Constant-Length Labeling Schemes for Deterministic Radio Broadcast
aa r X i v : . [ c s . D C ] A ug Constant-Length Labeling Schemes forDeterministic Radio Broadcast
Faith Ellen § Barun Gorain ∗ Avery Miller † Andrzej Pelc ‡ August 6, 2019
Abstract
Broadcast is one of the fundamental network communication primitives. One node of anetwork, called the source , has a message that has to be learned by all other nodes. We considerbroadcast in radio networks, modeled as simple undirected connected graphs with a distinguishedsource. Nodes communicate in synchronous rounds. In each round, a node can either transmita message to all its neighbours, or stay silent and listen. At the receiving end, a node v hearsa message from a neighbour w in a given round if v listens in this round and if w is its onlyneighbour that transmits in this round. If more than one neighbour of a node v transmits in agiven round, we say that a collision occurs at v . We do not assume collision detection: in caseof a collision, node v does not hear anything (except the background noise that it also hearswhen no neighbour transmits).We are interested in the feasibility of deterministic broadcast in radio networks. If nodesof the network do not have any labels, deterministic broadcast is impossible even in the four-cycle. On the other hand, if all nodes have distinct labels, then broadcast can be carried out,e.g., in a round-robin fashion, and hence O (log n )-bit labels are sufficient for this task in n -nodenetworks. In fact, O (log ∆)-bit labels, where ∆ is the maximum degree, are enough to broadcastsuccessfully. Hence, it is natural to ask if very short labels are sufficient for broadcast. Our mainresult is a positive answer to this question. We show that every radio network can be labeledusing 2 bits in such a way that broadcast can be accomplished by some universal deterministicalgorithm that does not know the network topology nor any bound on its size. Moreover, atthe expense of an extra bit in the labels, we can get the following additional strong propertyof our algorithm: there exists a common round in which all nodes know that broadcast hasbeen completed. Finally, we show that 3-bit labels are also sufficient to solve both versions ofbroadcast in the case where the labeling scheme does not know which node is the source. keywords: broadcast, radio network, labeling scheme, feasibility § Department of Computer Science, University of Toronto, [email protected] . Partially supported by NSERCDiscovery Grant RGPIN–2015–05080. ∗ Indian Institute of Information Technology Vadodara, [email protected] . † Department of Computer Science, University of Manitoba, [email protected] . Partially supported byNSERC Discovery Grant RGPIN–2017–05936. ‡ D´epartement d’informatique, Universit´e du Qu´ebec en Outaouais, [email protected] . Partially supported by NSERCDiscovery Grant RGPIN–2013–08136 and by the Research Chair in Distributed Computing at the Universit´e duQu´ebec en Outaouais.
Introduction
Broadcast is one of the fundamental and most extensively studied network communication primi-tives. One node of a network, called the source , has a message that has to be learned by all othernodes. We consider broadcast in radio networks, modeled as simple undirected connected graphswith a distinguished source. In the sequel, we use the word graph in this sense, and we consider thenotions of network and graph as synonyms. Nodes communicate in synchronous rounds. Through-out the paper, round numbers refer to the local time at the source, which can differ from the localtime at other nodes. In each round, a node can either transmit a message to all its neighbours, orstay silent and listen. At the receiving end, a node v hears a message from a neighbour w in a givenround if v listens in this round and if w is its only neighbour that transmits in this round. If morethan one neighbour of a node v transmits in a given round, we say that a collision occurs at v . Wedo not assume collision detection: in case of a collision, node v does not hear anything (except thebackground noise that it also hears when no neighbour transmits). If collision detection is available,broadcast is trivially feasible, even in anonymous networks: consecutive bits of the source messagecan be transmitted by a sequence of silent and noisy rounds, cf. [7], using silence as 0 and a messageor collision as 1.We are interested in the feasibility of deterministic broadcast in radio networks. If the nodesof the network do not have any labels (or all have the same label), then deterministic broadcast isimpossible even in the four-cycle. Indeed, the two neighbours of the source must behave identically,i.e., transmit in exactly the same rounds, and hence, due to collisions, the fourth node can neverhear a message. On the other hand, if all nodes have distinct labels, then broadcast can be carriedout, e.g., in a round-robin fashion, and hence O (log n )-bit labels are sufficient for this task in n -nodenetworks. It is easy to see that, by using a proper colouring of the square of the graph, O (log ∆)-bitlabels, where ∆ is the maximum degree, are enough to successfully broadcast. Hence, it is naturalto ask if very short labels are sufficient for deterministic broadcast. In particular, is it possible tobroadcast in every radio network using labels of constant length? Below we formalize our question.A labeling scheme for a network represented by a graph G = ( V, E ) is any function L from theset V of nodes into the set S of finite binary strings. The string L ( v ) is called the label of thenode v . Note that labels assigned by a labeling scheme are not necessarily distinct. The length ofa labeling scheme L is the maximum length of any label assigned by it.Consider all graphs G , each labeled by some labeling scheme, with a distinguished source s G . Initially, each node knows only its own label, and the source has a message. A universaldeterministic broadcast algorithm works in synchronous rounds as follows. In each round, everynode makes a decision if it should transmit or listen. This decision is based only on the currenthistory of the node, which consists of the label of the node and the sequence of messages heardby the node so far. In particular, the decision does not depend on any knowledge of the graph G , including its size. However, the labeling scheme can use complete knowledge of the