Ad-Hoc Networks Beyond Unit Disk Graphs

Transcription

1 A-Hoc Networks Beyon Unit Disk Graphs Fabian Kuhn, Roger Wattenhofer, Aaron Zollinger Department of Computer Science ETH Zurich 8092 Zurich, Switzerlan {kuhn, wattenhofer, ABSTRACT In this paper we stuy a moel for a-hoc networks close enough to reality as to represent existing networks, being at the same time concise enough to promote strong theoretical results. The Quasi Unit Disk Graph moel contains all eges shorter than a parameter between 0 an 1 an no eges longer than 1. We show that in comparison to the cost known on Unit Disk Graphs the complexity results in this moel contain the aitional factor 1/ 2. We prove that in Quasi Unit Disk Graphs flooing is an asymptotically message-optimal routing technique, provie a geometric routing algorithm being more efficient above all in ense networks, an show that classic geometric routing is possible with the same performance guarantees as for Unit Disk Graphs if 1/ 2. Categories an Subject Descriptors F.2.2 [Analysis of Algorithms an Problem Complexity]: Nonnumerical Algorithms an Problems geometrical problems an computations, routing an layout; G.2.2 [Discrete Mathematics]: Graph Theory network problems; C.2.2 [Computer-Communication Networks]: Network Protocols routing protocols General Terms Algorithms, Performance, Theory Keywors A-Hoc Networks, Face Routing, Flooing, Geometric Routing, Mobile Computing, Moeling, Unit Disk Graphs, Wireless Communication. The work presente in this paper was supporte (in part) by the National Competence Center in Research on Mobile Information an Communication Systems (NCCR-MICS), a center supporte by the Swiss National Science Founation uner grant number Permission to make igital or har copies of all or part of this work for personal or classroom use is grante without fee provie that copies are not mae or istribute for profit or commercial avantage an that copies bear this notice an the full citation on the first page. To copy otherwise, to republish, to post on servers or to reistribute to lists, requires prior specific permission an/or a fee. DIALM POMC 03, September 19, 2003, San Diego, California, USA. Copyright 2003 ACM /03/ $ INTRODUCTION A-hoc networks are forme by mobile evices communicating via raio. If two a-hoc noes are too far apart to communicate irectly, intermeiate noes can relay their messages. A wiely employe moel for the stuy of a-hoc networks is the so-calle Unit Disk Graph moel: Noes are locate in the Eucliean plane an are assume to have ientical (unit) transmission raii. Consequently an ege between two noes u an v representing that u an v are in mutual transmission range exists iff uv, their Eucliean istance, is not greater than one. On the one han, clearly, this is a glaring simplification of reality, since, even if all network noes are homogeneous, this moel oes not account for the presence of obstacles, such as walls, builings, mountains or also weather conitions, which might obstruct signal propagation. On the other han, Unit Disk Graphs are simple enough to promote strong theoretical results (such as the cost for routing [20, 21] or topology control [1, 11, 30]). In this paper we stuy a graph moel which is consierably closer to reality [4]. We maintain the assumption that all mobile noes are place in the plane (that is, they have coorinates in R 2 ). In a Quasi Unit Disk Graph, two noes are connecte by an ege if their istance is less than or equal to, being a parameter between 0 an 1. Furthermore, if the istance between two noes is greater than 1, there is no ege between them. In the range between an 1 the existence of an ege is not specifie. In this paper we first establish a constructive lower boun for Quasi Unit Disk Graphs showing that basically any algorithm without routing tables requires sening of Ω ( c )2 messages to route from a source s to a estination t, where c is the length of the shortest path between s an t. We show that, with the ai of a topology control graph structure, a restricte flooing algorithm is guarantee not to perform worse an that this technique is consequently asymptotically message-optimal. If we attribute the network noes with information about their own an their neighbors positions an assume that the message source knows the position of the estination the basic assumptions of geometric routing, a more subtle approach than flooing of the network is possible. We present a combination of greey routing an restricte flooing. This yiels a routing algorithm that is still asymptotically optimal in the worst case, but also efficient in the average case, as previous work on average-case efficiency of geometric ahoc routing algorithms suggests [6, 22]. Finally, if we assume to be at least 1/ 2, we show that it is possible to locally 69

