B. Haeupler, N. A. Lynch, and M. M´edard. Bounded-contention coding forwireless networks in the high snr regime. In
DISC , pages 91–105, 2012.[7] S. Daum, M. Ghaffari, S. Gilbert, F. Kuhn, and C. Newport. Maximal independent sets inmultichannel radio networks. In
PODC , pages 335–344, 2013.[8] S. Daum, M. Ghaffari, S. Gilbert, F. Kuhn, and C. Newport. Maximal independent sets inmultichannel radio networks.
Technical Report 275, University of Freiburg, Department ofComputer Science , 2013.[9] M. Farach-Colton, R. J. Fernandes, and M. A. Mosteiro. Lower bounds for clear transmissionsin radio networks. In
LATIN: Theoretical Informatics , pages 447–454, 2006.[10] B. Gfeller and E. Vicari. A randomized distributed algorithm for the maximal independentset problem in growth-bounded graphs. In
Proc. of the Twenty-Sixth Annual ACM Symp. onPrinciples of Distributed Computing, PODC , pages 53–60, 2007.[11] M. Ghaffari and B. Haeupler. Near optimal leader election in multi-hop radio networks. In
SODA , pages 748–766, 2013.[12] S. Gollakota and D. Katabi. Zigzag decoding: combating hidden terminals in wireless networks.In
SIGCOMM , pages 159–170, 2008.[13] A. Greenberg, P. Flajolet, and R. Ladner. Estimating the multiplicities of conflicts to speedtheir resolution in multiple access channels.
Journal of the ACM (JACM) , 34(2):289 – 325,1987.[14] P. Gupta and P. Kumar. The capacity of wireless networks.
IEEE Trans. on Info. Theory ,pages 388–404, 2000.[15] F. Kuhn, N. Lynch, and C. Newport. The abstract mac layer.
Distributed Computing , 24:187–206, 2011. 2116] F. Kuhn, T. Moscibroda, T. Nieberg, and R. Wattenhofer. Fast deterministic distributedmaximal independent set computation on growth-bounded graphs.
Distributed Computing ,pages 273–287, 2005.[17] F. Kuhn, T. Moscibroda, and R. Wattenhofer. What cannot be computed locally! In
Proc.PODC , pages 300–309, 2004.[18] F. Kuhn, T. Moscibroda, and R. Wattenhofer. On the locality of bounded growth. In
Proceed-ings of the Twenty-Fourth Annual ACM Symposium on Principles of Distributed Computing,PODC, 2005 , pages 60–68, 2005.[19] F. Kuhn and R. Wattenhofer. Constant-time distributed dominating set approximation.
Dis-tributed Computing , 17(4):303–310, 2005.[20] N. Linial. Locality in distributed graph algorithms.
SIAM Journal on Computing , 21(1):193–201, 1992.[21] Z. Liu and M. Herlihy. Approximate local sums and their applications in radio networks. In
DISC , pages 243–257, 2014.[22] M. Luby. A simple parallel algorithm for the maximal independent set problem.
SIAM Journalon Computing , 15:1036–1053, 1986.[23] Y. M´etivier, J. Robson, N. Saheb-Djahromi, and A. Zemmari. An optimal bit complexityrandomized distributed mis algorithm.
Distributed Computing , 23(5-6):331–340, 2011.[24] T. Moscibroda and R. Wattenhofer. Maximal independent sets in radio networks. In
PODC ,pages 148–157, 2005.[25] A. Ozgur, O. Leveque, and D. Tse. Hierarchical cooperation achieves optimal capacity scalingin ad hoc networks.
IEEE Trans. on Info. Theory , pages 3549–3572, 2007.[26] A. ParandehGheibi, J.-K. Sundararajan, and M. M´edard. Collision helps - algebraic collisionrecovery for wireless erasure networks. In
WiNC , 2010.[27] K. N. Ramachandran, E. M. Belding-Royer, K. C. Almeroth, and M. M. Buddhikot.Interference-aware channel assignment in multi-radio wireless mesh networks. In
INFOCOM ,pages 1–12, 2006.[28] J. Schneider and R. Wattenhofer. Coloring unstructured wireless multi-hop networks. In
PODC , pages 210–219, 2009.[29] J. Schneider and R. Wattenhofer. An optimal maximal independent setalgorithm for bounded-independence graphs.
