Analysis of a Cone Based Distributed Topology Control Algorithm for Wireless Multi Hop Networks | CS 8803, Papers of Computer Science

Download Analysis of a Cone Based Distributed Topology Control Algorithm for Wireless Multi Hop Networks | CS 8803 and more Papers Computer Science in PDF only on Docsity! Analysis of a Cone-Based Distributed Topology Control Algorithm for Wireless Multi-hop Networks Li Li Joseph Y. Halpern Department of Computer Science Department of Computer Science Cornell University Cornell University lili@cs.cornell.edu halpern@cs.cornell.edu Paramvir Bahl Yi-Min Wang Roger Wattenhofer Microsoft Research Microsoft Research Microsoft Research bahl@microsoft.com ymwang@microsoft.com rogerwa@microsoft.com ABSTRACT The topology of a wireless multi-hop network can be con- trolled by varying the transmission power at each node. In this paper, we give a detailed analysis of a cone-based dis- tributed topology control algorithm. This algorithm, intro- duced in [16], does not assume that nodes have GPS in- formation available; rather it depends only on directional information. Roughly speaking, the basic idea of the algo- rithm is that a node u transmits with the minimum power pu,α required to ensure that in every cone of degree α around u, there is some node that u can reach with power pu,α. We show that taking α = 5π/6 is a necessary and sufficient con- dition to guarantee that network connectivity is preserved. More precisely, if there is a path from s to t when every node communicates at maximum power then, if α ≤ 5π/6, there is still a path in the smallest symmetric graph Gα con- taining all edges (u, v) such that u can communicate with v using power pu,α. On the other hand, if α > 5π/6, connec- tivity is not necessarily preserved. We also propose a set of optimizations that further reduce power consumption and prove that they retain network connectivity. Dynamic re- configuration in the presence of failures and mobility is also discussed. Simulation results are presented to demonstrate the effectiveness of the algorithm and the optimizations. 1. INTRODUCTION Multi-hop wireless networks, such as radio networks [6], ad- hoc networks [10] and sensor networks [2, 11], are networks where communication between two nodes may go through multiple consecutive wireless links. Unlike wired networks, which typically have a fixed network topology (except in case of failures), each node in a wireless network can potentially change the network topology by adjusting its transmission power to control its set of neighbors. The primary goal of topology control is to design power-efficient algorithms that maintain network connectivity and optimize performance metrics such as network lifetime and throughput. As pointed out by Chandrakasan et. al [1], network protocols that min- imize energy consumption are key to the successful usage of wireless sensor networks. To simplify deployment and recon- figuration upon failures and mobility, distributed topology control algorithms that utilize only local information and allow asynchronous operations are particularly attractive. The topology control problem can be formalized as follows: We are given a set V of possibly mobile nodes located in the plane. Each node u ∈ V is specified by its coordi- nates, (x(u), y(u)) at any given point in time. Each node u has a power function p where p(d) gives the minimum power needed to establish a communication link to a node v at distance d away from u. Assume that the maximum transmission power P is the same for every node, and the maximum distance for any two nodes to communicate di- rectly is R, i.e. p(R) = P . If every node transmits with power P , then we have an induced graph GR = (V,E) where E = {(u, v)| d(u, v) ≤ R} (where d(u, v) is the Euclidean distance between u and v). It is undesirable to have nodes transmit with maximum power for two reasons. First, since the power required to transmit between nodes increases as the nth power of the distance between them, for some n ≥ 2 [13], it may re- quire less power for a node u to relay messages through a series of intermediate nodes to v than to transmit directly to v. In addition, the greater the power with which a node transmits, the greater the likelihood of the transmission in- terfering with other transmissions. Our goal in performing topology control is to find a subgraph G of GR such that (1) G consists of all the nodes in GR but has fewer edges, (2) if u and v are connected in GR, they are still connected in G, and (3) a node u can transmit to all its neighbors