graph.Upon completion of the algorithm, all nodes should have the source message. For simplicity, weassume that when a node transmits, it can transmit its entire history (which may include the sourcemessage). However, in our algorithm, much smaller messages will suffice: they consist of either thesource message or a constant-size “stay” message.We also consider a variant of the above problem called acknowledged broadcast , which requiresthat the source node eventually knows that all nodes have received the source message. In our2lgorithm for this version of the problem, each transmitted message additionally contains a binarystring of length O (log n ), where n is the size of the graph. One of the roles of this string is toimplement a global clock. More specifically, in our algorithms, a node transmits only in responseto receiving a message, and hence the current round number (which is the current local roundnumber at the source node) can be maintained by including it in each transmitted message andincrementing it appropriately. Using our algorithm for acknowledged broadcast, we can ensure thatthere is a common round in which all nodes know that the broadcast of the source’s message hasbeen completed.Using the above terminology, our central question can be formulated as follows:Does there exist a universal deterministic (acknowledged) broadcast algorithm usinglabeling schemes of constant length for all radio networks?The above question can be seen in the framework of algorithms using informative labelingschemes , or equivalently, algorithms with advice [1, 11, 14, 16–21, 26, 30–32, 38]. When advice isgiven to nodes, two variations are considered: either the binary string given to nodes is the samefor all of them [27] or different strings are given to different nodes [20, 21], as in the case of thepresent paper. If strings may be different, they can be considered as labels assigned to nodes.Several authors have studied the minimum amount of advice (i.e. label length) required to solvecertain network problems. The framework of advice or labeling schemes is useful for quantifyingthe amount of information used to solve a network problem, regardless of the type of informationthat is provided. Our main contribution is a positive answer to our central question. For every radio network, weconstruct labeling schemes of constant length, and we design universal deterministic broadcastand acknowledged broadcast algorithms using these schemes. For the broadcast task, our labelingschemes have length 2, while for acknowledged broadcast, our labeling schemes have length 3. Inthe more difficult situation where the source node is not known at the time of labeling, our labelingscheme has length 3 (for both versions of broadcast).The importance of our result can be shown in the following scenario. Suppose that transmittingdevices that form a radio network are already deployed, and only a central monitor knows thelocation and the transmitting range of each of them, thus knowing the topology of the resultingnetwork. This could be applicable in an Internet of Things network in a business or industrialcomplex. One node of this network has to broadcast many consecutive messages to all other nodes.Then the monitor can assign very short labels to the devices, enabling multiple executions of theuniversal broadcast. The fact that labels can be very short may be crucial in situations whennodes of the network are weak and simple devices with very limited memory. Moreover, the factthat we can also do acknowledged broadcast in this situation permits the source to send the nextmessage only after all nodes received the preceding one. Our work is also relevant in the contextof Software-Defined Networks (SDNs) where the central controller assigns to each network devicea role, i.e., a forwarding behaviour. Our solution gives an efficient implementation for broadcastthat requires very few roles as well as simple forwarding rules.3 .3 Related work
Algorithmic problems in radio networks modeled as graphs were studied for such tasks as broadcast[9, 24], gossiping [9, 23] and leader election [36]. In some cases [9, 23], the topology of the networkwas unknown, in others [24], nodes were assumed to have a labeled map of the network and couldsituate themselves in it.For the broadcast task, most of the papers represented radio networks as arbitrary (undirectedor directed) graphs. Models used in the literature about algorithmic aspects of radio communi-cation, starting from [5], differ mostly in the amount of information about the network that isassumed available to nodes. However, assumptions about this knowledge concern particular itemsof information, such as the knowledge of the size of the network, its diameter, maximum degree, orsome neighbourhood around the nodes. In this paper, we adopt the approach of limiting the totalnumber of bits available to nodes, regardless of their meaning.Deterministic centralized broadcast assuming complete knowledge of the network was consideredin [6], where a polynomial-time algorithm constructing a O ( D log n )-time broadcast scheme wasgiven for all n -node networks of radius D . Subsequent improvements by many authors [15, 22,24] were followed by the polynomial-time algorithm from [35] constructing a O ( D + log n )-timebroadcast scheme, which is optimal. On the other hand, in [2], the authors proved the existence ofa family of n -node networks of radius 2 for which any broadcast algorithm requires time Ω(log n ).The “minimal dominating sets” that appear in our work were used under the name “minimalcovering sets” in the context of deterministic centralized broadcast and gossiping assuming completeknowledge of the network [24, 25].One of the first papers to study deterministic distributed broadcast in radio networks whosenodes have only limited knowledge of the topology was [3]. The authors assumed that nodes knowonly their own identifier and the identifiers of their neighbours. Many authors [4, 7–10] studieddeterministic distributed broadcast in radio networks under the assumption that nodes know onlytheir own identifier (but not the identifiers of their neighbours). In [7], the authors gave a broadcastalgorithm working in time O ( n ) for undirected n -node graphs, assuming that the nodes can transmitspontaneously before getting the source message. For this model, a matching lower bound Ω( n ) ondeterministic