Distributed Computing , 22(5-6):349–361, 2010.[30] J. Schneider and R. Wattenhofer. What is the use of collision detection (in wireless networks)?In
DISC , pages 133–147, 2010.[31] R. Wattenhofer. Lecture notes in the course: Principles of distributed computing. In .22
PPENDIX
A Additional Related Work
In the wake-up problem, nodes can communicate only after successfully receiving a message. Farach-Colton et al. [9] show a lower bound of Ω(log n ) (more precisely, Ω(log n log (1 /ǫ )) for successprobability ǫ ) for the number of rounds required for solving the wake-up problem in the standardradio network model. Since sending a single message implies solving the wake-up problem, thisgives the same lower bound for MIS. This result holds for a single-hop network with half-duplexradios, no collision-detection, adversarial wake-up, and given only an upper bound on the size of thenetwork. They also give an Ω(log log n log (1 /ǫ )) lower bound in random geometric graphs wherenodes are placed uniformly at random in some area [0 , ℓ ] . The number of rounds is measuredstarting from the time at which the first node is woken.Moscibroda and Wattenhofer [24] show an MIS algorithm that requires O (log n ) rounds w.h.p.for unit disk graphs in the standard radio network model with half-duplex radios under asyn-chronous wake-up and no collision-detection. For each node, the number of rounds it requires ismeasured from the time it is woken until the time it produces an output. The complexity of thealgorithm is the maximum taken over all nodes of the number of rounds that they require.When collision detection is available, Schneider and Wattenhofer [30] show that Θ(log n ) roundsare required and sufficient for computing an MIS, as well as results about coloring and broadcasting.Schneider and Wattenhofer [29] show an MIS algorithm in O (log ∗ n ) rounds w.h.p in the classicmessage-passing model for bounded-independence graphs. These are graphs for which the numberof independent nodes in any r -neighborhood is bounded by some function f ( r ). A graph is ofpolynomially bounded-independence if f ( r ) is polynomial in r . This paper also shows ∆ + 1-coloring and maximal matching in O (log ∗ n ) rounds w.h.p. This matches the Ω(log ∗ n ) lower boundof Linial [20] for MIS in the classic message-passing model, which holds for a ring and thereforealso for bounded-independence graphs in general.For general graphs in the classic message-passing model, the best MIS algorithms are due toLuby [22] and to Alon et al. [1]. All algorithms require O (log n ) rounds w.h.p., while the bestknown lower bound is of Ω(log ∆ + √ log n ) due to Kuhn et al. [17].For standard radio networks another model that was studied is when F channels are available,and a node can choose which channel to transmit on or listen to at any given round. Daum etal. [7] showed an MIS algorithm that requires O ( log nF ) + ˜ O (log n ) rounds w.h.p., and use it tobuild a constant-degree connected dominating set. They then show how to solve leader electionand global broadcast in O ( D + log nF ) + ˜ O (log n ) rounds w.h.p., where D is the diameter of thegraph, and k -message broadcast in O ( D + k + log nF ) + ˜ O (log n ) rounds w.h.p. The assumptionsare that the underlying graph has polynomial bounded-independence, the radios are half-duplex,and no collision-detection is available. The authors also show [8] a lower bound of O ( log nF ) + log n for the number of rounds required for solving MIS in this model.The best known algorithms for leader election in the standard radio network model are dueto [11]. When collision detection is available, they provide an algorithm that runs in O (( D +log n log log n ) · min { log log n, log nD } ) rounds, and when collision detection is not available theyprovide an algorithm that runs in O (( D log nD + log n ) · min { log log n, log nD } ) rounds.Liu and Herlihy [21] give algorithms for approximating local sum and global sum in radionetworks. Their estimation technique employs the well-known decay-strategy [13]. Our algorithmsior approximating the degree and the size of the network are special case of the local-sum andglobal-sum respectively. In fact, our algorithms can be slightly modified to solve these tasks. B Additional Symmetry Breaking Tasks
B.1 Construction of BFS Trees