in G using less power than is required to transmit to all its neighbors in GR. Since minimizing power consumption is so important, it is desirable to find a graph G satisfying these three properties that minimizes the amount of power that a node needs to use to communicate with all its neighbors. For a topology control algorithm to be useful in practice, it must be possible for each node u in the network to construct its neighbor set N(u) = {v|(u, v) ∈ G} in a distributed fashion. Finally, if GR changes to G ′ R due to node failures or mobility, it must be possible to reconstruct a connected G′ without global coordination. In this paper we consider a cone-based topology-control al- gorithm introduced in [16], and show that it satisfies all these desiderata. Most previous papers on topology control have utilized position information, which usually requires the availability of GPS at each node. There are a number of disadvantages with using GPS. In particular, the acqui- sition of GPS location information incurs a high delay, and GPS does not work in indoor environments or cities. By way of contrast, the cone-based algorithm requires only the availability of directional information. That is, it must be possible to estimate the direction from which another node is transmitting. Techniques for estimating direction without requiring position information are available, and discussed in the IEEE antenna and propagation community as the Angle-of-Arrival problem. The standard way of doing this is by using more than one directional antenna (see [8]).1 The cone-based algorithm takes as a parameter an angle α. A node u then tries to find the minimum power pu,α such that transmitting with pu,α ensures that in every cone of degree α around u, there is some node that u can reach with power pu,α. In [16], it is shown that taking α ≤ 2π/3 is sufficient to preserve network connectivity. That is, let Gα be the symmetric closure of the communication graph that results when every node transmits with power pu,α (so that the neighbors of u in Gα are exactly those nodes that u can reach when transmitting with power pu,α together with those nodes v that can reach u by transmitting with power pv,α). Then it is shown that if there is a path from u to v in GR, then there is also such a path in G2π/3. Moreover, it is also shown that for a reasonable class of power cost func- tions and for α ≤ π/2, the network has competitive power consumption. More precisely, given arbitrary nodes u and v, it is shown that the power used in the most power-efficient route between u and v in Gα is no worse than k+2k sin(α/2) times the power used in the most power-efficient route in GR (where k is a constant that depends on the power consump- tion model; if only transmission power is considered and the transmission power p(d) is proportional to the nth power of the distance d, we have k = 1). Finally, some optimizations to the basic algorithm are presented. In the present paper, we show that taking α = 5π/6 is necessary and sufficient to preserve connectivity. That is, we show that if α ≤ 5π/6, then there is a path from u to v in GR iff there is such a path in Gα (for all possible node locations) and that if α > 5π/6, then there exists a graph GR that is connected while Gα is not. Moreover, we propose new optimizations and show that they preserve connectivity. Finally, we discuss how the algo- rithm can be extended to deal with dynamic reconfiguration and asynchronous operations. There are a number of other papers in the literature on topology control; as we said earlier, all assume that position information is available. Hu [4] describes an algorithm that 1Of course, if GPS information is available, a node can sim- ply piggyback its location to its message and the required directional information can be calculated from that. does topology control using heuristics based on a Delauney triangulation of the graph. There seems to be no guarantee that the heuristics preserve connectivity. Ramanathan and Rosales-Hain [12] describe a centralized spanning tree algo- rithm for achieving connected and biconnected static net- works, while minimizing the maximum transmission power. (They also describe distributed algorithms that are based on heuristics and are not guaranteed to preserve connectivity.) Rodoplu and Meng [14] propose a distributed position-based topology control algorithm that preserves connectivity; their algorithm is improved by Li and Halpern [9]. In a different vein is the work described in [3, 7]; although it does not deal directly