Random walk with long jumps for wireless ad hoc networks

Ad Hoc Networks (2009) 294–306 Contents lists available at ScienceDirect Ad Hoc Networks journal homepage: www.elsevier.com/locate/adhoc Random walk with long jumps for wireless ad hoc networks Roberto Beraldi * Dipartimento di Informatica e Sistemistica, Università di Roma ‘‘La Sapienza”, via Ariosto 25, 00184 Roma, Italy a r t i c l e i n f o Article history: Received July 2007 Received in revised form November 2007 Accepted 15 March 2008 Available online 26 March 2008 Keywords: Random walk Wireless networks Probabilistic search algorithms Random walk a b s t r a c t This paper considers a random walk-based search algorithm in which the random walk occasionally makes longer jumps The algorithm is tailored to work over wireless networks with uniform node distribution In a classical random walk each jump has the same mean length On the contrary, in the proposed algorithm a node may decide to double the expected jump length by increasing the nominal transmission power and picking a neighbor beyond the nominal range The aim of these long jumps is reducing the spatial correlation among short term subsequent node selections, thus improving the search performance, namely the hitting time Two versions of the algorithm are studied, with and without lookahead A protocol for implementing each version is also proposed When there is no lookahead the proposed protocol allows for a finer transmission power transmission regulation The paper studies, for three network topologies, the impact of the long jump probability on the hitting time and on the average total power required before the target is found Ó 2008 Elsevier B.V All rights reserved Introduction 1.1 Context of this study Searching is a common problem arising in distributed computer systems In general, a searching problem is to be solved when an element in a given set (the search space) needs to know the element(s) of the same set enjoying a known property Among the elements of the set a neighbor relationship exits, so that the search space is best modelled as a graph Searching is performed from the ‘‘inside” of the space, i.e., the search is described as a distributed algorithm Instantiations of the above problem can be recognized in several situations which include reactive protocols for mobile ad hoc networks [14], service discovery in service oriented architectures (SOA), e.g., [6], querying in sensor wireless networks, e.g., [17], and file discovery in peer-to-peer architectures, e.g., [7] Depending on the specific case, the property of the searched element can be as simple as an IP address, like for routing protocols, or it is given as a more sophisticated data struc* Tel.: +39 06 77274018 E-mail address: beraldi@dis.uniroma1.it 1570-8705/$ - see front matter Ó 2008 Elsevier B.V All rights reserved doi:10.1016/j.adhoc.2008.03.001 ture; e.g., in the SOA paradigm, a tree encoded as an XML text file describing the service (its public interface, message types, bindings, etc.) The approaches for addressing a search problem can be categorized as structured or unstructured.1 In the first case, the search space has some form of deterministic organization For example, a subset of nodes may form a virtual backbone and provide virtual access points for a search [13] More elaborated structures are implemented via overlay networks that are organized as a Distributed Hash Table (DHT), e.g., [9] In the unstructured approach the search space is not organized The lack of organization can be simply inherited from the network topology, or even superimposed via a random overlay network, [8,5] In this paper we consider the problem of searching in a wireless network with no structure or overlay support Due to the lack of a structure, a search needs to explore the whole network Among the two opposite solutions, namely flooding and random walk, we guess that latter is more suitable As pointed out in [5], compared to flooding a random walk search has in fact the advantages of having a more fine-grained control of We not consider centralized solutions, e.g., central service directory, since not suitable for a dynamic distributed environment R Beraldi / Ad Hoc Networks (2009) 294–306 295 the search space, a higher adaptiveness to termination conditions and can naturally cope with failures or voluntary disconnections of nodes jumps is more intuitive Long jumps allow to move faster in the network and this is useful when the target node is far from the source one 1.2 Motivation and basic idea 1.3 Contribution of this work The hitting time is the main performance metric of a random walk-based search algorithm It is defined as the average number of elements that has be visited, starting from a given source, before a target is reached Having a low hitting time is important, because this translates into lower response time to searches and, potentially, lower search costs, in terms of bandwidth and/or energy requirements To obtain a low hitting time, we propose a random walk that occasionally makes long jumps; this is achieved regulating the nodes’ transmission range A normal (short) jump is performed by setting the transmission range to R and picking one node at random among those at distance r R, whereas a long jump requires to set the transmission range to R0 > R and picking a node at distance R < r R0 , i.e., from a ring The percentage of long jumps is determined by the (long) jumping probability q, which is a protocol’s parameter The basic idea is that shuffling long and short jumps should help to explore the searching space more effectively, thus reducing the hitting time of a search (recall that any selection is blind) In particular, by occasionally making long jumps the spatial correlations among subsequent choices, i.e., the chances of visiting a same node, is reduced To provide the reader with a rough explanation of the idea, let’us consider a set of uniformly distributed nodes and assume that the random walk always makes short jumps, see Fig 1a This figure shows two nodes, A and B, visited in sequence by the walk If node B sends the walk to any node other than the ones located in the dashed area, then such a new node, say C (not shown in the figure), can again send the walk to A because it’s near enough to it Node C belongs to the dashed area with probability given by the ratio of the surface of the dashed area with pR2 , and then the event of reselecting A is determined by such a ratio However, we can reduce such a probability by allowing B to perform a long jump, see Fig 1b In this case, in fact, C is in the A’s transmission range with probability given by the ratio of the dashed area with pðR02 À R2 Þ, and this probability can be made lower than the previous one by properly setting R0 Another advantage of using long In this study the long jump length is twice the normal one, in