Select-and-Protest-based Beaconless Georouting with Guaranteed Delivery in Wireless Sensor Networks

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1 Select-and-Protest-based Beaconless Georoting ith Garanteed Deliery in Wireless Sensor Netorks Hanna Kalosha, Amiya Nayak, Stefan Rührp, Ian Stojmenoić School of Information Technology and Engineering (SITE), Uniersity of Ottaa, Canada {hkalosha, anayak, srehrp, Uniersity of Birmingham, UK Abstract Recently proposed beaconless georoting algorithms are flly reactie, ith nodes forarding packets ithot prior knoledge of their neighbors. Hoeer, existing approaches for recoery from local minima can either not garantee deliery or they reqire the exchange of complete neighborhood information. We describe to general methods that enable completely reactie face roting ith garanteed deliery. The Beaconless Forarder Planarization (BFP) scheme finds correct edges of a local planar sbgraph at the forarder node ithot hearing from all neighbors. Face roting then contines properly. Anglar Relaying determines directly the next hop of a face traersal. Both schemes are based on the Select and Protest principle. Neighbors respond according to a delay fnction, if they do not iolate the condition for a planar sbgraph constrction. Protest messages are sed to remoe falsely selected neighbors that are not in the planar sbgraph. We sho that a correct beaconless planar sbgraph constrction is not possible ithot protests. We also sho the impact of the chosen planar sbgraph constrction on the message complexity. This leads to the definition of the Circlnar Neighborhood Graph (CNG), a ne proximity graph, that enables BFP ith a bonded nmber of messages in the orst case, hich is not possible hen sing the Gabriel graph (GG). The CNG is sparser than the GG, bt this does not lead to a performance degradation. Simlation reslts sho similar message complexities in the aerage case hen sing CNG and GG. Anglar Relaying ses a delay fnction that is based on the anglar distance to the preios hop. Simlation reslts sho that in comparison to BFP more protests are sed, bt oerall message complexity can be frther redced. I. INTRODUCTION Beaconless georoting algorithms ork completely reactie and redce the oerhead for exchanging topology and roting information to a minimm. They follo the principle of geographic roting, here a message is roted to the location of the destination instead of a netork address. This is based on the assmptions that each node can determine its on geographic position and that the sorce knos the position of the destination. The se of position data enables roting ithot roting tables or prior rote discoery. Conentional geographic roting algorithms se to basic forarding principles: greedy forarding and face roting. Greedy forarding means to select a neighbor that minimizes the distance to the target. This strategy fails in case of a local minimm, i.e. if no neighbor is closer to the destination. Then, face roting can be sed in order to recoer from this sitation. The message is roted along the incident face of the commnication graph sing the right-hand rle ntil a position is fond that is closer to the destination than the local minimm. Face traersals ork only on a planar sbgraph, otherise crossing edges might case a roting loop. Ths, a local planarization strategy is needed, hich determines the edges of a planar sbgraph. Beaconless Roting: Conentional geographic roting algorithms rely on the position information of their 1-hopneighbors. This information can be gathered by a periodic exchange of beacon messages. Beaconless roting algorithms try to aoid this message exchange and proide a completely reactie roting. The basic principle of beaconless forarding is the folloing: The forarder, i.e. the node that crrently holds the packet, broadcasts it to its neighbors. The nodes ithin the forarder s transmission range receie the packet, bt only the nodes in the forarding area are eligible for forarding it frther (see Fig. 1). These nodes are called candidates. The most sitable candidate is determined by a contention mechanism: After receiing the packet, each candidate starts a timer. The timer is determined by a delay fnction that faors the most promising node, e.g., the node closest to the destination has the shortest timeot. This node forards the packet again, hen its timer expires. The other candidates notice that the packet is re-transmitted and cancel their timers. This strategy follos the greedy principle, becase it ses alays locally optimal decisions. The Beaconless Recoery Problem: As greedy roting fails in case of a local minimm, a recoery strategy is needed to garantee deliery. The preferred recoery method for conentional geographic roting is the face traersal on a planar sbgraph, hich is constrcted from neighborhood information. Bt in beaconless roting the fll knoledge of the neighborhood is not a priori aailable. Instead, part of this knoledge has to be gained by exchanging messages, if it is not implicitly gien by the location of the nodes. Therefore, e can describe the beaconless recoery problem by to qestions, hose anser is the key to garanteed deliery: 1) Ho to constrct a local planar sbgraph on the fly? 2) Ho to determine the next edge of a planar sbgraph traersal? The beaconless recoery problem has to be soled reactiely and ith as fe messages as possible. Existing approaches