broadcast time was proved in [34], even for the class of networks of constant diameter.Increasingly faster broadcast algorithms working for arbitrary radio networks were constructed, thecurrently fastest being the O ( n log D log log D ) algorithm from [12]. On the other hand, in [10], alower bound Ω( n log D ) on broadcast time was proved for n -node networks of radius D .Randomized broadcast algorithms in radio networks have also been studied [3, 37]. For thesealgorithms, no topological knowledge of the network and no labels of nodes were assumed. In[3], the authors showed a randomized broadcast algorithm running in expected time O ( D log n +log n ). In [37], it was shown that for any randomized broadcast algorithm and parameters D < n ,there exists an n -node network of radius D requiring expected time Ω( D log( n/D )) to execute thisalgorithm. It should be noted that the lower bound Ω(log n ) from [2], for some networks of radius2, holds for randomized algorithms as well. A randomized algorithm working in expected time O ( D log( n/D ) + log n ), and thus matching the above lower bounds, was presented in [13, 33].Many papers [1,11,14,16–21,26,31,32,38] have proposed algorithms to solve network tasks moreefficiently by providing arbitrary information to nodes of the network or mobile agents circulatingin it. These are known as algorithms using informative labeling schemes or algorithms with advice .Most relevant to this paper are those concerning radio networks. In [30], the authors consideredthe set of radio networks in which it is possible to perform broadcast in constant time when4ach node has complete knowledge of the network. They proved that O ( n ) bits of advice aresufficient for performing broadcast in constant time in such networks and Ω( n ) bits are necessary.Short labeling schemes have been found that can be used to perform topology recognition in radionetworks modeled by trees [28] and to perform size discovery in arbitrary radio networks withcollision detection [29]. In this section, we present a labeling scheme λ that labels each node with a 2-bit string, and givea deterministic algorithm B that solves broadcast on any graph G that has been labeled using λ .At a high level, broadcast is completed by having a set of “informed” nodes, i.e., those thatknow the source message, that grows every two rounds. In odd-numbered rounds, we considerthe set of “frontier” nodes, i.e., uninformed nodes that are each adjacent to at least one informednode. From among the informed nodes, a minimal set of nodes that dominates the frontier nodestransmits the source message. Some of the frontier nodes will become newly-informed via thesetransmissions, while others will not, due to collisions. In even-numbered rounds, some of the newly-informed nodes will transmit a “stay” message to inform certain nodes to stay in the dominatingset for the next round. The first bit, x , of the label of a newly-informed node is used to determinewhether or not it is added to the dominating set. The second bit, x , is used to determine whetheror not it sends a “stay” message. The formal description of our broadcast algorithm B with sourcemessage µ is provided in Algorithm 1. We assume that there is a special “stay” message that isdistinct from the source message. Figure 1 gives an example of an execution of B . s G f g (7) f g (1) f g (1,4,6) f g (3) f g (3) f g (3) f g (1,4) f g (5) f g (7) f g (5) f g (5) f g (5) f g (5) Figure 1:
Example of an execution of Algorithm B on a graph labeled by λ . Each node contains its 2-bitlabel. To the upper right of each node: numbers in curly brackets are the round numbers in which the nodetransmits, numbers in parentheses are round numbers in which the node receives a message. Messages sentor received in odd rounds contain the source message µ , and messages sent or received in even rounds contain“stay”. lgorithm 1 B ( µ ) executed at each node v % Each node has a variable sourcemsg . The source node has this variable initially set to µ , all othernodes have it initially set to null . for each round r do if (never sent or received a message) and ( sourcemsg = null ) then % the source node transmits µ in first round transmit sourcemsg else if ( sourcemsg = null ) then % v has not previously received µ , listen for transmission if (message m is received) and ( m = “stay”) then sourcemsg ← m end if else % v received µ before round r if v first received sourcemsg in round r − then if x = 1 then transmit sourcemsg end if else if v first received sourcemsg in round r − then if x = 1 then transmit “stay” end if else if v transmitted sourcemsg in round r − and received “stay” in round r − then transmit sourcemsg end if end if end for We now formally define the labeling scheme and prove the correctness of B . We rely heavily onfive carefully chosen sequences of node sets. The following notation will be used in the constructionof these sequences and throughout the remainder of this section.A set of nodes X dominates a set of nodes Y if, for each node y ∈ Y , there is a node x ∈ X that is adjacent to y . For any set of nodes X ⊆ V ( G ), denote by Γ( X ) the neighbourhood of X ,i.e., Γ( X ) = { v ∈ V ( G ) | ∃ w ∈ X, { v, w } ∈ E ( G ) } . We construct five sequences of sets, indexed by i ≥
1. At a high level, INF i will be the nodes thatare informed before round 2 i −
1, UNINF i will be the nodes that are not informed before round2 i −
1, FRONTIER i will be the uninformed nodes that are adjacent to at least one informed node inround 2 i −
1, NEW i will be the nodes that are newly-informed in round 2 i −
1, and DOM i will be thenodes that inform the nodes in NEW i in round 2 i −
1. Recalling that s G denotes the source node of G , we initialize the construction by setting INF = { s G } , UNINF = V ( G ) − { s G } , FRONTIER =Γ( s G ) , NEW = Γ( s G ) , DOM = { s G } . Our construction proceeds in stages, where stage i ≥ i = INF i − ∪ NEW i − .2. Define UNINF i = UNINF i − \ NEW i − . 6. Define FRONTIER i = UNINF i ∩ Γ(INF i ).4. Define DOM i to be a minimal subset of DOM i − ∪ NEW i − that dominates all nodes inFRONTIER i .5. Define NEW i to be the subset of nodes in FRONTIER i that are adjacent to exactly one nodein DOM i .The construction ends when INF i = V ( G ). We now provide some useful facts about thesequences. The first two observations are direct consequences of the construction. Fact 2.1.
