On the Complexity of Fixed-Schedule Neighbourhood Learning in Wireless Ad Hoc Radio Networks

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8243)

Abstract

Consider a synchronous static radio network of \(n\) nodes represented by an undirected graph with maximum degree \(\varDelta \). Suppose that each node has a unique ID from \(\{1,\ldots ,N\}\), where \(N \gg n\). In the complete neighbourhood learning task, each node \(p\) must produce a set \(L_p\) of IDs such that ID \(i \in L_p\) if and only if \(p\) has a neighbour with ID \(i\). We study the complexity of this task when it is assumed that each node fixes its entire transmission schedule at the start of the algorithm. We prove a \(\varOmega (\frac{\varDelta ^2}{\log \varDelta }\log {N})\)-slot lower bound on schedule length that holds in very general models, e.g., when nodes possess collision detectors, messages can be of arbitrary size, and nodes know the schedules being followed by all other nodes. We also prove a similar result for the SINR model of radio networks. To prove these results, we introduce a new generalization of cover-free families of sets, which may be of independent interest. We also show a separation between the class of fixed-schedule algorithms and the class of algorithms where nodes can choose to leave out some transmissions from their schedule.

1 Introduction

Neighbourhood learning is an important step in wireless network initialization and in algorithms for tasks such as routing, medium-access control, topology control, and gossiping. Further, in the study of local computation in distributed computing, it is assumed that information about neighbouring nodes has already been collected, which is non-trivial in the case of wireless radio networks. If it is not known how to collect neighbourhood information in an efficient way, or, if we are able to prove a strong lower bound for neighbourhood learning, then the actual running time of a solution that depends on this information can be significantly worse than its running-time analysis suggests. In many of these applications, it is important that each node learns their entire neighbourhood before proceeding to other tasks, so we focus our attention on deterministic algorithms that guarantee full neighbourhood discovery within a bounded amount of time that is known in advance.

The main challenge that is encountered when designing algorithms for wireless radio networks is the possibility of radio interference. When several nodes transmit during a single time slot, the signals from all transmissions may prevent a nearby listening node from receiving any message. This is known as a transmission collision. Unless otherwise specified, we consider the Unit-Disk Graph (UDG) radio network model in this paper: a collision occurs at \(p\) if two or more neighbours of \(p\) transmit in the same time slot, or, if a neighbour of \(p\) transmits during the same time slot that \(p\) does. A node \(p\) receives a message from a neighbour \(q\) if \(q\) transmits and no collision occurs at \(p\). In the basic model, \(p\) cannot tell the difference between a collision and the case where no neighbour transmits. With a weak collision detector, \(p\) can make this distinction, as long as \(p\) is not transmitting. With a strong collision detector, \(p\) can always distinguish between a collision and silence.

Our eventual goal is to determine the complexity of the neighbourhood learning in both static and mobile networks. Most of the results in the literature concentrate on determining upper bounds for this task in various models. However, good lower bounds are missing, even if we restrict attention to the class of algorithms where nodes follow a fixed schedule. Specifically, a \(t\)-slot fixed-schedule algorithm for the complete neighbourhood learning task is a deterministic algorithm run by each node \(p\) such that: \(p\) knows its entire \(t\)-slot schedule at time slot 0; \(p\) transmits a message during each slot for which the entry in its schedule is 1, and otherwise stays silent; and, at the end of slot \(t\), \(p\) outputs a list of all of its neighbours. We can classify existing fixed-schedule algorithms as one of three types:
  1. 1.

    Collision-Free Algorithms: In every execution, there is never a time slot during which a transmission collision occurs.

     
  2. 2.

    Local Broadcast Algorithms: For each node \(p\) in the network, there exists a time slot during which \(p\) transmits and no transmission collisions occur at \(p\)’s neighbours.

     
  3. 3.

    Direct-Discovery Algorithms: For each pair of nodes \((p,q)\) in the network, there exists a time slot during which \(p\) transmits and no transmission collision occurs at \(q\).

     
The best known fixed-schedule solution is a \(O(\varDelta ^2\log {N})\)-slot direct-discovery algorithm based on cover-free families (equivalently, strongly-selective families), which we describe in Sect. 3.2. However, it is not known if this is the optimal fixed-schedule solution. Of course, this might depend on the particular choice of model: direct-discovery algorithms work in models where nodes do not possess collision detectors, have no knowledge of the schedules used by other nodes, and can only send their own ID in every message. Perhaps, without these restrictions, there are algorithms that can do better. For example, in multi-hop networks, nodes can forward messages that they have received in previous slots, which can help other nodes infer who their neighbours are. As another example, with strong collision detection and knowledge of every node’s schedule, a node can learn from silence: all nodes scheduled to transmit in a slot where no message was received and no collision was detected can be eliminated as possible neighbours. Even if it is difficult to imagine how to use such additional information to devise algorithms that do significantly better than direct discovery, it is important to formally verify whether or not it is possible, and to understand the reasons why.

Our main contribution in this paper is a \(\varOmega (\frac{\varDelta ^2}{\log \varDelta }\log {N})\) lower bound for general fixed-schedule neighbourhood learning algorithms for UDG networks models, even when nodes possess strong collision detectors, have knowledge of every node’s schedule, and can send arbitrary messages (Sect. 4). Nodes with these features will be referred to as strong nodes. The fact that the lower bound holds for networks of strong nodes strengthens our result, since the lower bound automatically applies to models where nodes do not have these features. For networks of strong nodes, we also prove a separation between fixed-schedule algorithms and the class of algorithms where nodes can choose to leave out transmissions from their schedule (Sect. 5). Our lower bound for fixed-schedule neighbourhood learning depends on size bounds for a new generalization of cover-free families that we call thick cover-free families (Sects. 2.4 and 6). Further, we use our results about these combinatorial objects to prove, to our knowledge, the first non-trivial lower bound for fixed-schedule neighbourhood learning in SINR models (Sect. 7).

2 Models and Definitions

2.1 Network Model

A static ad hoc network consists of \(n\) nodes at arbitrary fixed locations. Nodes do not have information about their location, and nodes know the value of \(n\). Each node \(p\) possesses a unique identifier number \(ID(p)\) from the range \(\{1,\ldots , N\}\), where \(N \gg n\). All nodes know the value of \(N\). Denote by \(p(i)\) the node with identifier \(i\). The topology of a network is represented as an undirected graph, with a vertex for each node and an edge joining each pair of neighbours. The maximum degree of the network is denoted by \(\varDelta \) and we assume that nodes know the value of \(\varDelta \). For any node \(p\), the set of nodes that are neighbours of \(p\) is known as \(p\)’s neighbourhood. We consider the task of complete neighbourhood learning, where each node \(p\) must produce a set \(L_p\) of IDs such that ID \(i \in L_p\) if and only if \(p\) has a neighbour with ID \(i\).