2 introuce virtual eges an perform the classic variations of geometric routing while preserving performance guarantees known from Unit Disk Graphs. After iscussing relate work in the following section, we state the moel an provie efinitions in Section 3. In Section 4 we establish a lower boun for the message complexity of so-calle volatile memory routing algorithms. Section 5 contains the escription of the topology control structure forming the basis for the subsequent algorithms. Section 6 provies the analysis of flooing algorithms with respect to message an time complexity. Section 7 iscusses the combination of flooing with a greey approach for geometric routing, whereas Section 8 shows that for large enough, classic geometric routing can be employe. Section 9 raws the conclusions of the paper. 2. RELATED WORK So far, the most popular network structure to moel mobile a-hoc networks has been the Unit Disk Graph. The unerlying assumption of this moel is that the noes are place in the plane, all of them having the same normalize to one transmission range. A more general moel is provie by isk graphs where in contrast to Unit Disk Graphs, noes can have ifferent transmission ranges. Disk graphs have also been wiely use, but, while for Unit Disk Graphs a number of theoretical results have been achieve, most of the knowlege on isk graphs bases on simulations. Disk graphs provie a simple metho to analyze uniirectional links, however, it is not possible to moel any kin of obstacles. Barrière et al. [4] escribe a moel which is up to scaling ientical to our Quasi Unit Disk Graph moel. Similarly to the technique state in Section 8, [4] exploits local construction of virtual links to allow for correct (i.e. arrival of message guarantee) geometric routing on Quasi Unit Disk Graphs for 1/ 2. In Section 8, we exten this result towars efficiency. In this paper, we give ifferent complexity results concerning the Quasi-UDG moel. We show how to construct a subgraph of the network graph G which enables cost-optimal flooing an we show how the flooing overhea can be reuce (in practice) by using geometric routing for 1/ 2 an a combination of geometric routing an flooing for arbitrary. Constructing a sub-structure G of G such that G features some esirable properties is often terme topology control. Topology control is use to reuce the number of noes an the number of eges involve in protocols such as routing. An important issue of such a precomputation is the reuction of interference effects, message complexity, or energy consumption. Sometimes, algorithms nee network graphs with special properties; for example all facerouting base geometric routing algorithms [6, 16, 18, 20, 21, 22] nee a planar graph to operate correctly. For Unit Disk Graphs, a number of ifferent ieas in orer to reuce the complexity of the network topology have been propose. Many of them are base on ominating sets [1, 10, 11, 13, 14, 19, 29] or angle of arrival [30]. For fining planar subgraphs, various constructions of ifferent quality an complexity have been conceive [9, 11, 28]. A recent survey on topology control algorithms for a-hoc networks can be foun in [26]. Flooing is an essential ingreient of many a-hoc routing algorithms [15, 25]. It is therefore crucial to reuce the number of messages sent. One way to reuce the cost of flooing is to lower the complexity of the network by using appropriate topology control mechanisms. Apart from this there are other approaches which try to optimize flooing performance by using geometric information about the estination [5, 17]. These algorithms iffer from our greey routing/flooing approach in that they only try to floo into the right irection but they o not apply real geometric routing whenever possible. Geometric routing (also known as location-base, positionbase or geographic routing) has also mainly been stuie on Unit Disk Graphs. Greey routing algorithms have been stuie in [8, 11, 12, 18, 27]. Greey routing behaves well in practice, but no guarantee can be given about the arrival of messages for all of them. The first algorithms with guarantee elivery were [6, 18] followe by a slightly change approach in [16]. The first geometric routing algorithm whose cost is boune by a function of the cost 2 c of an optimal path was given in [21]. The cost O c was shown to be optimal an the routing scheme of [21] was later extene to also achieve practical efficiency [20, 22]. 3. MODEL This section provies efinitions of the moel employe in this paper. We first give a formal efinition of our a-hoc network moel: Definition 3.1. (Quasi Unit Disk Graph) Let V R 2 be a set of points in the 2-imensional plane an [0, 1] be a parameter. The symmetric Eucliean graph (V,E), such that for any pair of points u, v V - (u, v) E if uv an - (u, v) / E if uv > 1, is calle a Quasi Unit Disk Graph (Quasi-UDG) with parameter. In the subsequent section we establish a lower boun for the message complexity of so-calle volatile memory routing algorithms. With this moel noes are attribute with a short-term memory in which for each message a constant number of bits may be store temporarily. Definition 3.2. (Volatile Memory Routing Algorithm) The task of a volatile memory routing algorithm is to transmit a message from a source s to a estination t on a graph, where each noe of the graph hols a memory in which O(log n) bits 1 may be store as long as the message is en route. In particular this moel allows to store message ientifiers require for flooing (cf. Section 6). The secon important algorithm moel iscusse in this paper is geometric routing [21]: Definition 3.3. (Geometric Routing Algorithm) The task of a geometric routing algorithm is to transmit a message from a source s to a estination t on a graph while observing the following rules: - Every noe is informe about its own an all of its neighbors positions. 1 We nee a logarithmic number of bits to enable all noes to concurrently floo the network. 70