NEW i ⊆ FRONTIER i ⊆ UNINF i for all i ≥ . Fact 2.2.
INF i = INF ∪ S i − j =1 NEW j and UNINF i = UNINF \ S i − j =1 NEW j . Lemma 2.3.
For i = i ′ , we have NEW i ∩ NEW i ′ = ∅ .Proof. Without loss of generality, assume that i > i ′ . By Facts 2.1 and 2.2, it follows that NEW i ⊆ UNINF i = UNINF \ S i − j =1 NEW j . In particular, NEW i ⊆ UNINF \ NEW i ′ , so NEW i ∩ NEW i ′ = ∅ . The following result can be viewed as a guarantee of progress in each stage: if there are anyremaining uninformed nodes at stage i , then at least one node will be newly informed in stage i . Lemma 2.4.
For each i ≥ , if INF i = V ( G ) , then NEW i = ∅ .Proof. If INF = { s G } 6 = V ( G ), then NEW = Γ( s G ) = ∅ . So assume that i ≥
2. Since the graphis connected and V ( G ) is the disjoint union of INF i and UNINF i , it follows that FRONTIER i = ∅ .Consider any v ∈ DOM i . If each node w ∈ FRONTIER i that is adjacent to v is also adjacentto at least one other node in DOM i , then DOM i \ { v } also dominates all nodes in FRONTIER i ,contradicting the minimality of DOM i . Thus, there is at least one node w ∈ FRONTIER i that isadjacent to v and not adjacent to any other node in DOM i . Hence, by definition, NEW i = ∅ .The following result shows that the DOM i is well-defined. Lemma 2.5.
For all i ≥ , there exists a subset of DOM i − ∪ NEW i − that dominates all nodesin FRONTIER i .Proof. Consider any node v ∈ FRONTIER i and suppose that v does not have a neighbour inDOM i − . By definition, DOM i − dominates all nodes in FRONTIER i − , so it follows that v / ∈ FRONTIER i − . By Fact 2.1, v ∈ FRONTIER i ⊆ UNINF i , and by construction, UNINF i ⊆ UNINF i − , so v ∈ UNINF i − . By the definition of FRONTIER i − , it follows that v / ∈ Γ(INF i − ).But, v ∈ FRONTIER i implies that v ∈ Γ(INF i ). It follows that v has a neighbour in INF i \ INF i − =NEW i − . Therefore, we have shown that every node v ∈ FRONTIER i has at least one neighbourin DOM i − ∪ NEW i − , which implies the desired result.Let ℓ be the smallest value of i such that INF i = V ( G ). We now give an upper bound on thevalue of ℓ . Lemma 2.6. ℓ ≤ n . roof. The proof is by induction on i . By definition, | INF | = 1, and, by Fact 2.2 and Lemma 2.4,it follows that | INF i | ≥ i . Hence, INF n = V ( G ).It follows from Lemmas 2.3 and 2.6 that every node in G \ { s G } is contained in exactly one ofthe NEW i sets. We will later use this to ensure that all nodes in G \ { s G } are eventually informed. Corollary 2.7.
The sets NEW , . . . , NEW ℓ − form a partition of G \ { s G } . λ Formally, our labeling scheme λ ( G ) assigns a label x x to each node in G as follows: • For each node v , if there exists i ≥ v ∈ DOM i , then set x = 1 at node v .Otherwise, set x = 0 at node v . • For each i ≥
1, for each node v ∈ DOM i +1 ∩ DOM i , arbitrarily pick one node w ∈ NEW i thatis adjacent to v , and set x = 1 at node w . At all other nodes, set x = 0. B Our approach to showing that all nodes are eventually informed is to fully characterize which nodestransmit and which nodes are newly-informed in each round of the broadcast algorithm. Roughlyspeaking, we will show that, in an odd round 2 i −
1, the nodes in DOM i transmit and all nodes inNEW i receive the source message for the first time. Then, in round 2 i , a certain subset of NEW i transmits, which results in the nodes of DOM i +1 receiving “stay”. This will prompt the nodes ofDOM i +1 to transmit in round 2 i + 1, which informs all nodes in NEW i +1 . In this way, we willshow that, for all i ∈ { , . . . , ℓ − } , all nodes in NEW i are informed in round 2 i −
1. Since we havealready shown that the sets NEW , . . . , NEW ℓ − partition G \ { s G } , this will show that broadcastis completed. Lemma 2.8.