At any given time, a node can either transmit or listen, but not both. The signal transmitted by a node \(p\) reaches \(p\) and all neighbours of \(p\). We consider networks where the nodes share a single radio channel, which means that two signals that reach the same point at the same time interfere with one another. A listening node that receives two or more signals during a single slot \(t\) only hears noise, and we say that a collision has occurred during slot \(t\). A collision also occurs if a transmitting node receives one or more other signals. In the case when exactly one neighbour of a listening node transmits, we say that the listening node receives a message from the transmitting neighbour. In one radio model, we will assume that nodes cannot distinguish between a collision and silence. In another model, we will assume that nodes possess strong collision detectors, so that, whether they are listening or transmitting, they can detect that a collision has occurred.

Each node possesses a clock that divides time into equal-length slots, \((t_0,t_1,\ldots )\). Each slot is long enough to allow the complete transmission of any message. We consider synchronous models, in which it is assumed that all clocks run at the same rate, that slot boundaries coincide across all nodes, and that each node begins its local algorithm at time slot 1. Time slot 0 represents the initial state of the system. The model allows any set of nodes to transmit during a single slot.

We say that a network consists of weak nodes if: nodes cannot distinguish between a collision and silence; each node can only send its own ID in each message; and, no node has any knowledge about the schedules of other nodes. A network consists of strong nodes if: nodes possess strong collision detectors; nodes can send arbitrary messages; and, each node initially knows the schedule associated with each node ID.

2.2 Schedules and Algorithms

A node’s schedule\(T\) is a \(\{0,1\}\)-vector that indicates during which slots it will transmit. Entry \(T[i]\) of a node’s schedule is 1 if and only if the node transmits during time slot \(i\). A schedule matrix\(\mathcal {S}\) is a \(\{0,1\}\)-matrix with \(t\) rows and \(N\) columns. Column \(j\) is the schedule of the node with ID \(j\). A family of sets \(\{S_1,\ldots ,S_N\}\) over \(\{1,\ldots ,t\}\) can be represented by a \(\{0,1\}\)-matrix with \(t\) rows and \(N\) columns: entry \(M_{i,j}\) is 1 if and only if \(i \in S_j\). We will use this fact to relate families of sets with node transmission schedules.

A \(t\)-slot fixed-schedule algorithm for the complete neighbourhood learning task is a deterministic algorithm run by each node \(p\) such that: \(p\) knows its entire \(t\)-slot schedule at time slot 0; \(p\) transmits a message during each slot for which the entry in its schedule is 1, and otherwise stays silent; and, at the end of slot \(t\), \(p\) outputs a list of all of its neighbours. A non-adaptive algorithm is a fixed-schedule algorithm in which the sequence of messages sent by each node is the same in every execution.

2.3 Cover-Free Families

For any set \(S\), an \(r\)-cover for\(S\) is a collection of \(r\) sets other than \(S\), whose union contains \(S\). For \(r \ge 1\), an \(r\)-cover-free family\(\mathfrak {F}\) is a collection of \(N\) subsets of \(\{1,\ldots ,t\}\) such that, for each \(S \in \mathfrak {F}\), there is no \(r\)-cover for \(S\) consisting of sets from \(\mathfrak {F}\). We will say that \(t\) is the length of \(\mathfrak {F}\) and \(N\) is the size of \(\mathfrak {F}\). This reflects the fact that the schedules that we construct from cover-free families will have \(t\) time slots and will provide schedules for up to \(N\) nodes. Note that it is easy to find cover-free families of small size (e.g., any family consisting of exactly one set is \(r\)-cover-free for all \(r \ge 1\)) and that it is difficult to construct cover-free families of large size but small length.

Many related results in the literature refer to families of sets that are called strongly-selective families. A family \(\mathcal {G}\) of subsets of \(\{1,\ldots ,u\}\) is \(k\)-strongly-selective if, for each \(Z \subseteq \{1,\ldots ,u\}\) with \(|Z| \le k\) and for each \(z \in Z\), there exists a set \(G \in \mathcal {G}\) such that \(G \cap Z = \{z\}\). Strongly-selective families and cover-free families can be viewed as ‘duals’ of one another, as remarked in [3]: when represented in matrix form (see Sect. 2.2), the columns of a matrix form an \(r\)-cover-free family consisting of \(N\) subsets of \(\{1,\ldots ,t\}\) if and only if the rows of the matrix form an \((r+1)\)-strongly-selective family consisting of \(t\) subsets of \(\{1,\ldots ,N\}\). It follows that asymptotic bounds on the size of cover-free families also apply to strongly-selective families (and vice versa). The definition of strongly-selective families first appears in Clementi et al. [3], where they were used to solve the multi-broadcast problem in ad hoc wireless radio networks.

2.4 Thick Cover-Free Families

Our new generalization of cover-free families captures the ‘thickness’ of a cover. Namely, we would like to specify how many times the elements of a set \(S\) appear in the sets \(S_1,\ldots ,S_r\). This information is lost if we use the usual definitions of sets and unions. A multiset is a generalization of a set in which there can be multiple copies of the same element. For any multisets \(F=[f_1,\ldots ,f_m]\) and \(G=[g_1,\ldots ,g_\ell ]\), let \(F \uplus G = [f_1,\ldots ,f_m,g_1,\ldots ,g_\ell ]\) be the multiset union of \(F\) and \(G\). For \(c\) sets, we denote the multiset union by \(\biguplus _{i=1}^{c} S_i\). For any set \(F\), denote by \(\biguplus ^{b}F\) the multiset consisting of \(b\) copies of each element of \(F\). Note that, for any multisets \(F\),\(G\), \(|F \uplus G| = |F| + |G|\).

Definition 1

For any set \(S\), an \(r\)-cover of thickness\(b\)for\(S\) is a family of \(r\) sets other than \(S\), whose multiset union contains at least \(b\) copies of each element in \(S\).

Definition 2

A family \(\mathfrak {F}\) of sets is \(r\)-cover-free for thickness\(b\) if, for every set \(S \in \mathfrak {F}\), there does not exist an \(r\)-cover of thickness \(b\) for \(S\) consisting of sets from \(\mathfrak {F}\).

Note that an \(r\)-cover of thickness 1 is equivalent to a traditional \(r\)-cover, and that an \(r\)-cover-free family for thickness 1 is equivalent to a traditional \(r\)-cover-free family. Also, any cover-free family for thickness \(b\) is also a cover-free family for any thickness \(b' > b\).

