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500 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 4, APRIL 2009 Biased Random Walks in Uniform Wireless Networks Roberto Beraldi Abstract—A recurrent problem when designing distributed applications is to search for a node with known property. File searching in peer-to-peer (P2P) applications, resource discovery in service-oriented architectures (SOAs), and path discovery in routing can all be cast as a search problem. Random walk-based search algorithms are often suggested for tackling the search problem, especially in very dynamic systems-like mobile wireless networks. The cost and the effectiveness of a random walk-based search algorithm are measured by the excepted number of transmissions required before hitting the target. Hence, to have a low hitting time is a critical goal. This paper studies the effect of biasing random walk toward the target on the hitting time. For a walk running over a network with uniform node distribution, a simple upper bound that connects the hitting time to the bias level is obtained. The key result is that even a modest bias level is able to reduce the hitting time significantly. This paper also proposes a search protocol for mobile wireless networks, whose results are interpreted in the light of the theoretical study. The proposed solution is for unstructured wireless mobile networks. Index Terms—Algorithm/protocol design and analysis, random walks, mobile ad hoc networks, search algorithms. Ç 1 INTRODUCTION C ONTEXT of this study. To search for a node with known property is a basic recurrent problem arising in many distributed applications. For example, in routing protocols Compared to flooding, a random walk search has a more fine-grained control of the search space, a higher adap- tiveness to termination conditions, and can naturally cope for mobile wireless networks, e.g., [12] and [24], the searched with failures or voluntary disconnections of nodes [22]. node is identified by its IP address, while in peer-to-peer Examples of concrete exploitations of random walks in (P2P) architectures by a key, associated to the object the node wireless networks are found in the context of routing stores [19], [16]. Searching is also a central functionality in a protocols for MANET—e.g., ANT [23], Hint-Based routing service-oriented architecture (SOA), e.g., see [13]. With their [6], and most recently, in P2P over MANET, e.g., ROAN shift toward wireless communication support, distributed [14]. Graph theoretical studies on random walks that are systems are becoming more dynamic, and the search problem is consequently becoming even more challenging. This paper relevant for wireless networks have also recently appeared focuses on the search problem in the context of mobile in the literature, e.g., [5] and [2]. wireless networks, i.e., autonomous self-organizing net- Biased random walks are random walks in which nodes works composed of a set of wireless devices. have statistical preference to forward the walker toward the Broadly speaking, there are two approaches to face a target. The clear advantage of a biased random walk is that search problem: structured and unstructured. The former it reduces the excepted number of steps before the target is exploits a logical structure for guiding searches, e.g., reached, called the hitting time, significatively. However, routing tables stored at nodes, Distributed Hash Tables the bias level achievable in a real setting is limited, while (DHTs), or centralized/distributed directories, while the the implementation of any biasing mechanism comes at latter does not leverage any logical organization in the some additional cost. Thus, to understand the effect of bias search space. To maintain the structure used to support a on the hitting time is an important preliminary step for search may become challenging in mobile networks since deciding the practical benefit of a random walk-based the mobility of nodes makes the topology of the network search algorithm. The effect of bias on the hitting time when also variable. For this reason, the unstructured approach is the random walk is executed over a wireless network is the regarded as an attractive alternative, as more deeply subject of this paper. discussed, for example, in [20]. Contributions of this work. The random walk we An unstructured search has to potentially explore the consider exploits look-ahead one. The walker (a packet) is whole network; as such, it is generally carried out by forwarded from one node to a randomly chosen neighbor flooding. Alternatively, random walks can be used. until a neighbor of the target is found. To study the effect of bias analytically, we exploit a network model composed of . The author is with the Dipartimento di Informatica e Sistemistica, infinitely many nodes located at uniformly random posi- ` Universita di Roma “La Sapienza,” Via Ariosto 25, 00100 Rome, Italy. tions inside a circle. This model is useful to study the effect E-mail: beraldi@dis.uniroma1.it. of bias on a real network because during the lifetime of a Manuscript received 17 Oct. 2007; revised 30 May 2008; accepted 29 Sept. walk the variation in the network topology can, at first 2008; published online 16 Oct. 2008. glance, be neglected. For information on obtaining reprints of this article, please send e-mail to: tmc@computer.org, and reference IEEECS Log Number TMC-2007-10-0315. The contribution of this paper is twofold. First, we found Digital Object Identifier no. 10.1109/TMC.2008.151. that for the uniform model the hitting time is very sensible 1536-1233/09/$25.00 ß 2009 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on September 4, 2009 at 13:40 from IEEE Xplore. Restrictions apply. BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS 501 Fig. 2. A random walk with bias over [0, 6]. Definition 4. Gain. The gain g ði; jÞ of a -biased random walk is the percentage reduction of the hitting time of j starting from i with respect to the same hitting time for the h ði;jÞÀh ði;jÞ unbiased random walk, namely, g ði; jÞ ¼ 0:5 h0:5 ði;jÞ . Definition 5. Return time. The return time is the expected number of steps in a random walk starting at a given node, before the same node is reached again. Fig. 1. List of the main parameters and symbols used throughout this Let us start by considering a -biased random over the finite paper. line ½0::N. Without loss of generality, let N be the target point. At each time step, the walker moves closer to the to bias. In particular, we show that the hitting time is target with probability and away from the target with conveniently upper bonded by N 1= , where N is propor- probability 1 À . The extreme points are reflective barriers, tional to the radius of the circle and , 0:5 1, is a see Fig. 2. It is well known, e.g., [17], that for the unbiased number expressing the bias level of the walk. case we have h0:5 ð0; NÞ ¼ N 2 while for ¼ 1 it is trivial to Second, we present a random walk-based search see that h1 ð0; NÞ ¼ N. Some obvious question then arises. protocol designed for mobile networks that runs atop According to which law the hitting time passes from the the data link layer. The random walk is biased via a quadratic to the linear behavior? Is there some simple rule simple distance estimation mechanism, which generates a of thumb that can be useful to characterize the maximum