Searching strategy for multi-target discovery in wireless networks

Transcription

1 Searching strategy for ulti-target discovery in wireless networks Zhao Cheng, Wendi B. Heinzelan Departent of Electrical and Coputer Engineering University of Rochester Rochester, NY 467 (585) 75-{878, 453} {zhcheng, Abstract In this paper, we address a fundaental proble concerning the optial searching strategy in ters of searching for the ulti-target discovery proble in wireless networks. In order to find the nearest k targets fro a total of ebers with the least, how any searching attepts should we use, and how large should each searching area be? After providing the applications that otivate our research, we odel the proble and derive a general forula for the expected as a function of the paraeters of the nuber of searching attepts n and the searching area for each attept, A i. Based on this forula, we propose several algoriths to deterine the optial paraeters to achieve the inial, either pre-calculated or perfored online. Using the optial paraeters derived fro analysis, we experient with these algoriths on general wireless network scenarios. The results show that our algoriths perfor consistently close to optial, and they exhibit uch better perforance than other heuristic schees. The desired perforance is achieved by adapting the searching radius to estiates of network paraeters such as the total nuber of nodes and the total nuber of targets. I. INTRODUCTION The target discovery process exists extensively in applications of nowadays fast growing wireless ad hoc networks and sensor networks. Usually, query packets are propagated inside the network to search for the targets. The target nodes will respond upon receiving the query packets. Unlike ost unicast traffic, the query process usually includes a flooding process. Thus, it is crucial to deterine the best way to search for a target to iniize the searching. More specifically, the question is: what are the nuber of searching attepts and the searching area for each attept for the optial searching strategy? Of course, the siplest searching strategy is to search the entire interested area only once. Soe other heuristic solutions are proposed and ipleented as well. In [], the one-hop neighbors are first queried and the entire area is searched if the target is not aong the one-hop neighbors. In AODV [], an exponential expansion ring () schee is applied, which is to start searching fro one hop and increase the searching radius exponentially upon each failure. Depending on the nuber of targets available, the target discovery proble can be divided into single-target discovery and ulti-target discovery. The single-target discovery process is oriented for unique inforation such as the node ID in routing protocols [], [] or a unique service provided by a specific service provider [3], [4]. Previously, we showed that for singletarget discovery, the expansion ring schee cannot reduce the expected searching [7]. Instead, it increases both the and the latency draatically. We also showed that the saving of even the optial schee is negligible, and the siplest searching schee, which is to search the entire area only once, actually is the best schee fro both the and the latency s perspective. The reason that using ultiple searches does not reduce is that the saving fro a successful search in the local area cannot cover the extra when the local search fails since in wireless networks, the previously searched area has to be covered again for the new query to reach the farther area. In this paper, we study a ore general case, the ulti-target discovery proble. Unlike the single-target discovery proble, using ultiple searching attepts with increasing searching areas can reduce the for the ulti-target discovery proble. The chance of finding a target in the local area increases as the total nuber of targets increases, and this increase is exponentially related to the total nuber of available targets. Thus, the saving fro the local searching ay eventually cover the extra fro the possible failure of the local search. Deterining the optial schee for the ulti-target discovery proble is crucial for applications in large wireless networks, especially those whose coponents are battery-supplied and power sensitive. For soe applications, k servers ust be found in order to function. For exaple, in NTP (Network Tie Protocol) [5], the three closest servers are needed to synchronize a node s clock. In sensor networks, a node ay need to find out the hop-distance to the nearest k anchors in order to perfor location estiation []. Also in sensor networks, a obile host ay need to collect, say, teperature saples fro the nearby sensors to have an accurate overview of the local teperature situation. There ay be soe other applications that perfor ulti-target discovery in order to distribute the load evenly aong the network. For exaple, in a peer-to-peer file sharing network [9], a peer ay locate a nuber of nearby peers and distribute the load aong the. Another exaple is to discover an enseble of special nodes nearby to distribute the coputa-

2 tion aong the. Also, ulti-target discovery ay be intentionally perfored for robustness. A siple exaple is to locate ore service providers than necessary. When the priary service provider cannot function well, there will be soe backup to take the place to avoid interruption without initializing another search. For security sensitive applications such as NTP [5] and NIS (Network Inforation Syste) [6], ultiple-target discovery is alost a necessity, both for security and robustness concerns. Despite the extensive existence and iportance of the ultitarget discovery proble in wireless networks, the study of this field is alost non-existent. The schees being used are erely fro intuition without analytical support. To the best of our knowledge, this paper is the first foral study undertaken to generalize the proble and solve it both analytically and experientally. The rest of this paper is organized as follows. Section II provides an overview on the previous