2 2 r F greedy area C d D forarding area Fig. 1: Forarder (F), candidate (C) and destination (D). Eligible candidates are ithin the forarding area, hich is part of the greedy area (i.e. closer to the destination than the forarder). se a reactie message exchange in hich all neighbors are inoled in the orst case. This rises the qestion, hether e can redce this message oerhead and ths achiee a significant message redction in comparison to conentional protocols that rely on beaconing. In this paper e anser this qestion and proide soltions for both ariants of the beaconless recoery problem: Beaconless Forarder Planarization (BFP) first constrcts an approximation of the planar sbgraph and then sorts ot nodes that are not neighbors in a planar sbgraph. We se proximity graphs sch as Gabriel graph and relatie neighborhood graph for the planar sbgraph constrction, becase edges in these graphs can be determined locally. We propose the Circlnar Neighborhood Graph (CNG), a planar proximity graph that can be constrcted ith less messages than the Gabriel graph and that has a better connectiity than the relatie neighborhood graph. The second soltion of the beaconless recoery problem is Anglar Relaying, hich first tries to find the next neighbor of a right-hand face traersal and then sitches to another neighbor, if the selected neighbor is not adjacent in the planar Gabriel sbgraph. Oerie of the paper: In Section II e reie related ork. Section III describes the Beaconless Forarder Planarization method, hich proides the general frameork of creating planar sbgraphs reactiely for face roting. In Section IV e take a closer look at planar sbgraph constrctions and determine the crcial properties that affect the efficiency of BFP. In Section V e introdce the Circlnar Neighborhood Graph, a ne proximity graph hich has adantageos properties for local sbgraph constrction in beaconless protocols. It redces the message oerhead to a constant nmber hile proiding better connectiity than the relatie neighborhood graph. Section VI describes the Anglar Relaying method, an alternatie soltion to the beaconless recoery problem. In Section VII e present simlation reslts for the aforementioned protocols. II. RELATED WORK One bilding block of geographic roting strategies are greedy forarding strategies. They are based on positionbased progress criterions sch as MFR [22] or the greedy method [10]. Progress in terms of MFR means to decrease the distance of the projection on the straight line to the target, hile the greedy method simply refers to the Eclidean distance. The first beaconless roting algorithms, BLR [15], CBF [12], and IGF [1], se these greedy criterions to define the delay fnctions, hich determine the candidate ith the most progress by giing him the shortest timeot. There are frther protocols addressing specific problems of the initial approaches. Blind Geographic Roting (BGR) [23] contains a strategy to aoid simltaneos transmissions. Geographic Random Forarding (GeRaF) [25] diides the forarding area into zones and selects the next forarder by contention among the nodes ithin these zones. All these approaches ork ell in dense netorks, here there is alays a neighbor closer to the destination. If this is not the case and the greedy algorithm faces a local minimm, deliery can only be garanteed, if a recoery from that sitation is possible. Recoery strategies hae been deeloped for geographic roting algorithms (see [7] for a srey) and many of them are based on face traersals sing a planar sbgraph. Prominent sbgraph constrctions are the Gabriel graph (GG) [13] and the relatie neighborhood graph (RNG) [17], bt also localized ariants of the Delanay trianglation hae been proposed [14], [18], [20]. Face roting on a planar sbgraph in combination ith greedy forarding is the idea behind the Greedy-Face-Greedy algorithm (GFG) [4], hich became a standard techniqe for geographic roting. A. Beaconless Recoery While the recoery problem is ell stdied for geographic roting algorithms, the beaconless approaches leae room for improement. In beaconless roting, the term recoery is often sed in connection ith heristics, that enlarge the set of possible candidates, if the forarding area is empty, bt do not garantee deliery. BLR, CBF and BGR se this kind of heristic. PSGR [24] contains a more sophisticated recoery mechanism, hoeer the deliery is qestionable, as no crossing-free sbgraph is considered. The folloing beaconless protocols contain a real recoery strategy and can ths gie deliery garantees (cf. Table I). Hoeer, all these strategies reqire position information of the complete neighborhood to be exchanged in the orst case. BLR Backp mode [16] (also called Reqest-response approach in [15]): The forarder broadcasts a reqest and all neighboring nodes respond. If a node is closer to the destination, it becomes the next hop. Otherise the forarder constrcts a local planar sbgraph (GG) from the position information of the neighbors and forards the packet sing the right-hand rle. The position hen entering backp mode is stored in the packet. Greedy forarding is resmed hen a node is closer to the destination. Reqest-Response can be regarded as reactie beaconing, becase all neighbors are inoled in exchanging position information. The folloing protocols se an approach, that e classify as Select and Protest: they determine possible