with topology control, the notion of θ-graph used in these papers bears some resemblance to the cone- based idea described in this paper. Relative neighborhood graphs [15] and their relatives (such as Gabriel graphs, or Gβ graphs [5]) are similar in spirit to the graphs produced by the cone-based algorithm. The rest of the paper is organized as follows. Section 2 presents the basic cone-based algorithm and shows that α = 5π/6 is necessary and sufficient for connectivity. Section 3 describes several optimizations to the basic algorithm and proves their correctness. Section 4 extends the basic algo- rithm so that it can handle the reconfiguration necessary to deal with failures and mobility. Section 5 briefly describes some network simulation results that show the effectiveness of the basic approach and the optimizations. Section 6 con- cludes the paper. 2. THE BASIC CONE-BASED TOPOLOGY CONTROL (CBTC) ALGORITHM We consider three communication primitives: broadcast, send, and receive, defined as follows: • bcast(u, p,m) is invoked by node u to send message m with power p; it results in all nodes in the set {v|p(d(u, v)) ≤ p} receiving m. • send(u, p,m, v) is invoked by node u to sent message m to v with power p. This primitive is used to send unicast messages, i.e. point-to-point messages. • recv(u,m, v) is used by u to receive message m from v. We assume that when v receives a message m from u, it knows the reception power p′ of message m. This is, in gen- eral, less than the power p with which u sent the message, because of radio signal attenuation in space. Moreover, we assume that, given the transmission power p and the recep- tion power p′, u can estimate p(d(u, v)). This assumption is reasonable in practice. For ease of presentation, we first assume a synchronous model; that is, we assume that communication proceeds in rounds, governed by a global clock, with each round taking one time unit. (We deal with asynchrony in Section 4.) In each round each node u can examine the messages sent to it, compute, and send messages using the bcast and send communication primitives. The communication channel is reliable. We later relax this assumption, and show that the algorithm is cor- rect even in an asynchronous setting. From (1) and (2), we have \wuv + \xuv > (2π/3− \zvu/2) + (4π/3− 2× \yvu) = 2π −\zvu/2− 2× \yvu Since \wuv + \xuv ≤ α ≤ 5π/6, we have that 5π/6 > 2π − \zvu/2 − 2× \yvu. Thus, \zvu/2+2×\yvu = ((\zvu+\yvu)+3×\yvu)/2 > 7π/6. Since \zvu + \yvu ≤ α ≤ 5π/6, it easily follows that \yvu > π/2. As we showed earlier, \zvu ≥ \zvt > π/3. Therefore, \zvu + \yvu > 5π/6. This is a contradic- tion. The proof of Theorem 2.1 now follows easily. Order the edges in E by length. We proceed by induction on the the rank of the edge in the ordering, using Lemma 2.2, to show that if (u, v) ∈ E, then there is a path from u to v in Gα. It immediately follows that if u and v are connected in GR, then there is a path from u to v in Gα. The proof of Theorem 2.1 gives some extra information, which we cull out as a separate corollary: Corollary 2.3. If α ≤ 5π/6, and u and v are nodes in V such that (u, v) ∈ E, then either (u, v) ∈ Eα or there exists a path u0 . . . uk such that u0 = u, uk = v, (ui, ui+1) ∈ Eα, and d(ui, ui+1) < d(u, v), for i = 0, . . . , k − 1. Next we prove that degree 5π/6 is a tight upper bound; if α > 5π/6, then CBTC(α) does not necessarily preserve connectivity. Theorem 2.4. If α > 5π/6, then CBTC(α) does not nec- essarily preserve connectivity.























All circles have radius R Only black points are actual nodes. s s’ v1 v0 u1 u0 3u v3 v2 u2 Figure 5: A disconnected graph if α = 5π/6 + . Proof. Suppose α = 5π/6 +  for some  > 0. We construct a graph GR = (V, E) such that CBTC(α) does not preserve the connectivity of this graph. V has eight nodes: u0, u1, u2, u3, v0, v1, v2, v3. (See Figure 5.) We call u0, u1, u2, u3 the u-cluster, and v0, v1, v2, v3 the v-cluster. The construction has the property that d(u0, v0) = R and for i, j = 0, 1, 2, 3, we have d(u0, ui) < R, d(v0, vi) < R, and d(ui, vj) > R if i + j ≥ 1. That is, the only edge between the u-cluster and the v-cluster in GR is (u0, v0). However, in Gα, the (u0, v0) edge disappears, so that the u-cluster and the v-cluster are disconnected. In Figure 5, s and s′ are the intersection points of the cir- cles of radius R centered at u0 and v0, respectively. Node u1 is chosen so that \u1u0v0 = π/2. Similarly, v1 is cho- sen so that \v1v0u0 = π/2 and u1 and v1 are on oppo- site sides of the line u0v0. Because of the right angle, it