efforts in reducing discovery overhead and soe other related work. Section III odels the ulti-target discovery proble in an infinite network and proposes several algoriths to deterine the optial nuber of searching attepts and the searching area of each attept. In Section IV, we turn to realistic sall-scale networks and illustrate how our previous analysis and algoriths can be applied into these scenarios. Extensive siulations are perfored to copare our algoriths with existing schees. Section V concludes the paper and discusses potential future work. II. RELATED WORK In [7], we analyzed the single-target discovery proble and showed that searching the entire area only once is actually the best schee in ters of both and latency. In this work, we will study the ulti-target discovery proble, which can be seen as an extension to the single-target discovery proble. The ulti-target discovery proble can be further divided into two branches. The first branch is to find at least target fro a total of targets. The ost coon use of the this one-outof- discovery is in routing protocol ipleentations. Typical exaples are [] and AODV []. Although the target is a specific node ID, there ay be caches aong the other nodes and the searching becoes a ulti-target proble. Also, the searching schees adopted by and AODV, especially the expansion ring schee, can be used for coparison with our approaches. The other branch is a ore general case, which is to find at least k targets fro ebers. The k-out-of- ulti-target discovery proble also has extensive applications, as we entioned earlier. Exaples that require a andatory ulti-target discovery are NTP [5], ITTC (Intrusion Tolerance via Threshold Cryptography) [8], sensor localization [], and sensor inforation collecting []. Exaples that require a ulti-target discovery for robustness are NIS, NTP and any application requiring auxiliary backups. Exaples that require a ulti-target discovery for load distribution are peer-to-peer systes [9] and distributed coputing systes []. Depending on various application requireents, different portions out of the total targets are to be found. For NTP, only three servers are required. For teperature onitoring sensor networks, quite a few sensors are required. For peer-to-peer systes or distributed coputation systes, as any as possible peers are usually preferred. In [3], the concepts of anycast and anycast are introduced to ad hoc networks. The so-called anycast is close to our - out-of- proble in nature, and the anycast is coparable to our k-out-of- proble. Although the authors propose several echaniss to perfor anycast delivery, their priary goal is to provide an investigation on the trade-offs of these echaniss between perforance, reliability and ease of ipleentation. In this paper, we only focus on the discovery phase, and we provide optial solutions based on analysis. III. MULTI-TARGET DISCOVERY IN INFINITE NETWORKS: MODELING AND ALGORITHMS A. Proble odeling, assuptions and terinology Without loss of generality, we assue a large nuber of nodes are placed randoly and independently in a two-diensional space R. A source node wants to find at least one target within a unit area of interest. Suppose that targets are distributed uniforly within this unit area. What is the optial schee to search this unit area to have the iniu? In other words, how any searching attepts n should be perfored and what should be the searching area set A (n) = {A, A,, A n } for these n searching attepts? Using this odel, every searching strategy, including those entioned earlier, can be exclusively expressed by A (n). For exaple, for the siplest searching strategy, which is to search the entire interested area only once, it is A () = {}. For s searching strategy, which is to query the one-hop neighbors first and then search the entire area, it is A () = { M, } if we denote M as the axiu hop liit allowed. For the exponential expansion ring schee applied in AODV, the paraeter set becoes A ( log (M) +) = { M, M, 4 M,, ( log (M) ) M, } if we assue that the searching area is on the order of the searching hop squared. Here, we define the as the total area that has been searched. This general assuption does not contradict the traditional definition as the nuber of transissions. In wireless networks, a node usually needs to forward packets for other nodes, and in order to search a certain area, the nodes within this area have to forward the queries. Thus, the nuber of query transissions to search an area of A is proportional to A by a constant coefficient deterined by the forwarding echanis such as flooding and gossiping. Also, by defining the directly as the searching area, we iniize the nuber of variables and siplify our analysis without loss of generality. The conclusions drawn fro this definition can be specified for different applications siply by apping the area to realistic application paraeters. Also, we ignore the potential increase of the packet length and the it brings during packet propagation. For siplicity, we also ignore potential packet collisions, which can be effectively decreased by inserting a rando delay tie before for-

3 TABLE I NOTATIONS USED THROUGHOUT THIS PAPER. k n C n D A i A (n) the total nuber of targets the nuber of targets to be found the nuber of attepts perfored of an n-ring schee difference between two schees searching area of the ith attept optial searching set for n-ring search A A Ai Ai+ An= warding. Routing protocols are not needed for target discovery since broadcasted query packets will set up paths towards the source node that reply packets can follow. During our analysis, we assue we are studying a snapshot of the network and nodes are static during the analysis. However, even if nodes are obile, there are several reasons that our analysis is still valid. First, the flooding search tie is short and nodes will not ove too far away. Second, since nodes are oving randoly and independently, the nuber of nodes in a certain region is stable and will not have adverse effects on our analysis. The odel we are going to