3 - The source of a message knows the position of the message estination. s h - A message may contain control information about at most O(1) noes. - A noe is only allowe to temporarily store a message before retransmission; no other memory is available. As state above, in original geometric routing a noe is allowe to store messages only temporarily before relaying them. In orer to enable an algorithm to employ flooing, this assumption can be relaxe: +ε 1 Definition 3.4. (Geometric Volatile Memory Routing Algorithm) A geometric volatile memory routing algorithm is a volatile memory routing algorithm aitionally observing the first three rules of the efinition of geometric routing algorithms. In the following we provie a concise overview of basic concepts of istribute computing vital for the unerstaning of this paper. More etaile escriptions can be foun in textbooks, such as in [23]. At certain points of the paper we have to istinguish between the synchronous an the asynchronous moel of istribute computation. In the synchronous moel, communication elays are assume to be boune. As a consequence it can be assume that all processes running on ifferent network noes perform their message sening an receiving operations in simultaneous an globally clocke rouns. In the asynchronous moel, message elays are unboune. No assumptions can be mae on the uration of single process operations. Two funamental measures in istribute computing are message an time complexity. The message complexity of a istribute algorithm is the total number of messages sent uring its execution. The efinition of time complexity epens on the synchrony moel: In the synchronous moel, time complexity is the total number of rouns elapse between algorithm start an algorithm termination. In the asynchronous moel such a simple time moel cannot naturally be obtaine, since the transmission elay of a message is unboune. The common solution to this is the assumption that the message elay is at most one time unit. Finally, since we consier message complexities in this paper, we efine the cost of a path accoring to the link istance metric, that is, the cost of a path is the number of eges on the path. Similarly, we consier spanner graphs with respect to the link istance metric: A graph G =(V,E ) is a spanner of a graph G =(V,E) with stretch factor k iff for any pair of noes (u, v) the cost of the shortest path on G is at most k times the cost of the shortest path on G. 4. LOWER BOUND In this section, we present a lower boun on the message complexity of any volatile memory routing algorithm. Theorem 4.1. Let c be the cost of a shortest path from s to t. There exist graphs on which any (ranomize) volatile memory routing algorithm has (expecte) message complexity Ω(( c )2 ). Figure 1: Message Complexity Lower Boun for Volatile Memory Routing Algorithms on Quasi- UDGs Proof. We provie a constructive proof by escribing a class of graphs for which the theorem hols. The basic element use for the construction of these graphs is forme by k noes (k to be etermine later) equiistantly place on a line, such that the istance between two ajacent noes is + ε for a small ε>0 (cf. vertical chains in Figure 1). There exists an ege between every pair of noes (u, v), such that ( 1 1) < uv 1, that is, the noes are connecte by all the eges with maximum Eucliean length not greater than 1. In aition there is a hea noe having an ege to each one of the first 1 1 noes on the line (the hea noe has to be locate such that all aitional eges have length at most 1). As shown in Figure 1, k such vertical chains are place sie by sie with istance + ε such that the noes form a matrix. The hea noes of these chains now are interconnecte in a way that they among themselves have the same chain structure (uppermost row in Figure 1) with their hea noe (of secon orer) enote by s. The noe t locate near the bottom right corner of the noe matrix is connecte to one of the en noes of exactly one of the vertical chains by a simple chain of noes. Note that the constructe graph is a Quasi Unit Disk Graph. The main property posing a problem for a routing algorithm is that a matrix column consists of 1 1 interleave chains which are only connecte via the hea noe. (The same also hols for the first matrix row.) Consequently only one of s s neighbors leas to h, the hea noe of the column connecte to t, an only one of h s neighbors leas to the bottom noe connecte to t. Since a volatile memory routing algorithm has no a priori information about the graph structure, a eterministic algorithm has to explore every matrix noe before fining the path to t. (For a ranomize algorithm t can be connecte to the matrix such that the t 71

4 algorithm has to explore roughly half of the matrix noes in expectation.) A volatile memory routing algorithm therefore has to sen Ω(n) messages, where n is the total number of noes. The optimal path on the other han almost exclusively using eges of length nearly 1 has cost about 2 k, which together with k n establishes the theorem. 5. TOPOLOGY CONTROL In the previous section we introuce a lower boun graph class, on which any volatile memory routing algorithm cannot fin the estination with message complexity less than Ω(( c )2 ). In this section we now escribe how to obtain a subgraph of a given Quasi Unit Disk Graph which forms the basis for our algorithms matching the lower boun. This Backbone Graph features two important properties exploite A for routing: (1) It contains in a given area A at most O 2 noes an (2) it is a O log( 1 ) -spanner. Given a Quasi Unit Disk Graph G, the Backbone Graph is constructe in three steps. Steps 1 an 2 can be performe by a stanar istribute algorithm (as escribe in [1]) by having the noes sen ominator an connector messages. We o not iscuss the etails of this algorithm for space reasons. 