bias % 0:6 when the search is for a single target. We give gain one can achieve under a given bias ? We now will experimental evidence of the validity of our theoretical give an answer to these questions. study by simulating nodes moving inside a circle. Then, To compute the hitting h ð0; NÞ, we follow the method we present a simulation study for a more general setting, illustrated in [17]. The hitting time can be written as whose results can be interpreted in the light of the theoretical expectations. X N Structure of this paper. The reminder of this paper is h ð0; NÞ ¼ h ðk; k À 1Þ; ð1Þ k¼1 organized as follows: Section 2 studies the bias effect in one dimension, while Section 3 considers a 2D circular deploy- while the hitting time h ðk; k À 1Þ is one less than the expected ment; Section 4 discusses the possible options for imple- return time at position k of a -biased random walk over the segment ½0::k. The return time is the inverse of the stationary menting a biased random walk; the proposed protocol is probability of observing the walker at position k, see [17]; presented in Section 5 and the simulation results are given thus, if i;k is the stationary probability that a walker over in Section 6. Section 7 discusses the related work; conclu- ½0; k is observed at position i, we have sions are finally given in Section 8. h ðk; k À 1Þ ¼ À1 À 1 k;k 2 EFFECT OF BIAS ON A RANDOM WALK, and SIMPLE CASE Xh N i A summary of the main symbols used throughout this ht ð0; NÞ ¼ 1 þ À1 À 1 : k;k paper is given in Fig. 1, while the definitions of the key k¼2 terms are now given. The probability k;k is readily obtained by solving an Definition 1. Bias level. A random walk has bias level with elementary Discrete Time Markov Chain (DTMC) with state respect to a given target, and it is called -biased random walk, space ½0; k and transition probabilities if after a step the walker gets closer to the target with pi;iþ1 ¼ ; pi;iÀ1 ¼ 1 À 1 i k À 1; probability . p0;1 ¼ pk;kÀ1 ¼ 1: Definition 2. Unbiased random walk. A random walk with bias level ¼ 0:5 is said to be unbiased. From the balance equation of such a DMTC, we can write Definition 3. Hitting time. The hitting time h ði; jÞ of a -biased 1 À kÀ1 random walk is the expected number of steps before node j is À1 k;k ¼ À1 0;k visited for the first time starting from node i. Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on September 4, 2009 at 13:40 from IEEE Xplore. Restrictions apply. 502 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 4, APRIL 2009 Fig. 3. Hitting time of a biased random walk over N ¼ 9, exact value and upper bound: (a) hitting time and (b) gain. and and then ht ð0; NÞ is closer to N 2 , while for N > N 0 , kÀ1 ht ð0; NÞ is closer to N=ð2 À 1Þ. For example, for 0 ¼ 0:505, kÀ1 1 À ð1ÀÞkÀ1 N 0 ¼ 100. The value N 0 is always finite; thus, eventually À1 0;k ¼1þ þ ; hitting time behaves like N=ð2 À 1Þ. This last aspect has the ð1 À ÞkÀ1 2 À 1 following physical interpretation. hence, Consider an infinite line and a particle (walker) kÀ1 starting at point 0 and moving along the line, stepping ð1 À ÞkÀ1 ð1ÀÞkÀ1 À 1 of one unit to the right with probability ! 0:5 and with ¼1þ À1 k;k þ : kÀ1 2 À 1 probability 1 À to the left, i.e., the particle is subject to By substituting the above probability into (2), we obtain an external field that pushes it right. On the average, the particle arrives at point N after the hitting time, ht ð0; NÞ. k The ratio N=ht ð0; NÞ can then be interpreted as the X N À1 ð1 À Þk ð1ÀÞk À 1 ht ð0; NÞ ¼ 1 þ þ average drift speed in the direction of the field. k¼1 k 2 À 1 Now, using the function N=ð2 À 1Þ of Property 1, we can that after some manipulation can be rewritten as see that the speed becomes 2 À 1. This value, 2 À 1, is interpreted as the characteristic speed, which is acquired by " # N À1 À 1 2 1 À NÀ1 the particle when it is observed at a distance sufficiently far ht ð0; NÞ ¼ 1 þ þ2 À1 : ð2Þ from the origin (for N sufficiently high). When becomes 2 À 1 2 À 1 closer and closer to 0.5, in order to observe the “character” of the walker we need to go farther and farther from the origin. 2.1 Discussion With this interpretation, the speed of an unbiased particle Let us now summarize the properties of the biased is 1=N. That is, the speed goes to 0 as N goes to infinity. In random walk over the segment in terms of hitting time other terms, for ¼ 0:5, the random walk is unbiased and and gain. there is no drift in the direction of the external field. Property 1 (Character of the hitting time). ht ð0; NÞ ¼ Property 2 (Upper bound on the hitting time). Âð2À1Þ, for such that 0:5 < 1.1 N ht ð0; NÞ N 1= , for such that 0:5 1 and N > N0 , Proof. In fact, for 0:5 < 1, the fraction 1À < 1; thus, for where N0 is a constant. N ! 1, the third term of (2) tends to be a constant. t u Proof. For 0:5 < < 1, N 1= increases with N more rapidly Under a formal point of view, the hitting time ht ð0; NÞ is than 2À1 does; thus, N 1= > 2À1 , provided that N > N 0 , N N lower than the approximating values N=ð2 À 1Þ as well as for some N 0 > 0. But, from Property 1 the hitting time, the unbiased value, N 2 . The last term in (2) is in fact N for 0:5 < 1, indeed grows as 2À1 , while for ¼ 0:5 negative. Now, the bounding function and the correct hitting time are N=ð2 À 1Þ > N 2 the same [17]. Since N0 can be derived from N 0 , the property holds. u t until N reaches the value N 0 ¼ 1=ð20 À 1Þ. This is a The constant value N0 is derived numerically and is crossover point. For N < N 0 , N0 ¼ 5. As an example, Fig. 3a reports ht ð0; 9Þ and the N=ð2 À 1Þ À HT > N 2 À ht ð0; NÞ bounding function as is varied. 1À2 Property 3 (Lower bound on the gain). g ð0; NÞ ! 1 À N , 1. Recall that f ¼ OðgÞ if and only if lim supt!1 jfðtÞj=gðtÞ ¼ k, for some for such that 0:5 1 and N > N0 , where N0 is a k ! 0, and f ¼ ÂðgÞ if f ¼ OðgÞ and g ¼ OðfÞ. constant. Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on September 4, 2009 at 13:40 from IEEE Xplore. Restrictions apply. BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS 503 Proof. The property comes from the definition of the gain and Property 2. u t The results for N ¼ 9 are reported in Fig. 3b. A bias as small as ¼ 0:6 reduces the hitting time by approximatively a half. Property 4 (The minimum gain is obtained when starting from the farthest point). g ð0; NÞ < g ð0; iÞ, for i such that 0 < i < N. Proof. We have that h ði; NÞ ¼ h ð0; NÞ À h ð0; iÞ, e.g., see [17]. Since h0:5 ði; NÞ ¼ N 2 À i2 , we have to show that 1 À h ð0;NÞ h ð0;NÞÀh ð0;iÞ N2 < 1 À N 2 Ài2 . i2 h ði;0Þ The inequality can be rewritten as 1 À N 2 > 1 À hðN;0Þ , h ði;0Þ h ðN;0Þ which holds if and only if i2 > N 2 . But, h ð0; NÞ is given by (2), which varies linearly with N; thus, the h ð0;NÞ function fðNÞ ¼ N 2 decreases with N and then the property holds. u t Fig. 4. An example of a random walk on the symmetric setting we are From Properties 3 and 4, we can derive the following studying. lemma: The random walk studied in the previous section is a Lemma. The gain of random walk with bias , which starts from 1À2 particular case of our geometric random walk; in fact, it is an arbitrary point of a segment ½0; N, is at least 1 À N , obtained for RÃ ¼ NR and discrete steps of length R. Thus, provided that N > N0 ¼ 5. we can hope that the hitting time is also subject to a behavior similar to the one summarized by the properties