expectation Doubling the jump length appears to be a good compromise between the need of speeding up the search and avoiding a high increase in the transmission power, which – among other things – will increase interference (collisions) among concurrent transmissions and reduce the network capacity Moreover, this assumption simplifies the analysis of the protocol The contribution of the paper can be summarized as follows The main contribution is a study about the relationship between frequency of double jumps and hitting time We have considered two types of random walks, characterized by the lack or presence of lookahead The hitting time, as a function of the long jumping probability, is derived for three different topologies, namely a line, 2D grid, and random geometric graph (RGG) For the first topology we have derived an analytical upper bound, while for the grid one we present numerical results based on Markov chains RGG are studied via simulations In wireless networks it is often important to reduce the total energy spent to run a protocol Although long jumps reduce the hitting time, the energy spent to perform each long jump is higher than the one spent for a normal jump The net effect of long jumps on the total energy spent to reach the target is then also studied in the paper The last contribution is the description of two distributed and efficient implementations of the random walk algorithms These protocols can run directly atop the data link layers and are characterized by the presence or lack of look-ahead In particular, the implementation of the random walk with lookahead is suitable for searching objects whose description is short enough to fit into a single packet, as opposite to the implementation of the random walk search with no lookahead, which however allows for a finer power transmission regulation a b Fig Basic idea for speeding up random walk using long jumps Constant jump length (left), variable length (right) 1.4 Related works Random walks (RWs) have been studied as a query/ searching or gathering mechanism over ad hoc or sensor wireless networks and many variants of the basic algorithm have been proposed, [19] The aim of these variants is to improve the overall effectiveness of the algorithm A first way to achieve this result is to provide the RW with a memory of the visited nodes For example, in [2] Avin and Britto used a biased random walk that gives priority to unvisited neighbors instead of choosing uniformly at random The strength of bias is a protocol’s parameter that can be regulated according to the required needs Since the random walk is forced to visit new parts of the network, the hitting time is reduced For a bias level equal to one, a same node is never selected again, unless this is the only option; thus, the random walk aims at behaving as a selfavoiding walk (SAW), see [4] Memory is also used in random walks with choice, RWC, proposed in [1] At each step of RWCðdÞ, instead of selecting 296 R Beraldi / Ad Hoc Networks (2009) 294–306 just one neighbor, the walk moves to the next node after examining a small number d of neighbors sampled at random Again, the random walk process is enriched with memory In our work memory is not considered Another strategy for reducing the hitting time is to exploit lookahead In this case the RW uses information about the neighbors at distance L hops from the current position For example, in the ACQUIRE protocol [18] L is a tunable lookahead parameter to combine random walks with controlled floods In our work, lookahead L ¼ is considered However, the technique we propose to reduce the hitting time is orthogonal to lookahead Still other strategies for improving the search performance are based on concurrent RWs For example, Shakkottai in [16] has analyzed different variants of random walk-based query mechanisms that include source and sink-driven sticky searches Our solution can be extended to multiple RWs as well The core methodology employed throughout this paper is borrowed from the a recent paper from Zuniga et al., see [22], where it has been used to study a push–pull mechanism for enhancing the performance of random walk-based querying in heterogeneous sensor networks Our work is also related to the aforementioned paper because both works leverage heterogeneity to improve the performance More precisely, it considers a sensor network with two kinds of nodes, i.e., normal nodes with low communication capability (low degree) and cluster head nodes with higher communication capability (high degree) When an event is generated at a node in the network, the event is forwarded to a cluster head A query for that event is implemented as a simple random walk The presence of cluster heads introduces heterogeneity in the topology; and, this reduces the hitting time considerably Although not detailed in the paper, for a reliable communication between a low and high degree node to occur, the transmission power of both nodes should be increased; thus, transmission power regulation at each node is implicity assumed in the work There are three main differences with our work First, our work assumes heterogeneity in the behavior of each node, whereas the paper assumes that a fixed subset of nodes, the cluster heads, has different communication capabilities Second, we search for an object that – unlike the event – cannot be moved from a node to another; finally, in our study the cost is not limited to the hitting time, but it also includes the average power per hit Other important applications of the RW are as a sampling technique [3] and routing [5,11,17] However, the goal of these random walks is different from ours The rest of the paper is organized as follows The next section describes the proposed protocol; Section gives a background on the used methods; Section discusses the performance results for the deterministic topologies and Section for random geometric graph Conclusions are summarized in Section the transmission power in order to modify its transmission range In particular, the nominal transmission range, R, is covered using power Pmax1 while the extended range, R0 > R, using power P max2 The neighbors of a node u vary according to the range The nodes located at distance r R are the u’s close neighbors, whereas the ones at distance R < r R0 are called the far neighbor of u The wireless link between two nodes is always bidirectional A node of the network stores uniquely identified objects, so that a node is associated with two IDs: the network address, which is a low level ID used to send a message, and a high level more abstract one, representing the stored objects For example, in the context of a SOA, o can be a software service whose ID is a verbose text file describing the service in XML The searching problem arises when node s is interested in discovering the node currently hosting a given object o