In this section, we consider the construction of a Breadth-First-Search (BFS) tree rooted at a givensource vertex s . Towards the end of this section, we show the following. Theorem B.1
For every network G = ( V, E ) of diameter D and a source vertex s ∈ V , a BFStree rooted at s can be constructed with high probability within O ( D ) communication rounds. Note that one can combine Theorem B.1 and the leader-election protocol to construct a spanningtree of radius O ( D ) within O ( D ) rounds w.h.p. (i.e., by computing first a leader in O ( D ) roundsand then constructing a BFS tree with respect to this leader).Let D s = max v ∈ V dist( s, v, G ) be the radius of the BFS tree and define L t = { v ∈ V | dist( s, v, G ) = t } as the vertices at distance t from s , for every t ∈ { , . . . , D s } . Recall that we assume that eachvertex v has a unique identifier id v in the range of [1 , . . . , n c ] for some constant c ≥ ConstructBFS consists of two phases. The first phase consists of O ( D ) stages, duringwhich the vertices compute their level in the BFS tree rooted at s (i.e., distance from s ). Todetect termination, the vertices also compute the diameter D s . The second phase consists of 3communication rounds, and is devoted for selecting a unique parent for each vertex. Each of the O ( D ) stages of the first phase consists of two alternating rounds in which two types of messages, MAXLEVEL and
MYLEVEL are sent by the vertices. In the odd round of stage t ≥
0, the vertices of L t , transmit a MYLEVEL message consisting of the C It,v, codeword of their level where each C It,v, isa randomly sampled instance of [ N, log N, log N ]-BIC code. Initially, every vertex v , sets d v = ∞ .Upon receiving the first MYLEVEL message d ′ it lets d v ← d ′ + 1 and it becomes active in the oddround of the next stage. In the even round of stage t ≥ every vertex v uses C It,v, , a randomlysampled instance of [ N, log N, log N ]-BIC code. It transmits a MAXLEVEL message consisting ofthe C It,v, codeword of the value d ∗ t ( v ) where d ∗ t ( v ) is the maximum value of all previous MAXLEVEL messages, where d ∗ ( v ) is initialized to 0. If the root vertex s did not receive a MAXLEVEL messagewith an increased value for more than two stages, it initiates a termination message. Upon receivinga termination message, a vertex v ∈ L t , waits for D s − t rounds before beginning the second phase.The second phase aims to break the symmetry between potential parents of a given vertex.To provide a separation between conflicting levels in the BFS tree, the phase consists of threecommunication rounds. The vertices of the level L t transmit an O (log n )-bit message m v in round t mod 3. Let C be an [ n, log n, log n ]-BCC code that is used to encode the IDs of the vertices.The O (log n ) bits message m v is divided into 2 log n blocks each of size O (log n ). The vertex v writes the codeword of its ID in the j ’th block with probability of 2 − j . Formally, let r ( v ) be the r -value computed by Protocol SL . Then, v writes C ( id v ) in the r ( v )’th block.Upon receiving a message in round ( t −
1) mod 3, a vertex u ∈ L t decodes the last occupiedblock of the received message (in the analysis we show that the decoding is successful); it thenselects one of the decoded IDs as its parent. This completes the description of the algorithm.ii nalysis. We begin by analyzing the first phase of the algorithm and show that the verticessuccessfully compute their level in the BFS tree and the diameter.
Claim B.2
With high probability, the following hold:1. After the odd round of stage t , d v = t for every v ∈ L t , t ∈ { , . . . , D s } .2. After the even round of stage D s − t , d ∗ t ( v ) = D s for every vertex v ∈ L t .3. The root s initiates a termination message in stage D s + 2 .Proof. Part (1) is shown by induction on t . For the base of the induction, consider t = 1. In theodd round of stage t = 1, the root vertex s transmits 0 and thus the vertices of L successfullydecode this value and by letting d v = 1 they hold the correct distance. Assume the claim holds upto stage t − t . The only active vertices in stage t are L t − and by the inductionassumption d v = t − v ∈ L t − at the beginning of stage t . Hence, by the properties ofthe BIC code, the vertices of L t successfully decode the message MYLEVEL that contains the value t − d v = t for every v ∈ L t as desired.Consider Part (2). It is easy to see to that d ∗ t ( v ) = d ∗ t ( v ′ ) for every v, v ′ ∈ L t for every level L t . In addition, observe that within every two stages, the root s gets a MAXLEVEL message withan increased value. By Part (1), in stage D s , the vertices of level D s know their level. Since thevertices of each level transmit the same MAXLEVEL message, every vertex receives at most threedistinct