use in this section is based on the assuption that the source node is at the center of the searching area and the searching areas are concentric circles within the unit area as shown in Fig.. This assuption is valid for infinite networks, or practical large-scale networks. This siplified odel expedites our current analysis and is easy to extend for realistic sall-scale networks, as we will illustrate in Section IV. We also assue that the targets are uniforly distributed throughout the searching area. For applications such as sensor networks, even if soe sensors run out of energy, this general assuption should still be valid. This is because for a good sensor network architecture, sensors should deplete their energy in a balanced and unifor anner. However, there do exist soe scenarios where the target distribution is non-unifor. We will discuss these non-unifor target distributions in Section V. For quick reference, we use the ter n-ring as a strategy that nodes attept at ost n ties to discover the targets. Other notations are listed in table I. B. Finding out of targets Let us first look at the siplest case, finding only one target out of a total of targets. Let us restate this -out-of- proble briefly. Now, there are targets distributed randoly and uniforly in the unit area. The source node located at the center wants to find at least one target fro these targets with the least by using the optial n searching attepts. ) A two-ring approach: Suppose a two-ring approach is applied, and for the first searching attept, the searching area is A. For the second searching attept, the searching area A is, of course, the entire area and hence equals. As long as not all the targets are located outside the A area, the target will be found within the first attept. Therefore, the probability P to Fig.. The siplified odel of the ultiple target discovery. The searching areas are concentric circles. Both the target nodes (black dots) and non-target nodes (not shown) are uniforly distributed in the searching area. discover at least one target in the first attept and the for the first searching attept are P = ( A ), C = A () However, if the first attept fails, another search has to be perfored, and the total searching for these two searches C is C = A + A = A + () Note that if a second search needs to be perfored, the total is not only just the second searching area, but includes the fro the previous failed searching attept. If a second search is required, it eans that all the targets are located in the second ring outside the A area, and the probability P for this case to happen is P = ( A ) (3) Thus, the expected C for a two-ring schee to coplete the -out-of- target discovery is C = P C + P C = ( ( A ) )A + ( A ) (A + ) = A + ( A ) (4) It is easy to deterine the iniu C for A [, ] by solving C A =, which results in A = (5) ) An n-ring approach: To aid the expression, let us define a virtual th attept search for the area of A =. If the ith search attept succeeds, the total C i is siply the suation of the first i attepts C i = i A j (6) j= Siilarly, in order to perfor an ith search attept and coplete the task, there ust be no targets in the area A i and there ust be at least one target in the area A i. Thus, the probability P i for the task to be copleted in the ith attept is P i = ( A i ) ( A i ) (7)

4 Therefore, the expected C n for a general n-ring searching approach is C n = n P ic i = i= n i (( A i ) ( A i) ))( A j) i= j= (8) n = A i+( A i) i= The final equality above can be easily proven through atheatical induction. Due to space constraints, we skip the interediate steps. C. Finding k out of targets Now, we can easily extend the study to a general case of finding at least k targets out of a total of targets. Again, let us start fro a two-ring approach. ) A two-ring approach: Given the first searching area A, the probability p i for exactly i nodes to be located within the A area is actually of binoial distribution p i = C i A i ( A ) i (9) In order to find at least k nodes within the first attept, there ust be greater than or equal to k nodes within the first area A. The probability P for this case to happen is the suation of the probabilities p i for i k. P = p i = CA i i ( A ) i () i=k i=k Of course, the probability P is the suation of the probabilities that there are less than k nodes within A. k P = CA i i ( A ) i () i= To siplify the expression, let us first define I(p;, k) = Cp i i ( p) i () i=k For a given (, k) pair, we ay further siplify I(p;, k) as I(p) without causing confusion. Eventually, we can write the for a two-ring searching schee in a sipler for C = P C + P C = I(A )A + ( I(A ))(A + ) = + A I(A ) (3) ) An n-ring approach: In order to find k targets in the ith searching attept, there ust be ore than k nodes within the area A i. Also, there ust be fewer than k nodes within the area A i, or else the search would end in the (i )th attept. The probability P i for the ith search to coplete the searching task is P i = I(A i ) I(A i ) (4) The of the ith search, the sae as before, is i C i = A j (5) j= Thus, we have the expected for a general n-ring search C n = n P ic i = i= n i ((I(A i) I(A i )( A j)) i= n = A i+( I(A i)) i= j= (6) In the next section, we will propose several algoriths to deterine the optial searching area set A (n) to iniize C n based on equations 8 and 6. D. Algoriths We classify the algoriths into pre-planned algoriths and online algoriths, depending on when the paraeters are deterined. For pre-planned algoriths, A (n) for various n are all calculated before the search starts. The source node will refer to these precalculated values during the searching process. For online algoriths, the source node only calculates the current searching area exactly before this search starts. Online algoriths need less coputation than pre-planned algoriths, while they ay perfor less than optial due to the lack of global knowledge. ) Brute force (BF): Given n, there are n searching area variables fro A to A n (A n is set to one). BF searches every possible A i [, ] and calculates