To this end, s issues an asynchronous searchðObject : oÞ primitive that triggers a search for o Once the node t storing o is reached, t is in charge of notifying s that it is the owner of o For example, t can use some routing protocol to notify s or even trigger another search for s carrying the reply How the notification is actually performed is however out of the scope of our analysis Proposed protocol 2.2 Implementation description In the following, the terms device and node are used interchangeably We assume that each device can vary The key point when implementing the search primitive is how the neighbor node is selected Perhaps the most 2.1 Protocol description We consider two algorithms for the searchðÞ primitive, which are based on a random walk The first one – Algo1 – assumes that a node only knows the low level ID of the neighbors The second one – Algo2 - assumes that a node is also aware of their high level ID The pseudo-code is reported in Fig A more detailed description of how to implement the algorithms is given later in this section Random walk is performed by a searching message m carrying o The random walk terminates when the node storing the object receives m, i.e., we assume that the searched object exists In Algo1, if the node receiving the message does not store the object, then with probability q½1 À qŠ it sends the message m to a far [close] neighbor at random Algo2 exploits lookahead If an intermediate node, say i, decides to forward m via a long jump it first checks if a close or far node stores o If such is the case, i sends m to the node storing the object, otherwise it acts as in the previous algorithm We will now drill down to the implementation of the two search algorithms The key aspect is the use an efficient distributed selection protocol for the next node to visit We describe two protocol implementations The first one can be adopted for both algorithms and it is characterized by two transmission power levels, namely P max1 and Pmax2 , corresponding to the normal transmission range R and extended one, R0 ; the second protocol allows for a finer transmission power regulation, but it is only suitable for Algo1 R Beraldi / Ad Hoc Networks (2009) 294–306 Fig The random walk algorithms for the SearchðÞ primitive straightforward solution is for each node i to announce itself via a periodic beacon packet, sent at the maximum power Pmax2 , i.e., range R0 A node j classifies i either as a close or far node, according to the strength of the beacon signal received.2 Since j is aware of the ID of all the current neighbors and their distance attribute, it can easily make the right selection Beside the clear drawback of sending beacons at an adequate rate, this solution is not efficient to implement lookahead Beacons sent by the target node are in fact required to carry the high level description of objects it stores, e.g., XML files.3 Since beacons are sent at the maximum power Pmax2 the energy drain out for their transmissions can become quite high, especially when the node handles many objects We now describe two distributed selection protocols that leverage the broadcast nature of wireless transmissions and not require beacons 2.2.1 Solution with two fixed Power Levels (2PL) This protocol is suitable to implement both algorithms, Algo1 and Algo2 It is a variant of the classical RTS/CTS/DATA/ACK message exchange protocol, in which DATA is sent first to ala The received signal Sj decays as Sj ¼ kP i =dij , where dij is the distance among i and j, a a the path loss parameter and k a constant (it can be estimated when a signal S % S0 is received, S0 being the minimum value required for a correct signal detection) Hence, j classifies i as a far node if a kP i Sj > R and close otherwise Usually, a ¼ XML uses a text encoding; even the description of a simple service can reach roughly 1500 characters, e.g., see [21] 297 low, if necessary, lookahead (recall the high level ID is stored inside the packet which is going to be forwarded) The protocol works as follows The selecting node, say i, sends the packet m by a broadcast primitive, either at the power P max1 or P max2 (range R or R0 ) The packet carries a flag D, indicating wherever the far (D ¼ 1) of close (D ¼ 0) nodes are allowed to process m On receiving m, a node j first determines if it is a far or close neighbor (this is achieved as explained before) If j is a i’s close (far) neighbor and D ¼ (D ¼ 1) j can process m This consists in storing m and scheduling the transmission of a short control packet, Request To Forward (RTF), after a random delay dt With the transmission of this control packet, j candidates itself as the next node that has to forward m As soon as i hears the first control packet associated to m, say from node k, the node sends another control packet, clear to forward CTS, carrying k The aim of this new packet is to inform that the selected node is k All nodes but k delete m and de-schedule the transmission of their RTS The strength of such a solution is that it allows for the lookahead implementation If a node stores the searched object, the sends the RTS packet immediately, i.e., dt ¼ Note that when the transmission range is R0 , both close and far nodes are allowed to reply Time-outs are included to protect the above protocol against collisions For, example if no CTF/RTF are received after a given time interval, the node repeats the same actions, resp sending an RTS or m Finally, we remark that the transmission of m by k acts as an implicit ACK for i 2.2.2 Solution with full Transmission Power Control (TPC) This protocol is only suitable to implement Algo1 and it allows for transmission power control, TPC It can be used when the ID of the searched object fits a single packet TPC means that the selecting node is not constrained to set its power either to Pmax1 or P max2 ; rather, it can also tune the power level to the minimum one required for a correct packet detection Such a finer transmission power regulation allows to reduce the average power required by the whole random walk The next section analyzes such a reduction in more details The protocol is similar to the previous one, but it encompasses a simple variant of the power control MAC protocol (PCM), see [20] It works as follows The selecting node i sends a short request to send control packet (RTS), using either P max1 or P max2 (for range R and R0 , respectively) The RTS packet carries the power level P tx used for its transmission and a flag D indicating wherever far (D ¼ 1) or close (D ¼ 0) nodes are enabled to reply On receiving an RTS packet, a node j acts as follows If j is a close neighbor and D ¼ 1, then j ignores the control packet Otherwise, it schedules the transmission of a clear to send packet (CTS) at the maximum transmission power, after a random delay dt.4 The CTS packet contains the power level, P , with