As far as we know, there is no ‘dual’ definition for thick cover-free families in the literature. So, we propose \(k\)-strongly-\(b\)-selective families with the following definition: for each \(Z\) with \(|Z| \le k\) and for each \(z \in Z\), there exists a set \(G \in \mathcal {G}\) such that \(z \in G \cap Z\) and \(|G \cap Z| \le b\). When represented in matrix form, it is not hard to see that the columns form an \(r\)-cover-free family for thickness \(b\) consisting of \(N\) subsets of \(\{1,\ldots ,t\}\) if and only if the rows form an \((r+1)\)-strongly-\(b\)-selective family consisting of \(t\) subsets of \(\{1,\ldots ,N\}\).

3 Known Results

3.1 Cover-Free Families

Cover-free families were defined in Erdös, Frankl, and Füredi [12], but the concept was first introduced by Kautz and Singleton [17] in their study of superimposed binary codes. In particular, they were called zero-false-drop codes of order\(r\). When the columns of a matrix are taken to be codewords of a zero-false-drop code of order \(r\), the resulting matrix is called \(r\)-disjunct. These matrices play a central role in non-adaptive solutions to combinatorial group testing problems, where the goal is to identify a small number of defective items from a large set by performing tests on groups of items (see Du and Hwang [7] for a survey on this topic). There are also several generalizations of cover-free families (and strongly-selective families) in the literature [4, 5, 8, 10, 24].

Erdös et al. [12] provide a non-constructive proof that there exist \(r\)-cover-free families consisting of \(N\) subsets of \(\{1,\ldots ,t\}\) such that \(t \in O(r^2\log {N})\). Porat and Rothschild [21] provided the first construction of cover-free families that meet this asymptotic bound. Further, Erdös et al. [12] showed that if \(N < {r+2 \atopwithdelims ()2}\) (e.g., if \(r \ge 2\sqrt{N}-1\)), then \(N \le t\). The family consisting of \(\{1\},\ldots ,\{t\}\) meets this bound with equality. For any \(r\)-cover-free family consisting of \(N\) subsets of \(\{1,\ldots ,t\}\), a proof that \(t \in \varOmega ((r^2/\log {r})\log {N})\) has been provided by D’yachkov and Rykov [9], Ruszinkó [22], Chaudhuri and Radhakrishnan [1], and Füredi [13].

3.2 Neighbourhood Learning

Previous work about neighbourhood learning has focused mainly on upper bounds. A trivial upper bound of \(N\) slots is achieved by a round-robin algorithm where a node with ID \(i\) transmits during time slot \(i\). More efficient algorithms use schedules in which some transmissions may be lost due to collisions. In networks of weak nodes, one solution is to use a schedule based on cover-free families: given a \(\varDelta \)-cover-free family \(\mathfrak {F} = \{S_1,\ldots ,S_N\}\) where each \(S_j \subseteq \{1,\ldots ,t\}\), the node with ID \(j\) transmits its ID during time slot \(s\) if and only if \(s \in S_j\). Such a schedule guarantees that every node successfully transmits its ID to all of its neighbours. To see why, suppose to the contrary that there is a node \(p_i\) whose ID is not received by some neighbour \(p_j\). This means that during every slot that \(p_i\) transmitted, it must be the case that either \(p_j\) or one of \(p_j\)’s other neighbours (there are at most \(\varDelta -1\) of these) transmitted. It follows that the set of \(p_i\)’s transmission slots is contained in the union of \(\varDelta \) other sets of transmissions slots, that is, there is a \(\varDelta \)-cover for \(S_i\) in the family of transmission schedules. The length of the algorithm is at most \(t\), which, using the construction of cover-free families by Porat and Rothschild [21], leads to an algorithm that uses \(O(\varDelta ^2\log {N})\) slots (or \(N\) slots if \(\varDelta \ge 2\sqrt{N}-1\)).

The algorithm based on cover-free families is relatively easy to implement since it is non-adaptive: each execution of the algorithm follows the same schedule and sends the same messages, regardless of the network topology. A fully-adaptive algorithm for networks of weak nodes was provided by Gasieniec et al. [14] and uses \(O(\varDelta ^2\log {N})\) slots when \(\varDelta < \sqrt{n}\), or \(O(n\log ^2{n}\log ^2{N})\) slots otherwise. When nodes can distinguish between a collision and silence, Mittal et al. [20] devised a fully-adaptive algorithm for neighbourhood learning that uses \(O(n\log {N})\) slots. This algorithm works even if nodes do not know an upper bound for \(\varDelta \) (or, for that matter, \(n\)). In these fully-adaptive algorithms, nodes decide during which slots they will transmit based on messages that they receive during the execution. Though these two algorithms have better asymptotic running times than the solution based on cover-free families when \(\varDelta \) is large, they are much more difficult to describe and implement.

In networks of weak nodes, several lower bounds for neighbourhood learning algorithms are known. Krishnamurthy et al. [19] showed that the best-case running time of any collision-free algorithm for neighbourhood learning is at least \(N-n\) slots. A straightforward adversary argument shows that the worst-case running time of any collision-free algorithm is at least \(N-1\) slots if nodes know their degree in advance and \(N\) slots if they do not. It is not too difficult to show that any direct-discovery fixed-schedule algorithm that solves the neighbourhood learning task must use a schedule that corresponds to a \(\varDelta \)-cover-free family. Using the known size bounds from Sect. 3.1, this gives a lower bound of \(\varOmega ((\varDelta ^2/\log {\varDelta })\log {N})\) time slots.

For general fixed-schedule neighbourhood learning in networks of strong nodes, we know of only one non-trivial lower bound. Note that, for any neighbourhood learning algorithm, we require that each node is able to distinguish between the case where it has 0 neighbours and the case where it has at least 1 neighbour. This means that the schedule used by a fixed-schedule neighbourhood learning algorithm must correspond to a \(\varDelta \)-selective family, as defined in [2]. Clementi et al. [3] show that any such schedule has at least \((\varDelta /24)\log {(N/\varDelta )}\) time slots (when \(\varDelta \le N/64\)). Our main result in this paper gives a better lower bound for general fixed-schedule neighbourhood learning algorithms that matches the \(\varOmega ((\varDelta ^2/\log {\varDelta })\log {N})\)-slot lower bound for direct-discovery algorithms.

4 A Lower Bound for Neighbourhood Learning with Strong Nodes

In this section, we show that any fixed-schedule algorithm for the neighbourhood learning task in UDG networks of strong nodes uses at least \(\varOmega ((\varDelta ^2/\log {\varDelta })\log {N})\) slots.

