Broadcast is the most studied communication primitive in networks and distributed systems. Broadcast primitive ensures that once a source node (a.k.a. the broadcast initiator) sends message to all other nodes in the network should receive this message in a finite time. Limited by the transmission range, messages may not be able to be sent directly from one node to some other arbitrary node in the network. Therefore relay nodes need to assist the source node during the message propagation by re-propagating it. Deterministic centralized broadcast, where nodes have complete network knowledge has been studied by Kowalski et al. in . The authors propose an optimal solution broadcast that completes within rounds, where is the number of nodes in network and is the largest distance from the source to any node of the network. The time lower bound for broadcast, , has been proved in  for a family of radius-2 networks. For deterministic distributed broadcast, assuming that nodes only know their IDs (i.e. they do not know the IDs of their neighbors nor the network topology),  propose the fastest broadcast within rounds, where is the diameter of network. The lower bound in this case proposed in  is , where is the largest distance from the source to any node of the network.
In wireless networks, when a message is sent from a node it goes into the wireless channel in the form of a wireless signal which can be received by all the nodes within the transmission range of the sender. However, when a node is located in the range of more than one node that send messages simultaneously the multiple wireless signals may generate collisions at the receiver. The receiver cannot decode any useful information from the superimposed interference signals. At the MAC layer several solutions have been proposed in the last two decades in order to reduce the collisions. All of them offer probabilistic guarantees. Our study follows the recent work that addresses this problem at the application layer. More specifically, we are interested in deterministic solutions for broadcasting messages based on the use of extra information or advise (also referred as labelling) precomputed before the broadcast is invoked.
Labelling schemes have been designed to compute network size, the father-son relationship and the geographic distance between arbitrary nodes in the network (e.g. ,  and ). Labelling schemes have been also used in  and  in order to improve the efficiency of Minimum Spanning Tree or Leader Election algorithms. Furthermore,  and  exploit labelling in order to improve the existing solutions for network exploration by a robot/agent moving in the network.
Very few works ( e.g.  and ) exploit labelling schemes to efficiently design broadcast primitives. When using labelling schemes nodes record less information than in the case of centralized broadcast, where nodes need to know complete network information. Compared with the existing solutions for deterministic distributed broadcast the time complexity is improved. In  the authors prove that for an arbitrary network to archive broadcast within constant number of rounds a bits of advice is sufficient but not . Very recently, a labelling scheme with 2-bits advice (3 bits for broadcast with acknowledgement) is proposed in . The authors prove that their algorithms need rounds for the broadcast without acknowledgement and rounds for broadcast with acknowledgement in arbitrary network.
Contribution: Our work is in the line of research described in  and . We first introduce a new family of networks, called level-separable networks issued from our recent studies in wireless body area networks , , ,  and . We then propose an acknowledgement-free broadcast strategy using 1-bit labelling and a broadcast scheme with acknowledgement using a 2-bit labelling. In the class of level-separable networks our algorithms are memory optimal and finish within rounds for both types of broadcast primitives, where is the eccentricity of the broadcast source. Second, we address the arbitrary networks and improve the broadcast scheme with acknowledgement proposed in  in terms of memory and time complexity by efficiently exploiting the 3-bits labelling encoding. Our solution, use no local persistent variables except the 3-bits labelling.
Ii Model and Problem Definition
Ii-a Communication Model
We model the network as a graph where , the set of vertices, represents the set of nodes in the network and , the set of , is a set of unordered pairs , , that represents the communications links between nodes and . In the following denotes the set of neighbors of node .
We target wireless networks where due to the limitation of the transmission power, a node may not have connections with the other nodes in the network (i.e., ). However, we assume that the network is connected, i.e., there is a path between any two nodes in the network.
We assume that nodes execute the same algorithm and are time synchronized. The system execution is decomposed in . When a node sends a message at round , all nodes in receive the message at the end of round . Collisions occur at node in round if a set of nodes, and , send a message in round . In that case is considered that has not received any message successfully.
In the following we are interested in solving the Broadcast problem: when a source node sends a message it should be received by all the nodes in the network in finite bounded time.
Ii-B Level-Separable Network
In this section, we present a family of networks we called Level-Separable Network. We say an arbitrary network is a Level-Separable Network if the underlay communication graph of the network verifies the Level-Separable propriety defined below.
To define the Level-Separable propriety, we introduce some preliminary notations.
Let be a network and let , a predefined vertex , be the source node of the broadcast. Each vertex has a geometric distance with respect to denoted . is the eccentricity of vertex , that is the farthest distance from it to any other vertex. In the rest of the paper we denote by .