1. The first step consists of a clustering process. We construct a Maximal Inepenent Set MIS of noes on G. Note that since MIS is an inepenent set on G, any two noes in MIS have istance greater than an A consequently a given area A contains at most O noes in MIS. For the purpose of routing, the noes 2 in MIS will later become cluster heas: Since the noes in MIS also form a Dominating Set, any noe in G will have at least one noe from MIS within its neighborhoo an will choose one of these as its cluster hea. 2. In a secon step the cluster heas are linke together by connector noes, resulting in the Complete Backbone Graph G C. Note that since MIS is a Dominating Set, the cluster heas can be connecte by briges consisting of at most two noes. Also note that G C is a constant-stretch spanner of G. 3. G C can contain Ω( A ) noes in a given area A, 4 1 which is too many by a factor of compare to the lower boun. The size of MIS matching 2 the lower boun, the thir step now reuces the number of connecting briges between cluster heas. Let G (v) C enote the graph with noe set MIS an (virtual) eges between all noes connecte by briges in G C. Our objective 2 is now to construct a subgraph G (v) of G(v) C A with O (virtual) eges within the area A. It eventually follows that the final Backbone Graph G where the (virtual) eges in G (v) have again been replace by connector noes an their ajacent eges contains at most O A 2 noes within the area A. In orer to obtain a graph G (v) with the esire property, the plane is ivie by a gri into square cells of sie length 6. In each cell z all noes an eges completely containe within z temporarily form a local network. (Note that we assume for this operation that the noes are informe about their positions.) The number of noes containe within z is at most Figure 2: Construction of a Sparse Spanner: Gri Structure 1 O 2. We now apply an algorithm constructing a sparse spanner [2, 23, 24] to reuce the number of 1 eges containe in z to O.2 This proceure is repeate three times on gris with 2 their origin shifte by (3, 0), (0, 3), an (3, 3) relative to the origin of the first gri (cf. Figure 2). (Note that these are local operations, since the subgraphs are of boune size.) The ege set of graph G (v) is finally forme by the union of all eges resulting from the ege reuction steps on all four gris. Lemma 5.1. In a given area A (with constant extension in each irection) the number of noes an the number of eges in the Backbone Graph are both boune by O A 2. Proof. The gris employe for the ege reuction steps are chosen to have two properties: (1) Every ege in G (v) C is completely containe in at least one cell an (2) any region (with constant extension in each irection) is intersecte by at most a constant number of gri cells (for instance a square of sie length 3 can be intersecte in total by at most 9 gri cells). Property (1) guarantees that every ege is consiere at least with one of the four gris: Together with the fact that the ege reuction step oes not alter the number of components in a cell subgraph, it follows that the number of components in the complete graph is not altere either. Since each resulting subgraph contains at most O 1 2 eges an together with Property (2), it follows that also the union of all remaining eges that 2 is the number of eges in G (v) 1 is not greater than O for a constant region. The fact that each ege in G (v) an three eges in G correspons to at most two noes an G (v) having at most O A 2 noes for an area A (the noes in MIS) finally leas to the lemma. Lemma 5.2. The Backbone Graph G is a spanner of G C with stretch factor O log( 1 ). 2 The mentione algorithm constructs for a constant κ 1 an O(κ)-spanner with at most n 1+1/κ eges. Setting κ = log n an since n = 1, we obtain a graph with the require properties. 2 72

5 Proof. Every ege in G (v) C is containe in at least one gri cell an consequently also consiere in at least one of the accoring subgraphs. Since the eges retaine in each subgraph form a O log( 1 ) -spanner (on the subgraph), this property also hols for the union of all subgraphs, G (v). Finally, each ege in G (v) resulting in at most three eges in G, the lemma follows. In istribute computing a istinction can be mae between the one-hop broacast moel an the point-to-point communication moel: In the one-hop broacast moel a noe can simultaneously sen a message to all its neighbors, whereas in the point-to-point communication moel a message is sent over an ege to one istinct neighbor. The algorithms escribe in the remaining sections are assume to execute on G. Since on this graph the number of noes an the number of eges are asymptotically equal in a given area, the two moels can be employe interchangeably, epening on whether we argue over the number of noes or eges in the graph. When routing a message m from a source s to a estination t, the noes s an t will in general not be cluster heas. The complete process of routing therefore consists of 1. s sening m to its associate cluster hea s, 2. routing m from s to t, the cluster hea associate to t, an 3. t sening m to t. Since steps 1 an 3 incur only