First, for any fixed-schedule algorithm for the neighbourhood learning task, we show that the algorithm’s schedule matrix represents a \(\varDelta \)-cover-free family for thickness \(2\). A node \(p\)’s history\(h_{p,\alpha }\) is a vector that stores the messages that it has received during the execution \(\alpha \) of an algorithm. Entry \(i\) of a node \(p\)’s history, \(h_{p,\alpha }[i]\), is equal to: the message that \(p\) received during slot \(i\), if at least one neighbour of \(p\) transmitted during slot \(i\) and no collision occurred at \(p\); \(*\), if a collision occurred at \(p\); \(\bot \), otherwise. Two executions \(\alpha ,\beta \) of an algorithm are indistinguishable to a node\(p\)up to time slot\(s\) if \(h_{p,\alpha }[1 \ldots s] = h_{p,\beta }[1 \ldots s]\).

Observation 1

For any \(t\)-slot algorithm, if two executions are indistinguishable to a node \(p\) up to time slot \(t\), \(p\) outputs the same list of neighbours in both executions.

For any column \(j\) of a schedule matrix \(\mathcal {S}\), let \(S_j\) be the set of time slots during which the node with ID \(j\) is scheduled to transmit. Consider any \(k \in \{1,\ldots ,N\}\) and any set \(\mathcal {C} = \{k_1,\ldots ,k_r\} \subseteq \{1,\ldots ,N\}-\{k\}\). Then, the family \(\mathfrak {F} =\{S_{k_1},\ldots ,S_{k_r}\}\) is an \(r\)-cover of thickness \(2\) for \(S_k\) if and only if, for every time slot \(t'\) during which the node with ID \(k\) is scheduled to transmit, at least \(2\) nodes with IDs in \(\mathcal {C}\) are scheduled to transmit during \(t'\). The next result shows that the existence of such a \(\varDelta \)-cover of thickness \(2\) in an algorithm’s schedule means that there are networks in which the algorithm fails to solve the neighbourhood learning task.

Theorem 2

For any fixed-schedule algorithm \(\mathcal {A}\) with schedule matrix \(\mathcal {S}\) for the complete neighbourhood learning task in any UDG network of strong nodes with maximum degree \(\varDelta \), the family of sets \(\{S_1,\ldots ,S_N\}\) is a \(\varDelta \)-cover-free family for thickness \(2\).

Proof

To obtain a contradiction, assume that there exists a \(k \in \{1,\ldots ,N\}\) and a set \(\mathcal {C} = \{k_1,\ldots ,k_\varDelta \} \subseteq \{1,\ldots ,N\}-\{k\}\) such that the family \(\mathfrak {F} = \{S_{k_1},\ldots ,S_{k_{\varDelta }}\}\) is a \(\varDelta \)-cover of thickness \(2\) for \(S_k\).

Let \(\mathfrak {C}\) be a \(\varDelta \)-clique of nodes \(\{p(k_1),\ldots ,p(k_{\varDelta })\}\), and let \(V = \mathfrak {C} \cup \{p(k)\}\). Construct \(G_1=(V,E_1)\) with \(E_1 = \{p(k),p(k_1)\} \cup E(\mathfrak {C})\). Next, construct \(G_2 = (V,E_2)\) with \(E_2 = \{p(k),p(k_2)\} \cup E(\mathfrak {C})\). We show that all nodes in \(V-\{p(k)\}\) output the same list of neighbours in the executions of \(\mathcal {A}\) on \(G_1\) and \(G_2\), which contradicts the correctness of \(\mathcal {A}\).

Let \(\alpha _1\) be the execution of \(\mathcal {A}\) on \(G_1\), and let \(\alpha _2\) be the execution of \(\mathcal {A}\) on \(G_2\). We prove, by induction on the length \(\ell \) of the executions \(\alpha _1,\alpha _2\), that \(h_{p,\alpha _1}[1\ldots \ell ] = h_{p,\alpha _2}[1\ldots \ell ]\) for all \(p \in V-\{p(k)\}\).

In the base case, all nodes possess the same initial information in both networks. As induction hypothesis, assume that, for each \(p \in V-\{p(k)\}\), \(h_{p,\alpha _1}[1\ldots \ell ] = h_{p,\alpha _2}[1\ldots \ell ]\). We show that, for each \(p \in V-\{p(k)\}\), \(h_{p,\alpha _1}[1\ldots (\ell +1)] = h_{p,\alpha _2}[1\ldots (\ell +1)]\). Since \(\mathcal {A}\) is a fixed-schedule algorithm, each node \(p\) in \(V\) transmits in slot \(\ell +1\) of execution \(\alpha _1\) iff \(p\) transmits in slot \(\ell +1\) of execution \(\alpha _2\). Further, by the induction hypothesis, the nodes in \(V-\{p(k)\}\) send the same messages in slot \(\ell +1\) of execution \(\alpha _1\) as they do in slot \(\ell +1\) of execution \(\alpha _2\). Each node \(q\) in \(V-\{p(k),p(k_1),p(k_2)\}\) only has neighbours in \(V-\{p(k)\}\), so \(h_{q,\alpha _1}[\ell +1] = h_{q,\alpha _2}[\ell +1]\).

Finally, consider whether or not \(p(k)\) transmits in slot \(\ell +1\). If \(p(k)\) does not transmit in slot \(\ell +1\), then \(h_{p(k_1),\alpha _1}[\ell +1] = h_{p(k_1),\alpha _2}[\ell +1]\) and \(h_{p(k_2),\alpha _1}[\ell +1] = h_{p(k_2),\alpha _2}[\ell +1]\). If \(p(k)\) does transmit in slot \(\ell +1\), then at least \(2\) nodes in \(p(k_1),\ldots ,p(k_\varDelta )\) transmit during slot \(\ell +1\), since \(\{S_{k_1},\ldots ,S_{k_{\varDelta }}\}\) is a \(\varDelta \)-cover of thickness \(2\) for \(S_k\). Therefore, \(h_{p(k_1),\alpha _1}[\ell +1] = * = h_{p(k_1),\alpha _2}[\ell +1]\) and \(h_{p(k_2),\alpha _1}[\ell +1] = * = h_{p(k_2),\alpha _2}[\ell +1]\). Thus, for each node \(q\) in \(V-\{p(k)\}\), \(h_{q,\alpha _1}[\ell +1] = h_{q,\alpha _2}[\ell +1]\). By Observation 1, all nodes in \(V-\{p(k)\}\) output the same neighbour list in both executions, contradicting the correctness of \(\mathcal {A}\). \(\square \)

Theorem 2 tells us that the columns of any schedule matrix used by a fixed-schedule neighbourhood learning algorithm must represent a \(\varDelta \)-cover-free family for thickness \(2\). Therefore, a lower bound on the number of rows of any such matrix gives us a lower bound on the number of slots used by any fixed-schedule neighbourhood learning algorithm. To prove such a lower bound, we actually fix the number of rows \(t\) in the schedule matrix, and then prove an upper bound on the number of columns \(N\) in terms of \(t\) (see Theorem 7 in Sect. 6). Re-arranging the expression for the upper bound on \(N\) gives a lower bound on \(t\) in terms of \(N\) (see Corollary 8). Thus, we get the following lower bound for fixed-schedule neighbourhood learning algorithms.