3 3 Protocol Empty Forarding Recoery Garant. Area (from local minima) deliery BLR se MFR area Beaconing + face roting yes CBF se greedy area (left open)?? IGF no BGR rotate fd. area no GeRaF * no PSGR * Bypass?? NB-FACE ** Clockise timeot and yes Gabriel neighbor selection GDBF ** Distance-based timeot, yes Gabriel neighbor selection *) Fd. area coers the complete greedy area **) Fd. area coers the complete transmission area TABLE I: Beaconless roting protocols and their recoery methods Fig. 2: BFP: Nodes respond in the order 1, 2, 3, 6 ; 4 and 5 are hidden. 4 protests against 6, and 5 remains silent after 4 s protest 2 5 neighbors of a planar sbgraph by a contention process and allo protests afterards to correct rong decisions. NB-FACE [21] is a beaconless ariant of the face roting algorithm. The delay fnction depends on the angle beteen candidate, forarder and preios hop sch that the first candidate in (conter-)clockise order responds first. If this node is not a neighbor in the Gabriel graph, then other nodes may protest. The NB-FACE algorithm is similar to a ariant of or Anglar Relaying scheme (Section VI). Hoeer, e ill see that NB-FACE yields not alays optimal reslts. GDBF [5], [6] proides a beaconless Gabriel graph constrction and seres as basis for face roting algorithms sch as GFG. The local Gabriel sbgraph is constrcted in to phases, sing a timer-based contention mechanism: First, the candidates anser ith a delay proportional to their distance to the forarder, bt only if no other neighbor located ithin their Gabriel circle has responded earlier. The ths constrcted sbgraph contains directed (asymmetric) edges and is not necessarily planar. Therefore, after the face roting algorithm has selected a candidate that iolates the Gabriel graph condition, frther nodes may protest against the decision in a second phase. We ill see that in the orst case all neighbors hae to respond hen sing the Gabriel graph. GDBF is a ariant of or more general BFP scheme. III. BEACONLESS FORWARDER PLANARIZATION The basic problem of beaconless protocols is that they cannot rely on 1-hop-knoledge. Bt this knoledge is necessary to bild a planar sbgraph. Ths, in a recoery sitation, the forarder has to gather information and this is connected ith the exchange of messages. In contrast to the Reqest-Response approach of BLR [15], here all neighbors annonce their positions pon reqest, e follo the idea of GDBF [6] to redce the message oerhead. Beaconless Forarder Planarization (BFP) is a general scheme, that can be sed to constrct different proximity graphs, sch as Gabriel graph and RNG. The BFP algorithm is described in the folloing. It s message complexity depends on the chosen sbgraph. We ill later discss appropriate sbgraph constrctions and analyze the message complexity. Fig. 3: Proximity regions of GG and RNG: An edge (, ) exists only if the proximity region (shaded) is empty. The BFP Algorithm The BFP algorithm consists of to phases, the selection and the protest phase. N(, ) denotes the proximity region of the chosen sbgraph, e.g. the Gabriel circle or the RNG lne oer (, ) (cf. Fig. 3). 1. Selection Phase The forarder broadcasts an RTS (inclding its on position) and sets its timer to t max. Each candidate sets its contention timer, sing the folloing delay fnction: t(d) = d r t max (1) (d = distance to forarder =, r = transmission radis, t max = maximm timeot). When the contention timer expires, a candidate ansers ith a CTS. If a candidate receies the CTS of another node that lies in the proximity region N(, ), then cancels its timer and remains qiet. We call this mechanism sppression and the candidate being sppressed a hidden node. Hidden nodes listen to other nodes after their timer expired. If a hidden node receies the CTS of another node ith N(, ), then iolates the proximity condition and adds to the set of iolating nodes S. We call (, ) a iolating edge. See also Fig Protest phase In the second phase, the hidden nodes protest against iolating edges. If the set of iolating nodes S is not empty, the hidden node starts its timer, sing the same delay fnction as in the first phase (closest candidates protest first). If oerhears a protest from another hidden node, then the set of iolating nodes has to be checked: A node x can be remoed from S, if N(, x). When the timer expires and S is not empty, sends the protest message. The forarder remoes iolating edges hen it receies protests and finally obtains a planar sbgraph.