Definition 1 (Level).
Let be a network. For any vertex in , where is the source node, the of the node is
i.e., the level of is its geometric distance to . Let
denote the set containing all the vertices at level .
Definition 2 (Parents and Sons).
Let be a network. A vertex is parent of vertex (a vertex is son of vertex ) in graph with the root source node : if
Let () be the set of sons (parents) of (). If (), we say that () has () as son (parent).
Level-Separable propriety focus on how to separate nodes in the same level into two disjoint subsets of . We therefore define Level-Separable Subsets below.
Definition 3 (Level-Separable Subsets).
Given a network and the set (the set of all vertices in the same level of ), the level-separable subsets of are and , such that
There may be many possible pairs of and for a level . Let be the set of all possible pairs of Level-Separable Subsets:
where represent the index of pairs in .
Definition 4 (Multi Parents Set).
Let be network and let and two successive levels in () and let the sets , be such that contains all parents of all vertices in , and contains all sons of all vertices in . The Multi Parents Set at level is a subset of . It contains all the vertices at level , who have more than one parents in level . For any level , we define as:
Definition 5 (Level-Separable Propriety).
Given an arbitrary graph , for all level , where is the eccentricity of source node, verifies the Level-Separable property, if there exists pairs for every (the set of all possible pairs of Level-Separable Subsets at level ), , such that:
i.e., for every vertex at level having multi-parents at level , has only one parent in .
Note that if , then . When is fixed, is .
Definition 6 (Level-Separable Network).
A network is a Level-Separable Network, if its underlay graph verifies the Level-Separable property.
Note that Level-Separable Graph has similar flavor with Bipartite Graph . A graph is said to be Bipartite if and only if there exists a partition and . So that all edges share a vertex from both sets and , and there is no edge containing two vertices in the same set. A bipartite graph separates nodes into two independent sets. In a level-separable network we aim at separating nodes of the same level. Moreover, we are interested in the relation between the two separated sets at level and nodes in level , i.e., node’s father-son relationship.
Note that a level-separable network is not necessary a tree network. However a tree is a level-separable network. A simple example of level-separable network is a tree network, where the root of the tree is the source node who begins the broadcast. In a tree topology all non-source nodes have only one parent, i.e. , so that in each level, the . So that all and .
Figure 1 shows an example of a level-separable network that is not a tree. In this network, 16 nodes are connected: one source node (i.e. the node that starts the broadcast) and 15 non-source nodes. Note that this network is not a tree: nodes may have more than one parent (e.g., node has two parents: node and node ). This network is represented by levels for easy of the observation. For any level all three nodes at that level can be separated into two level-separable sets: and . That is true because the Multi Parents Set and the parents set of node is . Therefore holds. According to Definition 5, and verify the level-separable propriety. From the same reason, at level , and also verify the level-separable propriety.
Studies conducted recently in wireless body area networks , , ,  and  show that various postural mobilities can be model as graphs that fit our definition of level-separable network.
In , authors studied the cross-layer broadcast in wireless body area network and model the network as graphs for different human postures. In this case each graph is a level-separable network, see Figure 2.
In the next section we propose a broadcast algorithm without acknowledgement with optimal labelling in separable networks. Then, we improve the broadcast algorithm proposed in . Finally, we propose a solution for broadcast with acknowledgement using only 2 bits in level-separable networks.
Iii Broadcast in Level-Separable Network
In this section we propose a 1-bit constant-length labelling broadcast Algorithm detailed in Algorithm 1. The algorithm needs rounds, where is the eccentricity of the broadcast source node.
Iii-a Broadcast with 1-bit Labelling
Given a level-separable network whose root is the source of the broadcast, we propose Algorithm (shown as Algorithm 1) to archive the wireless broadcast, when a 1-bit labelling scheme is used. Each node in the network has a 1-bit label, . is set to or following the labelling scheme described below. The idea of the broadcast algorithm is to separate nodes at each level into two independent sets. Nodes in the first set transmit at round and nodes in the second set transmit at round (the next round), so that they will not generate valid collisions. Note that collisions that occur at a node who has already received the message successfully are not considered valid collisions. The broadcast Algorithm using the labelling scheme is as follows: the source node sends the message, , at round . Nodes at level receive at the end of round .
When nodes with receive at round or (), they send at round . When nodes with receive at round or , then they send at rounds . That is, nodes at level will receive from their parents (nodes at level ) at round or , and they will send at round or . In other words, at each level , nodes take two rounds to propagate to all nodes at level .