Theorem 3

Any fixed-schedule algorithm for the complete neighbourhood learning task for any UDG network of strong nodes with maximum degree \(\varDelta \) uses at least \(\varOmega ((\varDelta ^2/\log {\varDelta })\log N)\) slots.

5 The Power of Choice

A \(t\)-slot optional-schedule algorithm for the complete neighbourhood learning task is a deterministic algorithm run by each node \(p\) such that: \(p\) knows its entire \(t\)-slot schedule at time slot 0; \(p\) does not transmit during each slot for which the corresponding entry in its schedule is 0, and otherwise can choose to transmit or stay silent; and, at the end of slot \(t\), \(p\) outputs a list of all of its neighbours. We present an optional-schedule algorithm for complete neighbourhood learning that uses \(O(n+n\log {(N/n)})\) slots, which beats our \(\varOmega ((\varDelta ^2/\log {\varDelta })\log {N})\) fixed-schedule lower bound in networks of high degree (e.g. \(\varDelta \in \varOmega (n)\)). This algorithm relies heavily on an algorithm for conflict resolution described by Komlós and Greenberg (KG) [18]. In the conflict resolution task, there is a set of \(k\) nodes that have all transmitted to a channel simultaneously and we must schedule these nodes such that each node eventually gets to transmit on its own. Their conflict resolution algorithm fixes each node’s entire schedule at time slot 0 and each node \(p\) follows its schedule until there is a slot \(t\) in which \(p\) transmits and all others stay silent. This slot \(t\) is guaranteed to exist due to the combinatorial properties of the set of schedules. After slot \(t\), \(p\) chooses to remain silent for the remainder of the algorithm. This is possible in their network model since node \(p\) is able to detect whether or not it was the only transmitter on the channel. The authors prove that there exists a schedule matrix \(S\) of length \(O(k + k\log {(N/k)})\) slots such that the conflict resolution task is solved using their algorithm.

We now show how to use the KG algorithm to construct a neighbourhood learning algorithm \(\mathcal {A}\) for arbitrary networks. Starting with a schedule matrix \(S\) used by the KG algorithm to solve conflict resolution for \(n\) nodes, we construct a schedule matrix \(S'\) as follows: for each \(i \ge 1\), set row \(2i-1\) of \(S'\) to be equal to row \(i\) of \(S\), and set row \(2i\) to be the row of all 1’s. We refer to the odd-numbered slots as KG slots, and the even-numbered slots as feedback slots. In each KG slot \(s\), the nodes behave as they do in the KG algorithm: if node \(p\)’s schedule has a 1 in slot \(s\), \(p\) transmits its ID if and only if there is no previous KG slot in which \(p\) transmitted and \(p\)’s ID was received by all of \(p\)’s neighbours. In each feedback slot \(f\), a node \(q\) transmits its ID if and only if \(q\) detected a collision in slot \(f-1\). The number of slots used by \(\mathcal {A}\) is twice the number of slots used by the KG algorithm, which is \(O(n+n\log {(N/n)})\).

Recall that nodes possess strong collision detectors, so they can detect collisions whether they are transmitting or listening. Each feedback slot \(f\) allows any node \(p\) that transmitted in KG slot \(f-1\) to determine if its message was received by all of its neighbours: a collision has occurred at a neighbour of \(p\) during KG slot \(f-1\) if and only \(p\) receives a message or detects a collision during feedback slot \(f\). Hence, before the beginning of any KG slot \(f+1\), \(p\) can determine if there is a KG slot \(f' \le f-1\) during which all of its neighbours received its ID, as required by the algorithm. The correctness of the algorithm follows directly from the correctness of the KG algorithm. To see why, we know that the KG algorithm guarantees that each node \(p\) in an \(n\)-clique transmits on its own, say during slot \(t_p\). When our algorithm is run on a network of \(n\) nodes but with fewer edges than the \(n\)-clique, \(p\) will successfully transmit its ID to all of its neighbours during the KG slot corresponding to \(t_p\), if not before.

6 An Upper Bound on the Size of Thick Cover-Free Families

In this section, we provide an upper bound on the number of sets in \(r\)-cover-free families for thickness \(b\). First, we will need a result about shadows due to Sperner, as well as Dilworth’s Theorem.

Consider a family \(\mathfrak {F}\) of \((h-1)\)-element subsets of \(\{1,\ldots ,t\}\). The upper shadow of\(\mathfrak {F}\) [11] is the family consisting of all \(h\)-element supersets (over \(\{1,\ldots ,t\}\)) of members of \(\mathfrak {F}\). Sperner [23] (see [11]) proved the following bound on the upper shadow of \(\mathfrak {F}\).

Theorem 4

(Sperner [23]) If \(h \le (t+1)/2\), then the cardinality of the upper shadow of \(\mathfrak {F}\) is at least \(|\mathfrak {F}|\).

Corollary 5

For any \(h \le \lceil t/2 \rceil \), consider a set family \(\mathfrak {G}\) consisting of subsets of \(\{1,\ldots ,t\}\), and suppose that each set in \(\mathfrak {G}\) has cardinality less than \(h\). If \(\mathfrak {G}\) is an antichain with respect to inclusion, then \(|\mathfrak {G}| \le {t \atopwithdelims ()h}\).

Proof

Suppose that \(\mathfrak {G}\) is an antichain, and let \(\ell \) be the cardinality of the smallest set in \(\mathfrak {G}\). Construct the family of all sets in \(\mathfrak {G}\) of cardinality greater than \(\ell \) and the upper shadow of the subfamily of all \(\ell \)-element sets in \(\mathfrak {G}\). By Proposition 4, we know that the cardinality of this family is at least \(|\mathfrak {G}|\). We repeat this process a total of \(h-\ell \) times to obtain the family \(\mathcal {B}\) of all \(h\)-element supersets of members of \(\mathfrak {G}\). Then, \(|\mathfrak {G}| \le |\mathcal {B}| \le {t \atopwithdelims ()h}\). \(\square \)

Theorem 6

(Dilworth’s Theorem [6]) Let \(P\) be a partially ordered set. The cardinality of largest antichain in \(P\) is equal to the minimum number of chains needed to cover \(P\).