which i is required to send the data packet m, where Pmin ¼ PPrxtx S0 P rx is the received signal strength and S0 is the minimum power required for a correct signal detection A far node cannot receive an RTS with D = R Beraldi / Ad Hoc Networks (2009) 294–306 298 With the transmission of the CTS control packet, j candidates itself as the next node that has to forward m (this is similar to the candidature done in the previous protocol by the RTS packet) Node i sends m to the node, say k, form which the the first CTS packet is received, using the minimum power indicated in the packet.5 All nodes that hear such a transmission de-schedule their CTS; any reception of other CTS packets, if any, is ignored Node i uses the transmission of a CTS packet received from k as an implicit ACK of its previous transmission (if k is the target of m, k acknowledges i explicitly) As for the previous protocol, the critical operations are made guarded via timers 2.3 Energy consideration Let Pj be the average power required to perform a step (jump) of the walk The mean energy spent by the whole random walk before the target is reached can be written as E ¼ hDTPj , where h is the average hitting time and DT the physical time taken to perform a step, assumed equal for each step (the hitting time is independent from the powers used at each step) To compare Algo1 against Algo2 under the energy efficiency point of view we compute E ¼ hPj , namely the average power per hit The hitPhit ¼ DT ting time h will be derived in the next sections for the deterministic topologies The average power per step, P j , depends on the actual protocol implementation used and can be roughly computed as follows The energy spent in both protocols, 2PL and TCP, is dominated by the energy spent to send the data packet, m (the size of the control packets is, in fact, usually very small compared to the data packet size) We set the nominal transmission range to R ¼ and assume that no retransmissions due to collisions occur; furthermore, the transmitting power P and the maximum range r are assumed to be related through the well known decay function P $ r a , where a is the path attenuation factor Fading is not modelled in our analysis Let P1 [P ] be the average power required for making a short [long] jump Then, Pj ẳ qịP1 ỵ qP2 For the sake of simplicity, we neglect border effects, i.e., the selecting node is not at the edge of the topology As far as 2PL is concerned, for all the topologies considered in the paper, P1 ¼ and P2 ẳ R0a ; thus, Pj ẳ ỵ q R0a À 1Þ For the TPC protocol, P j is computed as follows First, the line topology does not allow for power regulation and thus P j ẳ ỵ qR0a 1ị For the grid topology, P1 ẳ 1, while P2 ẳ a p a ỵ 2Þ 2 pffiffiffi In fact, half of the far neighbors are at distance Thus, the average power per step in the grid topology is aÀ2 Pj ¼ ỵ q2a1 ỵ 1ị Finally, for the random geometric graph model, the selected node is at distance r; r ỵ drị with probability 2prdr; hence As explained in [20], the sending node may periodically raises the transmission power for a short amount of time Table Average power per step, R ¼ 1, a is path attenuation factor Topology Two power levels Transmission power control Line Grid ỵ q2a 1ị ỵ q2a 1ị ỵ q2a 1ị a2 ỵ q2a1 þ 2 À 1Þ h 0aþ2 i ðR À1Þ 2q 2ỵaị ỵ 2ỵaị R02 1ị 1 ỵ qR0a 1ị RGG P1 ẳ p Z 2prra dr ẳ aỵ2 Similarly, we have Z R0 R0aỵ2 1ị 2prra dr ẳ P2 ẳ 02 a ỵ 2ị R02 1ị pR 1ị from which Pj ẳ 2qị 2q R0aỵ2 1ị ỵ ỵ aị ỵ aÞ ðR02 À 1Þ Table summarizes the results 2.3.1 Discussion It is worth measuring the benefit of a transmission power control per se More precisely, let suppose we want to implement a natural random walk (q ¼ 0) with no lookahead and assume that RGG is a reasonable model for our network What is the energy we can save by regulating the transmission power? Since the hitting time is independent from the transmission power nodes use, the net reduction in the energy spent by the whole random walk is given by the ratio of the power per step under the TPC with the power per step under 2PL, calculated for q ¼ in the a For a ¼ the reducRGG case (see Table 1) This ratio is 2ỵa 1 tion is then and it raises to for a ¼ In the rest of this paper, we assume that Algo1 is implemented via TPC and Algo2 by 2PL Background on the used methods The hitting time huv of a random walk executed on a graph G ẳ V; Eị is defined as the expected number of steps before node v is visited starting from node u Before to present the results, we summarize the method used to derive the hitting time 3.1 Resistance method This method is borrowed from graph theory, [10], and it will be used to derive an upper bound on the hitting time for the line topology The commute time is by definition the sum C uv ¼ huv þ hvu , which represents the expected number of steps in a random walk starting from u before node v is visited and then node u is reached again Although in general huv –hvu , we restrict ourself to consider the special class of symmetric graphs, for which huv ¼ hvu and thus C uv ¼ 2huv (a graph is symmetric if it is both vertex transitive and edge transitive) The reason of using the commute time C uv is that it is related to the effective resistance r uv of the electrical network of G in which each edge represents a 1-X resistor In particular, we have: R Beraldi / Ad Hoc Networks (2009) 294–306 C uv ¼ 2jEjruv 299 a Thus, for a symmetric graph: huv ¼ jEjruv from which the problem of computing the hitting time is reduced to the computation of ruv This method is valid for a natural random walk To accommodate long jumps as a natural behavior of the walk we will connect nodes with multiple arcs If fact, the token performing the walk follows one edge at random In this way, if muv is the number of edges connecting u to v the transition probability from u to v becomes muv puv ẳ P muk k2duị where d(u) is the set of neighbors of node u The transition probability can be modified as long as the constrain muk ¼ mku is satisfied, while the above resistance result is still valid 3.2 Markov chains This method is more general than the previous one It exploits the definition of a random walk as a Markov chain, [12] Let N ¼ jVj be the number of nodes of the network and P a N  N transition matrix of a Markov chain, whose entry pij is the probability that the walk moves from node i to j Let now t be our target node and construct the matrix Q from P by removing row t and column t The matrix ðI À Q ÞÀ1 , where I is identity matrix, exists and it is called the fundamental matrix The hitting time of state t starting from i is the ith element of the column vector w w ẳ I Q ị1 where is a column vector all of whose entries are The computation of the hitting time is usually done numerically Results for deterministic topology In this section we compute the hitting time for the proposed algorithm when it is executed on