the based on equation 8 or 6. It picks the sallest as the optial and the corresponding area set as the optial solution. Despite its siplicity, this brute force technique requires excessive coputation tie and ay be infeasible. We perfor it offline just to provide a lower bound on achievable for the other algoriths. During realization, the interval of [,] for each A i has to be discretized, and the results fro this discretization ay not be optial. With a granularity of δ for each diension A i, the coputational coplexity is on the order of ( δ )n for an n-ring schee. ) Ring-splitting (): Since BF cannot find the optial solution within tolerable tie, we ay better focus on an alternative algorith that is able to find good solutions using fewer coputations. An intuitive solution is to insert a new searching ring between existing searching rings to reduce the as uch as possible. We ipleent this idea in the Ring-splitting schee. Suppose we already have an n-ring schee. By inserting another searching attept with searching area A j between the ith attept and the (i + )th attept, an (n + )-ring schee can be derived fro the original n-ring schee. Fro equation 6, the difference D between the old n-ring schee and the new (n + )-ring schee is D = C n C n+ = A i+( I(A i)) A j( I(A i)) A i+( I(A j)) (7)

5 starts fro the one-ring searching schee with [A =, A = ] and splits the ring that provides the largest reduction aong all the possible ring splitting choices. This continues until there are no possible choices to split a ring to achieve any ore savings. The procedure is as follows. ) Start with the ring [, ]. ) With an existing n-ring schee, a given ring set of {[, a ], [a, a ],, [a n, ]} already exists. Check all these n rings and find out the candidates that can be split to further reduce the. 3) Terinate if there are no ore candidates. Else, go to Step 4. 4) Pick the candidate that will reduce the ost and split it. Go back to Step. Whether a ring between [A i, A i+ ] should be split and becoe a candidate is a axiization proble of D and is deterined as follows. ) By solving D A j =, we have the potential splitting point A j. Nuerical ethods for root-finding are required to find A j. ) First, check if A j is within [A k, A k+ ]. Second, check if D(A j ) is larger than zero. Only when both requireents are satisfied, should A j be a ring splitting candidate for [A k, A k+ ]. Since each splitting only brings two rings for calculations in the next step, if the optial schee is found at the n ring, the total coputation is only n 3. The nuber of coparisons is i for the i-ring schee, so the total nuber of coparisons is only n n(n ) i= (i ) =. Although does not guarantee the final solution set to be optial, it reduces the coputation tie draatically copared with the BF schee. Also, it is scalable to n by providing a sub-optial solution for all n-rings within one sequence of calculation, while BF has to calculate the solution for each n-ring schee separately. This property akes ore desirable for realistic ipleentations than schee BF. 3) Online ring-splitting (O): BF and are pre-planned algoriths. The optial nuber of searching attepts and the optial searching area set are deterined before the first search begins. O, instead, calculates the searching area only for the current search and only when necessary, either when the search is just beginning or the last search failed and a new search has to be perfored. In this algorith, the source node always plans to finish the search within two attepts by splitting the reaining area. However, once it fails, it perfors another splitting on the reaining searching area and perfors another attept. This process continues until the target is found, or there will be no ore saving in splitting the reaining area. Suppose the source node has already searched the area of S and k targets have been found. The new goal is to find k k targets fro the reaining k targets in the reaining S area. If the source node plans to finish the searching within two attepts by using A as the first searching area, the new would be C e = I( A S S ; k, k k ))A + ( I( A S S ; k, k k ))(A + ) = + A I( A S S ; k, k k ) (8) Again, soe nuerical ethods for root-finding are required to solve Ce A =. Also, the root à has to pass the following two checks to provide the axiu saving: à [S, ] and C e <. If the check fails, just use A = to finish the last searching attept. If not, use à to perfor the next search. O is siilar to but is perfored online. Upon each failure, it only deterines how to split the reaining area, although there ay be soe better splitting ethods in the previously searched area. Thus, it perfors even less than optial copared to. However, it requires even less coputation. There is only one coputation for each additional searching attept, and there is no wasted coputation. E. Nuerical results ) Algorith analysis: In the previous section, we proposed several algoriths that vary in their coputational requireents and their perforance. Let us first deterine the optial searching area set A (n) using these algoriths and copare the expected of these algoriths. In Fig., the expected s for the solution of the -out-of proble calculated by each algorith are shown. BF and have such close perforance that their curves overlap with each other. O perfors at ost 5% worse than the other two algoriths. As entioned earlier, this is because O is an online algorith and lacks global knowledge. However, its perforance is still very close to that of the pre-planned schees. For the pre-planned schees, although a different nuber of rings and different area paraeters ay be required to achieve their own optial point (see colun 3 in Table II), these algoriths perfor nearly identically in ters of (see colun in Table II). The BF perforance shown in Fig. is on a liited brute force search on up to 4-ring schees with a granularity of.. It uses over 65 illion coputations to achieve the of.5685, while achieves a very close.5675 using only 9 coputations. Fro this view, is uch ore practical for realistic ipleentations. In Fig., we also show the perforance of the optial - ring and 3-ring schees. For -ring schees, all the algoriths perfor alost the sae. For 3-ring schees, O perfors a little worse than the pre-planned algoriths, but it is still close. Notice that despite the fact that the real optial solution ay occur at a large value of n, the two-ring schees have a ajor ipact on the reduction copared with the -ring schee whose is, but the three-ring schees only further reduce the by around a trivial -5%. This infors us that it is very iportant to find a good searching area at the first attept. We also show the results for the k-out-of- proble using (,k) pairs of (6,), (6,3), (6,4), (,), (,), (, 8), (6,),