Figure 3 presents the propagation of the message. The left side shows the of a level-separable network, from level to . It shows three rounds during the execution from to . The right side shows that at which round nodes at a level receive (denoted ) or transmit (denoted ) a message. At round , source sends message to all the nodes at level . Nodes at level have been already separated into two sets, blue ones and white ones by the labelling scheme . At round , nodes in the white set send , and two nodes at level receive the message. At round , the nodes in the blue set send and the remaining nodes at level receive the message.
1-bit Labelling Scheme . To archive collision free transmission, 1-bit Labelling Scheme of all nodes in for level is , and of all nodes in for level is where and are the sets identified in Definition 5.
Iii-B Correctness and Complexity of Algorithm
In the following we prove that Algorithm is correct. First we show that the previously described scheme when used by Algorithm do not generate collisions.
Let be a level-separable network such that each node has a label according to the labelling scheme . If nodes with at the same level , where is the eccentricity of the source node, send a message concurrently they do not generate collisions at nodes at level . If nodes with at the same level , send message concurrently they do not generate collisions at nodes having only one parent at level .
At level , nodes with are the nodes in the subset . According to Definition 5, each node in level has at most one parent in . Therefore, when nodes in send a message, none of nodes in level will receive more than one message. So sending of nodes holding will not generate any collision at level . Nodes with are nodes in . As contains at least one parent for all the nodes at level having multi-parents, contains therefore all parents of each node at level having only one parent. That is, when nodes with send, all nodes having only one parent can receive the message without collisions. ∎
Note that 1-bit labelling scheme is optimal for broadcast in a level-separable network. That is, with 0-bit labelling (i.e. without using any labelling) it is possible that some node in the network does not receive the broadcasted message due to the collisions since nodes are synchronized and transmit in the same time.
Algorithm with 1-bit constant Labelling Scheme implements broadcast in a level-separable network. within rounds.
Given a level-separable network whose root is the source node by applying and , nodes in level receive message at round .
We begin from the base case where , nodes at level means nodes that are only one hop away from the source node. At round , the source sends the message. All nodes at level will receive the message at the end of round . For , as nodes at level can receive message at round , they will begin to send at round and round for nods in and , respectively. According to Lemma 1, no collision occurs at level . Therefore all the nodes in level can receive the message at round . For the general case, we assume that all nodes at level , receive the message at round . So that nodes begin to send the received message at round and , and nodes at level receive the message at and . ∎
Given a level-separable network whose root is the source node by applying and , the broadcast finishes in rounds.
From Lemma 2, nodes having the longest distance to the source will receive the message at round , where is the source eccentricity. After receiving the message, these nodes will send it according to the broadcast algorithm, even though they are already the ending nodes in the network which takes two more rounds. So the broadcast finishes at round . ∎
Given a level-separable network whose root is the source node by applying and , the algorithm finishes within rounds
According to Lemma 3, the broadcast finishes at round . Therefore, our algorithm terminates at round . ∎
The idea of the proof is as follow. Consider the execution of the Algorithm in a level-separable network with labelling scheme , where nodes in level have been separated into two sets and verifying level-separable propriety at level . Nodes in have , and nodes in have . The main idea of is that, nodes in each level separated into two different sets transmit their received messages in different execution rounds to reduce the collisions impact at nodes in level .
According to Algorithm , the message will be propagated from level to level. Each propagation from a level to the next one takes two execution rounds. In the first round all nodes in send the received message . At the end of this round all the nodes that are the sons of nodes in receive , without collision, see Lemma 1. As sons of nodes in contain all the nodes at level who have multi-parents, that means it remains only nodes at level who have only one parent that haven’t received message yet. In the second round, all nodes in send , and the remaining part of the nodes at level can therefore receive from their unique parent. So that after these two rounds of transmission from level , all the nodes at can successfully receive the message . It takes therefore rounds to finish the broadcast. Note that nodes will only send once according to . Therefore the algorithm terminates.
Iii-C Labeling Preinstall
In this section we propose a strategy to select for each level in a level-separable network. Note that this strategy is executed off line before the execution of the broadcast algorithm. Given two arbitrary successive levels, and , let be the set of all nodes at level that have multi parents at level (see Definition 4). Let be the set of all fathers of nodes in , such that:
The main idea to select is to select from the Power Set of , i.e., the set of all the subset of . The set we should chose from the power set of should verify: 1) nodes in do not have the same son nodes; 2) nodes in contain all parents of nodes in . Assume that the mean number of nodes in each level is , then in each level, we need to chose from at most possible choices. The offline time complexity of choosing at each level is .