three different topologies, namely a line, square grid and random geometric graph 4.1 One dimension case In this topology n ỵ wireless devices, numbered 0; 1; ; n, are deployed along a straight line, at distance R from each other while the extended transmission range is R0 ¼ 2R, see Fig Our aim is to calculate the maximum hitting time h0n as a function of q, denoted hn ðqÞ b c Fig (a) Multi edge infinite linear graph (m represents the number of multiple links); (b) elimination of multiple links; (c) elementary recurrent subgraph Consider the graph with n ỵ nodes such that i is connected to its one hop neighbors with one edge and to the two-hops neighbors with m edges, where m is such that m , see Fig 4a Unless for nodes and n À 1, the natuq ẳ mỵ1 ral random walk on such a muti-edge graph makes a long 2m m ẳ mỵ1 ẳ q and a short one with jump with probability 2mỵ2 ¼ p ¼ À q The random walk will in fact probability mỵ1 select one edge at random The probability of a long (short) m p) The same is valid for node n This means that (mỵ2 long jumps at such a node are emulated with a lower probability This error becomes negligible as n increases Let now calculate the effective resistance, see Fig We assume n ¼ 4k The multiple edges connecting 4i to 4i ỵ are eliminated so that the graph in (a) is transformed in (b), which in turns is a series of elementary subgraphs (c) The effective resistance of the graph (b) is higher than the original one Fig shows how to compute the effective resistance of an elementary graph First, we observe that the electrical network associated to the elementary graph can be drawn as in Fig 5a, where the unlabelled edges are 1-X resistors, and r is the resistance associated to the m multiple edges This resistance corresponds to m 1-X resistors in parallel, By applying the Kennelly’s Delta–Star i.e., r ¼ m1 ¼ 1Àq q transformation, the network is now transformed into Fig 5b, and from here into the one plotted in Fig 5c, which is obtained exploiting the series/parallel resistor rules It is now easy to see that ỵ rịRAE ẳ 2r ỵ 2r==2r ỵ r2 ỵ 2ị; where a==b denotes the effective resistance of resistors a and b connected in parallel We have 2r==ð2r ỵ r2 ỵ 2ị ẳ 2r2r ỵ r2 ỵ 2ị : 2r ỵ 2r ỵ r ỵ Thus, Fig An example of linear deployment of 15 nodes The transmission range can be set either to R or 2R RAE ẳ 2r 2rịr2 ỵ 2r ỵ 2ị : þ þ r ð2 þ rÞðr þ 4r þ 2Þ We have R Beraldi / Ad Hoc Networks (2009) 294–306 300 a b c Fig Computation of the equivalent resistance of the electrical network Edges with no labels are 1-X resistors n ẵmn 1ị ỵ n n2 nq ẳ ỵ 2q q2 ỵ 2q q2 2r 1q ẳ2 2ỵr 1ỵq hn qị hn qị ẳ and r ỵ 2r ỵ 2ị ỵ q2 ẳ ; r ỵ 4r ỵ 2ị q2 ỵ 2q Note that for q ¼ the analysis provides the exact hitting time of n2 Thus, the percentage reduction in the hitting time compared to a natural walk is À hnnðqÞ > À hnnðqÞ 2 from which RAE ẳ 41 qị : ỵ 2q q2 Consider now k > subgraphs in series The original graph has mðn À 1Þ þ n links, while its resistance is kRAE Then: 4.1.1 Hitting time under Algo1 Fig summarizes the search performance of random walk with no lookahead The first plot (top, left) shows the hitting time for n ¼ 12 as a function of q The hitting time first decreases with q, and then it increases again This 140 120 Hitting time reduction 0.8 Hitting time 100 80 60 Upper bound Markov 40 0.6 0.4 n=16,Bound n=16,Markov n=32,Bound n=32,Markov n=64,Bound n=64,Markov 0.2 20 0 0.2 0.4 0.6 0.8 0 0.2 0.4 Jumping probability 0.6 0.8 Jumping proabability 140 q=0 q=0.2 q=0.5 q=0.7 q=0.9 120 Hitting time 100 80 60 40 20 0 10 12 Starting point Fig Performance of Algo1 Absolute hitting time for 13 nodes as a function of q (top, left), relative reduction in the hitting time as a function of q (top, right); effect of the initial position (bottom) R Beraldi / Ad Hoc Networks (2009) 294–306 is due to the fact that when the packet falls in an odd nodes,i.e., it occupies an even node and then makes a short jump, it cannot reach the target unless it makes a short jump again The value of q regulates the frequency of such events – a switch from odd to even nodes and vice versa occurs with probability À q Once the walk falls into an odd node, the expected number of steps before the token returns back to an even position is 1=ð1 À qÞ, a penalty on the hitting time that increases with q On the other hand, having q high is beneficial because the random walk moves faster towards to target The combined effect explains the variation of the plot The relative reduction as a function of q is given in the next plot (top, right) The line size is a parameter The analysis now provides a lower bound For q ¼ the probability to fall into an odd node is zero For any value of q ¼ À  such a probability is however higher than zero, affecting the hitting time negatively This is more evident in this plot, in which a discontinuity at q ¼ is visible The lost plot of Fig (bottom) shows the hitting time as a function of the starting point of the walk Results are obtained using Markov chains One can wondering whenever making long jumps is always beneficial regardless from where the walk starts The plot answer to this question As expected the highest reduction in the hitting time due to long jumps is obtained when the source is far from the 301 target since the walker move ‘‘faster” in the line The reduction decreases as the walk starts nearby the target In this case in fact long jumps also helps the walk to move away from the target This behavior is clearly visible for q ¼ 0:2 However, as q is further increased, starting from nearby nodes can even be worst than avoiding long jumps Odd positions are particularly penalized since the target cannot be reached before a short jump is made, an event that happens with probability À q For example, starting from node 11 and setting q ¼ 0:9 provides the highest hitting time 4.1.2 Hitting time under Algo2 Let now consider the case when lookahead is in place The presence of lookahead reduces the actual length of the line The target is in fact reached the one step after the random walk visits node n À Also, with probability q, the walk moves from node n À to the target We can then exploit the previous bound on the maximum hitting time by simply reducing the line length of one unit The exact values are obtained via Markov chains Fig summarizes the results The first plot (top, left) shows the hitting time as a function of q Lookahead avoids that the walk remains stacked visiting odd nodes The hitting time thus always decreases with q In particular, for q ¼ 1, the walker runs over points and it hits the target 140 0.8 Markov Upper bound Hitting time reduction 120 Hitting time 100 80 60 0.6 0.4 40 0.2 n=16 n=32 n=64 20 0 0.2 0.4 0.6 0.8 0 0.2 0.4 Jumping probability 120 0.8 q=0 q=0.2 q=0.5 q=0.7 q=0.9 100 Hitting time 0.6 Jumping proabability 80 60 40 20 0 10 12 Starting point Fig Performance of Algo2 