For each t > ,1. If t = 2 i − , the following hold:(a) Node v transmits µ in round t if and only if v ∈ DOM i .(b) Node w receives µ for the first time in round t if and only if w ∈ NEW i .2. If t = 2 i , the following holds:(a) Node v transmits “stay” in round t if and only if v ∈ NEW i and v ’s label has x = 1 .Proof. The proof proceeds by induction on t . In the base case, t = 1, we see that the source s G isthe only node that transmits in round 1, it transmits µ , and the set of nodes that receive µ for thefirst time in round 1 is Γ( s G ). Since DOM = { s G } and NEW = Γ( s G ), this proves the base case.For a fixed t ≥
2, assume that the result holds for all rounds t ′ < t . The induction step has twocases: 8 t = 2 i for some i ≥ v ∈ NEW i such that v ’s label has x = 1. By the induction hypothesis, v receives µ for the first time in round 2 i −
1. By the definition of the broadcast algorithm, v transmits “stay” in round 2 i at line 15.Conversely, suppose that v transmits “stay” in round 2 i . By the algorithm, v must havetransmitted “stay” at line 15. From the code, it follows that v ’s label has x = 1 and v received µ for the first time in round 2 i −
1. By the induction hypothesis, v ∈ NEW i . Thiscompletes the proof of 2(a). • t = 2 i − i ≥ – Proof of 1(a):First, suppose that v ∈ DOM i . By the definition of DOM i , we know that DOM i ⊆ DOM i − ∪ NEW i − . If v ∈ NEW i − , then, by the induction hypothesis, v received µ forthe first time in round 2 i −
3. By the definition of the labeling scheme, we know that v ’s label has x = 1. Hence, by lines 9-11, v transmits µ in round 2 i −
1. So, suppose v ∈ DOM i − . By the induction hypothesis, v transmitted µ in round 2 i −
3. By thedefinition of the labeling scheme, there is exactly one node in NEW i − that is adjacentto v and is labeled with x = 1. Therefore, exactly one neighbour of v transmits “stay”in round 2 i −
2, so v receives “stay” in round 2 i −
2. Hence, from lines 17-18, v transmitsin round 2 i − v transmits µ in round 2 i −
1. There are two cases to consider,depending on whether v ’s transmission of µ in round 2 i − v ’s label has x = 1, so, by the definition oflabeling scheme, v ∈ DOM j for some minimal j . Since DOM j ⊆ DOM j − ∪ NEW j − ,the minimality of j implies that v ∈ NEW j − . Further, by line 9, v received µ forthe first time in round 2 i −
3. Hence, by the induction hypothesis, v ∈ NEW i − . ByLemma 2.3, it follows that i = j . Thus, v ∈ DOM i . Now, assume that v ’s transmissionoccurred at line 18. By line 17, we know that v received “stay” in round 2 i −
2. By theinduction hypothesis, the nodes in NEW i − with x = 1 are the nodes that transmit“stay” in round 2 i −
2. It follows that v is adjacent to exactly one node w ∈ NEW i − whose label has x = 1. By the definition of the labeling scheme, w is adjacent to anode v ′ ∈ DOM i ∩ DOM i − . By the induction hypothesis, since v ′ ∈ DOM i − , we knowthat v ′ transmitted in round 2 i −
3. By line 17, v transmitted in round 2 i −
3. Since w ∈ NEW i − , the induction hypothesis implies that w received a message in round 2 i − v = v ′ ∈ DOM i . – Proof of 1(b):First, suppose that w receives µ for the first time in round 2 i −
1. Since w receives µ inround 2 i −
1, it must be adjacent to exactly one node that transmits in round 2 i − i is the set of nodes that transmit in round 2 i −
1, whichimplies that w is adjacent to exactly one node in DOM i . By the definition of NEW i , itfollows that w ∈ NEW i .Conversely, suppose that w ∈ NEW i . Then, by definition, w is adjacent to exactly onenode in DOM i . By 1(a), DOM i is the set of nodes that transmit in round 2 i −
1. Itfollows that w receives message µ in round 2 i −
1. If w received µ for the first time in9ome round t ′ < i −
1, then, by the induction hypothesis, w is contained in some NEW i ′ where i ′ < i . This is impossible, by Lemma 2.3. Hence, w received µ for the first timein round 2 i − G \ { s G } are informed within 2 n rounds. Theorem 2.9.