4 4 IV. PROXIMITY GRAPHS AND BEACONLESS SUBGRAPH CONSTRUCTION The BFP algorithm can be based on different proximity graph constrctions, in order to obtain a planar commnication graph (here, it means that the graph is a planar embedding). Most prominent sbgraph constrctions are Gabriel graph and RNG (cf. [8]): Definition 1: The Gabriel graph (GG) of a node set V contains an edge (, ), iff for all V,,. Definition 2: The relatie neighborhood graph (RNG) of a node set V contains an edge (, ), iff max{, } for all V,,. The definition implies that to nodes and are adjacent, if the so-called proximity region oer (, ) is empty (proximity condition). We denote the proximity region ith N(, ). In case of the Gabriel graph, the N GG (, ) is a circle haing as diameter, in case of the RNG, N RNG (, ) is a lne oer (see Fig. 3). In this paper e assme that all distances are different in order to aoid degenerated cases. Hoeer, eqal distances can be handled by sing =( 2,key(),key()) as distance measre [20], here key( ) is based on the node ID or on a lexicographic order of the geographic coordinates. In a similar ay, a modified RNG ith a constant maximm node degree can be obtained that is still connected on degenerated node sets [19]. The choice of the sbgraph determines the message efficiency of the BFP algorithm. In the folloing e ill identify the crcial properties to constrct a planar and connected sbgraph ith as fe messages as possible. A. Basic Reqirements We consider only ndirected, planar, and connected proximity graphs. The proximity region of these graphs is symmetric, it contains at least the Gabriel circle, and it is not larger than the RNG lne. Lemma 1: The RNG lne is the maximm proximity region to presere connectiity. Proof: Let,, be nodes of an ndirected proximity graph, and let N RNG (, ) denote the RNG lne oer (, ), i.e. the intersection of to circles ith radis centered at and. Sppose the proximity region of (, ) is larger than N RNG (, ). Then there is a point otside N RNG (, ) (i.e. > or > ) that belongs to the proximity region and ths inalidates the edge (, ). If < then N RNG (, ), hich disconnects. Otherise, N RNG (, ), hich disconnects. Lemma 2: The Gabriel circle is the minimm proximity region to obtain planarity. Proof: Let N GG (, ) denote the Gabriel circle oer (, ), i.e. the circle haing as diameter ith its interior. Let m be the midpoint of (, ). Sppose the proximity region is smaller than N GG (, ). Then there is a node inside N GG (, ) ith m < m, hile (, ) is a alid edge. As G is ndirected, the proximity region is symmetric; and this Fig. 4: Sppression region for GG and RNG: A node in the shaded area is not a alid neighbor of, becase old be inside the Gabriel circle or the RNG lne. implies that there is another point hich can be constrcted by rotating by 180 arond the midpoint m. Then the circle N GG (, ) is inside N GG (, ) and empty (becase of m = m < m = m ). Therefore, (, ) is a alid edge, and it intersects (, ) in the midpoint, hich is a contradiction. The graph is planar, if the proximity region contains N GG (, ): IfN GG (, ) is empty, then the empty circle rle of the Delanay Trianglation is also flfilled for any three nodes. Ths, G is a sbgraph of the Delanay Trianglation, hich is planar. B. Hidden Nodes and Sppression The constrction of Gabriel graph or RNG is based on the proximity region, hich is an empty circle or an empty lne. BFP makes se of this fact to redce messages: Candidate nodes are sppressed, i.e. they remain qiet, if they old iolate this condition. Definition 3: The sppression region of a node ith respect to contains all points ith N(, ), here N(, ) denotes the proximity region of an edge (, ). Fig. 4 shos the sppression region for Gabriel graph and RNG. In case of the Gabriel graph, is sppressed, if < 90, and this implies that the border of the sppression region is orthogonal on (, ). In case of the RNG, <, and this means that the perpendiclar bisector of (, ) marks the border of the sppression region. C. Ordered Neighborhoods and Protest Messages In beaconless protocols, the location of the neighbors are not knon in adance, bt they are reealed one by one hen they reply to the forarder s reqest. From a graph theoretic point of ie, the candidate nodes are inserted into the set of neighbors, and the insertion order is gien by the delay fnction. This determines the reslting neighborhood, becase after one node responds, others may be sppressed and remain qiet. In order to formalize this mechanism, e introdce the definition of an ordered neighborhood. Let G denote a graph and Γ() the set of neighbors of a node in G. For a node, define a total order π so that π () is the rank of Γ(). Definition 4: A node Γ() is hidden, if it is sppressed by a non-hidden node ith smaller rank, i.e. Γ() ith π () <π () and N(, ).