constant cost with respect to both message an time complexity, we exclusively consier step 2 in the remaining part of the paper. Whenever mentioning a source s or a estination t we therefore assume that s an t are cluster heas. 6. MESSAGE-OPTIMAL FLOODING In this section we iscuss the message an time complexities of the Echo algorithm on Quasi Unit Disk Graphs. Due to space reasons we only give a short outline of the algorithm execution; more etaile information can be foun in [7, 23]. The Echo algorithm consist of a flooing phase an an echo phase. - The flooing phase is initiate by the source s by sening a flooing message containing a Time To Live (TTL) counter τ to all its neighbors. Each noe receiving the flooing message for the first time ecrements the TTL counter by one an retransmits the message to all its neighbors (with the exception of the neighbor it receive the message from). In the synchronous moel this flooing phase constructs a Breath First Search (BFS) tree. - From the leaves of this tree the noes where the τ counter reaches 0 echo messages are sent back to to the source along the BFS tree constructe uring the flooing phase. An inner noe in the BFS tree can ecie locally when to sen an echo message to its parent in the tree by awaiting the receipt of an echo message from all of its chilren. By initiating the first flooing phase with τ set to 1 an relaunching a flooing phase with ouble τ whenever the echo messages inicate that the estination has not yet been reache, both time an message complexities can be boune: Theorem 6.1. Employe on G in the synchronous moel, the Echo algorithm reaches the estination with message complexity O ( c )2 an time complexity O c log( 1 ), where c is the cost of a shortest path between s an t. This is asymptotically optimal with respect to message complexity. Proof. The Echo algorithm floos the complete network with message complexity O(m) an time complexity O(D), where m is the number of eges in the network an D is the iameter of the network. Since no ege in G is longer than 1, all noes reache with a certain τ lie within the circle centere at s with raius τ. The number of eges within this circle is boune by O ( τ )2. Note that the estination is reache at the latest for τ =2 c. Since τ is ouble after each failure, the total number of visite eges is ominate by the number of eges in the circle with maximum τ, from which the message complexity follows. Asymptotic optimality follows from the lower boun establishe in Section 4. The time complexity follows from the fact that the BFS tree constructe uring the flooing phase contains a shortest path from s to t. Since G is a log( 1 )-spanner of G, the shortest path on G, on which the algorithm is execute, is c log( 1 ). The time complexity of a single flooing-echo roun being proportional to τ an again the total time complexity asymptotically being ominate by the maximum τ use, the time complexity follows. In the asynchronous moel the synchronizer construction introuce in [3] can be employe. Theorem 6.2. When employe on G in the asynchronous moel, the Echo algorithm reaches the estination with message ) complexity O ( c )2 log 3 ( c an time complexity O c log( 1 ) log3 ( c ), where c is the cost of the shortest path between s an t. Proof. The synchronizer construction ) introuce in [3] incurs a cost factor of O log 3 ( 1 with respect to both message an time complexity. Plugging in the above Echo algo- rithm for the synchronous moel yiels the lemma. For geometric routing as iscusse in the following section, a variant of Echo can be efine by replacing the Time To Live counter by a geometric argument: The flooing message is retransmitte only by noes locate within a circle centere at s with a certain raius r. Lemma 6.3. The geometric Echo algorithm reaches t with message an time complexity O ( c )2, where c is the link cost of the shortest path. This hols for both the synchronous an the asynchronous moel an is asymptotically optimal with respect to message complexity. Proof. In contrast to the above Echo algorithm using TTL, all noes locate within the restricting circle centere at s with raius r participate in the execution of the geometric algorithm. This circle containing at most O ( r )2 noes, the message complexity follows, where the remaining reasoning is analogous to the one in the proof of Theorem 6.1. The time complexity follows from the fact that time complexity cannot be greater than message complexity. 73

6 7. GREEDY ECHO ROUTING Although asymptotically message-optimal, a flooing-base algorithm is prohibitively expensive in most networks for practical purposes. Previous work showe that this problem can often be tackle by combining a correct routing algorithm (that is guarantee to fin the estination) with a greey routing scheme [6, 22]. In this section we therefore escribe a geometric volatile memory routing algorithm that tries to leverage the avantages of a greey routing approach with respect to both conceptual simplicity an message-efficiency: In orer to route a message, a noe simply forwars it to its neighbor closest to the estination. Greey routing can however run into a local minimum with respect to the istance to the estination, that is a noe without any neighbors closer to t. In our case such a local minimum is circumvente by employment of restricte flooing, in particular by the ai of the geometric Echo algorithm as escribe in the previous