Consider any n -node unlabeled graph G with a designated source node s G withsource message µ . By applying the 2-bit labeling scheme λ and then executing algorithm B , allnodes in G \ { s G } are informed within n − rounds.Proof. Consider an arbitrary node w ∈ G \ { s G } . By Corollary 2.7, w is contained in NEW i forexactly one i ∈ { , . . . , ℓ − } . By Lemma 2.8, w receives µ for the first time in round 2 i − ≤ ℓ − ℓ − ≤ n −
3, as desired.
To solve acknowledged broadcast, we provide an algorithm B ack in which the source node s G receivesan “ack” message in some round t after all nodes in G \ { s G } have received µ . At a high level, B ack is obtained from B by considering a particular node z that receives µ last when B is executedon G . An additional bit x in each node’s label is used to identify z . Once it receives µ , node z initiates the acknowledgement process by immediately transmitting an “ack” message that containsthe round number k in which it first received µ . The (unique) neighbour of z that transmitted inround k will receive this message, and it immediately transmits an “ack” message that containsthe round number k ′ in which it first received µ . This process continues until the source nodereceives an “ack” message. The difficulty is that each node must know the round number in whichit received µ , and the round numbers in which it transmits. This is implemented as follows. Thesource node appends “1” to its first transmitted message. Every other node determines the roundnumber by recording the number that is appended to the first received message containing µ , andappends the round number (appropriately increased) whenever it transmits. The formal descriptionof our acknowledged broadcast algorithm B ack with source message µ is provided in Algorithm 2.We assume that there is a special “ack” message that is distinct from the source message and the“stay” message. λ ack The labeling scheme is identical to λ except that one node z will have a new label. This can berepresented using an additional bit, x , which is 1 for z and 0 for all other nodes. The node z ischosen as follows: label G using labeling scheme λ and execute B on the resulting labeled graph,then determine the first round r after which there are no uninformed nodes, and choose z to bea node that receives µ in round r . If there is more than one such node, choose z arbitrarily fromamong them.We note that the labeling scheme λ ack will never assign certain labels to any node, which meanswe may safely use these labels in later schemes that are built on top of λ ack . At a high level, thisis because z is the only node with bit x set to 1, and, as there are no remaining uninformed nodes10fter z receives µ , our labeling scheme will set z ’s bits x and x to 0 to indicate that z should nottransmit after receiving µ . Fact 3.1.
For any graph G , when the labeling scheme λ ack is applied to G , no node is labeled with101 or 111 or 011.Proof. By definition, node z is the only node that has bit x = 1, so it is sufficient to prove thatnode z has bit x = x = 0. By the definition of λ in Section 2.2, it is sufficient to prove that thereis no value of i ≥ z ∈ DOM i . To obtain a contradiction, assume that there exists an i ≥ z ∈ DOM i , and let j be the smallest such i . Then, by the definition of DOM j , thefact that z ∈ DOM j implies that z ∈ NEW j − . By the choice of z by λ ack and Lemma 2.8, node z receives µ for the first time in round 2( j − −
1, and there are no remaining uninformed nodesafter this round. In particular, by Lemma 2.8, this means that NEW i = ∅ for all i ≥ j , and soINF j = V ( G ) by Lemma 2.4. This implies that UNINF j = ∅ , so FRONTIER j = ∅ , which meansDOM j = ∅ , which contradicts the fact that z ∈ DOM j . B ack To prove the correctness of B ack , we first observe that all transmissions of “ack” messages occurafter all transmissions of µ and “stay” messages, i.e., the broadcast and the acknowledgementprocess do not interfere with one another. The first two observations follow from Lemma 2.8 andthe fact that NEW i = DOM i = ∅ for all i ≥ ℓ . Observation 3.2.
The last round in which a node receives µ for the first time is ℓ − . Observation 3.3.
No transmissions of µ nor “stay” occur after round ℓ − . The next observation follows from Observation 3.2 and the definitions of algorithms λ ack and B ack . Observation 3.4.
The first transmission of “ack” occurs in round ℓ − , and is transmitted bythe unique node z whose label has x = 1 . lgorithm 2 B ack ( µ ) executed at each node v % Each node has a variable sourcemsg . The source node has this variable initially set to µ ,all other nodes have it initially set to null . Each node maintains a variable informedRound that keeps track of the first round in which it received µ . Each non-source node maintains avariable transmitRounds that keeps track of the set of rounds in which it transmitted µ . informedRound ← null transmitRounds ← null for each round r do if (never sent or received a message) and ( sourcemsg = null ) then % source node transmits µ in first round transmit ( sourcemsg , else if ( sourcemsg = null) then % has not previously received µ , listen for transmission if (message ( m, k ) is received) and ( m = “stay”) then sourcemsg ← m informedRound ← k end if else % the node received µ before round r if v first received sourcemsg in round r − then if x = 1 then transmit ( sourcemsg , informedRound + 2) insert informedRound + 2 into transmitRounds end if else if v first received sourcemsg in round r − then if x = 1 then % start acknowledgement process transmit (“ack” , informedRound ) else if x = 1 then transmit (“stay” , informedRound + 1) end if else if v received (“stay” , k ) in round r − then if v transmitted sourcemsg in round r − then transmit ( sourcemsg , k + 1) insert k + 1 into transmitRounds end if else if v received (“ack” , k ) in round r − then if k is contained in transmitRounds then transmit (“ack” , informedRound ) end if end if end if end for
12e now prove that the correct round number is appended to each message containing µ , whichis necessary for the correctness of the acknowledgement process. Lemma 3.5.