6 TABLE II A COMPARISON OF DIFFERENT ALGORITHMS FOR FINDING K= OUT OF = 3 TARGETS. Schee Cost n A (n) Coputations BF {.5,.594,.85,.} 65,667, {.96,.465,.7467,.9647, ,.} 9 O {.465,.7467,.9647, ,.} the overall iniu BF O the optial ring the optial 3 ring.9.8 : find k out of BF: k= BF: k=/ BF: k= : k= : k=/ : k= total nuber of targets Fig.. The optial expected of finding one target for each algorith and the optial -ring and 3-ring for each algorith. The x-axis indicates the targets available in the searching area. The y-axis indicates the expected. Although the nuber of rings n to achieve the overall optial is usually larger than 3, the optial 3-ring schee already perfors very close to the real optial. (6,3), (6,58). By investigating the results of these discovery requests, we can have an idea of the trend of the searching and the searching radius for different total nubers of targets and for cases of searching few/half/ost out of these targets. Only the results fro BF and are shown. For O, after finding k targets, the goal of the next search becoes finding k k out of k. Therefore, the expected of O is dependent on each searching result; hence it is hard to deterine analytically. The perforance of O will be shown later through siulations. As we can see fro Fig. 3, the perforance of these algoriths is still very close to each other and the curves overlap with each other. The larger the nuber of targets that need to be found, the less the can be reduced. Although the details are not shown here, the -ring and 3-ring schees are still doinant in the reduction and ore than 3-ring is unnecessary, which is the sae conclusion as in the -out-of- case. In suary, we find that the two-ring schee can provide close to optial perforance, and the three-ring schee can further reduce the by at ost 5%. More searching attepts can only reduce the by a negligible aount of less than % and are unnecessary. When only a few nuber of targets are to be found, or when k <<, the saving is significant. When ost of the targets are to be found, the is close to the siple flooding searching schee nuber of targets Fig. 3. The optial expected for k-out-of- discovery by different algoriths. The x-axis indicates the targets available in the searching area. The y-axis indicates the expected.,, targets are to be searched for each algorith. ) Algorith verification: Since our algoriths are all stochastically based and our odel is built over an infinite network, we experient with these algoriths in a large-scale geography-based network to verify that the experiental results atch our analytical expected. Also, we would like to exaine how these algoriths affect the discovery latencies copared to the one-ring searching schee, which is a issing part in our analysis. Hence, we place a large nuber of nodes, N T, in a disk area randoly and independently. Each node has the sae transission range of R t and the density is large enough for a well-connected network. The source node is located at the center of the unit area. The targets are chosen randoly fro these N T nodes, and the nuber of targets << N T. In this geography-based scenario, the source node controls its searching area by appending a searching radius liit on the query packet, and only nodes inside the distance liit will forward the query packet. Thus, it is assued that nodes know about their own geographic location. In this scenario, latency is defined as the round trip distance fro the source node to the target. For exaple, for the source node to hear the response fro the border, the latency is =. We experient on 3-ring BF, 3-ring and 3-ring O using the area set obtained fro analysis for BF and and record their and latency. The is copared to the expected of the 3-ring schee fro analysis. In the top row of Fig. 4, we show the results of the -out-of- discovery, and on

7 .8.6 : out of analysis 3 ring siu 3 ring BF siu 3 ring siu 3 ring O.5 latency: out of siu 3 ring BF siu 3 ring siu 3 ring O 6 5 N T :, R t :., x: CI:.39 experient estiation 5 N T :, R t :., x: CI: latency.5 N i : 4 3 N i : : / out of.8 analysis 3 ring siu 3 ring BF.75 siu 3 ring siu 3 ring O latency latency: / out of.6 siu 3 ring BF siu 3 ring.55 siu 3 ring O i hops i hops Fig. 5. The nuber of nodes N i,x at i hops away fro a node that is x away fro the border and its estiation Ñi,x. The results are shown for nodes with transission range.. The left plot is for the border node x =, and the right plot is for the center node x =. Fig. 4. The average and latency perforance for each algorith for geographical scenarios. The x-axis indicates the targets available in the searching area. The top row shows the results of -out-of- discovery, and the botto row shows the results of -out-of- discovery. the botto row, we show the results of the -out-of- discovery. In both cases, the of these algoriths is very close to the expected of 3-ring. This verifies our odel and analysis. For latency, O perfors a little better than the other algoriths. This is because it is ore aggressive in searching a larger area and tends to take fewer attepts to coplete the task. Thus, the corresponding latency tends to be saller. IV. MULTI-TARGET DISCOVERY IN SMALL-SCALE NETWORKS Previously, we studied the target discovery proble based on a siplified odel for infinite networks. In this odel, we do not differentiate source nodes since source nodes are always located