Controlled Flooding Search with Delay Constraints

Transcription

1 Controlled Flooding Search with Delay Constraints Nicholas B. Chang and Mingyan Liu Departent of Electrical Engineering and Coputer Science University of Michigan, Ann Arbor, MI Eail: Abstract In this paper we consider the proble of query and search in a network, e.g., searching for a specific node or a piece of data. We liit our attention to the class of TTL tie-to-live) based controlled flooding search strategies where query/search packets are broadcast and relayed in the network until a preset TTL value carried in the packet expires. Every unsuccessful search attept results in an increased TTL value i.e., larger search area) and the sae process is repeated. Every search attept also incurs a cost in ters of packet transissions and receptions) and a delay tie till tieout or till the target is found). The priary goal is to derive search strategies i.e., sequences of TTL values) that iniize a worstcase cost easure subject to a worst-case delay constraint. We present a constrained optiization fraework and derive a class of optial strategies, shown to be randoized strategies, and obtain their perforance as a function of the delay constraint. We also use these results to discuss the trade-off between search cost and delay within the context of flooding search. Index Ters data query and search, TTL, controlled flooding search, wireless sensor and ad hoc networks, constrained optiization, randoized strategy, copetitive analysis I. INTRODUCTION Query and search for an iportant functionality for any network applications. Searching for a destination node whose location is unknown is a prie exaple frequently encountered by ad hoc network routing protocols and services, e.g., [1, [2, [3. Other exaples include the search for certain data of interest in an environental onitoring sensor network [4, and ore broadly, the search for a shared file in a peer-topeer P2P) network. A good search echanis should have a short response tie, e.g. the tie it takes to find the object of interest, and should do so with inial cost. Such cost refers to the aount of processing and transissions incurred by the search. This is particularly iportant in a wireless context where cost is associated with energy consuption, easured by the aount of packet transissions and receptions. There are a variety of echaniss one ay use to conduct search. These include aintaining a centralized directory service, or by sending out a query packet that traverses the network in a certain way, e.g., the ruor routing proposed in [4 and the rando walk ethod in [5. In this paper we focus on the class of TTL-based controlled flooding search. This decentralized search echanis This work is ported by NSF award ANI , ONR grant N , and through collaborative participation in the Counications and Networks Consortiu sponsored by the U. S. Ary Research Laboratory under the Collaborative Technology Alliance Progra, Cooperative Agreeent DAAD is widely used, e.g., in ad hoc routing protocols [6 as well as in P2P networks [7. Under this schee the node originating the search also referred to as the source node) sends out a query packet that carries an integer TTL tie-to-live) value. If the underlying network is wired, this query will be transitted once along each outgoing link of the source. In a wireless network, this packet reaches all iediate neighbors in a single broadcast transission. If the search target is found at a neighboring node, it will reply to the source. Otherwise it decreents the TTL value by one and retransits the query packet containing the new TTL value. This continues until TTL reaches zero. Thus how uch of the network has been queried is controlled by the TTL value. If the target is not found in this search area, the source node will eventually tie out and initiate another round of search covering a bigger area using a larger TTL value, and presuably setting a larger tieout value for that round of search. This process continues until either the object is found or the source gives up. Hence the perforance of a search strategy both in ters of cost and delay is deterined by the sequence of TTL values used. Controlled flooding search has previously been studied in [7, [8, [6, [9. In [8 it was shown that if the target location distribution is known, then a dynaic prograing forulation can be used to find TTL sequences that iniize the average search cost. When this distribution is not known, [7 derived the optial deterinistic TTL sequence that iniizes a worst-case cost easure. For the sae cost easure [8 showed that the best strategies are randoized TTL sequences. [9 further derived the optial randoized strategy. All these studies focused only on the cost of search and did not consider the delay of search. The priary goal of this paper is to derive TTL controlled flooding search strategies that perfor well both in ters of the search cost and the search delay. While there are a variety of ways to handle ultiple potentially conflicting) objectives, in this study we will approach this by forulating a constrained optiization proble. Specifically, we will attept to iniize a cost easure subject to a delay constraint. The solution to this proble results in search strategies that iniize the search cost and locate the target within a specified tie constraint. Iposing a delay constraint also allows us to study the trade-off between search cost and search delay. This trade-off can be seen by considering the strategy of flooding the entire network e.g., setting the TTL to be the axiu allowed value, assued sufficient to cover the whole network). Such