The messages ( µ, t ) and (“ stay ” , t ) are transmitted only in round t .Proof. The proof is by induction on t . For the base case, t = 0, we see that the source node sends( µ,
0) in its first transmission, and no other nodes transmit before receiving µ for the first time.As induction hypothesis, assume that, for all 0 ≤ t ′ < t , a message ( µ, t ′ ) or (“stay” , t ′ ) is onlytransmitted in round t ′ .First, suppose that a node v transmits a message (“stay” , t ). This occurs at line 21, which, byline 17, implies that v received µ for the first time in round t − µ, t ′ ). By theinduction hypothesis, t ′ = t −
1. Therefore, v sets informedRound equal to t − v transmits (“stay” , informedRound + 1), it follows that informedRound + 1 = t , as desired.Next, suppose that a node v transmits a message ( µ, t ). If this transmission occurs at line 14,then, by line 12, we know that v received µ for the first time in round t − µ, t ′ ). By the induction hypothesis, t ′ = t −
2. Therefore, v sets informedRound equal to t − v transmits (“stay” , informedRound + 2), it follows that informedRound + 2 = t ,as desired. The other possibility is that the transmission by v occurs at line 25, which, by line 23,implies that v received a (“stay” , t ′ ) message in round t −
1. By the induction hypothesis, t ′ = t − v transmits ( µ, t ′ + 1), it follows that t ′ + 1 = t , as desired.From Lemma 3.5, it follows that if a node v = s G first receives µ in round t , then the informedRound variable at node v is equal to t in all subsequent rounds. Similarly, if a node v = s G transmits a message containing µ in round t , then the transmitRounds variable at node v contains t in all subsequent rounds.We complete the proof of correctness of B ack by showing that the source node will eventuallyreceive an “ack” message. First, we show that at most one node transmits “ack” in any round,which implies that no collisions occur during the acknowledgement procedure. Lemma 3.6.
After round ℓ − , at most one node v transmits in each round.Proof. The proof is by induction on the round number t . For the base case, Observations 3.2-3.4imply that the unique node z with x = 1 in its label transmits (“ack” , ℓ −
3) in round 2 ℓ − t ≥ ℓ −
2. If no nodetransmits in round t , then, from Observation 3.3 and the code, no node transmits in round t + 1.Otherwise, suppose that exactly one node v transmits in round t . By Observation 3.3, this messageis of the form (“ack” , k ). At most one neighbour w of v contains k in its transmitRounds variablesince v received µ in round k . From Observation 3.3 and the code, no other node transmits inround t + 1.We now show that the “ack” message propagates through a sequence of nodes, where each nodeis contained in some DOM i . Further, the indices of the corresponding DOM i sets form a decreasingsequence, which implies that { s G } = DOM will eventually receive an “ack” message. Lemma 3.7.
For each i ∈ { , . . . , ℓ − } , in round ℓ − i , some node in DOM j with j ≤ ℓ − i − receives ( “ack” , j − . roof. The proof is by induction on i . For the base case i = 0, Observations 3.2 and 3.4 imply that z transmits an (“ack” , ℓ −
3) message in round 2 ℓ −
2. By Lemma 3.6, no other node transmitsin round 2 ℓ −
2, so all of z ’s neighbours receive the transmitted “ack” message. Since z received µ in round 2 ℓ −
3, it follows that a neighbour z ′ of z transmitted µ in round 2 ℓ −
3. By Lemma 2.8, z ′ ∈ DOM ℓ − . Therefore, the statement is satisfied with j = ℓ −
1, which completes the base case.As induction hypothesis, assume that, for some i ∈ { , . . . , ℓ − } , in round 2 ℓ − i , somenode w ∈ DOM j with j ≤ ℓ − i − , j − w ∈ DOM j , Lemma 2.8 impliesthat w transmitted µ in round 2 j −
1. Therefore, its transmitRounds variable contains 2 j −
1. Bylines 28-30 of B ack , it follows that w transmits (“ack” , informedRound ) in round 2 ℓ − i . Wenote that the value of informedRound at w must be less than 2 j −
1, since w must have received µ for the first time before it transmitted µ in round 2 j −
1. From Lemma 2.8, we conclude that informedRound = 2 j ′ − j ′ < j . So we have shown that w transmits (“ack” , j ′ −
1) forsome j ′ < j in round 2 ℓ − i . By Lemma 3.6, no other node transmits in round 2 ℓ − i , so allof w ’s neighbours receive the transmitted “ack” message in round 2 ℓ − i . Since w received µ inround 2 j ′ −
1, it follows that a neighbour w ′ of w transmitted µ in round 2 j ′ −
1. By Lemma 2.8, w ′ ∈ DOM j ′ . To summarize, we have shown that in round 2 ℓ − i + 1), some node w ′ ∈ DOM j ′ with j ′ ≤ j − ≤ ℓ − i − ℓ − ( i +1) − , j ′ − Corollary 3.8.