Absolute hitting time for 13 nodes as a function of q (top, left), relative reduction in the hitting time as a function of q (top, right); effect of the initial position (bottom) R Beraldi / Ad Hoc Networks (2009) 294–306 302 300 Algo1 Algo2 200 150 100 0.6 0.4 0.2 50 0 X 6, 10 X 10 20 X 20 X 6, Lookaheaed 10 X 10, Lookaheaed 20 X 20, Lookaheaed 0.8 Hitting time reduction Average power per hit 250 0.2 0.4 0.6 0.8 0 0.2 Fig Average power per hit, for the line topology; 13 nodes, a ¼ one step after it has reached point Thus, the hitting time becomes 26ẳ 52 ỵ 1ị The improvement over the natural random walk is displayed in the next plot (top, right) for different line sizes Lookahead also allows the hitting time to be reduced regardless the initial starting point, see the last plot (bottom) 4.1.3 Power Fig shows the average total power per hit as a function of the jumping probability, for the two algorithms The attenuation factor is a ¼ and the number of edges is n ¼ 12 Such a power is then equal to hqị1 ỵ 3qị, where hðqÞ is the hitting time for the jumping probability q (see Table 1) In Algo1 the power increases with q whereas in Algo2 it decreases This means that for Algo2 the reduction in the hitting time completely compensates the higher cost per step of long jumps Note that for q ¼ and n an even num2 ber, the hitting time of Algo1 is n2 ị2 ẳ n4 On the other hand, the average power per step is Thus, the total power spent making always long jumps (q ¼ 1) is the same of the total power spent making always short jumps (q ¼ 0), i.e., n2 However, as soon as q becomes less than the hitting time increases and, consequently, the average power increases too 4.2 Two dimension case To derive the hitting time for the grid topology, we exploit the Markov chain approach, [15] To this end we need to define the transition probability matrix Points of the grid are assigned coordinates w.r.t a Cartesian axis, with origin at the bottom-leftmost point of the grid Let s ẳ i; jị be a grid point and ks1 ; s2 k ¼ ji1 À i2 j ỵ jj1 j2 j the distance between s1 and s2 They are: > < ð1 À qị=d1 s1 ị ks1 ; s2 k ẳ ps1 s2 ẳ q=d2 s1 ị 1ị ks1 ; s2 k ¼ > : otherwise where di ðuÞ is the number of u’s i-hops neighbors For Algo2 they also include: 0.4 0.6 0.8 Jumping probability Jumping probability Fig Reduction in the hitting time as a function of the jumping probability, different L  L grids; source placed at ð1; L=2Þ target at ðL À 1; L=2Þ ps1 s2 ¼  q ks1 ; s2 k ¼ 1; s2 ¼ t ks1 ; s2 k ¼ 2; s2 ẳ t 2ị This approach is also used to obtain the exact numerical results for the line topology 4.2.1 Hitting time under Algo1 and Algo2 Fig shows the hitting time percentage reduction w.r.t a natural random walk (q ¼ 0) as a function of the jumping probability for the two algorithms, for different grid sizes L A L L grid contains L ỵ 1ị2 nodes The source is placed at ð1; L=2Þ while the target to ðL À 1; L=2Þ With no lookahead the improvement increases until roughly q % 0:8 and then it decreases The reason is that since the distance between the source and the destination is L À 2, i.e., an even number, the walk needs to perform an even number of short jumps for the target being reached And, the probability of such an event decreases with q On the other hand, long jumps alone reduces the hitting time because nodes at odd distance from the target are not visited, i.e., the searching space is halved The combination of these two effects explains the presence of a maximum in the plot Note that for q ¼ 1, nodes at an odd distance from the target are never visited, i.e., the searching space is halved The advantage of using lookahead is that the target is reached as soon as one of its four one hop neighbor is reached (one hop lookahead) Moreover, depending on q the random walk can sometimes even exploit two-hops lookahead (recall that when a two-hops target’s neighbor performs a long jump, the packet is delivered to the target) Thus, the lookahead capability increases with q As the grid size increases, lookahead becomes less effective The number of target’s neighbors will in fact be lower compared to the total number of nodes in the network This explain why the improvement decreases as L increases As L ! the lookahead advantage becomes negligible, so that the hitting time reduction is the same as Algo-1 Fig 10 shows the hitting time as a function of the starting point x, for L ¼ 10, the target placed at ðL À 1; L=2Þ and the source at ðx; L=2Þ, q given as a parameter For a given q, the hitting time decreases with the starting point for both R Beraldi / Ad Hoc Networks (2009) 294–306 350 303 250 300 q=0 q=0.2 q=0.5 q=0.8 q=0 q=0.2 q=0.5 q=0.8 200 Hitting time Hitting time 250 200 150 150 100 100 50 50 0 Starting point Starting point Fig 10 Hitting time as a function of the starting point Target at (L À 1; L=2), Initial point at x; L=2ị L ẳ 10 Algo1 (left), Algo2 (right) 600 R are connected via a short range wireless link, while two points at distance r, R < r < R0 are connected through a long range link A short [long] wireless link is used with probability À q½qŠ Without loss of generality, to calculate R0 we set R ¼ The expected length of a short range link and then of a short jump is Algo1 Algo2 Average Power per hit 500 400 300  n¼ p 200 100 0 0.2 0.4 0.6 0.8 Jumping probability Fig 11 Average power per hit Source at ð1; L=2Þ, target at L 1; L=2ị, a ẳ protocols As expected, this reduction is however much stronger in Algo1 (left plot) than Algo2 (right) Moreover, in Algo1 when starting very close to the target, i.e., x ¼ 8, an increase in q is beneficial only until a given value For example, the left side plot shows how the hitting time for q ¼ 0:8 is higher than for q ¼ 0:2 4.2.2 Power Fig 11 shows the average power per hit for a 10  10 grid, starting point (1,5), target point (9,5) and a ¼ Algo2 provides the lowest power per hit; its average power is almost constant The reduction in the hitting time then varies with q as ỵ 2qị1 (see Table 1) Thus, we can reduce the hitting time by keeping the total power constant However, for a > the power will increase with the jumping probability Simulation results for the random geometric graph The final set of results are given for a random geometric graph For this kind of topology we have used simulations A given number of points, n, are drawn at random inside a square region of edge L Any two points at distance at most Z 2pr2 dr ¼ while the one of a long range link is Z R0 ðR03 À 1ị n0 ẳ 2pr2 dr ẳ : 02 R02 1ị pR 1ị by setting n0 ẳ 2n we get R0 % 1:613 If we are observing the walk at a random instant of time, then its distance, given that a short jump occurs, varies of 2/3, whereas under a long jumps this variation is twice, i.e., 4/3 5.1 Results We have studied Algo1 and Algo2 by lunching 1000 independent runs of the algorithms over 300 nodes, qffiffiffiffiffiffi scat(node tered