given in the 3 EFFECT OF THE BIAS ON A UNIFORM WIRELESS previous section. To see if such is the case, we now compute NETWORK the exact hitting time numerically and will then try to find a In this section, we study the effect of bias on two dimensions. connection between the two random walks. To study this problem analytically, we examine a circular area Let ri be a random variable representing the distance of of radius RÃ , which contains infinitely many nodes deployed the walker at the ith step and fi ðrÞ its probability density at random, i.e., the probability that a node occupies the function (pdf). The target is reached when ri becomes zero, infinitesimal squared area centered at ðx; yÞ with respect to a i.e., the target is point 0. After hitting the target, the walk 1 pair of Cartesian axis is RÃ2 dxdy. The target node occupies the remains alive but its distance does not change. This means center of the circle. The transmission range of each node is R. that fi ð0Þ increases monotonically with i and f1 ð0Þ ¼ 1 (the The neighbors of a node are the nodes located at distance at walker eventually hits the target). most R from itself. Nodes have no physical extension; thus, in The probability that the distance becomes zero the first the following the terms point and node are used inter- time at the ith step is given by fiþ1 ð0Þ À fi ð0Þ; thus, the changeably. hitting time of point 0 starting from r0 is The random walk we study exploits look-ahead one, i.e., X the neighbors of the target will send the walker directly to hðr0 ; 0Þ ¼ i½fiþ1 ð0Þ À fi ð0Þ: the target rather than making a random choice. Lookahead i!0 is simple to implement in wireless transmissions (as The functions fi ðÞ can be computed through the detailed in our proposal) while it makes easier to compute following recursive relationship: the hitting time mathematically. Due to the circular symmetry, the random walk over the ZþR r circle induces a geometric random walk on the finite fi ðrÞ ¼ fiÀ1 ðr0 Þpðrjr0 Þdr0 ; segment ½0; RÃ . This new random walk describes how the rÀR distance of the walker from the target varies. The walker makes variable steps in the range ½ÀR; R.2 For example, where f0 ðrÞ ¼ ðr À r0 Þ is the pdf associated to the initial Fig. 4 shows a walker started at distance RÃ . Steps are position, r0 , and pðrjr0 Þ ¼ P rfri ¼ rjriÀ1 ¼ r0 g is the transi- numbered progressively starting from zero; the distance tion probability from point r0 to r. Such a transition pdf is from the target at step i, ri , is reported on a vertical radius. only defined for jr0 À rj R. For the natural random walk, The walker hits the target when r becomes zero. The circle it can be expressed as (bias is introduced later in this of radius R is the look-ahead area. The hitting time in the section) example is then nine. 8 < ðrÞ; 0 r0 < R; 0 2. A geometric random walk starts at some point in Rn and, at each step, pðrjr Þ ¼ fðrjr0 Þ; R r0 < RÃ À R; ð3Þ : moves to a neighboring point chosen according to some distribution that fðrjr0 Þ þ fð2RÃ À rjr0 Þ þ; RÃ À R r0 RÃ : depends only on the current point, see [26]. Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on September 4, 2009 at 13:40 from IEEE Xplore. Restrictions apply. 504 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 4, APRIL 2009 Fig. 6. Analysis versus simulation, a variable number of nodes deployed in a circular-shaped area of radius RÃ ¼ 1; the target node occupies the center, the random walk starts at distance RÃ . Fig. 5. Variation of the euclidian distance after retransmitting a packet. by simulations. The number of nodes is varied. The The function fðrjr0 Þ will be called the progress pdf. It parameters are r0 ¼ RÃ , RÃ ¼ 1, and R ¼ 0:3. The analysis expresses the probability that, after a retransmission, the provides accurate results when the total number of nodes distance of the walker from the target varies from r0 to r. is higher than 300, which corresponds to approximatively The value r À r0 is called the progress. 24 neighbors per node. Note that, due to lookahead, in The last expression in (3) takes the border effect into order to hit a target it is sufficient to reach any of its account, i.e., the fact that the region is limited. To deal with neighbors. Thus, the hitting time decreases as the network such a limitation, we allow to select virtual points beyond becomes more dense. RÃ and map point RÃ þ x to RÃ À x, i.e., a virtual transition Introducing bias. To include bias in our model, it is to RÃ þ ðRÃ À rÞ ¼ 2RÃ À r generates the transition to r. enough to modify the transition probability by a scaling Let us now compute the progress pdf. Consider the circle factor. The new transition probability is p0 ðrjr0 Þ ¼ k1 pðrjr0 Þ if c of radius R centered at the selecting node and let C1 ðC2 Þ r < r0 and p0 ðrjr0 Þ ¼ k2 pðrjr0 Þ otherwise, where be the circle of radius r0 ðrÞ centered at the target node (see Fig. 5). When the selected node belongs to the arch 1À k1 ¼ R r0 dr; k2 ¼ R r0 þR : ð6Þ delimited by the intersection of C2 with c (bold in the r0 ÀR pðrjr0 Þ r0 pðrjr0 Þdr figure), the distance of the packet varies from r0 to r. Let ðr0 ; rÞ be the length of such an arch. Since nodes are 3.1 Discussion uniformly deployed we can write The random walk studied in the previous section is a ðr0 ; rÞ particular random walk on ½0; RÃ , which makes only fðrjr0 Þ ¼ ¼ 2c r; discrete steps of fixed size ÆR. On the other hand, due to R2 the circular symmetry, a random walk in 2D also induces a where c is the angle under which the arch is seen from the random on the segment ½0; RÃ but with variable steps in the target. Therefore, combining range ½ÀR; R. Is there some connection between the performance of the two random walks? To answer to this ðr0 þ Rcosc Þ2 þ ðRsinc Þ2 ¼ r2 question, we follow a fairly pragmatic approach. The value with N in the bounding function for the discrete case is the hitting time for ¼ 1. We will compute the “equivalent” N rcosc ¼ r0 þ Rcosc ; for the continuous random walk and then, a posteriori, test we obtain if the function is still useful. RR 02 Let then Ri ¼ r0 ÀR rfðrjr0 Þdr be the average positive 1 0 r2 À r02 À R2 r þ r2 À R2 progress at the ith step of the walk, i.e., the expected c ¼ arccos r þ ¼ arccos r 2r0 2r0 r progress when the selected point is closer than the selecting ð4Þ is N ¼ PK The equivalent N PKÀ1 K þ 1, where K is such that one. Ã Ã i¼1 Ri > R À R and i¼1 Ri < R À R; in other words, so that K is the minimum number of steps after which the distance 02 becomes less than R (recall that due to lookahead the 2r r þ r2 À R2 fðrjr0 Þ ¼ arccos : ð5Þ walker hits the target after its distance becomes less than R). R2 2r0 r To simplify the computation, we consider Ri ¼ R, where R Model validation. Fig. 6 compares the hitting time is the average progress observed when the packet is very far Ã ÀR calculated by the model against the hitting time estimated from the target. Hence, K ¼ dR R e. For r ! 