There exists a round t ∈ { ℓ − , . . . , ℓ − } in which the source node receives an“ack” message. The correctness of B ack follows directly from Corollary 3.8, which gives us the main result ofthis section. Theorem 3.9.
Consider any n -node unlabeled graph G with a designated source node s G withsource message µ . By applying the 3-bit labeling scheme λ ack and then executing algorithm B ack ,all nodes in G \ { s G } are informed by round t ≤ n − , and s G receives an “ack” message by round t ′ ∈ { t + 1 , . . . , t + n − } . Finally, we note that B and B ack can be used to ensure that there is a common round in whichall nodes know that the broadcast of the source’s message µ has been completed. First, run B ack ,and have the source node record the round number m in which it first receives an “ack” message.Then, the source executes B with message m . All nodes will receive the value of m before round2 m . So, in round 2 m , all nodes know that the original broadcast of µ has been completed. In this section, we consider the more difficult scenario in which the source node is not designatedin G when the labeling scheme is applied. We provide a labeling scheme of length 3 and a universaldeterministic algorithm B arb that solves (acknowledged) broadcast regardless of which node initiallyknows the source message. λ arb Choose an arbitrary node r and label this node using the string 111. Apply the labeling scheme λ ack to the remaining nodes in the network, but use r as the source node (as there is no designatedsource s G ). By Fact 3.1, note that λ ack does not assign the label 111 to any node, so the node r is14 unique node in the network that our algorithm can use to play a special role in coordinating thebroadcast, regardless of which node is the actual source s G . Let z be the node labeled 001 by λ ack ,i.e., the node that initiates the acknowledgement process in an execution of B ack . B arb
1. Perform an acknowledged broadcast using B ack with node r as source and with message“initialize”. Each node v stores in a variable t v the timestamp value contained in the first“initialize” message it received. In particular, node r sets t r to 0. When starting the acknowl-edgement process, node z appends to the “ack” message the timestamp value T = t z . Thisstep of the algorithm ends when r receives the “ack” message, at which point it knows thevalue of T and it knows that all nodes have received “initialize”.2. Perform an acknowledged broadcast using a modified version of B ack with node r as the sourceand with message (“ready”, T ). The modification to the algorithm is that the node z does notinitiate the acknowledgement process. Instead, when the source node s G receives the “ready”message, it waits T rounds, then initiates the acknowledgement process (as described in B ack ),but with the source message µ appended to the “ack” message. (Waiting T rounds ensuresthat this acknowledgement process started by s G does not begin until the “ready” broadcasthas completed.) This step of the algorithm ends when r receives the “ack” message, at whichpoint it knows the source message µ . Further, all nodes know the value of T .3. Perform a broadcast using B with node r as source and with message µ . At the end of thisbroadcast, all nodes know the source message µ . If each node v waits T − t v rounds afterreceiving µ in this step, then the algorithm solves acknowledged broadcast, as all nodes canbe sure that all nodes have received µ . We presented a universal deterministic broadcast algorithm using labeling schemes of constantlength that works for arbitrary radio networks. Our schemes are of length 2, and we showed howto solved acknowledged broadcast with schemes of length 3 (but only 5 different labels). In thecase where the source node is not designated when the labeling scheme is applied, our scheme alsohas length 3, but uses 6 different labels. It would be interesting to determine if schemes usingfewer than 4 different labels are sufficient for broadcast. We do not have any impossibility resultsbeyond the trivial 1-bit lower bound (2 different labels), and we are intrigued by the possibility thatthere exists a scheme of length 1 for broadcast. A positive answer can be obtained for broadcastin graphs where each node’s distance to the source is at most 2: use λ and B from Section 2, butuse only the bit x , and modify the definitions of FRONTIER i and DOM i by changing instances ofDOM i − ∪ NEW i − to DOM i − . We can also show that it is possible to perform broadcast in series-parallel graphs and grid graphs using single-bit labels. In both cases, using the same techniquefrom Section 3, acknowledged broadcast is possible using 3 labels. It would also be interesting todetermine whether or not acknowledged broadcast is possible in all graphs using 1-bit labels, and,if not, if it is possible using 2-bit labels. Another open question is whether acknowledged broadcastcan be performed with only constant-length messages, instead of O (log n ) bits. The above open15uestions can also be asked for the case where the source node is not designated when the labelingscheme is applied.In this paper, we focused on the feasibility of radio broadcast with short labels, and we didnot try to optimize the time complexity. Our algorithm works in time O ( n ). This yields thefollowing open problem. What is the fastest universal deterministic broadcast algorithm usinglabeling schemes of constant length? References [1] Serge Abiteboul, Stephen Alstrup, Haim Kaplan, Tova Milo, and Theis Rauhe. Compactlabeling scheme for ancestor queries.
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