Theorem 7

Consider any family \(\mathfrak {F}\) of subsets of \(\{1,\ldots ,t\}\), and let \(h = \lceil (tb(b+1))/(r(r+1)-b(b-1)-1)\rceil \). For \(r \ge 2b + 1\), every family \(\mathfrak {F}\) that is \(r\)-cover-free for thickness \(b\) has size at most \((r-b) + 2b{t \atopwithdelims ()h}\).

Proof

Note that \(0< h < \lceil t/3 \rceil \le t/2\) when \(r \ge 2b+1\). Let \(\mathfrak {R}\) be the family of all \(h\)-element subsets of \(\{1,\ldots ,t\}\) that are contained in at least one and at most \(b\) sets in \(\mathfrak {F}\). We partition \(\mathfrak {F}\) into three disjoint sub-families. Let \(\mathfrak {F}(\mathfrak {R}) = \{S \in \mathfrak {F} : S \supseteq A \text { for some}~ A \in \mathfrak {R}\}\), let \(\mathfrak {F}_{<h} = \{S \in \mathfrak {F} : |S| < h\}\), and, let \(\mathfrak {F}' = \mathfrak {F} - \{\mathfrak {F}(\mathfrak {R}) \cup \mathfrak {F}_{<h} \}\). Equivalently, \(\mathfrak {F}' = \{S \in \mathfrak {F}\ |\ |S| \ge h \mathrm{~and~ } \forall C \subseteq S \mathrm{with } |C| = h, \text { there exist at least}~ b~\text {sets in}~\mathfrak {F}-\{S\} ~\text {that contain}~C\}\).

To obtain the desired result, we will prove that: (1) \(|\mathfrak {F}(\mathfrak {R})| \le b{t \atopwithdelims ()h}\), (2) \(|\mathfrak {F}_{<h}| \le b{t \atopwithdelims ()h}\), and (3) \(|\mathfrak {F}'| \le r-b\).

To prove (1), note that, for each set \(T \in \mathfrak {R}\), there are at most \(b\) sets in \(\mathfrak {F}(\mathfrak {R})\) that contain \(T\). This implies that \(|\mathfrak {F}(\mathfrak {R})| \le b|\mathfrak {R}| \le b{t \atopwithdelims ()h}\). To prove (2), consider the partial ordering of the elements of \(\mathfrak {F}_{<h}\) by inclusion. Since \(\mathfrak {F}\) is \(r\)-cover-free for thickness \(b\), every chain of \(\mathfrak {F}_{<h}\) has cardinality at most \(b\) (otherwise, every element in the minimal set in the chain would appear at least \(b\) times in the multiset union of the other sets in the chain). Also, by Corollary 5, every antichain of \(\mathfrak {F}_{<h}\) has cardinality at most \({t \atopwithdelims ()h}\). By Theorem 6, \(\mathfrak {F}_{<h}\) can be covered using \({t \atopwithdelims ()h}\) chains. Therefore, \(|\mathfrak {F}_{<h}| \le b{t \atopwithdelims ()h}\). Finally, to prove (3), assume that there are at least \(w=r-b+1\) sets \(F_1,\ldots ,F_w \in \mathfrak {F}'\). We show that the union of these sets contains more than \(t\) elements. This is a contradiction, since sets in \(\mathfrak {F}'\) are subsets of \(\{1,\ldots ,t\}\).

Let \(Q = \biguplus _{q=1}^{w} [(\biguplus ^{b}F_q) - \biguplus _{i=1}^{q-1}F_i]\). For each \(q \in \{1,\ldots ,w\}\), the multiset \((\biguplus ^{b}F_q) - \biguplus _{i=1}^{q-1} F_i\) consists of all elements of \(F_q\) that do not appear at least \(b\) times in \(\biguplus _{i=1}^{q-1} F_i\). We proceed by proving upper and lower bounds on the size of \(Q\) (Claims 1 and 3, respectively), and, by comparing the two expressions, we will reach the desired contradiction.

Claim 1\(|Q| \le (b(b+1)/2)| F_1 \cup \ldots \cup F_{w} |\)

Proof

For any fixed \(q \in \{1,\ldots ,w\}\), let \(X_{q,k}\) be the set consisting of all elements \(x \in F_q\) such that \(x\) is in exactly \(k\) of \(F_1,\ldots ,F_{q-1}\). Then, \(( \biguplus ^{b}F_q) - \biguplus _{i=1}^{q-1} F_i\) consists of exactly \(b-k\) copies of each \(x \in X_{q,k}\), for all \(k \in \{0,1,\ldots ,b-1\}\). Hence, we can re-write \(|Q| = |\biguplus _{q=1}^{w} [(\biguplus ^{b}F_q) - \biguplus _{j=1}^{q-1} F_j]| = \sum _{q=1}^{w}\sum _{k=0}^{b-1} (b-k)|X_{q,k}| = \sum _{k=0}^{b-1} (b-k) \sum _{q=1}^{w}|X_{q,k}|\). The desired result follows once we show that, for each \(k \in \{0,\ldots ,b-1\}\), \(\sum _{q=1}^{w}|X_{q,k}| \le |F_1 \cup \ldots \cup F_{w}|\).

Fix an arbitrary \(k \in \{0,\ldots ,b-1\}\). Since \(X_{q,k} \subseteq F_q\) for \(q = 1,\ldots ,w\), it suffices to show that, for each element \(x \in F_1 \cup \ldots \cup F_{w}\), there is at most one \(q \in \{1,\ldots ,w\}\) such that \(x \in X_{q,k}\). Choose the largest \(\ell \) such that \(x \in X_{\ell ,k}\). By the definition of \(X_{\ell ,k}\), \(x \in F_\ell \) and \(x\) is in exactly \(k\) of \(F_1,\ldots ,F_{\ell -1}\). Let \(i_1,\ldots ,i_k \in \{1,\ldots ,\ell -1\}\) be such that \(x \in F_{i_1},\ldots ,x \in F_{i_k}\). For each \(\alpha \in \{i_1,\ldots ,i_k\}\), \(x\) is in at most \(k-1\) of \(F_1,\ldots ,F_{\alpha -1}\), so \(x \not \in X_{\alpha ,k}\). Further, for each \(\beta \in \{1,\ldots ,\ell -1\} - \{i_1,\ldots ,i_k\}\), \(x \not \in F_\beta \supseteq X_{\beta ,k}\).\(\blacksquare \)

Claim 2 For \(F \in \mathfrak {F}'\) and \(F_1,\ldots ,F_z \in \mathfrak {F}\) with \(z \le r-b\), \(| (\biguplus ^{b}F) - \biguplus _{i = 1}^{z} F_i | > h(r-z)\).