MAXLEVEL values in a single round, and by the properties of the BIC code, the message isdecoded successfully. The claim can be shown by a backwards induction on the stage t , in a similarmanner to Part (1). Finally, by Part (2), s receives D s in stage 2 D s and hence terminates in stage2 D s + 2, during these two rounds no MAXLEVEL message with an improved value is received andhence it initiates termination. (cid:3)
We now consider the second phase. Note that by the waiting time defined for every vertex uponreceiving the termination message from s , the vertices are synchronized at the second phase. Toestablish correctness, it is left to show that with high probability, the message size is sufficient andthe for every vertex v , the last occupied block of each received message is decodable. Recall thatwe define j SL max = max { r ( v ) | v ∈ V } and S SL max = { v ∈ V | r ( v ) = j SL max } . Claim B.3
Consider the vertices of L t for some t ∈ [0 , . . . , D s ] . W.h.p., the following hold:(a) The only transmitting neighbors of v ∈ L t in round ( t −
1) mod 3 are in L t − .(b) j SL max ≤ n ;(c) The decoding of the last occupied block is successful;Proof. Part (a) follows by definition where in round i such that ( t −
1) mod 3 = i only theparenting level L t − transmits and the levels L t and L t +1 are silence. This implies that the messagesreceived by the vertices of L t in this round are sent by the parenting level. Consider (b). By Lemma4.1, with high probability, j SL max ≤ n and | S SL max | ≤ n . Finally, consider (c). By the proofof Lemma 4.1(b), the number of parents that wrote into the last occupied block is at most 2 log n ,and by the properties of the BCC code (and the uniqueness of the ID’s), the decoding is successful. (cid:3) iii .2 MIS Computation In this section we discuss algorithms for finding an MIS in the network. That is, each node hasto output a value in { , } such that the set of nodes that output 1 is a maximal independent setin the graph. We address both general graphs and graphs with bounded-growth. A graph withbounded growth is a graph for which there is a function f ( r ) such that the number of independentnodes in every r -neighborhood is at most f ( r ). Graphs of bounded growth have been studied inthe literature for the standard radio network model since intuitively one expects a real wirelessnetwork to be such that stations that are close to some transmitter are also relatively close to eachother. Algorithms for computing an MIS in a message passing model were given in [16,18,29], withan optimal algorithm requiring O (log ∗ n ) rounds [29]. In the standard radio network model withcollision detection, an MIS algorithm using O (log n ) rounds was given [30].We claim that for graphs of bounded growth, one can compute an MIS in the additive networkmodel within O (poly log log n ) rounds with full-duplex radios. The main tool that is required bythe algorithm is the degree approximation procedure of Section 5, and then one can essentiallysimulate the algorithm of [10], with a similar analysis. We omit the details from this extendedabstract.
Computing MIS for general graphs in full-duplex model
We show that for general graphs, it is possible to find an MIS in an additive wireless network within O (log n ) rounds, w.h.p. This matches the best known algorithm for a message-passing setting, dueto Luby [22] and Alon et al. [1]. In fact, we show that in the additive network model we can simulateLuby’s algorithm efficiently. We begin by recalling Luby’s algorithm, and afterwards we explainthe challenges for implementing it in a wireless network. Finally, we describe how we overcomethese challenges and present our implementation and its proof.Luby’s cornerstone algorithm works in phases, where in each phase every node v chooses tomark itself with probability 1 / d ( v ). If a marked node has the