2 a strategy would ost likely result in a short search delay, as the target node is likely to be found during the first round. On the other hand, this strategy is not the ost cost effective [7, [8, as it results in a large aount of packet transissions and retransissions. Conversely, it ay be the case that soe strategy incurs extreely low cost but suffers fro high delay. Under the constrained optiization fraework studied in this paper, we will be able to conveniently address the above perforance trade-off. The ain contributions of this paper are suarized as follows: 1) We provide an analytical fraework within which the delay of search strategies can be studied along with their cost. The previously studied unconstrained proble siply iniizing a cost easure) [7, [9 becoes a special case under this fraework when the delay constraint is not binding. In this sense, this ethod is a key generalization of prior work and presents a uch ore powerful analytical tool. To the best of our knowledge, delay has not been studied in this context before. 2) When a worst-case delay constraint is iposed, we derive a class of optial strategies that iniize worstcase cost easure aong all strategies that satisfy the delay constraint. In [8, it was shown that randoized strategies outperfor deterinistic strategies in the unconstrained proble. With this study we show that the sae holds when a worst-case delay constraint is iposed. 3) We establish an understanding of the trade-off between delay constraints and corresponding optial achievable cost, and show specifically how the two conflicting objectives can affect each other. In addition to the above, our proble forulation and abstraction generalize to a uch larger class of optiization probles involving constrained resource allocation. This is discussed at the end of this paper. The rest of the paper is organized as follows. In Section II we present the network odel and assuptions used in this study. We then introduce our perforance easures and objectives, along with the ain results of this paper in Section III. In Section IV we derive the optial worst-case strategies satisfying a delay constraint. These results are discussed and exained in Section V. Section VI concludes the paper. II. NETWORK MODEL A. Model and Assuptions We will liit our analysis to the case of searching for a single target, which is assued to exist in the network. For the rest of our discussion we will use the ter object to indicate the target of a search, be it a node, a piece of data or a file. Within the context of controlled flooding search, the distance between two nodes is easured in nuber of hops, assuing that the network is connected. Two nodes being one hop away eans they can reach each other in one transission. We easure the position of an object by its distance to the source initiating the searching. We will use the ter object location to indicate the iniu TTL value needed to locate the object, denoted by X. Theternetwork diension refers to the iniu TTL required to reach every node in the network, denoted by L. Also, FX u) P X >u) denotes the tail distribution of the rando variable X. We will assue that a TTL value of u will reach all nodes within u hops of the source and will find the object with probability 1 if it is located within u hops, when the tier expires. This assuption iplies that a tieout event is equivalent to not finding the object in the u-hop neighborhood, and that flooding the entire network will for sure locate the object. This is reasonable in a wired network, as long as packet loss is low and tieout values are properly set to sufficiently account for delay in the network. On the other hand, this assuption is a siplification in a wireless network because packet collisions and corruption losses can cause the query propagation process to be uch ore rando and less reliable. This assuption nevertheless allows us to reveal soe very interesting fundaental features of the proble and obtain valuable insights. A search strategy u is a TTL sequence of certain length N, u [u 1,u 2,,u N. It can be either fixed/deterinistic where u i,i 1,,N, are deterinistic values, or rando where u i are drawn fro probability distributions. For a fixed strategy we assue that u is an increasing sequence. For randoized strategies, we assue all realizations are increasing sequences. The requireent for the sequence to be increasing is a natural one under the assuption that search with TTL u will always find the object if it is indeed within u of the source. Note that in a specific search experient we ay not need to use the entire sequence; the search stops whenever the object is found. In practice, it is natural to only consider integer-valued discrete) policies. However, considering real-valued sequences proves to be helpful in deriving optial integer-valued strategies. For this reason we will also consider continuous realvalued) strategies, denoted by v, where v [v 1,v 2,,v N, and v i is either fixed or a continuous rando variable that takes real values. When considering discrete strategies, TTL values are integers and the object location X is assued to be a positive integer taking values between 1 and L. In analyzing continuous strategies, X is assued to be a real nuber in the interval [1,L. A strategy is adissible if it locates any object of finite location with probability 1. For a fixed strategy this iplies u N L. For a rando strategy, this iplies Pru i L for soe 1 i N) 1. In the asyptotic case as L, a strategy u is adissible if x 1, Pru n x for soe n Z + )1. This iplies that in the asyptotic case, u is an inite-length vector. We let V denote the set of all real-valued adissible strategies rando and fixed). U denotes the set of all integer-valued adissible strategies rando and fixed).