5 5 Definition 5: The π-ordered neighborhood Γ π () contains all nodes for hich there is no non-hidden node ith N(, ). An ordered neighborhood can be constrcted by inserting nodes one by one, if they flfill the proximity condition (e.g. empty Gabriel circle). In contrast to the original proximity graph, this condition is only checked for the nodes hich hae been already added to the neighborhood. Note that in contrast to ordered θ-graphs [3], π defines a local order for each node. In BFP a distance-based delay fnction is sed (eqation 1) hich defines the insertion order and determines the neighborhood. The reslt of Phase 1 of the BFP algorithm is a distanceordered neighborhood, hich contains at least the edges of the desired sbgraph. Theorem 1: In a proximity graph, the ordered neighborhood of a node is a sperset of the original neighborhood, i.e. Γ π () Γ(). Proof: Let be a neighbor of, i.e. Γ(). Then, the proximity region N(, ) is empty and remains empty, regardless of the rank of. Ths, Γ π (). When constrcting the ordered neighborhood, e can be sre, that the nodes of the desired sbgraph are inclded, bt there may be iolating edges depending on the insertion order. Therefore, Phase 2 of the BFP algorithm is reqired, here the hidden nodes send protest messages to indicate edges iolating the proximity condition. In the orst case, there is one protest message reqired for each iolating edge. D. Distance-ordered neighborhoods The orst case nmber of iolating edges depends on the order (i.e. the delay fnction) and also on the chosen sbgraph constrction. In case of the Gabriel graph this nmber is nbonded, hereas it is constant in case of the RNG. Theorem 2: A distance-ordered Gabriel neighborhood contains an nbonded nmber of iolating edges. Proof: The constrction in Figre 5 shos that a node can hae Θ(n) neighbors in its distance-ordered Gabriel neighborhood hile it has only one alid Gabriel neighbor. Nodes 1,..., 5 are placed arond ith increasing distance and partially oerlapping Gabriel circles as shon in the figre. In the Gabriel neighborhood 1 inhibits an edge (, 2 ), 2 inhibits an edge (, 3 ) etc., so that has only one alid edge. In the distance-ordered neighborhood 1 is inserted first and 2 is hidden, becase node 1 is in its Gabriel circle. Node 3 becomes a neighbor, becase 2 is hidden and not part of the neighbor set. Eery second node in the chain ill become a neighbor of, i.e. Γ π () hasasizeof (n 1)/2. Corollary 1: The beaconless Gabriel graph constrction ith a distance-based delay fnction reqires an nbonded nmber of protests in the orst case. The crcial property to bond the nmber of protests is that a circlar sector has to be part of the proximity region. Theorem 3: If the proximity region contains a circlar sector of angle θ, then the node degree at most 4π/θ Fig. 5: Gabriel graph (left) and distance-ordered neighborhood (right) ith hidden nodes (hite) and iolating edges (x) Fig. 6: A proximity region containing a sector bonds the nmber of iolating edges θ x region A region B x Fig. 7: Hidden node scenario for Theorem 5 a distance-ordered neighborhood has at most 4π/θ 1 iolating edges. Proof: Let θ (, ) be a sector of the circle C(, ) ith angle θ and as bisecting line (see Fig. 6), and assme that it is contained in the proximity region. A node is only inclded in the neighbor set of, if > θ/2, becase of the folloing reason: If <, mst be otside θ (, ); otherise, mst be otside θ (, ). Therefore, e can insert alid neighbors in Γ π () only at an anglar distance of more than θ/2 to an existing neighbor. Then the maximm node degree of is 4π/θ. Thisisthelimitfor the nmber of iolating edges and this limit can be reached in the orst case: The example in the figre shos that for a pair of nodes ith oerlapping proximity regions there can alays be a hidden node x, ith higher rank than and N(, x) and x N(, ), that renders (, ) a iolating edge. This theorem shos that e can limit the nmber of iolating edges by choosing an appropriate proximity region. The relatie neighborhood graph flfills this criterion. Theorem 4: A distance-ordered relatie neighborhood contains at most 4 iolating edges. Proof: The RNG lne contains a circlar sector of θ< 120. From this fact and Theorem 3 follos the reslt. Corollary 2: The beaconless RNG constrction ith a distance-based delay fnction reqires a constant nmber of protests in the orst case. Hoeer, the proximity region of the RNG is qite large, sch that more edges are forbidden than in the Gabriel graph. The RNG has (length/poer) stretch factor Θ(n), the Gabriel