is clear that, whatever d(u0, u1) is, we must have d(v0, u1) > d(v0, u0) = R; similarly, d(u0, v1) > R whatever d(v0, v1) is. Next, choose u2 so that \u1u0u2 = min(α, π) and u0u2 comes after u0u1 as a ray sweeps around counterclockwise from u0v0. It is easy to see that d(v0, u2) > R, whatever d(u0, u2) is, since \v0u0u2 ≥ π/2. For definiteness, choose u2 so that d(u0, u2) = R/2. Node v2 is chosen similarly. The key step in the construction is the choice of u3 and v3. Note that \s′u0u1 = 5π/6. Let u3 be a point on the line through s′ parallel to u0v0 slightly to the left of s′ such that \u3u0u1 < α. Since α = 5π/6 + , it is possible to find such a node u3. Since d(u0, s ′) = d(v0, s′) = R by construc- tion, it follows that d(u0, u3) < R and d(v0, u3) > R. It is clearly possible to choose d(v0, v1) sufficiently small so that d(u3, v1) > R. The choice of v3 is similar. It is now easy to check that when u0 runs CBTC(α), it will terminate with pu0,α = max(d(u0, u3), R/2) < R; similarly for v0. Thus, this construction has all the required proper- ties. 3. OPTIMIZATIONS In this section, we describe three optimizations to the basic algorithm. We prove that these optimizations allow some of the edges to be removed while still preserving connectivity. 3.1 The shrink-back operation In the basic CBTC(α) algorithm, u is said to be a boundary node if, at the end of the algorithm, u still has an α-gap. Note that this means that, at the end of the algorithm, a boundary node broadcasts with maximum power. An opti- mization, sketched in [16], would be to add a shrinking phase at the end of the growing phase to allow each boundary node to broadcast with less power, if it can do so without reduc- ing its cone coverage. To make this precise, given a set dir of directions (angles) and an angle α, define coverα(dir) = {θ : for some θ′ ∈ dir, |θ − θ′| mod 2π ≤ α/2}. We modify CBTC(α) so that, at each iteration, a node in Nu is tagged with the power used the first time it was discovered. Sup- pose that the power levels used by node u during the algo- rithm were p1, . . . , pk. If u is a boundary node, pk is the maximum power P . A boundary node successively removes nodes tagged with power pk, then pk−1, and so on, as long as their removal does not change the coverage. That is, let diri, i = 1, . . . , k, be the set of directions found with all power levels pi or less, then the minimum i such that coverα(diri) = coverα(dirk) is found. Let N s α(u) consist of all the nodes in Nα(u) tagged with power pi or less. Let Nsα = {(u, v) : v ∈ Nsα(u)}, and let Esα be the symmetric closure of Nsα. Finally, let G s α = (V,E s α). Theorem 3.1. If α ≤ 5π/6, then Gsα preserves the con- nectivity of GR. Proof. It is easy to check that the proof of Theorem 2.1 depended only on the cone coverage of each node, so it goes through without change. Note that this argument actually shows that we can remove any nodes from Nu that do not contribute to the cone cov- erage. However, our interest here lies in minimizing power, not in minimizing the number of nodes in Nu. There may be some applications where it helps to reduce the degree of a node; in this case, removing further nodes may be a useful optimization. 3.2 Asymmetric edge removal As shown by Example 2.1, in order to preserve connectivity, it is necessary to add an edge (u, v) to Eα if (v, u) ∈ Nα, even if (u, v) /∈ Nα. In Example 2.1, α > 2π/3. This is not an accident. As we now show, if α ≤ 2π/3, not only don’t we have to add an edge (u, v) if (v, u) ∈ Nα, we can remove an edge (v, u) if (v, u) ∈ Nα but (u, v) /∈ Nα. Let E−α = {(u, v) : (u, v) ∈ Nα and (v, u) ∈ Nα}. Thus, while Eα is the smallest symmetric set containing Nα, E − α is the largest symmetric set contained in Nα. Let G − α = (V,E − α ). Theorem 3.2. If α ≤ 2π/3, then G−α preserves the con- nectivity of GR. Proof. We start by proving the following lemma, which strengthens Corollary 2.3. Lemma 3.3. If α ≤ 2π/3, and u and v are nodes in V such that (u, v) ∈ E, then either (u, v) ∈ Nα or there exists a path u0 . . . uk such that u0 = u, uk = v, (ui, ui+1) ∈ Nα, and d(ui, ui+1) < d(u, v), for i = 0, . . . , k − 1. Proof. Order the edges in E by length. We proceed by strong induction on the rank of an edge in the ordering. Given an edge (u, v) ∈ E of rank k in the ordering, if (u, v) ∈ Nα, we are done. If not, as argued in the proof of Lemma 2.2, there must be a node w ∈ cone(u, α, v) ∩Nα(u). Since α ≤ 2π/3, the argument in the proof of Lemma 2.2 also shows that d(w, v) < d(u, v). Thus, (w, v) ∈ E and has lower rank in the