largest degree within its markedneighborhood (ties broken arbitrarily, say, by ID), then it enters the MIS in this phase and isremoved from the graph along with all of its neighbors. The following phase is executed with theremaining graph (and remaining degrees). It is straightforward that this algorithm produces anMIS, since no two neighbors can enter the MIS at the same phase, and all neighbors of an MIS nodeare removed along with it in the phase in which it entered the MIS. The beauty of the algorithm liesin its ability to remove a constant fraction of the edges from the graph in every phases, implyingthat it completes after O (log n ) rounds, w.h.p.To implement Luby’s algorithm in an additive network, we need the following tools. First, sincea node marks itself with probability inversely proportional to its degree, it has to able to computeits degree in the remaining subgraph. For this, we use our degree-approximation technique, andprove that working with approximate degrees is sufficient. Second, a node has to know whetherit has the maximal estimated degree within its marked neighborhood. While we can computean approximation to the maximal value in a neighborhood efficiently, here an approximation isinsufficient: it may be that there is more than a single node with the maximal estimated degree ina neighborhood, and we need to somehow be able to break symmetry. If the number of neighborswith the maximal estimated degree is not too large, i.e., O (log n ), then we can break symmetry by Using our simulation, this implies ˜ O (log n ) rounds with half-duplex radios, but the state-of-the-art algorithm forthe standard radio network model with half-duplex radios requires only O (log n ) rounds [30]. ivending IDs, using the simple BCC framework. However, if the number of neighbors with maximalestimated degree is larger, employing BCC will require too many rounds. Our crucial observationis that this event occurs with low probability, and hence we can simply disregard it. In more detail,a marked node that has the maximal estimated degree in its marked neighborhood estimates thenumber of its marked neighbors with maximal estimated degree. If this number is O (log n ) thensymmetry is broken using IDs, sent using BCC. Otherwise, the node does not enter the MIS.The pseudocode of our implementation is given in Algorithm 2. The proof of correctness isstraight forward using one additional verification round for each phase, where each node that isabout to enter the MIS transmits this information to its neighbors, and if a conflict is detectedthen the conflicting nodes both give up their attempt to enter the MIS. Theorem B.4 shows thatw.h.p., the algorithm terminates after O (log n ) rounds. Let C be an [ N, O (log N ) , O (log N )]-BCC code Initially: V ′ = V , M = ∅ , S = ∅ Locally: For every node v , α ( v, V ′ ) = ( δ ( v, V ′ ) , ID v ) Repeat until V ′ = ∅ : For every v ∈ V ′ do: δ ( v, V ′ ) ← AppDegree ( v, V ′ ) If δ ( v, V ′ ) = 0 then b ( v ) ← Else, b ( v ) ← / cδ ( v, V ′ ); 0 , otherwise. S ← { u ∈ V ′ | b ( u ) = 1 } For every v ∈ S do: δ ( v, S ) ← AppDegree ( v, S ) S ′ ← { u ∈ S | δ ( u, S ) ≤ log n } For every v ∈ S ′ do: Send C [ h v, α ( v, V ′ ) i ] If α ( v, V ′ ) > max { α ( u, V ′ ) | u ∈ ( S ′ ∩ Γ( v )) \ { v }} , then m ( v ) ← Else, m ( v ) ← If m ( v ) = 1 and { u ∈ Γ( v ) | m ( u ) = 1 } 6 = ∅ then m ( v ) ← A ← { u ∈ S ′ | m ( u ) = 1 } M ← M ∪ A S ← ∅ V ′ ← V ′ \ ( M ∪ Γ( M )) For every u ∈ A , Send 1 For every u ∈ Γ( A ) \ A , Send 0 Algorithm 2:
An MIS algorithm for general graphs. The parameter c is the constant given by AppDegree . Theorem B.4
The set M returned by Algorithm 2 is an MIS. The algorithm completes in O (log n ) rounds, w.h.p. To prove that M is an independent set, we claim that no two neighbors enter it in the samephase, which is guaranteed by Line 17. If a node was added to M in a certain phase, then becauseonly a single bit of information can be sent to it by all of its neighbors, it is removed along withall of its neighbors from any following phases, which gives that M is indeed an independent set.vt also holds that M is maximal, since any node removed in a certain phase is either in M or aneighbor of a node in M .The main task is to prove the number of rounds required for the algorithm to complete. Wefollow the line of proof of Luby [22], and show that in each phase, a constant fraction of the edgestouch at least one removed node. This implies that there are no more edges after O (log n ) rounds,w.h.p., after which the algorithm completes. The details of the proof are adapted from the proofof [31] to Luby’s algorithm.One modification we have to address is Line 12, where a node that has too many neighborswhich are selected and have its maximal estimated degree simply drops out of the set of selectednodes. For a given node, the probability of being in S \ S ′ is at most 1 / (2 log n ) log n +1 , since itneeds to have a degree of at least log n and so do all of its at least log n selected neighbors. A unionbound over all nodes still gives that this happens only in a very low probability, and hence the restof the proof is conditioned on the evert that this does not occur. Lemma B.5
For every node v , the probability that v is added to M in a certain phase is at least / cδ ( v, V ′ ) .Proof. To join M , a node v has to first mark itself, and then be the maximal node that markeditself in its neighborhood. This implies that P [ v ∈ M ] = P [ v ∈ M | b ( v ) = 1] P [ b ( v ) = 1] = P [ v ∈ M | b ( v ) = 1] · / cδ ( v, V ′ ) . (B.1)To bound this probability, we calculate the probability that v does not enter M despite beingmarked. This happens if either it does not have the maximal degree and ID among its markedneighbors, or if it has too many marked neighbors which share the largest degree with it.We denote by D ( v ) the neighbors of v with the same approximate degree, that is D ( v ) = { u ∈ Γ( v ) | δ ( u, V ′ ) = δ ( v, V ′ ) } . The probability that v has too many marked neighbors which shareits degree is small since in expectation it is constant, and w.h.p. it is at most O (log n ), using astandard Chernoff bound. Formally, E (cid:2) |{ u ∈ D ( v ) | b ( u ) = 1 }| (cid:3) = X u ∈ D ( v ) / cδ ( u, V ′ ) = X u ∈ D ( v ) / cδ ( v, V ′ ) ≤ d ( v, V ′ ) · / cδ ( v, V ′ ) ≤ cδ ( v, V ′ ) / cδ ( v, V ′ ) = 1 / . Since the random choices for b ( u ) are independent for different nodes u , we have that P (cid:2) |{ u ∈ D ( v ) | b ( u ) = 1 }| > O (log n ) (cid:3) < /n t , for a constant t > v does not have the maximal degree amongits marked neighbors. Denote D ′ ( v ) = { u ∈ Γ( v ) | α ( u, V ′ ) > α ( v, V ′ ) } . It holds that P [ ∃ u ∈ S ′ ∩ D ′ ( v )] ≤ X u ∈ D ′ ( v ) P [ u ∈ S ′ ] ≤ X u ∈ D ′ ( v ) / cδ ( v, V ′ ) ≤ cδ ( v, V ′ ) / cδ ( v, V ′ ) = 1 / . Finally, for every two neighbors u, v , the probability that m ( v ) = m ( u ) = 1 at Line 17 is at most1 /n t for some constant t . This is because for this to occur, it must be that one of their degreeestimations failed. This procedure is used at most twice per node and hence with probability atvieast 1 − /n t all four invocations of this procedure were successful in obtaining a c -approximation,in which case either m ( v ) or m ( u ) are 0.Hence, by Equation B.1, the probability that v joins M is at least P [ v ∈ M ] = (1 − P [ v M | b ( v ) = 1]) · / cδ ( v, V ′ ) ≥ (1 − (1 /n t + 1 /n t + 1 / / cδ ( v, V ′ ) ≥ / cδ ( v, V ′ ) . (cid:3) To show that a constant fraction of edges are removed in each phase, we show that a constantfraction of edges have at least one endpoint that is removed. We say that a node v is good if P u ∈ Γ( v ) / cδ ( u, V ′ ) ≥ / c , and claim that good nodes are removed with constant probability. Lemma B.6
Let v be a good node. Then the probability that v gets removed in Line 21 is at least p r = 1 / c − / c .Proof. If there is a node u ∈ Γ( v ) such that δ ( u, V ′ ) ≤ / cδ ( u, V ′ ) ≥ / c ≥ p r the node u joins M , and v is removed in Line 21.Otherwise, all nodes u ∈ Γ( v ) are such that δ ( u, V ′ ) ≥