at the center of the interested area to be searched. While these assuptions siplify analysis and are approxiately valid for large-scale networks, they do not hold true for sall-scale networks. In sall-scale networks, different nodes located at different positions in the network have different views of the network, especially for those nodes that are close to the network borders. Also, in sall-scale networks, source nodes are ore likely to search the entire network to coplete the discovery task instead of searching only a part of the network. Furtherore, since hop liits are ore widely applied to restrict flooding in sall-scale networks than geographical liits, we ust deterine how to transfor the analytical paraeter searching area set A (n) into hop liit values to ake our results practical. Let us reodel the proble for sall-scale networks. Suppose the network is a circle of unit radius, and there are a total of N nodes and a total of targets uniforly distributed in this network. A node wants to find k targets out of these targets. What is the best strategy for this node if it has knowledge about its distance fro the network border x? What is the best strategy if nodes do not know their location and have to apply a consistent syste-wide searching strategy? A. Self location aware Although we odify our assuptions and network odel to fit sall-scale networks, it is interesting to find that equations 8 and 6 still hold true no atter where the source node is located. This is because we assue that both the nodes and targets are uniforly distributed in the entire area, which akes the deduction of the searching and the probability of finding targets follow exactly the sae procedures. Therefore, the algoriths proposed based on these equations can still be applied to deterine the searching areas. The only question incurred by the sall-scale network odel is how to utilize these calculated areas in practical applications by transforing the into hop liits to restrict flooding. ) Mapping areas into hops: Previously, we provided detailed analysis on how to coplete this apping [7]. To avoid redundancy, we will just briefly discuss it here. The authors in [4] point out that each node has to be connected to ore than log N neighbors to have the network asyptotically connected with probability approaching one. In other words, in a well connected network, each node s transission range R t ust be large enough to satisfy this requireent. By investigating the nuber of nodes at different hops fro a node located x fro the border, we provided a ethod to estiate the nuber of nodes Ñi,x at i hops away fro the source node based on the value of N and the transission range R t. To concentrate on our analysis, we assue that we have the estiated sequence Ñi,x for a node located at x. Fig. 5 shows N i,x in a -node network with x = and x =, along with their estiates Ñ i,x. Since the calculated area A can be seen as the percentage of nodes to be searched, we only need to find out which hop liit leads to a ratio of nodes covered close to the value of A. Therefore, each node first calculates the ratio of nodes within k hops k j= Ñj,x N. Given the calculated area A i, T k,x using T k,x = the corresponding nuber of hop h i can be found fro [,M] where T hi,x is closest to the value of A i. M is the network diaeter, which can be estiated using the ethods provided in [7]. We show a apping exaple in Fig. 6. In a network with a total of nodes and transission radius of R t =., a node

8 6 An exaple on how to find the atching hop liit ~ step: find N i,x 5 : out of 3 latency: out of ~ 4 N i,x = step: transfored to Ti,x : out of latency: out of 8 T i,x = T i,x = step3: find atching points A () ={.48945,.}==>{5, } A (3) ={.48945,.7773,.}==>{5,9, } Hop liit : 8 out of x latency: 8 out of x Fig. 6. An exaple on how to transfor calculated areas into hop liits. First, N i,x is estiated. Second, noralized T i,x is deterined. Finally, hop liits can be found by atching the areas with T i,x. Fig. 7. Searching and latency coparison in sall-scale networks for selflocation knowledgable nodes. The x-axis indicates the source node location. perfors consistently better than schees and in ters of both and latency for all searching tasks. located at the network border with x = is able to estiate the nuber of nodes at exactly i hops Ñi,x as shown in the first plot. Then, it calculates T i,x as shown in the second plot. T i,x indicates what portion of nodes are located within i hops fro the source node. For a task of finding 3 targets out of, we have A () = {.48945,.} for a two-ring schee, and A (3) = {.48945,.7773,.} for a 3-ring schee. In the last plot, we find that T 5,x A = and therefore the first hop is 5 for the two-ring schee. Siilarly, T 9,x A =.7773, and 9 should be the second hop liit for the 3-ring searching schee. Therefore, the hop liit set should be {5, } for the two-ring schee and {5, 9, } for the three-ring schee. By using as the last hop liit, we ean that any large enough integer can be used just to flood the entire network. Fro this exaple, we can see that with the estiation of N i,x, we can easily ap the calculated area set A (n) into hop liits and apply these hop liits in sall-scale networks. ) Siulation results: Using the above apping ethod, we experient in a sall network coposed of N T = nodes with transission radius R t =. in a unit radius circle area. We copare our two-ring schee with existing schees and in a scenario where nodes know their distance to the border x. is actually a two-ring schee with searching radius {, M}. uses a searching radius set {,, 4., M}. For our schee, two-ring is applied due to its siplicity and its ajor ipact in reduction. Fig. 7 shows that when is large and k is sall, perfors close to despite the latency being nearly twice that of. Although we cannot explain this phenoena analytically, one possible explanation is due to the discretization when apping areas into hops. shows a close perforance to the one-ring searching, whose is. Fig. 7 shows that for nodes at different locations, perfors consistently