1, the Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on September 4, 2009 at 13:40 from IEEE Xplore. Restrictions apply. BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS 505 Fig. 7. Computation of the average progress for a far point. Fig. 9. Validity of the bounding function for the gain. Gain versus bias, n ¼ 400 nodes, transmission range R given as a parameter. Fig. 8. Validity of the bounding function for the hitting time. Hitting time versus bias, n ¼ 400 nodes, R ¼ 0:3. probability that the progress is x is given by the length of Fig. 10. Gain versus starting point (distance from the target); N ¼ 400, theﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ at distance x from the selecting node, that is, R ¼ 0:3. p chordﬃ 2 R2 À x2 , divided half of the area of the circle, see Fig. 7; thus, our model. Second, the function upper bounds the hitting time correctly. ZR pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ Fig. 9 shows the gain as a function of the bias level for 4 4R R¼ x R2 À x2 dx ¼ : RÃ ¼ 1 and R ¼ 0:3, 0.25, 0.2. The average progress is R2 3 R ¼ 0:127, 0.106, 0.084, which corresponds to K ¼ 6, 8, 10. 0 Since simulation results are equal to the analytical one, they Then, we conjecture the hitting time and the gain are are not reported to simplify the plot. Again, the bounding bounded, respectively, by function provides a useful approximation. 1 Finally, Fig. 10 reports the gain estimated by simulation RÃ À R as a function of the starting point for 400 nodes and R ¼ 0:3. h ðr; 0Þ 1þ ; ð7Þ R When the starting point gets closer to the target, the gain increases, as predicted by Property 4 of the previous Ã 1À2 section. R ÀR g ðr; 0Þ ! 1 À 1 þ : ð8Þ R 4 HOW TO IMPLEMENT A BIASED RANDOM WALK These bounds are the equivalent of Property 2 and Property 3. Fig. 8 shows the hitting time as a function of The previous sections provide us with a theoretical flavor about the goodness of biased random walks for searching. the bias level for RÃ ¼ 1 and R ¼ 0:3. The figure reports: the As our analysis has shown, the hitting time is very sensible upper bound function—with these values, K ¼ 6; the hitting to the bias level. Motivated by this encouraging result, we time computed with the biased transition probabilities—see will now drill down to a practical implementation of a (6); and the hitting time estimated via simulations for search algorithm. Before that, we discuss the possible n ¼ 400 nodes. Confidence intervals are not reported. This options for achieving bias and present general implementa- figure shows two things. First, the simulation results validate tion frameworks that adhere to these frameworks. Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on September 4, 2009 at 13:40 from IEEE Xplore. Restrictions apply. 506 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 4, APRIL 2009 In a random walk, the walker moves by making blind random among all nodes. A given bias level is obtained by and memoryless random selections. The decision of the next regulating the estimation correctness, . node to visit is blind because the walker does not use any The expected bias level of this framework can be external information to decide, and it is memoryless since if computed as follows: Let C be the average number of near the walker visits two times the same node, it behaves the nodes of a randomly observed selecting node and F the same. The bias arises when such a fairly simple decision average number of far nodes; then, the expected number of mechanism is somehow altered in a way that the walker is near nodes that the oracle correctly tags as such is C, while statistically “pushed” toward the target. In this paper, we the expected number of far nodes, wrongly tagged as near explore two orthogonal ways to bias the walk, dubbed as nodes, is ð1 À ÞF . Hence, the probability that the selected bias-by-information and bias-by-memory. node is actually closer than the selecting one to the target, In Bias-by-information, the walker exploits information i.e., the bias level, is available at the currently visited node, which indicates the most appropriate decision to take for reaching the target. C ¼ : The information used for deciding is maintained by some ðC À F Þ þ F protocol, which basically corresponds to the oracle used in Note that if the neighbors of a node are equally likely closer our model. or farther to the target, i.e., F ¼ C, we obtain ¼ . This In the literature, many examples fall in this category. means that the bias level corresponds to the estimation In P2P architectures, where this technique is widely used, correctness of the oracle. the information can consist of some topological knowl- The second framework is called PARTIALðI; kÞ. This edge, e.g., the connectivity degree of the neighbors of the implementation strategy requires only a subset I of nodes to selecting node; another option is leveraging on a learning be equipped with the oracle, which correctly tags protocol that estimates the goodness of the candidate nodes, according to previous searches, see [10]. A routing minfk; NNeigh g near nodes—NNeigh being the total number protocol can also be considered as a special case of this of near neighbors. When the walker arrives at an informed class. The routing tables represent the information stored node, i.e., one equipped with the oracle, it makes an at nodes while the routing protocol, which is in charge of informed step by selecting one of the recognized near maintaining the table up to date, corresponds to the nodes, at random. In this case, the bias level is oracle. In this particular case, the “search” becomes C deterministic. ¼iþ ð1 À iÞ; CþF In Bias-by-memory, the walker maintains memory of its previous selections, so that bias merely consists of forcing to where i is the probability that the selecting node is visit new portions of the network. No sense of direction is informed. Assuming C ¼ F , we obtain exploited. For example, in [3], visited nodes are hidden with 1þi a given probability. ¼ : ð9Þ 2 Bias-by-information is potentially more effective than bias-by-memory, especially if the number of target is low. For example, the best one can expect from bias-by-memory 4.1.1 Evaluation is that at each step a new node is visited. If we assume that a To assess the performance of these frameworks, we have newly visited node has the same probability of being the simulated a random walk over 400 nodes, placed at target, then the lowest average hitting is nþ1 , where n is total 2 random inside a unit circle. The target occupies the center the number of candidate nodes (see Appendices A and B). of the area while the source node lies on the circumfer- The hitting time may be much shorter under bias-by- ence. In order to also study the effect of node density, the information, the lowest value being the average network transmission range R is varied and takes the values of 0.2, distance of a requesting node from the target. 0.25, or 0.3, which correspond to roughly 15, 23, and We now discuss two implementation frameworks that 32 average neighbors per node, respectively; the number allow one to design random walks of these two classes, of independent trials was 10,000 times. All the protocols along with their performance assessment. use lookahead. The oracle is simulated according to the framework studied. 