section. In this section we therefore refer by Echo to the geometric Echo algorithm. We enote with Echo r the subalgorithm of geometric Echo consisting of the flooing an the corresponing echo phase for the raius r. Our algorithm GEcho combines both greey routing an flooing in two moes: Generally the message is forware in greey moe as long as possible. Whenever running into a local minimum, the algorithm switches to echo moe. In orer to keep the cost of flooing-base echo low, the algorithm tries to fall back to greey moe as early as possible. The fallback criterion is chosen such that the combine routing algorithm is asymptotically optimal with respect to message complexity. In particular, the Echo algorithm oes not terminate only when fining t, but alreay when fininganoev which is significantly closer to t than the local minimum, as escribe in Step 2 of the GEcho algorithm: GEcho. The value q is a constant parameter chosen prior to algorithm execution such that 0 <q Start at s. 1. (Greey Moe) Forwar the message to the neighbor in G closest to t. If t is reache, terminate. If a local minimum is reache, continue with step 2, otherwise repeat step 1 at the next noe. 2. (Echo Moe) Execute algorithm Echo starting at the local minimum u until either reaching t in which case the algorithm terminates or fining a noe v, such that ut vt q r, where r is the currently chosen raius in Echo s subalgorithm Echo r. Procee to v an continue with step 1. In the following we obtain a statement on the asymptotic complexity of the algorithm. We first show that the number of messages sent in Greey moe is boune: Lemma 7.1. The number of messages sent in Greey moe is boune by O ( c )2. Proof. Let us exclusively consier the sequence U of noes sening messages in Greey moe or receiving messages sent in greey moe uring the execution of the algorithm. Note that the istance to t is strictly ecreasing within U. Since the algorithm stays in greey moe until reaching a local minimum, U is partitione into subsequences U 1,U 2,...,U k,k 1 of noes by the occurrence of local minima: A local minimum only receives a Greey message without being able to sen it to a subsequent noe in Greey moe. Within a subsequence U i = u 1,u 2,...,u li, l i 2anytwonoesu j,u j+2, 1 j l i 2 have istance greater than (otherwise u j woul have sent the Greey message irectly to u j+2). On the other han also the istance between a local minimum u li an the first noe in the following subsequence U i+1 have istance greater than (otherwise u li woul be no local minimum). Together with the fact that all noes in U are locate within the circle C centere at t with raius st, the number of noes in the total sequence U is therefore boune by two times the maximum number of noes with relative istance greater than or likewise the maximum number of nonintersecting isks of raius /2 that can be place within C. With st c, the lemma follows. We now confine ourselves to the number of messages sent in Echo moe. Note that after each roun, efine to be one execution of Step 1 or Step 2, the algorithm is strictly closer to t than before that roun. Lemma 7.2. For a given r the subalgorithm Echo r is execute at most st 1 times. qr Proof. Accoring to the criterion escribe in Step 2, an Echo roun initiate at noe u terminates unless arriving at t only if it fins a noe v, such that ut vt q r. For any r (also if at a particular noe Echo r fails an r is ouble) such a progress can be mae at most st 1 qr times, since after each roun the algorithm is strictly closer to t than before. With this property we can obtain the total number of messages sent in Echo moe uring algorithm execution. Lemma 7.3. The total number of messages sent in Echo moe is at most O ( c )2. Proof. We obtain the total number of messages sent in Echo moe by summing up over all noes ever containe in a circle bouning Echo r. Since the number of noes containe in a given circular area is asymptotically proportional to the size of the area, it is sufficient to compute the total area covere by all Echo r bouning circles. Let r i =2 i,i= 1, 2, 3,...enote the raii of the Echo bouning circles. The maximum r i can be foun by the observation that (1) all Echo restricting circles have their centers at a noe closer to t than s an (2) the circle centere at any noe closer to t than s having raius 2 c completely contains the shortest path. Since the value of r in Echo r is obtaine by oubling, the maximum r i use over all is less than 4 c; the maximum i reache is consequently log(4 c). With n i being the total number of bouning circles use with raius r i, we obtain A = X log(4 c) i=0 n i πr 2 i for the total covere area A. Using Lemma 7.2 we obtain log(4 c) X A π st 1 ri 2 qr i i=0 X log(4 c) < π i=0 st q ri ( st c) = πc q 2 log(4 c) O c 2. X log(4 c) πc q i=0 2 i 74