Iv Broadcast with ACK for arbitrary networks
In  the authors propose a broadcast with ACK algorithm using a 3-bits labelling scheme. At the end of the broadcasting, the last informed node generates and sends back to the source node an ACK message. In a 3-bits labelling, there are 8 states: 000, 001, 010, 011, 100, 101, 110 and 111 available. The algorithm in  uses only 5 of them: 000, 001, 010, 100 and 110. In this section, we propose an optimal labelling scheme, and a broadcast with ACK that uses all the 8 states of the 3-bits labelling in order to improve the memory complexity of the solution proposed in .
The optimization of our solution is as follows: instated of only using the last bit (the third bit) as a mark to point who is (one of) the last informed node(s) during the broadcast, we use also this third bit to show a path back to the source node from the last informed node. Differently from the solution proposed in  nodes do not need to keep additional variables in order to send back to the source the during the execution. Our proposition can therefore save node’s memory and computational power. In the following we present our labelling scheme.
Iv-a 3-bit Labelling Scheme
The first two bits of the labelling scheme and have the same function as in the scheme of  (see  for more details and proof). The intuitive idea is as follows: for nodes who should propagate the message when they receive it; 2) for nodes that need to send message back to their parent to continue send the message in the next round; 3) for one of the last receiving node to generate and send it back to the source node. In our scheme we set (the third bit) to 1 for all nodes on the path back from the last informed node (who holds ) to the source node. Note that, nodes on that path could have four kinds of different labels: 101, 011, 111 and 001, where 001 is the label of the last informed node. As the three states 101, 011 and 111 have not been used in the original , nodes can easily recognize if they are on the path to transmit the ACK message back to the source node.
Iv-B Broadcast Algorithm
Our broadcast algorithm that uses the is described in Algorithm 2.
Given an arbitrary network applying the labelling scheme execute . Nodes with receiving a message at round send it at round . Then nodes who sent at round wait the message, at round , from other nodes with . If nodes who sent at round receive at round , then they continue to send at round . Otherwise, they will stay silent. When nodes with label receive the message, they generate the message and send it. Since already marked the path back from this node to the source node, in Algorithm , the message will only be re-propagated by nodes with . i.e., node with label , and .
Note that our proposed Algorithm does not need additional variables to reconstruct the path back to the source during the broadcast execution. In Algorithm , two additional variables (type ) and (type table of int) are needed to rebuild the back-way path. is used to record the round number in which a node received ; is a table used to record all the round numbers in which one node transmits . However, by using , the message transfer processing can be completed only by checking the third bit, . Our Algorithm does not need any extra local storage for directing the message.
Iv-C Labeling Preinstall
In the following we propose a strategy to decide the back-way path in arbitrary network. According to the idea of in , the last informed node, the 001 node, can be detected easily. If is the last informed node, let is the father node of from whom received . Since the computation is done offline, the of any node (if it exists) can always be computed offline. The members of the back-way path belong to the set:
where is the last informed node and is the source node. To mark the back-way path, sets the bit of the labels of all nodes in to .
V Broadcast with ACK in Level-Separable Network
In this section, we combine the Broadcast algorithm and the labelling scheme to propose an algorithm of broadcast with ACK, , and the Labelling Scheme, , for level-separable networks. Our algorithm (Algorithm 3) uses only 2-bits labelling and the broadcast finishes within rounds. In our solution goes back to the source node at rounds or , where is the eccentricity of (the broadcast source node).
V-a 2-bit Labelling Broadcast with ACK
According to Theorem 1 the broadcast finishes in a level-separable network within rounds where is the eccentricity of the source node. If the source node has the knowledge of , then it automatically can decide if the broadcast is finished. However, when an message is necessary to inform the source node to trigger some additional functions then the source waits for the reception of this message. In order to avoid that the message takes addition time after the end of the broadcast, we propose to send in advance the message at the halfway of the transmission during the broadcast execution. Since in a level-separable network, informing nodes from level to level takes exactly rounds, then also takes 2 rounds to inform one level above. Therefore, when the last node receives , the source node receives the message at the same round. Interestingly, compared with non-ACK broadcasting, our solution uses one extra bit for labelling and no additional rounds for forwarding back to the source the message.
V-B 2-bit Labelling Scheme
We use to set in in order to verify Lemma 1. Let be the second bit of the labelling scheme. for a set of nodes if they are on the way back path from a node at level to the source node, where is the eccentricity of and is the broadcast source. For the other nodes, . In Section V-C, we explain why we chose node at level to begin the sending of ACK.