4.1 Bias-by-Information Implementation The results report the bias level, namely the number of Frameworks times that, after a step, the walker reduced its distance from The first implementation strategy for an informed random the target and total number of steps done before the target is walk is called ALLðÞ. In this case, all nodes are equipped reached as well as the correlation between bias level and with an oracle functionality, which tags all neighbors either hitting time/gain. as near or far. A near (far) neighbor is a node whose ALLðÞ framework. Fig. 11 shows the bias level as a distance from the target is less (higher) than the tagging function of . The relationship is almost linear. The one. The oracle is required to provide correct discrimina- discrepancy from a line is due to the border effect and to tions with probability ; namely, a near (far) node is the fact that F > C.3 Note that for R ¼ 0:2 and ¼ 1 we correctly tagged as near (far) with probability and obtain < 1, since some nodes have no near neighbors at all. wrongly as far (near) with probability 1 À . The next node to visit is selected at random among the estimated near 3. Far nodes fall into an area higher than the area where near nodes nodes. If no nodes are tagged as near, then the selection is at should fall, see Fig. 5. Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on September 4, 2009 at 13:40 from IEEE Xplore. Restrictions apply. BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS 507 Fig. 11. Bias level versus estimation correctness. Fig. 13. Bias level versus number of informed nodes. Fig. 12. Hitting time and gain versus bias. Fig. 14. Hitting time and gain versus bias. By approximating the relationship with a line, we obtain recognizes some near nodes, or by equipping some node a useful rule of thumb. The bias level directly maps to the with a precise oracle that recognizes all its near nodes. As an correctness in the estimations performed by the oracle. example, to obtain ¼ 0:6 either all nodes should have an Fig. 12 shows the correlation among the hitting time/ oracle that correctly tags near nodes with probability gain and the bias level. We can see that for all the ¼ 0:6 or 80 nodes must have a precise oracle, which transmission range, the bias level greatly affects the search detects all near neighbors. performance. The upper bounds predicted by the analysis PARTIALðI; kÞ framework. This implementation strat- are also reported as solid lines in the plot. Since the egy requires a less “powerful” oracle, in the sense only a analytical model assumes infinitely many nodes, the bound subset of near nodes must be known, not all. In the derived from the model becomes more and more correct as extreme case, we may ask the oracle to recognize just one the node density increases, i.e., R increases. near node, i.e., k ¼ 1. PARTIALðI; 1Þ framework. Fig. 13 reports the bias as a Fig. 15 shows the bias as a function of the number of function of I. The bias level varies almost linearly with the informed nodes for such a framework. The bias first number of informed nodes, meaning that, as a rule of increases rapidly with I and then more slowly, meaning I thumb, we can assume i ¼ N and use (9) to calculate the that the usefulness of each new informed node depends on number of informed nodes required to achieve a given bias how many informed nodes are already in the system. As far level . More specifically, to achieve a bias level the as the hitting time is concerned, see Fig. 16, the same percentage of informed nodes must be 2 À 1. For example, relationship previously found holds. to obtain ¼ 0:6, the fraction of informed nodes must be Reducing k ¼ 1 provides somehow a counterintuitive 20 percent. Fig. 14 shows how the hitting time and the gain result. For a fixed I, having just one option is better than are correlated to the bias level. having more chances. Having no options among which to Discussion. Figs. 12 and 14 put in evidence that the bias choose has a positive effect on the bias level. By inspecting level has the same effect on the search performance, the simulation results, we saw that, on the average, the regardless the implementation strategy used to obtain it. informed nodes are encountered more frequently than Thus, the same search performance are achieved either by when all near nodes are detected. And, this increases the equipping all nodes with an imprecise oracle, which bias level, according to (9). Roughly speaking, the fact that Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on September 4, 2009 at 13:40 from IEEE Xplore. Restrictions apply. 508 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 4, APRIL 2009 Fig. 15. Bias level versus informed nodes, k ¼ 1. Fig. 16. Hitting time and gain versus bias. an informed node always selects the same neighbor has the number of nodes. In this case, each node maintains a flag effect of “hiding” to the walker the portion of the network associated to the walker that indicates if the node was behind the not selected neighbors. Although some informed visited or not. At selection time, a node probes the status of node is also hidden, the overall effect is that the walker all neighbors and sends the walker to one unvisited node, meets less uninformed nodes. selected uniformly at random. The overhead due to probing To better understand how the number of detected nodes can easily be avoided by leveraging the broadcast nature of affects the search performance, in Fig. 17, we report the bias transmissions (see Section 4.2.1). Fig. 17a and the hitting time Fig. 17b as a function of the detected near nodes, k, when R ¼ 0:3 (the same results are 4.2.1 Evaluation obtained for other transmission ranges). We can see how the Fig. 18 shows the bias induced by the first solution as a search performance becomes worst as k increases, especially function of H. The distributed memory option corresponds if the percentage of informed nodes is high. to H ¼ 400. Since we saw that the bias and the hitting time for H ! 100 are the same, the plot shows only results until 4.2 Bias-by-Memory Implementation Frameworks H ¼ 100. The bias level increases rapidly as a memory To implement bias-by-memory, the identifiers of the visited capacity is added, and then it stabilizes to a value. nodes must be available at the walker. The first implemen- Consider that in this case the bias cannot be regulated. tation is based on a Memory List, carried by the walker. Rather, it is just a way to model the side effect of the The list contains at most H different identifiers of the most memory. Nevertheless, see Fig. 19, a strong correlation recently visited nodes. between the hitting time and the bias exists. The next node to visit is selected at random among the neighbors that do not appear in the list. If all neighbors are in the list, the selection is at random among all neighbors. 5 PROPOSED PROTOCOL The other implementation option is called Distribu- After analyzing the characteristics of the frameworks, we are ted Memory and corresponds to H that is equal to the now ready to describe our protocol. The protocol is designed Fig. 17. (a) Bias level and (b) hitting time versus number of near nodes detected; transmission range R ¼ 0:3. Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on September 4, 2009 at 13:40 from IEEE Xplore. Restrictions apply. BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS 509 Fig. 18. Bias level versus history. Fig. 19. Search performance. for mobile settings; it combines bias-by-information with tLBj . When i misses K consecutive beacons, i.e., bias-by-memory and uses one hop lookahead. t À tLBj > KÁT , where t is the time when the Kth beacon The information is computed by each node, i.e., no should have been received, i sets a local variable tij to t. tij central authority is required, and consists of the estimated represents the (estimated) time when j exited from the i’s euclidian distance of a node from the target. Estimations are transmission range. At time t tLBj þ KÁT , dij ¼ 0, while carried out through a simple distributed algorithm, as for t > tLBj þ KÁT , it is detailed later in this section. The bias is embedded in the next hop’s selection logic in the form of an effective distributed algorithm, which also t À tij dij ðtÞ ¼ : encompasses lookahead. The protocol can be adapted to kij ÁT multitarget search straightforwardly. We will first present how the biasing information is achieved—both for single target and multitarget searches. Then, we present the The denominator kij ÁT represents the dwell time of j with forwarding protocol, which implements the steps of the respect to i, namely how long j remained in i’s transmission random walk. range. The above formula assumes that the dwell time is inversely proportional, in expectation, to the relative speed 5.1 The Biasing Information Source between the two nodes, see [12] and [6] for a deeper Basic idea. The information exploited for biasing is the discussion. Appendix A reports an experimental evidence estimated distance of the neighbor of the selecting node about the bias level the estimated distances may provide. from the target. Such a form of information is well suited for Bias in multitarget search. A multitarget search arises mobile networks, exactly due to mobility. During its when the target node is any node among a subset G of the lifetime, the target node comes frequently in contact with network nodes. The targets share a common unique ID, other nodes, where a contact of i with j occurs and i which is included into their beacons. For example, in SOA, receives a packet, e.g., a beacon from j. The time elapsed the ID could be the description of the same service. diG ðtÞ since the last contact, namely the contact time, provides a denotes the distance of node i from the set G. It is the node with a rough indication about how far the target could minimum distance of i from a member of G, formally currently be. To exemplify, suppose that a node needs to diG ðtÞ ¼ minfdij ðtÞjj 2 Gg. decide to which of two neighbors, say a and b, to send the 5.2 Packet Forwarding Algorithm with Lookahead packet; assume further that the target node sends a periodic beacon every second and that a received the last beacon The forwarding algorithm aims at selecting the neighbor 10 seconds ago while b received the last beacon 20 seconds closest to G, which is not yet visited. It assumes that the ago. It is clear that, unless movements are highly irregular, a walking packet, say m, is uniquely identified and carries the is more likely to be closer than b to the target; thus, bias may G’s ID. Moreover, each node i is required to manage a local be introduced by selecting a. The above idea of contact time list, F W Di , of the last packets it has forwarded. m 2 F W Di was originally proposed in [9] and exploited in the Last means that node i has forwarded packet m. Encounter Routing (LER) protocol, described in [25]. In this The idea of the protocol is simple. The selecting node paper, we adopt a slightly different estimation mechanism, probes the neighbors with Request To Send (RT S) control which also takes an estimation of the relative speed into packets and then sends m to the first neighbor that replies account, see [6]. back with a Clear To Send (CT S) packet. The CT S packet is Implementation details. To estimate the distance be- emitted by a node after a suitable delay, which takes bias and tween a target node, say j, and another node, say i, the lookahead into account. When a node hears that the selecting target node is required to send a beacon periodically, every node is sending m to another node, it aborts its own reply. ÁT s. The estimated distance from node i to j will be Implementation details. Let k be the selecting node. k denoted as dij . Initially, dij is set to unknown. As soon as i broadcasts an RT S control packet containing the G’s ID and receives a beacon from j, i sets dij ¼ 0 and starts counting m’s ID. On receiving such a control packet, a node i the number kij of consecutive beacons it is receiving. schedules the transmission of a CT S control packet after a Moreover, it stores the time of the last beacon received into delay t. The following four cases are considered: Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on September 4, 2009 at 13:40 from IEEE Xplore. Restrictions apply. 510 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 4, APRIL 2009 Fig. 20. Parameters of the experimental evaluation. i 2 G, i.e., i is a target. In this case, t ¼ 0. 1. The metrics of interests are: 1) the hitting time, measured i 62 G, diG ¼ 0, m 62 F W Di , i.e., i is not a target, but 2. as the ratio of the number of times the packet is the target is likely a neighbor of i; moreover, i is retransmitted before it hits the target with the total number going to forward the packet for the first time. In this of successful random walks; 2) the gain with respect to the case, t ¼ randomð0; 1 Þ, i.e.,t is a uniform random unbiased walk. They are estimated using five independent value in the range 0; 1 ½: replications and 95 percent confidence interval. 3. i 62 G, 0 < dij H, m 62 F W Di , i.e., i is not a target, but it has a valid estimation of its distance from the 6.1 Baseline Assessment on the Circular Area closest target; moreover, i is going to forward the In this first set of experiments, the target node is static and packet for the first time. In this case, t ¼ 1 þ positioned at the center of the area, while the other nodes dij ð2 À 1 Þ H , i.e., t is proportional to a value in the move. A search is initiated either by the Farthest node range ½1 ; 2 . (Biased-F), or by Any node (Biased-A). 4. None of the previous conditions are met. For this topology, the bounds given in (7) and (8) should t ¼ randomð2 ; 3 Þ. Ã ÀR apply. We have 1 þ K ¼ dR R e ¼ 9; thus, the hitting time 1 Node k sends the packet to the one from which the first should at most be 9 , where is the bias level. Fig. 21 RT S is received. Thus, node k tries to select a neighbor shows such a predicted hitting time along with the hitting according to the following order: time measured via simulations. The bias used in the a target node (lookahead); 1. expression was also estimated during the simulations. It a new node (bias-by-memory) whose neighbor is 2. was found that % 0:54; 0:65, 0:58, and 0:59 for vmax ¼ 0, very likely the target (bias-by-information); 10, 30, and 50 m/s, respectively. When the network is 3. a new node (bias-by-memory) with the highest static, the random walk is biased on the basis of the chance of being the closest one to the target (bias- memory, as discussed in Section 4.2. by-information); When mobility is added, the information starts also to 4. a node at random. bias the walk. The hitting time is then reduced until a All nodes that overhear such transmissions delete their minimum value. Increasing the speed further has two scheduled CT S transmissions. Also, k ignores any subse- effects. From one hand, nodes come more often in contact quent CT S packet it would receive. 