6 6 Graph only Θ( n) (both are not hop-spanners) [2]. E. Releance of Protest Messages We hae seen that in the presence of hidden nodes edges can be created that iolate the proximity condition. Therefore it is necessary to allo hidden nodes to protest against the selection of a neighbor. One might ask if there is any delay fnction or any practical sbgraph constrction that faors only the alid neighbors. Unfortnately this is not the case. Theorem 5: No ndirected, planar and connected proximity graph can be constrcted ithot protests. Proof: Consider the scenario in Figre 7 as a conterexample: Node is located in the sppression region of, is sppressed by, bt is not sppressed by. When considering the sppression region for arbitrary proximity graphs (that are ndirected, planar and connected), the region is at least the sppression region of the Gabriel graph and at most the sppression region of the RNG. This follos from Lemmata 1 and 2. Therefore, region A is part of the sppression region of and region B is not a sppression region of for all considered proximity graphs. No e bild the ordered-neighborhood of x for all permtations of,,. insertion order π neighborhood immediate protest of hidden nodes in () Γ π(x) protest hidden nodes () {,} () {,} () {} () {} () {,} {}, We can see from the table, that regardless of the insertion order, there is alays a protest, either becase the inserted node immediately knos that it iolates the proximity graph condition, or becase of a hidden node that protests later. V. THE CIRCLUNAR NEIGHBORHOOD GRAPH For the beaconless sbgraph constrction e ant to presere as mch edges as possible, bond the nmber of protests and obtain a planar graph. The planarity can be achieed by inclding the Gabriel circle in the proximity region. Protests can be bonded by inclding a circlar sector. The larger the angle of the sector, the smaller the maximm node degree, bt this also cancels more edges. Therefore, e propose the Circlnar Neighborhood Graph (CNG) as an alternatie to Gabriel graph and RNG. It is a planar graph ith constant degree; it s proximity region is only a small enhancement of the Gabriel circle and the proximity condition can be tested ith 1-hop-knoledge and simple arithmetics. Definition 6: The circlnar neighborhood N CNG (, ) of to nodes and is gien by the intersection of for disks of radis centered at the corners of a sqare of hich (, ) is the diagonal (cf. Fig. 8). The cirlnar neighborhood graph contains an edge (, ) if and only if N CNG (, ) is empty: Definition 7: The circlnar neighborhood graph of a node set V contains an edge (, ) iff V,, : < max{,, p 1,, p 2, }. p 1 p 2 N RNG N CNG N GG Fig. 8: The circlnar neighborhood N CNG (, ) ith RNG lne and Gabriel circle A. Properties of the Circlnar Neighborhood Graph The CNG has a strong relation to Gabriel graph and RNG and inherits planarity and connectiity. Theorem 6: The circlnar neighborhood graph of a node set V is planar and connected, if the nit disk graph of V is connected. Proof: Follos from the shape of the proximity region and Lemmata 1 and 2. The CNG inherits also a disadantage from the RNG, namely the nbonded spanning ratio of Θ(n) (maximm ratio of shortest path in CNG oer shortest path in the original graph). One can constrct the same loer bond example ( RNG toer [2]) for the CNG. In other ords, hen sing the CNG planarization, the maximm detor is nbonded in the orst-case. Apart from these orst-case considerations, e performed simlations on 200 random nit disk graphs ith 100 nodes for netork densities (aerage nmber of neighbors) beteen 4 and 12. Measrements of the spanning ratio sho that the CNG is closer to the Gabriel graph than to the RNG: The hop spanning ratio of the CNG is only 5%-7% larger than in the Gabriel graph, hile the RNG s spanning ratio is 36%- 61% larger (see Fig. 9). In general, the CNG has an expected node degree of 3.6 and is ths sparser then the Gabriel graph and denser than the RNG. Table II smmarizes these reslts (cf. [9]). Theorem 7: The circlnar neighborhood graph has an expected node degree of approx. 3.6 and a maximm node degree of 14. Proof: Folloing the considerations in [9], e can derie the expected degree of the CNG from the ratio of the circle C(, ) and the area A of the proximity region N CNG (, ). The area of the circlnar neighborhood is A r 2.This gies an expected degree of C(, )/A One can sho that the circlnar neighborhood contains a circlar sector of 48, 6. From this and Theorem 3 follos the maximm node degree of 14. B. Beaconless constrction With the CNG a beaconless planar sbgraph constrction ith a constant nmber of protests is possible. Corollary 3: A distance-ordered neighborhood in the CNG has at most 13 iolating edges.