low while the perforance of varies greatly. This is because chooses different hop liit values based on the source node s location to atch the sae optial searching area to achieve the sae optial, while chooses the sae searching strategy for every node. Finally, schee is only suitable for searching a sall nuber of targets in ters of. In addition, the excessive searching expansion procedure incurs unnecessary latency increase. For finding -out-of- targets, the schee ay have a larger than, even worse than the siple one-ring searching schee. B. Self location unaware In a ore general scenario, nodes do not know their positions in the network, and they have to apply the sae searching strategy. The optial searching strategy that iniizes the searching has to be reconsidered fro the syste s point of view. We liit our scope to the two-ring searching schee since the three-ring schee does not bring significant iproveent. Although this conclusion is drawn fro the idealized infinite network odel, we believe that it should still be valid for hop-based cases as well. Now, there is only one paraeter we need to deterine, the first searching hop k sys. ) Deterining k sys : It is easy to deterine that for a uniforly distributed network, the probability that a rando node is located x away fro the border is f X (x) = ( x) x (9) Given a sapling of x and an arbitrary k sys value, we can find the corresponding searching area by calculating A(k sys, x) = T ksys,x based on Ñi,x as in the last section. Then putting A(k sys, x) into equation 8 or 6, we can obtain the searching C(k sys, x) for this specific node with the searching hop liit at k sys. Therefore, the syste-wide expected that takes into account every node can be expressed as C sys (k sys ) = f X (x)c(k sys, x)dx δ f X (iδ)c(k sys, iδ) i= ()

9 Here we propose our two-ring schee based on Eq.. For each possible value of k less than the estiated network diaeter M, we saple x fro [,] using sapling interval δ and deterine the corresponding C(k, x). We then use equation to calculate the syste C sys (k), and deterine the optial first searching hop k opt where the inial C sys is obtained. ) Siulation results: We siulate this location unaware scenario following the above procedure. First, we need to clarify the sapling interval δ and its effects on the accuracy of the first hop liit and the coputational coplexity. Fro table III, we find that when δ decreases as the sequence {.,.5,.,.,.5}, the hop liit of the out of task is always 7, and the hop liit of the out of task is {4, 5, 4, 3, 3}, while the nuber of coputations increases linearly with δ. This infors us that although a sall interval ay bring about ore accurate hop count calculation, the iproveent is restricted since the hop liit ust be chosen as an integer. We believe that the interval of. is good enough for use, and we apply δ =. in the rest of our siulations. The for the 8-out-of- task indicates that there is no better schee to find 8 targets ore efficiently than just searching the entire area once by using a large enough hop liit. TABLE III THE IMPACT OF SAMPLING INTERVAL δ. δ Coputations First searching hop of {,, 8}-out-of-. {7, 4, }.5 {7, 5, }. 5 {7, 4, }.5 {7, 3, } In Fig. 8, we copare the syste-wide and latency perforance of different schees. In, the first hop liit of {7, 4, } are used for finding {,, 8} out of targets. Again, perfors consistently well for all the tasks. When k is sall as for the -out-of- task, perfors close to in ters of with a uch longer latency. The estiated network diaeter M, as seen fro the left plot of Fig. 5, is 9, and the largest possible network diaeter fro statistical results is 33. Therefore, using any nuber larger than 33 as the first hop liit is actually a one-ring searching schee. 3) Robustness validation: As entioned earlier, our algorith outperfors schees and because it utilizes knowledge of the network paraeters N and to choose the optial searching hop liits. The schee, on the other hand, also requires this topology inforation to a certain degree. First, needs to deterine if the task of k-out-of- is feasible by checking k <. Then, it requires N to estiate the network diaeter M so that it knows when it should stop the expansion searching. Failure to estiate M ay lead to redun- latnecy and latency perforance coparison in sall scale networks out of out of out of out of 8 out of 8 out of Fig. 8. Searching and latency coparison in sall-scale networks for selflocation unaware nodes. perfors consistently close to optial in ters of both and latency for all searching tasks. dant attepts to flood the entire network, especially when the task cannot be copleted. During the network design phase, the scale of the network is usually deterined and the value of N ay be roughly estiated. The inforation of the server nubers can be achieved by letting each server announce its existence through broadcasting when a service is available. Due to the dynaic nature of the network, knowing the existence of a service does not explicitly help in finding a server. Therefore, the announceent can be so infrequent that the extra of this announceent can be seen as negligible when the nuber of service requests is large. Also, nodes newly joining in the network can ask their neighbors for the nuber of services available and have a rough idea about the services available. Despite the fact that both and require knowledge of N and, needs it to be uch ore accurate. Erroneous N and ay lead to erroneous calculation of the first hop liit and thus affect the final searching. In this section, we will study the ipact of erroneous paraeters and test the robustness of our algoriths. First, for a network of N nodes, let us define the error of N as e N = Ñ N N. Ñ is the estiated total nuber of nodes in the network by a specific source node. Siilarly, we can define the error for the nuber of targets as e =, where is the estiated total nuber of targets in the network. Although e N and e are two different types of errors, when applying using these erroneous values, they eventually end up in an erroneous value of the first hop liit. For exaple, for the -out-of- task, the hop liits calculated based on erroneous e N or e are shown in table IV. An exaple of how these erroneous first hop liits affect the can be found in Fig. 9. Only when the error is very large, e.g., as large as e = %, does the increase fro the optial 65 transissions per search to 364 transissions per search. For not so large errors, the will be 79 or 35 transission, which is not so far away fro the of the optial -ring searching schee, 65 transissions. For sall-scale networks, siilar conclusions can be drawn as in the last section. When the knowledge of N and is accu-