Proof

Assume that \(| (\biguplus ^{b}F) - \biguplus _{i=1}^{z} F_i | \le h(r-z)\). We can cover the elements of \((\biguplus ^{b}F) - \biguplus _{i=1}^{z} F_i\) using \(v=r-z \ge b\) subsets of \(\mathfrak {F}\) with \(h\) elements each. Call these sets \(A_{1}, A_{2},\ldots , A_{v}\). By the definition of \(\mathfrak {F}'\), for each \(A_k\) with \(k \in \{1,\ldots ,v\}\), there exists a set \(S_{k} \in \mathfrak {F} - \{F\}\) that contains \(A_k\). However, this means that \(F_1 \uplus \ldots \uplus F_z \uplus S_{1} \uplus \ldots \uplus S_{v} \supseteq \biguplus ^{b}F\). In other words, \(\biguplus ^{b}F\) is covered by the multiset union of \(z + v = z + (r-z) = r\) sets. This contradicts the fact that \(\mathfrak {F}\) is \(r\)-cover-free for thickness \(b\).\(\blacksquare \)

Claim 3\(|Q| \ge (h/2)[r(r+1)-b(b-1)]\)

Proof

Since \(w=r-b+1\) and \(q \in \{1,\ldots ,w\}\), \(q-1 \le r-b\), so, by Claim 2 with \(z=q-1\), \(|Q| = \sum _{q=1}^{w} |(\biguplus ^{b}F_q) - \biguplus _{i=1}^{q-1}F_i| > \sum _{q=1}^{w} h(r-q+1)\). Substituting \(w = r-b+1\), this sum is equal to \(h\sum _{j=b}^{r} j = h[(r(r+1)/2)-(b(b-1)/2)]\).\(\blacksquare \)

Claims 1 and 3 imply that \(| F_1 \cup \ldots \cup F_{r-b+1} | \ge (h/b(b+1))[r(r+1) - b(b-1)]\). Since \(h = \lceil (tb(b+1))/(r(r+1)-b(b-1)-1)\rceil \), we get that \(| F_1 \cup \ldots \cup F_{r-b+1} | > t\), a contradiction. This implies that \(|\mathfrak {F}'| \le r-b\). \(\square \)

Fixing the size of \(\mathfrak {F}\) and re-arranging the upper bound from Theorem 7 gives us the following lower bound on \(t\), the size of the universe over which the sets in \(\mathfrak {F}\) are taken. The proof is omitted due to space constraints.

Corollary 8

For any \(r \ge 2b+1\) and any \(r\)-cover-free family \(\mathfrak {F}\) for thickness \(b\) consisting of subsets of \(\{1,\ldots ,t\}\), \(t \in \varOmega \left( \frac{r^2}{b^2\log {r}}\log {|\mathfrak {F}|}\right) \).

7 A Lower Bound for Neighbourhood Learning in the SINR Model

In the SINR model of wireless networks [15], a node \(p\) receives a message from node \(q\) if \(q\)’s signal is sufficiently stronger than the sum of all other signals received at \(p\) (plus some constant amount of background noise). At \(p\)’s physical location, the strength of \(q\)’s signal is calculated as \(P/(d(q,p)^\alpha )\), where \(P\) is \(q\)’s transmission power, \(d(q,p)\) is the Euclidean distance between nodes \(q\) and \(p\), and \(\alpha \) is the path-loss exponent (usually taken to be greater than 2, so that a signal degrades at least quadratically with respect to distance). Assuming that all nodes transmit with the same power \(P\) (known as a uniform power assignment), the sum of all other signals received at \(p\) is calculated as \(\sum _{q' \ne q} (P/d(q',p)^\alpha )\) for all transmitting nodes \(q'\) other than \(q\). Formally, if \(S\) is a set of transmitting nodes, then node \(p\) receives a message from node \(q\) if \(\frac{P/d(q,p)^\alpha }{\mathcal {N} + \sum _{q' \in S - \{q\}} (P/d(q',p)^\alpha )} \ge \beta \), where \(\mathcal {N}\) represents the constant amount of background noise, and \(\beta \) is a parameter known as the minimum signal to interference ratio. We define \(q\) to be a neighbour of \(p\) if a transmission by \(q\) alone is received by \(p\), namely, if \(P/(\mathcal {N}d(q,p)^\alpha ) \ge \beta \). We assume that each node knows a linear upper bound on the number of nodes in the network.

To prove a lower bound for fixed-schedule neighbourhood learning, we first observe that, if too many nodes transmit during the same time slot \(t\), no node in the network receives a message during \(t\), regardless of the network topology. We denote by \(D_{min}\) a fixed lower bound on the distance between any two nodes in the network, and we denote by \(D_{max}\) a fixed upper bound on the distance between any two nodes in the network.

Lemma 9

Suppose that all nodes transmit with the same power \(P\). Let \(a = 1+\lceil (D_{max}^\alpha /P)((P/\beta D_{min}^\alpha )-\mathcal {N})\rceil \). For any node \(p\) and any time slot \(t\), if exactly \(b > a\) nodes transmit during time slot \(t\), then \(p\) receives no message during time slot \(t\).

Proof

Suppose that \(q_1,\ldots ,q_b\) are the transmitting nodes during time slot \(t\), each transmitting with power \(P\). Re-arranging \(b > 1+ (D_{max}^\alpha /P)((P/\beta D_{min}^\alpha )-\mathcal {N})\), we get that \(\beta > \frac{P/D_{min}^\alpha }{((b-1)P/D_{max}^\alpha )+\mathcal {N}}\). Consider any transmitting node \(q_i\). Note that \(d(q_i,p) \ge D_{min}\) and that \(\sum _{q_j \ne q_i} P/d(q_j,p)^\alpha \ge \)\((b-1)P/D_{max}^\alpha \). Therefore, \(\beta > \frac{P/d(q_i,p)^\alpha }{\mathcal {N}+ \sum _{q_j \ne q_i} P/d(q_j,p)^\alpha }\). Hence, \(p\) does not receive \(q_i\)’s message. \(\square \)

We prove a lower bound for the neighbourhood learning task in a model SMAX that is at least as strong as the SINR model, which implies that the same lower bound holds for the SINR model. Specifically, SMAX is a model of wireless radio networks that includes a parameter, \(s_{max}\), that specifies the maximum number of simultaneous transmissions that can occur in a single slot without causing collisions at every node. In particular, a collision occurs at every node if greater than \(s_{max}\) nodes transmit, and, otherwise, each node receives the message contained in the strongest received signal. Further, we assume that each node can distinguish between a collision and the case where no nodes transmit.

Analogously to Theorem 2, we can show that, in the SMAX model, any fixed-schedule neighbourhood learning algorithm uses a schedule that corresponds to an \((n-1)\)-cover-free family for thickness \(s_{max}+1\).