at random inside a square of edge L ¼ 300 density 5) The nominal transmission range of each node is R ¼ 1; The source and the destination are placed at height L=2 and distance from the left and right borders, respectively The extended range is R0 The average power per hit is estimated as the ratio of the total power used during all the walks with the number of random walks Algo1 uses the TPC protocol, while Algo2 2PL The path attenuator factor is a ¼ The evaluation assumes an ideal transmission system with no collisions 5.1.1 Extending the nominal transmission range The first set of experiments are obtained for R0 ¼ 1:613 Fig 12 reports the random walk performance as a function of q for Algo1 (left) and Algo2 (right) The innermost graph is the average power per step estimated during the simulations, which matches the value of analysis (see Table 1) The other internal plot reports the total power per hit In Algo1 the hitting time decreases only due to the beneficial R Beraldi / Ad Hoc Networks (2009) 294–306 200 200 Average power per hit 500 400 Hitting time 600 300 500 Average power per hit Hitting time 150 200 100 400 150 2.5 1.5 0.5 Anal Sim 0.2 0.4 0.6 0.8 100 50 100 0 300 Average power per step 304 0.2 0.4 0.6 0.8 1.5 50 200 0.5 100 Anal Sim 0 0.2 0.4 0.6 0.8 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 0.2 0.4 Jumping probability 0.6 0.8 Jumping probability Fig 12 Performance for RGG, R ¼ 1; R0 ¼ 1:613; Algo1/TPC (left), Algo2/2PL (right) effect of mixing short with long jumps, whereas in Algo2 an increase in the frequency of long jumps also increases the lookahead advantage In the limit case of q ¼ 1, for the target to be reached the walk has to visit any node in the circle of radius R0 And this area is % 2:6 times higher than the normal lookahead area In Algo1 a reduction in the hitting time is seen since a small frequency of long jumps, but the hitting time remains unvaried when the frequency of long jumps is further increased The reduction in the hitting time is however not sufficient to reduce the total power, which then always increases with q A reasonable compromise between the increase in the power requirements and the reduction in the hitting time, appears to be q ¼ 0:2 With this value, the hitting time is almost 20% lower than the natural walk’s one while the power also increases of a same amount Algo2 performs much better than Algo1, the reason being the lookahead The best search performance are for q ¼ Note, however, that lookahead requires to send the data packet without any protection against collisions; thus, if a collision occurs at the target the lookahead capability is lost (in our study collisions are not taken into account) A more serious drawback due to frequent long jumps, is a po- 5.1.2 Keeping the nominal transmission range constant Our protocol can also be seen under another point of view Assume that for some reason we can’t increase the nominal transmission range, e.g., simply because our devices already operate at their maximum allowable power level or for avoiding the deficiencies outlined above We can still use our protocol by defining an inner circle of radius R0 < R, such that R is 1.613 times longer than R0 , i.e., the nominal range plays now the role of the extended range For R ¼ 1, we have R0 ¼ 0:62 The walker makes a double jump when it peaks a node inside the ring from R0 to R, whereas a short jump occurs when the selection is restricted to those nodes inside R0 Clearly, this protocol never behaves like the original random walk algorithm, because this equivalence occurs for R0 > R and q ¼ However, a reasonable question aries, i.e., is such a strategy somehow convenient w.r.t the classical implementation of the random walk? Fig 13 shows the hitting time of Algo1 (left) and its average power per hit (right) The performance of the 300 600 280 550 260 500 450 400 350 300 0.2 0.8 Average power per step Algo1 Natural RW,R=1 Avereage power per hit Hitting time 650 tential reduction in the network throughput During a long jump all nodes inside the ring from R to R0 have to defer their transmissions; and, this area is R2 À % 1:6 larger than the area covered with the nominal transmission range 240 Algo1 Natural RW Analysis Simulation 0.7 0.6 0.5 0.4 0.3 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.6 220 200 180 0.3 0.4 0.5 0.6 0.7 Jumping probability 0.8 0.9 160 0.2 0.4 0.5 0.7 0.8 0.9 Jumping probability Fig 13 Performance of Algo1/TPC under RGG as a function of q, nominal transmission range used as extended range; hitting time (left), average power per hit (right) R Beraldi / Ad Hoc Networks (2009) 294–306 400 305 200 Algo2 Natural RW Algo2 Natural RW 350 180 250 200 150 100 160 140 120 50 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Jumping probability Average power per step Avereage power per hit Hitting time 300 100 0.2 1.1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Analysis Simulation 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Jumping probability Fig 14 Performance of Algo2/2PL for RGG as a function of q, nominal transmission range used as extended range; hitting time (left), average power per hit (right) natural RW protocol (random walk), which picks nodes inside the whole nominal range at random, is also included in the plot The hitting time highly decreases with q (for q % 0:4 is has the same value of the natural random walk) and it reaches a minimum value for q ¼ 0:8 This can be explained saying that when Algo1 operates with these values for the internal and extended ranges, the reduction in the correlation among subsequent selections, due to probabilistically mixing short with long jumps, is stronger (this reduction was in fact not observed in Fig 12) What is also interesting to observe is that for q ¼ 0:8 the total power required by the protocol is the same amount of the natural random walk Thus, with a high frequency of long jumps, we are now able to reduce the hitting time without any additional cost in terms of power The inner plot of Fig 13 shows the average power per step as a function of q The value increases with q, and matches the one predicted by our previous analysis.6 Finally, Fig 14 shows the performance of Algo2 Short jumps attenuate the effect of lookahead; thus, for q small the hitting time is quite higher than the natural random walk For q ¼ 0:6, the hitting time reaches the same value of the natural walk, but a smaller amount of power is required As expected, the lowest value for hitting time is when q ¼ By making always long jumps the lookahead region is not reduced However, for q ¼ 0:8 a slightly less power is required Conclusion This paper has studied the search performs of a random walk-based algorithm for wireless networks in which the walker may decide to perform longer jumps In a classical random walk, the selection of the next node always occurs among all