7 The Area A containing at most O A 2 noes (cf. Section 5), the lemma follows. Lemma 7.4. The algorithm GEcho fins the estination with both message an time complexity O ( c )2, where c is the link cost of the shortest path. Proof. The message complexity boun follows irectly from the previous two lemmas. The time complexity boun follows from the fact that time complexity cannot be greater than message complexity. Theorem 7.5. The algorithm GEcho is asymptotically optimal with respect to message complexity. Proof. Follows from Lemma 7.4 an Section LARGE D-VALUES This section treats the special case where the parameter of the Quasi Unit Disk Graph G is 1/ 2. This case has alreay been consiere by Barrière et al. in [4]. There, it is shown that for 1/ 2 stanar geometric routing is possible. Here, we exten their results an present a geometric routing algorithm which is asymptotically optimal, i.e. whose cost is quaratic in the cost of an optimal path (cf. [21]). The structural ifference between Quasi-UDGs for < 1/ 2 an Quasi-UDGs for 1/ 2 lies in the local environment of intersecting eges. If 1/ 2, all intersections can be etecte locally. This is shown by the following two lemmas. Lemma 8.1. Let e =(u, v) be an ege an w beanoe which is in the isk with iameter (u, v). Either u an w or v an w are connecte by an ege. u w C Figure 3: w is either connecte to u or to v Proof. The following proof is illustrate by Figure 3. Because uv 1, the regions of the points whose istances to u an v are greater than 1/ 2 o not intersect insie C (hatche areas in Figure 1). Thus either uw 1/ 2or vw 1/ 2. In the picture, this hols for u an w an therefore there is an ege between the two noes in G. v Lemma 8.2. Let e 1 =(u 1,v 1) an e 2 =(u 2,v 2) be two intersecting eges in a Quasi-UDG G with parameter 1/ 2. Then at least one of the eges (u 1,u 2), (u 1,v 2), (v 1,u 2), an (v 1,v 2) exists in G. Proof. We have to show that one of the four sies of the quarangle (u 1,u 2,v 1,v 2) is shorter than 1/ 2. Because the sum of the interior angles of the quarangle is 2π, at least one of the angles has to be greater or equal to π/2. W.l.o.g. let it be the angle at noe u 2. In this case u 2 lies in the isk with iameter (u 1,v 1) an the lemma follows from Lemma 8.1. We will now give an overview of the results of [4]. The algorithm consists of three steps. In a first step, the Quasi- UDG G is extene by aing virtual eges. Whenever there is an ege (u, v) ananoew which is insie the circle with iameter (u, v), for at least one of the noes u an v w.l.o.g. let it be u the istance to w is smaller than or equal to 1/ 2 (Lemma 8.1) an therefore u has a connection to w. If there is no ege between v an w, a virtual ege is ae. Sening a message over this virtual ege is one by sening the message via noe u. This process is one recursively, i.e. also if (u, v) is a virtual ege. The graph obtaine by aing the virtual eges to G is calle super-graph S(G). Barrière et al. prove that on S(G) the Gabriel Graph GG(S(G)) can be constructe yieling a planar subgraph of S(G). In the Gabriel Graph construction, an ege (u, v) is remove if an only if there is a noe w in the isk with iameter (u, v) [9]. On GG(S(G)), any correct geometric routing algorithm is applie ([6, 18]). In orer to obtain an optimal geometric routing algorithm, we have to change the algorithm of [4] in two ways: i) the planar graph which we nee for geometric routing shoul be a spanner an the number of noes in a given area A shoul not be larger than O(A), an ii) we have to replace the geometric routing algorithm by a more elaborate variant such as AFR [21] or one of its successors GOAFR [22] an GOAFR+ [20]. One of the bouning factors for the spanning property is given by the recursive epth of the virtual ege construction, i.e. the length of paths corresponing to virtual eges. From [4] we have the following result. Lemma 8.3. Let λ be the minimum Eucliean istance between any two noes. If 1/ 2, the length of the route in G corresponing to a virtual ege in S(G) is at most λ 2 Proof. The lemma irectly follows from Property 1 in Section 5 of [4]. We cannot assume that there is a minimum Eucliean istance λ between any two noes. However, by using the Backbone Graph G (cf. Section 5) we obtain a Quasi- UDG with boune egree, a property which we prove to be equivalent to the minimum istance assumption. Precisely, we start by constructing G. This gives us a set of ominator noes D = MIS an a set of connector noes C. We transform G into a Quasi-UDG G = (V,E ) by setting V = D C an by incluing all possible eges of E in E (all eges between noes of V ). Lemma 8.4. The egree of each noe in the Quasi-UDG G is boune by a constant. Proof. Because the ominator noes D have istance at least 1/ 2 from each other, the number of ominators which 75

8 are within three hops from a noe v V is boune by a constant. Only those noes can a connector noes which are neighbors of v. Each of them can only a a constant number of connector noes an therefore, the egree of noe v has to be constant. A more etaile proof can be one analogously to the proof for the same lemma for Unit Disk Graphs [1, 29]. G is now use for the Gabriel Graph construction. First, virtual eges are ae as in the algorithm of [4] resulting in a super-graph S(G ). Then GG(S(G )) is constructe. Analogously to Lemma 8.3 we are able to get a boun on the maximum route length for any virtual ege. Lemma 8.5. Let G =(V,E) be a Quasi-UDG with maximum noe egree. If 1/ 2, the length of the route in G corresponing to a virtual ege in S(G) is at most O 2. λ 11 w 11 w 9 l 10 l 11 w 10 λ 10 λ 9 w 8 l l 9 8 w 7 λ l λ 7 l w w 6 4 λ λ 4 l 6 5 w w 5 2 l 3 l4 λ λ 5 2 w w 3 1 u λ l 1 l 2 λ 1 3 l 0 v Figure 4: Recursive Depth of Virtual Eges Proof. Let (u, v) E be an ege of G. Further let w 1,...,w k be a sequence of noes which recursively force the creation of new virtual eges e i for which the corresponing route contains (u, v). Let l 0 := uv an l i be the Eucliean length of the virtual ege e i (see Figure 4 as an explanation). λ i is the length