Lemma 10

For any fixed-schedule algorithm \(\mathcal {A}\) with schedule matrix \(\mathcal {S}\) for the complete neighbourhood learning task for SMAX networks of strong nodes, the family of sets \(\{S_1,\ldots ,S_N\}\) is an \((n-1)\)-cover-free family for thickness \(s_{max}+1\).

Proof

To obtain a contradiction, assume that there exists a \(k \in \{1,\ldots ,N\}\) and a set \(\mathcal {C} = \{k_1,\ldots ,k_{n-1}\} \subseteq \{1,\ldots ,N\}-\{k\}\) such that the family \(\mathfrak {F} = \{S_{k_1},\ldots ,S_{k_{n-1}}\}\) is an \((n-1)\)-cover of thickness \(s_{max}+1\) for \(S_k\).

Let \(G_1\) be any network consisting of nodes \(V = \{p(k_1),\ldots ,p(k_{n-1})\}\), and let \(G_2\) be any network consisting of nodes \(V \cup p(k)\) where \(p(k)\) is a neighbour of at least one node in \(V\). We show that all nodes in \(V-\{p(k)\}\) output the same list of neighbours in the executions of \(\mathcal {A}\) on \(G_1\) and \(G_2\), which contradicts the correctness of \(\mathcal {A}\).

Let \(\alpha _1\) be the execution of \(\mathcal {A}\) on \(G_1\), and let \(\alpha _2\) be the execution of \(\mathcal {A}\) on \(G_2\). We prove, by induction on the length \(\ell \) of the executions \(\alpha _1,\alpha _2\), that \(h_{p,\alpha _1}[1\ldots \ell ] = h_{p,\alpha _2}[1\ldots \ell ]\) for all \(p \in V-\{p(k)\}\).

In the base case, all nodes possess the same initial information in both networks. As induction hypothesis, assume that, for each \(p \in V-\{p(k)\}\), \(h_{p,\alpha _1}[1\ldots \ell ] = h_{p,\alpha _2}[1\ldots \ell ]\). We show that, for each \(p \in V-\{p(k)\}\), \(h_{p,\alpha _1}[1\ldots (\ell +1)] = h_{p,\alpha _2}[1\ldots (\ell +1)]\). Since \(\mathcal {A}\) is a fixed-schedule algorithm, each node \(p\) in \(V-\{p(k)\}\) transmits in slot \(\ell +1\) of execution \(\alpha _1\) iff \(p\) transmits in slot \(\ell +1\) of execution \(\alpha _2\). Further, by the induction hypothesis, the nodes in \(V-\{p(k)\}\) send the same messages in slot \(\ell +1\) of execution \(\alpha _1\) as they do in slot \(\ell +1\) of execution \(\alpha _2\).

Finally, we consider whether or not \(p(k)\) transmits in slot \(\ell +1\) of execution \(\alpha _2\). If \(p(k)\) does not transmit in slot \(\ell +1\) of execution \(\alpha _2\), then, for each \(p \in V - \{p(k)\}\), \(h_{p,\alpha _1}[\ell +1] = h_{p,\alpha _2}[\ell +1]\). If \(p(k)\) does transmit in slot \(\ell +1\) of execution \(\alpha _2\), then we know that at least \(s_{max}+1\) nodes in \(p(k_1),\ldots ,p(k_{n-1})\) transmit during slot \(\ell +1\) in both executions \(\alpha _1\) and \(\alpha _2\), since \(\{S_{k_1},\ldots ,S_{k_{n-1}}\}\) is an \((n-1)\)-cover of thickness \(s_{max}+1\) for \(S_k\). Therefore, by the definition of \(s_{max}\), no nodes receive any messages in slot \(\ell +1\) in both executions \(\alpha _1\) and \(\alpha _2\), so, for each \(p \in V-\{p(k)\}\), \(h_{p,\alpha _1}[\ell +1] = h_{p,\alpha _2}[\ell +1]\). Thus, we have shown that, for each \(p \in V-\{p(k)\}\), \(h_{p,\alpha _1}[1\ldots (\ell +1)]=h_{p,\alpha _2}[1\ldots (\ell +1)]\). By Observation 1, all nodes in \(V-\{p(k)\}\) output the same list of neighbours in executions \(\alpha _1\) and \(\alpha _2\), contradicting the correctness of \(\mathcal {A}\). \(\square \)

By Corollary 8, the resulting lower bound for any fixed-schedule algorithm in the SMAX model is \(\varOmega (\frac{n^2}{(s_{max}+1)^2\log {n}}\log N)\). It follows from Lemma 9 that any algorithm that solves neighbourhood learning in the SINR model will also be correct in the SMAX model when \(s_{max} = 1+\lceil (D_{max}^\alpha /P)((P/\beta D_{min}^\alpha )-\mathcal {N})\rceil \). Therefore, we can apply the lower bound for the SMAX model to SINR networks.

Theorem 11

In the SINR model with uniform power assignments, any fixed-schedule algorithm for the complete neighbourhood learning task uses at least \(\varOmega (\frac{n^2}{(s_{max}+1)^2\log {n}}\log N)\) slots, where \(s_{max} = 1+\lceil (D_{max}^\alpha /P)((P/\beta D_{min}^\alpha )-\mathcal {N})\rceil \).

While we are familiar with some existing upper bounds for neighbourhood learning and local broadcast in SINR models [16, 25], the above result seems to be the first non-trivial lower bound for complete neighbourhood learning.

8 Conclusions and Future Work

In this paper, we have demonstrated the fundamental limitations of fixed-schedule algorithms for neighbourhood learning in UDG networks. Even if nodes can send arbitrary messages, if they possess strong collision detectors, and if they know every node’s schedule, there is no fixed-schedule algorithm that does significantly better than the \(O(\varDelta ^2\log {N})\) algorithm based on cover-free families (equivalently, strongly-selective families.) This was shown by presenting a \(\varOmega ((\varDelta ^2/\log {\varDelta })\log {N})\)-slot lower bound for any such algorithm. However, if nodes are allowed to leave out some transmissions from their schedule, then there is a \(O(n+n\log {(N/n)})\)-slot algorithm for neighbourhood learning, which beats our lower bound in networks of high degree. We have also shown how to use thick cover-free families to prove a lower bound in the more realistic SINR model.

Our work leaves open several directions for future research. First, we would like to prove lower bounds for neighbourhood learning algorithms in networks where nodes are allowed to leave out some transmissions from their schedule. Second, we would like to find a non-trivial lower bound for fully-adaptive algorithms, that is, algorithms where nodes do not follow a pre-calculated schedule. Finally, we are very interested in better understanding the neighbourhood learning task in dynamic versions of our network model, i.e., by allowing the network topology to change over time.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

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