the neighbors, whereas in our proposal, the set of candidate nodes is made more variable Such a variability is a form of heterogeneity, which aims at reducing the correlation among selections of nodes visited A simple normalization due to the assumption that the outmost radius is now unitary is required by the walk, and it is obtained by regulating the transmission power We have proposed two algorithms, Algo1 and Algo2, characterized by the lack of presence of lookahead and suggested two implementations, suitable for searching objects with long or short descriptive IDs In particular, the lack of lookahead allows for transmission power regulation, an important feature when energy is an issue We have analyzed the hitting time and energy efficiency (power per hit) for three different topologies, namely, line, grid and geometric random graph For the line topology, Algo1 gets the lowest hitting time when the frequency of long jumps is fairly high (% 0:8 in our experiments) whereas for Algo2 the lowest hitting time was observed when the walker always makes long jumps, i.e., q ¼ Moreover, when the strength of the received signal decays quadratically, the power required by Algo1 increases with q, while in Algo2 it is constant In the grid topology the two algorithms followed a similar behavior As far the RGG is concerned, we have experimented two different modes of operations for the proposed protocols In the first one, the extended range is actually higher than the nominal one In this case for q as low as 0.2, Algo1 exhibited an appreciable reduction in the hitting time w.r.t the natural random walk, at the cost of a moderate increase in the power requirement Due to the huge lookahead advantage, Algo2 got the lowest hitting time and the lowest power for q ¼ Clearly, with a high frequency of long jumps the interference with other nodes, and the potential reduction in the overall network throughput, become an issue This reason has motivated our second operating mode, which doesn’t require any jump beyond the nominal transmission range The simulation results have shown how the hitting time for Algo1 is now reduced when long jumps often occur A reduction of almost 20% was measured for q ¼ 0:8, but without any increase in the power As far as Algo2 is concerned the lowest hitting time was measured again for q ¼ 1; however, the lowest power was observed for q ¼ 0:8 Thus, 80% of long jumps are now a good compromise for both protocols 306 R Beraldi / Ad Hoc Networks (2009) 294–306 Acknowledgement This work was supported by the ReSIST European Network of Excellence (Grant No 026764) [12] [13] References [14] [1] C Avin, B Krishnamachari, The power of choice in random walks: an empirical study, in: 9th ACM/IEEE International Symposium on Modeling, Analysis and Simulation of Wireless and Mobile Systems, (MSWiM), Malaga, Spain, October 2006 [2] C Avin, C Brito, Efficient and robust query processing in dynamic environments using random walk techniques, in: Proc of the third international symposium on Information processing in sensor networks, 2004, p 277286 [3] Z Bar-Yossef, R Friedoman, G Kliot, RaWMS – random walk based lightweight membership service for wireless ad hoc networks, in: Proc MobiHoc, 2006, pp 238–249 [4] Eric W, Self-avoiding walk, From MathWorld – A Wolfram Web Resource, [5] C Gkantisidis, M Mihall, A Saberi, Random walks in peer-to-peer networks: algorithms and evaluation, Performance Evaluation 63 (2006) 241–263 [6] J Kopena et al., Service-based computing on manets: enabling dynamic interoperability of first responders, IEEE Intelligent Systems, September/October 2005, in: 38th Annual Simulation Symposium, San Diego, California, USA, April 2005 [7] M Gerla, C Lindemann, A Rowstron, Perspectives workshop: peer-topeer mobile ad hoc networks – new research issues, in: Dagstuhl Seminar Proceedings, 2005 [8] G Lau, M Jaseemuddin, G.M Ravindran, RAON: a P2P network for MANET, in: Wireless and Optical Communications Networks, 2005, WOCN 2005, Second IFIP International Conference on 6–8 March, 2005 [9] X Li, J Wu, Searching Techniques in Peer-to-Peer Networks, Handbook of Theoretical and Algorithmic Aspects of Sensor, Auerbach Publications, 2006 [10] L Lovazs, Random walks on graphs: a survey, Combinatorics, Paul Erdos in Eighty, vol 2, J’anos Bolyai Mathematical Society Budapest, 1993 [11] M Günes, O Spaniol, Ant-routing-algorithm for mobile multi-hop ad-hoc networks, in: Proceedings of the 2002 ICPP Workshop on Ad [15] [16] [17] [18] [19] [20] [21] [22] Hoc Networks (IWAHN 2002), IEE Computer Society Press, 2002, pp 79–85 August J.R Norris, Markov Chains, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge, 1998 U Kozat, L Tassiulas, Network layer support for service discovery in mobile ad hoc networks, in: Proceedings of IEEE INFOCOM03, San Francisco, 2003 C Perkins, E Belding-Royer, S Das, Ad hoc On-Demand Distance Vector (AODV) Routing, RFC 3561, July 2003, http://www.ietf.org/ rfc/rfc3561.txt Sheldon M Ross, Introduction to Probability Models, Academic Press Inc., 1993 S.Shakkottai, Asymptotics of query strategies over a sensor network, in: Proceedings of IEEE Infocom, Hong Kong, March 2004 S.D Servetto, G Barrenechea, Constrained random walks on random graphs: routing algorithms for large scale wireless sensor networks, in: Proceedings of the First ACM WSNA, 2002 N Sadagopan, B Krishnamachari, A Helmy, Active query forwarding in sensor networks (acquire), Elsevier Journal on Ad Hoc Networks (2003) Santpal S Dhillon, P Van Mieghem, Comparison of random walk strategies for ad hoc networks, in: The Sixth Annual Mediterranean Ad Hoc Networking WorkShop, Corfu, Greece, June 12–15, 2007 E Jung, N Vaidya, A power control MAC protocol for ad-hoc networks, in: Proc of MOBICOM, 2002 M.Zuniga, C Avin, B Krishnamachari, Using heterogeneity to enhance random walk-based queries, USC Computer Engineering Technical Report CENG-2006-8, August 2006 Roberto Beraldi received the Laurea degree in computer science from the University of Calabria, Italy, in 1991, and the Ph.D degree in computer science in 1996 He is currently an assistant professor at the Department of Systems and Computer Science (DIS) at University ‘‘La Sapienza”, Rome His research interests include distributed systems and dynamic networks  ... has studied the search performs of a random walk- based algorithm for wireless networks in which the walker may decide to perform longer jumps In a classical random walk, the selection of the... networks (acquire), Elsevier Journal on Ad Hoc Networks (2003) Santpal S Dhillon, P Van Mieghem, Comparison of random walk strategies for ad hoc networks, in: The Sixth Annual Mediterranean Ad. .. valid for a natural random walk To accommodate long jumps as a natural behavior of the walk we will connect nodes with multiple arcs If fact, the token performing the walk follows one edge at random