6 AN EXPERIMENTAL EVALUATION To assess the suitability of our proposal, we have conducted a simulation study by exploiting a custom discrete event simulator, already used in [6]. The simulator has the following main characteristics. The transmission of a packet starts after the channel is sensed free for a Random Assessment Delay (RAD) randomly chosen in the range [0..500] ms; the packet reception is notified to a sender’s neighbor provided that it remained for the whole duration of the transmission within the transmission range and such that no collisions with other transmissions occurred; a FIFO buffer of 20 packets in size is used at each node. Nodes move according to the round trip mobility model with waypoints (RWP); we adapted the public code available Fig. 21. Hitting time versus speed. Biased-F is the biased random walk in [15]; the speed varies in the range ½1::vmax m/s, there is when the farthest node starts the walk; Biased-A is the biased random no pause time. The main simulation and protocol para- walk when any node can start the walk. The upper bound predicted by meters are reported in Fig. 20. our model is also reported. Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on September 4, 2009 at 13:40 from IEEE Xplore. Restrictions apply. BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS 511 Fig. 22. Gain time versus speed. Biased-F is the biased random walk Fig. 24. Impact of speed on the gain. when the farthest node starts the walk; Biased-A is the biased random walk when any node can start the walk. The lower bound predicted by our model is also reported. decreasing again. This can be explained considering that a higher speed helps the target node to come more frequently with the target while from the other hand estimations in contact with other nodes, which can thus refresh their become more evanescing. The net effect is that information estimations and make the bias stronger again. becomes less effective. The figure shows how the hitting For jGj > 1, a low correctness is less critical; rather, it turns time is upper bounded by the theoretical value. out that mobility helps in reducing the hitting time because Fig. 22 reports the gain for the same setting. The targets have more chances of becoming neighbors of more theoretical gain of (8) is now a valid lower bound. nodes. Since increasing the speed does not have a significant impact on the hitting time, we can deduce that the hitting time 6.2 Results for the Square Area is now determined by the bias-by-memory effect. Fig. 23 reports the average hitting time as a function of the Compared to the natural random walk, the hitting is maximum speed for the square area. In these experiments, reduced considerably, e.g., roughly the hitting time passes the target node is also mobile. For a single target search from 90 to 30 for 10 m/s and one target only. The ðjGj ¼ 1Þ, the bias level is highly affected by the informa- improvement over the natural walk is shown systematically tion. When the network is static, bias is obtained by in Fig. 24, in the form of gain. We can see how for all the remembering the previous choices. Memory alone is able mobility conditions the hitting time reduced by at least a half. to decrease the hitting time by a half. When mobility is added, it decreases the other half. Thus, mobility indeed allows one to gather information useful to bias the search. 7 RELATED WORK By increasing the maximum speed from 10 to 30 m/s, the Random walks are used in several algorithms proposed for hitting time for the singleton case increases; this is a wireless networks. In [4], the RAndom Walk-based Member- consequence of a reduced correctness of estimations. ship Service (RaWMS) for ad hoc networks is described. The Directing the packet away from the region where the target service provides each node with a partial uniformly chosen is currently located has a strong negative effect on the view of network nodes. The algorithm uses random walk as a hitting time; roughly speaking, the packet has to return sampling technique, whereas the aim of our protocol is to back. By further increasing the speed, however, we can locate a target. Dolev et al. [8] propose a randomized self- observe that the hitting time for the singleton case starts stabilizing full group membership service for ad hoc net- works. The group membership list is collected by a single random walk agent traversing the network. They apply a single random walk that covers the whole network, not for searching. An efficient token passing algorithm is exploited in NASCENT to provide a network layer service dedicated to group communication in ad hoc networks [18]. Again, the goal of the random walk is not to perform a search. Avin and Brito [3] apply what we have called bias-by- memory to query in sensor networks. A previously visited node is hidden to subsequent selections with a given probability, which is called the bias of the walk. The work exploits only one form of bias. Differently from our case, no information is used. In [1], nodes are allowed to choose the next hop among a small subset selected at random. The authors discuss the power of such a strategy for improving the performance of Fig. 23. Impact of speed on the hitting time. a random walk considerably. Their results are consistent Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on September 4, 2009 at 13:40 from IEEE Xplore. Restrictions apply. 512 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 4, APRIL 2009 with our findings, because making an informed choice is a way to achieve a strong bias. Random walk over wireless networks are also studied from a graph theoretical point of view in several papers, e.g., [2] and [5]. However, all these studies focus on unbiased walks. Biased random walks are widely adopted for search in unstructured P2P architectures, both in the form of bias-by- memory and bias-by-information. The interested reader can, for example, refer to [10] for a survey. However, several key aspects make search in P2P different from search in wireless networks. First, the topology of P2P networks is best modeled as a power-law graph, whereas wireless networks adhere to the random geometric graph model. Second, the channel model in the Fig. 25. Experimental evidence of the validity of the estimations. two networks is quite different. While in P2P nodes are connected via unicast channels (a TCP connection), in networks. For a network with uniform node distribution wireless networks a transmission is inherently broadcast. and a circular symmetry, we have presented an analytical The cost of implementing the same technique, like looka- study, which shows the sensibility of the hitting with head or next hop selection, is then quite different. Last, the respect to bias; in particular, the hitting time varies as N 1= , changes in the topology are strongly correlated; this makes where N is proportional to the radius of the circle and , some source of information, like the distance among nodes, 0:5 1, is a number expressing the biasing level of the meaningful only in mobile networks. walk. We have then suggested a protocol that exploits We remark that physicians often use random walks to biased random walk. A simple beacon-based biasing model numerous dynamical processes that occur in nature, mechanism is used. The simulation study shows that the the most notorious being the Brownian motion. The work hitting time is reduced by at least a half with respect to described in [21] is of particular interest for our work. It natural walks. The results are interpreted in the light of the