ordering of edges. Applying the induction hypothesis, the lemma holds for (u, v). This completes the proof. Lemma 3.3 shows that if (u, v) ∈ E, then there is a path consisting of edges in Nα from u to v. This is not good enough for our purposes; we need a path consisting of edges in E−α . The next lemma shows that this is also possible. Lemma 3.4. If α ≤ 2π/3, and u and v are nodes in V such that (u, v) ∈ Nα, then there exists a path u0 . . . uk such that u0 = u, uk = v, (ui, ui+1) ∈ E−α , for i = 0, . . . , k − 1. Proof. Order the edges in Nα by length. We proceed by strong induction on the rank of an edge in the ordering. Given an edge (u, v) ∈ Nα of rank k in the ordering, if (u, v) ∈ E−α , we are done. If not, we must have (v, u) ∈ Nα. Since (v, u) ∈ E, by Lemma 3.3, there is a path from v to u consisting of edges in Nα all of which have length smaller than d(v, u). If any of these edges is in Nα − E−α , we can apply the inductive hypothesis to replace the edge by a path consisting only of edges in E−α . By the symmetry of E − α , such a path from v to u implies a path from u to v. This completes the inductive step. The proof of Theorem 3.2 is now immediate from Lem- mas 3.3 and 3.4. To implement asymmetric edge removal, the basic CBTC needs to be enhanced slightly. After finishing CBTC(α), a node u must send a message to each node v to which it sent an Ack message that is not in Nα(u), telling v to remove u from Nα(v) when constructing E − α . It is easy to see that the shrink-back optimization discussed in Section 3.1 can be applied together with the removal of these asymmetric edges. It is clear that there is a tradeoff between using CBTC(5π/6) and using CBTC(2π/3) with asymmetric edge removal. In general, pu,5π/6 (i.e., p(rad − u,5π/6)) will be smaller than pu,2π/3. However, the power p(radu,5π/6) with which u needs to transmit may be greater than pu,5π/6 since u may need to reach nodes v such that u ∈ N5π/6(v) but v /∈ N5π/6(u). In contrast, if α = 2π/3, then asymmetric edge removal al- lows u to still use pu,2π/3 and may allow v to use power less than pv,2π/3. Our experimental results confirm this. See Section 5. 3.3 Pairwise edge removal The final optimization aims at further reducing the trans- mission power of each node. In addition to the directional information, this optimization requires two other pieces of information. First, each node u is assigned a unique in- teger ID denoted IDu, and that IDu is included in all of u’s messages. Second, although a node u does not need to know its exact distance from its neighbors, given any pair of neighbors v and w, node u needs to know which of them is closer. This can be achieved as follows. Recall that a node grows its radius in discrete steps. It includes its transmission power level in each of the “Hello” messages. Each discov- ered neighbor node also includes its transmission power level in the Ack. When u receives messages from nodes v1 and v2, it can deduce which of v1 and v2 is closer based on the transmission and reception powers of the messages. Even after the shrink-back operation and possibly asym- metric edge removal, there are many edges that can be removed while still preserving connectivity. For example, if three edges form a triangle, we can clearly remove any one of them while still maintaining connectivity. This op- timization (where the longest edge is removed) is used in [16]. In this section, we improve on this result by show- ing that if there is an edge from u to v1 and from u to v2, then we can remove the longer edge even if there is no edge from v1 to v2, as long as d(v1, v2) < max(d(u, v1), d(u, v2)). Note that a condition sufficient to guarantee that d(v1, v2) < max(d(u, v1), d(u, v2)) is that \v1uv2 < π/3 (since the longest edge will be opposite the largest angle). To make this precise, we use the notion of an edge ID. Each edge (u, v) is assigned an edge ID eid(u, v) = (i1, i2, i3), where i1 = d(u, v), i2 = max(IDu, IDv), and i3 = min(IDu, IDv). Edge IDs are compared lexicographically, so that (i, j, k) < (i′, j′, k′) iff either (a) i < i′, (b) i = i′ and j < j′, or (c) i = i′, j = j′, and k < k′. Definition 3.5. If v and w are neighbors of u, \vuw < π/3, and eid(u, v) > eid(u,w), then (u, v) is a redundant edge. As the name suggests, redundant edges are redundant, in that it is possible to remove them while still preserving con- nectivity. The following theorem proves this. Theorem 3.6. For α ≤ 5π/6, all redundant edges can be removed while still preserving connectivity. Proof. Let Enrα consist of all the