of the ege which together with e i 1 provies the route for e i (λ i ). For the length l i of the i th virtual ege e i we get q s l 2 i 1 λ2 i = l i s 1 λ2 i l 2 0 l i 1 1 λ2 i l l 2 i 1 i 1 q 1 λ 2 i li 1. The secon inequality follows from l 0 l i 1, the last inequality follows from l 0 1. We therefore get l k ky i=1 q 1 λ 2 i l0 k Y i=1 q 1 λ 2 i. (1) P k We efine λ := 1/k i=1 λi to be the average length of the eges corresponing to the λ i. It can be shown that the expression of Equation (1) can be upper-boune by replacing each λ i by λ: l k p k 1 λ 2 = 1 λ 2 k/2. (2) In a Quasi-UDG all noes in a isk with raius /2 are neighbors of each other. Therefore, when starting at a noe u, after at most + 1 hops, one has to leave the isk with raius /2 aroun u. Thus, the sum of the lengths of + 1 successive eges on a cycle-free path has to be greater than /2, i.e. for 1/ 2 this is a constant. The average ege length of any cycle-free path is thus O(1/ ). In Figure 4, we see that the λ i form two paths. Therefore, the average λ i has to be in the orer of λ =O(1/ ). Because (1 1/n) n 1/e, we set k =2/λ 2 =O 2 in (2) an get l k (1 λ 2 ) 1/λ2 1/e 1/ 2. Because the length of a virtual ege e k has to be l k 1/ 2, this conclues the proof. Lemma 8.6. The Gabriel Graph GG(S(G )) is a spanner for the Quasi-UDG G. Proof. By Lemma 8.4 an Lemma 8.5, we see that the virtual eges only impose a constant factor on the cost of a path. We can therefore procee as if all virtual eges were normal eges of G. Further, it is well known that the Gabriel Graph construction retains an energy-optimal path (ege cost = square of the Eucliean ege length) (see e.g. [21]). This is because whenever an ege is remove, there is an alternative (two-hop) path with lower or equal energy cost. Because the average ege length of S(G ) is a constant (cf. proof of Lemma 8.5), the number of hops an the energy cost of a path only iffer by a constant factor an therefore, the minimum energy path is only by a constant factor longer than the shortest path connecting two noes. For further etails, we refer to the analysis for Unit Disk Graphs [20]. Theorem 8.7. Let G be a Quasi Unit Disk Graph with 1/ 2. Applying AFR [21], GOAFR [22], or GOAFR+ [20] on GG(S(G )) yiels a geometric routing algorithm whose cost is O c 2, where c is the cost of an optimal path. This is asymptotically optimal. Proof. Because G can be the Unit Disk Graph, the lower boun follows from the lower boun for Unit Disk Graphs in [21]. The number of noes as well as the number of eges of GG(S(G )) in a given area A is proportional to A an therefore, the O c 2 cost also irectly follows from the respective analyses in [20, 21, 22]. 8.1 Alternative Construction We conclue the section on Quasi-UDG for 1/ 2 with the escription of an alternative construction of a planar graph which can be use to perform geometric routing. By Lemma 8.2, all ege intersections of a Quasi-UDG with 1/ 2 can be etecte locally (one communication roun). Instea of the virtual eges/gabriel Graph construction, we can efine virtual noes at all intersections of two eges. These virtual noes are manage by the en-points of the intersecting eges; sening a message from or to a virtual noe means sening a message from or to a neighbor (not virtual) of the virtual noe. If this is applie on G, we obtain a planar (by efinition!) graph with only O(A) noes in any given area A. Because this planar graph is a spanner, we get a geometric routing algorithm with cost O c 2 by applying AFR, GOAFR, or GOAFR+ [20, 21, 22]. 76

9 9. CONCLUSION What is the benefit of oversimplifie moels about which interesting properties can be proven, that have however barely anything in common with reality? But what if we ajust our moel to imitate reality to the least etail an obtain nothing but a system far too complex for stringent reasoning? These are the two extremes for which we stuy a potential way out in the fiel of a-hoc network moeling: A moel capturing the essence of a-hoc networks, yet concise enough to permit stringent theoretical results. For the Quasi Unit Disk Graph moel having eges between all noes with istance at most, lying between 0 an 1, an no ege of length greater than 1 we constructe a message complexity lower boun for any volatile memory routing algorithm. We showe that a flooing algorithm matches this lower boun an is consequently asymptotically optimal with respect to message complexity. We escribe a geometric routing algorithm combining greey routing an geometric flooing, resulting in a message-optimal algorithm in the worst case an message-efficient algorithm in the average case. We finally showe that classic geometric routing algorithms can be employe with the same performance guarantees as for Unit Disk Graphs if is at least 1/ REFERENCES [1] K. Alzoubi, P.-J. Wan, an O. Frieer. Message-Optimal Connecte Dominating Sets in Mobile A Hoc Networks. In Proc. of the 3 r ACM Int. Symposium on Mobile a hoc networking & computing (MOBIHOC), [2] B. Awerbuch. Complexity of Network Synchronization. Journal of the ACM (JACM), 32(4): , [3] B. Awerbuch an D. Peleg. Network synchronization with polylogarithmic overhea. In Proc. of the 31 st IEEE Symp. on Founcations of Computer Science (FOCS), pages , [4] L. Barrière, P. Fraigniau, an L. Narayanan. Robust Position-Base Routing in Wireless A Hoc Networks with Unstable Transmission Ranges. 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