3, and hence 1 / cδ ( u, V ′ ) ≤ / c . Let L ⊆ Γ( v ) be a subset of neighbors of v such that 1 / c ≤ P u ∈ L / cδ ( u, V ′ ) ≤ / c . The set L exists because if we take all of Γ( v ) then since v is good it holds that P u ∈ Γ( v ) / cδ ( u, V ′ ) ≥ / c ,and if this sum is larger than 1 / c then we can take out nodes until we reach such a set L (becausefor every u ∈ Γ( v ) we have 1 / cδ ( u, V ′ ) ≤ / c ).We can now calculate the probability that v is removed in Line 21, by being a neighbor of anode in M . P [ v ∈ Γ( M )] ≥ P [ ∃ u ∈ L ∩ M ] ≥ X u ∈ L P [ u ∈ M ] − X u,w ∈ L,u = w P [ u ∈ M ∧ w ∈ M ] ≥ X u ∈ L P [ u ∈ M ] − X u,w ∈ L P [ u ∈ S ∧ w ∈ S ] ≥ X u ∈ L / cδ ( u, V ′ ) − X u,w ∈ L / cδ ( u, V ′ ) · / cδ ( w, V ′ ) ≥ X u ∈ L / cδ ( u, V ′ ) / − X w ∈ L / cδ ( w, V ′ ) ! ≥ / c (1 / − / c ) = 1 / c − / c = p r . The third inequality above follows since the probability of a node to be in M is at most its probabilityof being in S . This completes the proof. (cid:3) Next, we consider a directed auxiliary graph G ′ over V ′ , where each edge of the graph inducedby V ′ is directed towards the endpoint with the larger α ( v, V ′ ) value. We claim that the outdegreein G ′ of any node v which is not good is at least twice its indegree. This holds because otherwise, P u ∈ Γ( v ) / cδ ( u, V ′ ) ≥ P u ∈ Γ( v ) / cδ ( v, V ′ ) ≥ d ( v ) / · / cδ ( v, V ′ ) ≥ /
6. It implies that at leasthalf of the edges of the graph induced by V ′ are good, in the sense that they have at least one goodendpoint, since the number of edges that are directed from a non-good node to a good node is atleast the number of edges in between two non-good nodes.vii roof of Theorem B.4: By Lemma B.6 each good node is removed in Line 21 with probabilityat least p r . Since at least half of the edges have a good endpoint, this implies that at least half ofthe edges have a probability of p r to be removed. Let X e be the characterizing random variable ofthe event that edge e is removed. Then E [ X e ] ≥ p r and by linearity of expectation we have thatthe expected number of edges that are removed is p r | E | . Since phases are independent this impliestermination in O (log n ) phases, in expectation. To show that this also holds with high probability,we let Y i be the characterizing random variable of the event that at least p r | E | edges were removedin phase i , and we denote Y = P Y i . The above argument implies that E [ Y ] = O (log n ). Sincephases are independent we can use a standard Chernoff bound to get that P [ Y > O ( E [ Y ])] < /n t for some constant t . Hence, with high probability, the number of phases required is O (log n ). B.3 Coloring
In this Section we address the problem of finding a (∆ + 1)-coloring of the underlying graph. Eachnode has to output a color in { , ..., ∆ + 1 } , such that no two neighbors share the same color. Webuild upon known techniques and embed the usage of BCC codes to them in order to obtain ourresults.Suppose A is an MIS algorithm that works in f ( n ) rounds in BGG graphs. We show how to getan algorithm for (∆ + 1)-coloring in O (∆ + f ( n )) rounds for BGG graphs. The algorithm followsthe line of the O (log ∗ n ) algorithm for (∆ + 1)-coloring in BGG graphs in the message-passingmodel, by Schneider and Wattenhofer [29]. We repeat the following procedure for the subgraph G i induced by the node set V , where initially V = V .1. Find an MIS S i in the graph G i .2. Denote by H i the graph ( S i , E S i ), where ( u, v ) ∈ E S i if d G i ( u, v ) ≤
3. Find an MIS S ′ i in H i .3. Each node in S ′ i colors itself and all of its neighbors.4. V i +1 = V i − Γ( S ′ i ) Analysis Sketch:
The analysis follows the analysis of previous work and hence we do not repeatit here in full, but rather sketch the idea. There are a constant number of phases because in eachneighborhood of radius 6 there is at least one node in S ′ i which gets colored in phase i , and therecan be at most a constant number of such nodes throughout the phases since they are independent.Steps 1 and 2 take at most f ( n ) rounds. Step 3 requires O (∆) rounds, as follows. Each node v ∈ S ′ i transmits its ID. Each uncolored neighbor u of v applies for a color, by sending its ID withsome probability. Once v knows an ID of a neighbor u it sends that ID and then u colors itself byannouncing its color (this is similarly to [28]). This way all nodes know about the colors that arealready used by their neighbors, and allows them to safely choose an unused color.It may be possible to go below O (∆) and obtain a solution in O (∆ / log n ) rounds, since one canuse BCC for the randomized attempts of coloring the neighbors. Moreover, we conjecture that onecan derive a lower bound of Ω(∆ / log n ) using an information theoretic argument that is similar tothat of Schneider and Wattenhofer [30]. B.4 Minimum Dominating Set
Using the algorithm of [19], a O (log n ) approximation for the minimum dominating set can becomputed within O (log nn