non-redundant edges in Eα. We show that if (u, v) ∈ Eα − Enrα , then there is a path from u to v consisting only of edges in Enrα . Clearly, this suffices to prove the theorem. Let e1, e2, · · · , em be a listing of the redundant edges (i.e, those in Eα−Enrα ) in increasing lexicographic order of edge ID. We prove, by induction on k, that for every redundant edge ek = (uk, vk) there is a path from uk to vk consisting of edges in Enrα . For the base case, consider e1 = (u1, v1). By definition, there must exist an edge (u1, w1) such that \v1u1w1 < π/3 and eid(u1, v1) > eid(u1, w1). Since e1 is the redundant edge with the smallest edge ID, (u1, w1) cannot be a redundant edge. Since \v1u1w1 < π/3, it follows that d(w1, v1) < d(u1, v1). If (w1, v1) ∈ Eα, then (w1, v1) ∈ Enrα (since (u1, v1) is the shortest redundant edge) and (u1, w1), (w1, v1) is the desired path of non-redundant edges. On the other hand, if (w1, v1) /∈ Eα then, since d(w1, v1) < d(u1, v1) ≤ R and α ≤ 5π/6, by Corollary 2.3, there exists a path from w1 to v1 consisting of edges in Eα all shorter than d(w1, v1). Since none of these edges can be redundant edge, this gives us the desired path. For the inductive step, suppose that for every ej = (uj , vj), 1 ≤ j ≤ i − 1, we have found a path H ′j between uj and vj , which contains no redundant edges. Now consider ei = (ui, vi). Again, by definition, there exists another edge (ui, wi) with eid(ui, vi) > eid(ui, wi) and \viuiwi < π/3. If (ui, wi) is a redundant edge, it must be one of ej ’s, where j ≤ i− 1. Moreover, if the pathHi (from Corollary 2.3) between vi and wi contains a redundant edge ej , we must have |ej | < |ei| and so j ≤ i−1. By connecting (ui, wi) with Hi and replac- ing every redundant edge ej on the path with H ′ j , we obtain a path H ′i between ui and vi that contains no redundant edges. This completes the proof. Although Theorem 3.6 shows that all redundant edges can be removed, this doesn’t mean that all of them should nec- essarily be removed. For example, if we remove some edges, the paths between nodes become longer, in general. Since some overhead is added for each link a message traverses, having fewer edges can affect network throughput. In ad- dition, if routes are known and many messages are being sent using point-to-point communication between different senders and receivers, having fewer edges is more likely to cause congestion. Since we would like to reduce the trans- mission power of each node, we remove only redundant edges with length greater than the longest non-redundant edges. We call this optimization the pairwise edge removal opti- mization. 4. DEALING WITH RECONFIGURATION, ASYNCHRONY, AND FAILURES In a multi-hop wireless network, nodes can be mobile. Even if nodes do not move, nodes may die if they run out of energy. In addition, new nodes may be added to the network. We need a mechanism to detect such changes in the network. This is done by the Neighbor Discovery Protocol (NDP). A NDP is usually a simple beaconing protocol for each node to tell its neighbor that it is still alive. The beacon includes the sending node’s ID and the transmission power of the beacon. A neighbor is considered failed if a pre-defined number of beacons are not received for a certain time interval τ . A node v is considered a new neighbor of u if a beacon is received from v and no beacon was received from v during the previous τ interval. The question is what power a node should use for beaconing. Certainly a node u should broadcast with sufficient power to reach all of its neighbors in Eα (or E − α , if α ≤ 2π/3). As we will show, if u uses a beacon with power p(radu,α) (recall that p(radu,α) is the power that u must use to reach all its neighbors in Eα), then this is sufficient for reconfiguration to work with the basic cone-based algorithm (possibly com- bined with asymmetric edge removal if α ≤ 2π/3, in which case we can use power p(rad−u,α)). We define three basic events: • A joinu(v) event happens when node u detects a bea- con from node v for the first time; • A leaveu(v) event happens when node u misses some predetermined number of beacons from node v; • An aChangeu(v) event happens when u detects that v’s angle with respect to u has changed. (Note this could be due to movement by either u or v.) Our reconfiguration algorithm is very simple. It is conve- nient to assume that each node is tagged with the power used when it was first discovered, as in the shrink-back op- eration. (This is not necessary, but it minimizes the number of times that CBTC needs to be rerun.)