presents a method for calculating the properties of biased theoretical study. random walks on complex networks in general, and for a segment in particular. Specifically, the Mean First passage APPENDIX A Time (MFT)—which is synonymous to the hitting time—is To give experimental evidence of the bias level we can computed for a particle (the walker) moving on a segment of achieve by estimating the distance from the target, we have size N under an external biasing field, discretely in space and simulated the following scenario. Four nodes, i, j, k, and g time (hopping). This paper shows how the behavior of the move into a unitary square region, according to the random walker, when it moves in the direction of the bias, changes waypoint mobility model with an average speed of 30 m/s from a diffusive regime to a drift one as a bias is applied. The and no pause time. The transmission range of the nodes is diffusive regime is characterized by the MFT, which grows as N 2 and it is observed in the limit of a weak bias (it basically R ¼ 0:3. Node j plays the role of target node, g is the corresponds to a natural random walk). As the bias is selecting node while i and k are test nodes. When these two increased, the MFT varies linearly with N, i.e., the so-called nodes are both neighbors of node g and they are both out of drift regime arises. The results discussed in that paper the j’s transmission range, their estimated and actual provide an interesting physical interpretation of our study. distance from j, dij , dkj and d0ij , d0kj , respectively, are Finally, the idea of using a random delay before sending observed. The following measurement are then performed: a packet has already been used in other protocols, including 1) probability P ðHÞ that both dij and dkj are below the counter-based, distance-based, and position-based broad- threshold H; 2) probability Pdwn ðHÞ that the estimations are casting schemes [11]. The basic idea is to collect duplicate correct, namely, dij < dkj and d0ij < d0kj , given that they are packets received from neighbors for a random period of both below H. Pdwn ðHÞ measures the bias level. If g would time after the first packet is received and to use knowledge select the node with the lowest estimation, i.e., node i, then from these packets to make a forwarding decision. The the packet will actually get closer to node j with probability counter-based scheme exploits the total number of received Pdwn ðHÞ. duplicates, and the packet is forwarded if it is below a The results are reported in Fig. 25. When H is very low, counter threshold. The distance-based scheme uses the the difference in the two estimations is not high and both minimum distance from the node to the sender of these nodes are equally close to the target j. This, however, packets, which is an estimation of the node’s additional happens with a very low probability—see P ðHÞ in the plot. (broadcast) coverage area, and the packet is forwarded if it As H increases, the potential bias level also increases. For is over a distance threshold. The location-based scheme very high H, the bias level becomes % 0.6. leverages the precise location information to provide a more accurate estimation of the additional coverage area. APPENDIX B Consider an ideal bias-by-memory search initiated by a 8 CONCLUSION node over n other nodes. Assume that the target replication In this paper, we have studied the effect of bias on the degree is k and that each node can be targeted with the hitting time for a random walk executed over wireless same probability of 1=n. The hitting time can be computed Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on September 4, 2009 at 13:40 from IEEE Xplore. Restrictions apply. BERALDI: BIASED RANDOM WALKS IN UNIFORM WIRELESS NETWORKS 513 [6] R. Beraldi, L. Querzoni, and R. Baldoni, “A Hint-Based Probabil- istic Protocol for Unicast Communications in MANETs,” Elsevier J. Ad Hoc Networks, 2006. [7] N. Chang and M. Liu, “Optimal Controlled Flooding Search in Large Wireless Networks,” Proc. Third Int’l Symp. Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WIOPT ’05), 2005. [8] S. Dolev, E. Schiller, and J. 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Rowstron, “Perspectives Work- is the probability that after j extractions none of the k black shop: Peer-to-Peer Mobile Ad Hoc Networks—New Research Issues,” Proc. Dagstuhl Seminar, 2005. balls are extracted, given n balls in total. The average [21] I. Goldhirsch and Y. Gefen, “Biased Random Walk on Networks,” number of extraction before a black ball is picked, i.e., our Physical Rev. A, vol. 35, no. 3, Feb. 1987. hitting time, is [22] C. Gkantisidis, M. Michail, and A. Saberi, “Random Walks in Peer-to-Peer Networks,” Proc. IEEE INFOCOM, 2004. X nÀkþ1 [23] ¨ M. Gunes and O. Spaniol, “Ant-Routing-Algorithm for Mobile htðn; kÞ ¼ j½F ðj À 1jn; kÞ À F ðijn; kÞ: Multi-Hop Ad-Hoc Networks,” Proc. Int’l Workshop Ad Hoc Networking (IWAHN ’02), 2002. j¼1 [24] C.E. Perkins, E.M. Royer, and S.R. Das, Ad-Hoc on Demand Distance Vector (AODV) Routing, IETF Internet draft, work in progress, July Fig. 26 shows htð300; kÞ as a function of k. For k ¼ 1, the 2008. effect of “bias” is quite light; it becomes significative as [25] N. Sarafijanovic-Djukic and M. Grossglauser, “Last Encounter k % 10. Routing under Random Waypoint Mobility,” Proc. Third Int’l IFIP-TC6 Networking Conf. (NETWORKING ’04), May 2004. [26] S. Vempala, Geometric Random Walks: A Survey, http://www-math. mit.edu/~vempala/papers/survey.pdf, 2008. REFERENCES [1] A. Avin and B. Krishnamachari, “The Power of Choice in Random Roberto Beraldi received the Laurea degree in Walks: An Empirical Study,” Proc. Ninth ACM/IEEE Int’l Symp. computer science from the University of Calabria, Modeling, Analysis, and Simulation of Wireless and Mobile Systems Cosenza, Italy, in 1991 and the PhD degree in (MSWiM ’06), 2006. computer science in 1996. He has been an [2] C. Avin and G. Ercal, “On the Cover Time of Random Geometric assistant professor in the Dipartimento di Infor- Graphs,” Proc. 32nd Int’l Colloquium Automata, Languages, and ` matica e Sistemistica (DIS), Universita di Roma Programming (ICALP ’05), 2005. “La Sapienza,” since 2002. From 1996 to 2002, [3] C. Avin and C. Brito, “Efficient and Robust Query Processing in he was an expert in computer networks at the Dynamic Environment Using Random Walk Techniques,” Proc. Italian’s National Institute of Statistica (ISTAT). Third Int’l Symp. Information Processing in Sensor Networks He has published more than 50 peer-reviewed (IPSN ’04), 2004. papers in various fields including computer networks, wireless networks, [4] Z. Bar-Yossef, R. Friedman, and G. Kliot, “RaWMS—Random and distributed systems. He participates in many research projects and Walk Based Lightweight Membership Service for Wireless regularly serves as a reviewer for international conferences and journals Ad Hoc Networks,” Proc. ACM MobiHoc, 2006. on the above areas. He was a program cochair of the First International [5] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah, “Gossip and Workshop on Dynamic Distributed Systems held in 2006. Mixing Times of Random Walks on Random Graphs,” Proc. SIAM Second Workshop Analytic Algorithmics and Combinatorics (ANALCO ’05), 2005. Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on September 4, 2009 at 13:40 from IEEE Xplore. Restrictions apply.