7 7 Spanning ratio Graph Exp. degree Max. degree Spanning ratio [9] [2] RNG Θ(n) CNG Θ(n) GG n 1 Θ( n) TABLE II: Properties of RNG, CNG and GG GG CNG RNG Netork density Fig. 9: Spanning ratio of RNG, CNG and GG (aerage ith 95% confidence error for 200 random graphs, 100 nodes) Proof: This follos from Theorem 3. One can sho that the circlnar neighborhood contains a circlar sector of 48, 6. Plgging this into Theorem 3 gies the reslt. C. Face Roting on the Circlnar Neighborhood Graph The circlnar neighborhood graph has the strctral graph properties that are necessary to garantee recoery. The folloing graph property holds for the Gabriel graph (Lemma 1 in [11]) and can be shon analogosly for the CNG. Lemma 3: For any edge (, ) crossing the s-t-line connecting sorce s and destination t in the circlnar neighborhood graph, at least one of the end points or is closer to the target than s. Proof: As the circlnar neighborhood contains the Gabriel circle, the Gabriel circle oer (, ) neither contains s nor t. It follos that s and t are less than π/2. Since the sm of the angles of the qadrangle st is 2π, at least one of the angles st or st is greater than π/2. This implies that at least one of the nodes or is closer to t than s. For garanteed deliery, face roting on the planar sbgraph has to proide progress toards the destination. This is shon by the folloing theorem (cf. Corollary 2 of [11]). Theorem 8: Let s and t be nodes in a circlnar neighborhood graph. When starting at s, face roting ill alays find a node that satisfies t < st. Proof: The CNG is planar and from Lemma 5 in [11] follos that face roting ill alays find an edge intersecting the s-t-line. With Lemma 3 e can conclde, that one of the edge s end points satisfies t < st. VI. ANGULAR RELAYING Anglar Relaying is a beaconless face roting strategy, hich can be sed as a method for recoery from local minima. While BFP orks independent of the roting protocol, Anglar Relaying needs the information of the preios hop and the recoery direction (right-hand or left-hand). It ses an angle-based delay fnction to determine a candidate for the next hop in combination ith the Select and Protest method for aoiding crossing edges. Here, e se the Gabriel graph condition as planarization criterion. By sing an angle-based delay fnction the first neighbor in conter-clockise order is selected. Other approaches, sch as NB-FACE, the clockise relaying approach in an earlier ersion of BLR [15], or the Bypass method of PSGR are also based on an angle-based fnction, bt they either cannot garantee deliery or the complete neighborhood is inoled in the message exchange. An simple angle-based delay fnction has the folloing form: t(θ) = θ 2π t max (2) The angle θ can be considered in clockise or conterclockise order, depending on the traersal direction (lefthand or right-hand). Selecting a candidate by this fnction is not sfficient to garantee deliery, becase it is not necessarily a neighbor of the forarder in the Gabriel sbgraph. Therefore, e se protest messages to preent crossing links. This is similar to the protest phase sed in the BFP algorithm, bt here, both the selection of the candidate as ell as protesting is done in consectie interals sing an anglebased delay fnction. A. The Anglar Relaying Algorithm The Anglar Relaying algorithm consists of to phases: 1. Selection phase After receiing a packet from the preios hop, the forarder sends an RTS (inclding preios hop and on position) and sets its timer to t max. Eery candidate sets its timer t(θ) sing the anglar distance θ = to the preios hop. Candidates anser ith a CTS in conter-clockise order, according to the delay fnction. We allo candidates to respond, if they hae the preios hop in the Gabriel circle (i.e. nodes in region B in Fig. 10). These nodes anser ith an inalid CTS, becase they iolate the Gabriel graph condition, bt other nodes shold be aare of their existence. Otherise they old be hidden and need a chance to protest later. After the first candidate ansers ith a alid CTS, the forarder immediately sends a SELECT message annoncing that is the first selected node. All candidates ith pending CTS ansers cancel their timers. 2. Protest phase After the selection of the first candidate, the protest phase begins. The forarder starts its protest timer that coers only the time hen protests can occr, hich is t pr = t( π 2 )= 1 4 t max for the Gabriel graph. No, no frther CTS ansers are alloed. Instead, each candidate x sets a ne timer t(θ) that determines the order of protests (θ = x ). First, only nodes in N GG (, ) are alloed to protest. If a node x protests, then it atomatically becomes the next hop. After that, only nodes in N GG (, x) are alloed to protest. Finally, if the forarder s timer expires (i.e. there are no more protests), the data packet is sent to the crrently selected (first alid or last protesting) candidate.