Note that 2-bit labelling scheme is optimal to archive broadcast with acknowledgement in a level-separable network. From Note 1 1-bit labelling for broadcast without acknowledgement is optimal. When an acknowledgement has to be sent back to the source node, at least one additional bit is necessary to indicate who should be the node to generate the acknowledgement message and send it back to the source node. Without this additional bit no node can decide (unless it uses extra local memory) if it is the last receiving node, and who should send back the ACK.
V-C Correctness and complexity of Algorithm
Theorem 2 below proves the correctness of Algorithm .
Algorithm with 2-bit labelling scheme implements broadcast in a level-separable network. The broadcast terminates in rounds. The ACK message is transmitted back to the source at round , if is odd or , if is even.
Given a level-separable network whose root is the source node by applying and , nodes in level receive message at round .
Given a level-separable network whose root is the source node by applying and , the broadcast finishes at round .
Given a level-separable network whose root is the source node by applying and , the message goes back to source node at round , if is odd; or , if is even.
When is odd, ACK and the message will begin to be sent to source and to the ending nodes from levels and , respectively. The distances from levels back to source is the same with that from to the ending nodes. arrives to the source at the same round as the broadcasted message arrives at the ending nodes. According to Lemma 5, this is round . When is even needs to go ne level farther compared with the broadcasted message. Therefore, it takes two extra rounds when is even. Therefore, when is even the message goes back to source node in rounds. ∎
Given a Level-Separable Network whose root is the source node by applying and , the algorithm finishes within
The idea of the correctness proof is as follows. Consider a level-separable network with the labelling scheme , where all nodes in level have been separated into two sets and . Nodes in have , and nodes in have . A way back path is marked with between source and an arbitrary node at level , where is the eccentricity of , i.e., we only mark the way back path from the half-way level of the network in this case.
The idea is that when the message propagates to the half-way level of the network, a node at that level will begin the transmission processing, so that when the reaches to the ending node(s) at level , the message reaches the source at (almost) the same round. As nodes cannot decide if they are the ones at the half-way of network who should generate and send message, we use a Waiting Period and an extra message.
According the , when a node with , receives and finishes the retransmission, it cannot decide its position in the way back path. Therefore, it sends a message and begins to wait message sent to him in the following rounds. When a node with receives a within the , that means it is not the ending node, because there is another node with that received and sent to him. When a node with does not receive any within its , this means no node in the next level has , i.e., it is the half-way ending node, so it generates and sends the message. All the nodes with will forward message from the ending node to the source according to the marked way back path. In the , the is delayed two rounds after a node sends message to avoid the collision between and .
A node with that receives at round , transmits at round , then it sends to its parents at round , then it waits a Waiting Period until round . If it doesn’t receive another , then it sends at round . That means, for the half-way ending node, it needs to wait 6 rounds to begin sending . What we want for this half-way mechanism is that the source node can receive as fast as possible, after the broadcast finishes. When (the eccentricity of the broadcast source ) is odd, then if we chose the node at level as the half-way ending node, then the can be received by source node at the same round as the end of the broadcast. Because after waiting 6 rounds at level , message has already been transmitted to level . The distance from node sending to source node is ; the distance from node sending to nodes at level is also . When is even, if we chose the node at level as the half-way ending node, then the can be received by the source node only two rounds after the round of the ending of broadcast.
Therefore it takes to finish Broadcast and the ACK can be transmitted back to the source node at round or round . Note that nodes will only send (both for data message and ACK message) once according to . Therefore the algorithm terminates.
V-D Labeling Preinstall
In the following we propose a strategy to decide the back-away path in a level-separable network from the halfway during the broadcast propagation. Similar to Section IV-C, instead of choosing the last informed node as the generator of message, we chose in level a node and build the set from to . To mark the way back path, one needs only to set of all nodes in to .
When using a 3-bits label instead of 2-bits, the last informed node can be marked directly by the labelling scheme using the third bit. That means that during the broadcast execution any Waiting Period or message is unnecessary, so that during the execution of , we can save the unnecessary message transmission.
We proposed solutions for implementing broadcast in wireless networks when the broadcast is helped by a labelling scheme. We studied broadcast without acknowledgement (i.e. the initiator of the broadcast is not notified at the end of broadcast) and broadcast with acknowledgement. We propose an optimal acknowledgement-free broadcast strategy using 1-bit labelling and a broadcast with acknowledgement using a 2-bit labelling in level-separable networks. The complexity of both algorithms is where is the eccentricity of the broadcast initiator. They use an optimal number of bits. Then, we improved in terms of memory and time complexity the labelling-based broadcast scheme with acknowledgement proposed in  for arbitrary networks. Our improvement better exploits the encoding of the labels in order to not use extra memory to carry back the acknowledgement to the source.
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