10 TABLE IV THE IMPACT OF ERRONEOUS N AND ON COST. e N -5% -4% -3% -% -% % % 3% 4% 5% st hop e -% -8% -6% -4% -% % 4% 6% 8% % st hop perforance of different first hops in a two ring searching optial by δ=.5 e =4~%, e N =~5% optial by δ=. e N = 3~ %, e = 4~% e N = 5~ 4%, e = 8~ 6% e = % 5 5 first hop Fig. 9. The for all the possible first hops using a two-ring searching in sall-scale networks. The x-axis indicates the first hop liit. rate enough, can be applied to save while reducing the latency by about 5% copared to. When N and cannot be accurately estiated but the nuber of nodes to be found k satisfies k <<, can be perfored to reduce while doubling the latency. When k is close to or when no inforation is known about the network topology, a siple flooding searching schee is not bad since its latency is the sallest and it ay perfor even better than an arbitrary n-ring schee. The schee shows a trivial iproveent and a trivial latency reduction copared to the -ring schee, and hence is of little practical value. In suary, we illustrate how to apply our forer analytical results to realistic hop-based sall-scale networks. If a node has knowledge about its current location, it can calculate the optial searching area A using and transfor it into the desired hop liits. If nodes do not have the knowledge of their location in the network, a consistent two-ring searching strategy can be utilized for all nodes to reduce the searching fro a syste-wide perspective. V. CONCLUSION AND DISCUSSIONS In this paper, we studied the ulti-target discovery proble in wireless networks. Through analysis, we provide several algoriths to deterine the optial searching strategies to reduce. We found that a -tier search usually perfors close to optial already and a 3-tier search can further reduce the searching by a trivial aount of around 3%. Our analysis is based on general assuptions, and the conclusions are universal to wireless networks. Siulations validate our analysis and show the practical value of our schees in realistic scenarios. One assuption of this paper is that nodes and targets are all uniforly distributed within the network. The liitation of this assuption is obvious in that our algoriths cannot be directly applied to non-unifor target distributions. One potential area for future work is to deterine the optial solution under the circustances where nodes and targets are not uniforly distributed. In soe scenarios, nearer areas ay contain ore targets of interest than farther areas. We expect that the first several searching rings should be even saller than that in the unifor distributed network. In soe other scenarios, farther areas ay contain ore targets of interest, e.g., caches are ore likely to be around the targets and be far fro the source nodes. In this case, a good strategy should be able to direct the query packets to the targets surroundings as fast as possible. Also, the effects of ore realistic wireless propagations and transissions ay be taken into account and investigated, although we expect the to have the sae effects on and. REFERENCES [] D. B. Johnson and D.A.Maltz. Mobile Coputing, Chapter Dynaic source routing in ad hoc wireless networks, pages Kluwer Acadeic Publishers, Iielinski and Korth edition, 996. [] C.Perkins and E.M.Royer. Ad hoc on-deand distance vector routing Proceedings of IEEE WMCSA 99, pp. 9, Feb [3] E. Woodrow and W. Heinzelan, SPIN-IT: A Data Centric Routing Protocol for Iage Retrieval in Wireless Networks, Proc. International Conference on Iage Processing (ICIP ), Sep. [4] C. Intanagonwiwat and R. Govindan and D. Estrin, Directed diffusion: a scalable and robust counication paradig for sensor networks, Mobile Coputing and Networking, pp ,. [5] Mills, D.L., Network Tie Protocol (Version 3), RFC (Request For Coents) 35, March 99. [6] Sun Mircorsystes. Syste and Netowrk Adinistration, March 99. [7] Z. Cheng, W. Heinzelan, Flooding strategy for target discovery in wireless networks, Proc. of the 8th international workshop on Modeling analysis and siulation of wireless and obile systes (MSWIM 3), Sep 3. [8] T. Wu, M. Malkin, and D. Boneh. Building intrusion tolerant application. Proc. of the 8th USENIX Security Syposiu, 999. [9] Gnutella peer-to-peer file sharing syste. [] M. Roan, C. K. Hess, R. Cerqueira, A. Ranganathan, R. H. Capell, and K. Nahrstedt. Gaia: A iddleware infrastructure to enable active spaces, IEEE Pervasive Coputing,pp , Oct-Dec. [] N. Bulusu and J. Heideann and D. Estrin. Adaptive beacon placeent, In Proceedings of the st International Conference on Distributed Coputing Systes (ICDCS-), pp , Phoenix, Arizona, USA, April. [] T. Oates, M. V. Nagendra Prasad and V. R. Lesser. Cooperative Inforation Gathering: A Distributed Proble Solving Approach, Coputer Science Technical Report version, University of Massachusetts, Aherst, MA. [3] C. Carter, S. Yi, P. Ratanchandani, and R. Kravets. Manycast: Exploring the Space Between Anycast and Multicast in Ad Hoc Networks, Proc. on Mobile Coputing and Networking archive Proceedings of the 9th annual international conference on Mobile coputing and networking (MO- BICOM) pp San Diego, CA, USA, 3. [4] F. Xue and P. R. Kuar. The nuber of neighbors needed for connectivity of wireless networks Manuscript,. Available fro files.htl [5] G. Engeln-Mullges, F. Uhlig. Nuerical Algoriths with C, Springer, 996. [6] L. L. Peterson, B. S. Davie. Coputer networks, a syste approach, Morgan Kaufann,.