8 C 3 A 2 Fig. 10: Anglar Relaying: 1 and 2 are inalid, 3 is selected, 4 and 5 protest. Finally, 5 is the next hop. Anglar Relaying orks similar to NB-FACE. In NB-FACE the forarder aits for a time span τ after the first candidate responded, in order to leae room for protests ( NAck ). After that, it sends a message ( Fin ) to stop the contention period and select the final candidate. If τ is a constant angle, then the case of cascading protests is not coered; otherise, if τ spans the hole rotation, then all neighbors respond, een if they are not protesting, and the adantage oer the Reqest- Response approach anishes. Also the details abot ho nodes are treated that hae the preios hop in their Gabriel circle (region B in Fig. 10) are left open. Anglar Relaying orks also ith CNG and RNG as sbgraph constrctions. Hoeer, in this scheme it is not adantageos to select a sbgraph ith a larger proximity region than the Gabriel graph, becase this old make protests more likely (after a candidate as selected). This old reslt in a larger nmber of messages. B. Correctness of the Anglar Relaying Algorithm Theorem 9: The Anglar Relaying algorithm selects the first edge of the Gabriel sbgraph in conter-clockise order. Proof: Let N + GG (, ) be the left part of the Gabriel circle of (, ), hich is left/ahead of the ray (in conterclockise direction) from forarder throgh candidate. (region C in Fig. 10). Analogosly, let N GG (, ) be the remaining part of the Gabriel circle. For the first selected candidate holds, that N GG (, ) is empty, becase of the folloing reasons: First, all nodes are alloed to respond, also the inalid ones. That ensres that there are no hidden nodes inalidating. Ths, has the smallest angle among the Gabriel neighbors (otherise another alid neighbor old hae responded before). There is only one region that e did not consider yet, namely the part of N GG (, ) beyond (region A in Fig. 10). Bt this region is empty, becase it is alays coered by N GG (, ); otherise, (, ) old not be an edge of the Gabriel graph. For similar reasons, N GG (, ) is empty for nodes that protest in the second phase. Protesting nodes are atomatically selected as tentatie next hop. For the crrently selected node holds (inariant of the algorithm): If N + GG (, ) is empty, then is the Gabriel neighbor of ith the smallest angle. This follos from the fact that no node ith smaller angle responds later 1 B and from the considerations aboe. The algorithm terminates if the forarder s timer expires. Then N + GG (, ) is empty, becase there are no frther protests. If a part of the Gabriel circle intersects ith the radial line from throgh, then this part lies ithin the Gabriel circle oer (, ) and therefore, this region is also empty. VII. SIMULATIONS We performed simlations of BFP and Anglar Relaying on 500 random graphs ith 100 nodes for netork densities (i.e. ag. nmber of neighbors) ranging from 4 to 12. Messages are sent from the leftmost to the rightmost node sing GFG roting [4]. The greedy part is performed by a beaconless greedy scheme sing RTS/CTS, the face roting part is performed by BFP on different sbgraphs or by Anglar Relaying. We se a simplified MAC layer model assming niform transmission radii and no collisions. We measre the nmber of messages in recoery mode sed for each rote. Messages in greedy mode are neglected in the statistics, becase the algorithms do not differ in the greedy part. In order to obtain a fair and consistent measre for different roting paths and sbgraphs, the ales are normalized, i.e. diided by the length of the shortest path (nmber of hops) in the original nit disk graph. Beaconless Forarder Planarization: The reslts for the nmber of protests (Fig. 12) and for the oerall message complexity (Fig. 11) sho a gap beteen Gabriel graph and RNG. The inferior performance hen sing RNG is de to the long detors cased by this planarization method. The CNG reaches the good performance of the Gabriel graph, hile garanteeing a orst-case bond for the nmber of protests, hich is not possible hen sing the Gabriel graph planarization. In comparison to BLR Reqest-Response (reactie beaconing) e obsere a redction of messages by more than 20%. Anglar Relaying: We can obsere, that BFP ses less protest messages than Anglar Relaying. Bt that does not imply that BFP is more efficient, becase some nodes that send a protest in Anglar Relaying old send a CTS hen sing BFP. This is reflected in the oerall message complexity, here Anglar Relaying ses less messages. Hoeer, BFP constrcts a complete local sbgraph, Anglar Relaying determines only the next hop. VIII. CONCLUSION We hae presented to soltions for the beaconless recoery problem and introdced a theoretical frameork to analyze the message complexity of beaconless face roting algorithms. We cold improe the orst-case message complexity by introdcing a ne planar sbgraph constrction. Frther improements cold be achieed by storing and sing oerheard transmissions in an RTS cache. Ftre research incldes extensions to handle discrete timeots and collisions hen sing a realistic MAC layer. ACKNOWLEDGMENTS This research is spported by NSERC Strategic Grant DefaltSENS, NSERC CRD Grant CRDPJ , and the UK Royal Society Wolfson Research Merit Aard.

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