3 B. Search Cost and Delay We will associate a cost Cu) with a single round of search using TTL value u. The functional for of this cost depends on the properties of the network as well as the underlying broadcast techniques used. In our analysis we will ignore these details and siply assue that such a function is obtainable, i.e., by estiating the nuber of transissions and receptions, etc. Cu) is therefore an abstraction of the lower layer properties, and for the rest of our discussion we will no longer regard network as wired or wireless, but only discuss in ters of the search cost Cu). Note that in general, a node receiving the search query will be unaware whether the object is found at another node in the sae round except perhaps when the object is found at one of its neighbors, or soe other ore sophisticated schees are eployed). Thus this node will continue the process by decreenting the TTL value and passing on the search query. We can therefore regard the search cost as being paid in advance, i.e., the search cost for each round is deterined by the TTL value and not by whether the object is located in that round. We next introduce the search delay function. We denote by D t u) the tieout value used when searching with TTL u. This is the delay incurred when the object is not found using u, i.e., when u<x. On the other hand, if u X, then the object will be found within this round of search. The delay incurred in this case is the aount of tie it takes for the query to propagate X hops and for the reply to reach back to the source. We will denote this delay by D r X) for object location X. Therefore atheatically the search delay of using TTL value u can be written as: Iu <X)D t u) +IX u)d r X), where I is the indicator function: IA) 1if A is true and 0 otherwise. For real-valued sequences, we require that the function Cv) and Dv) be defined for all v [1, ), while for integer-valued sequences we only require that the cost and delay functions be defined for positive integers. When the cost function is invertible, we write C 1 ) to denote its inverse. We will adopt the natural assuption that Cv 1 ) >Cv 2 ) and Dv 1 ) >Dv 2 ) if v 1 >v 2. We also denote by C the class of cost functions C :[1, ) [, ), that are increasing, differentiable, and have the property li v Cv). Finally, we let C 1 denote the set of functions C ) C such Cx+1) that li x Cx) 1. Note that even though C 1 is a subclass of C, it contains all polynoial cost functions and therefore reains very general. III. PROBLEM FORMULATION AND MAIN RESULTS A. Proble Forulation We will consider the search perforance in the asyptotic regie as L. This is because it is difficult if at all possible to obtain a general strategy that is optial for all finite-diension networks as the optial TTL sequence often depends on the specific value of L. In this sense, an asyptotically optial strategy ay provide uch ore insight into the intrinsic structure of the proble. It will becoe evident that asyptotically optial TTL sequences also perfor very well in a network of arbitrary finite diension. Let u denote the expected search cost of using strategy u when the object location is X. This quantity can be calculated as follows: u E u E X I X >u k 1 ) Cu k ) 1) E u F X u k 1 )Cu k ), 2) where u 0 0, E u and E X denote expectations with respect to u and X, respectively. The expectation and suation can be interchanged due to the Monotone Convergence Theore [10. We will drop the variable fro the subscript when it is clear which variable the expectation is taken with respect to. Siilarly, let DX u denote the expected search delay induced by strategy u for X. This quantity can be calculated as follows: DX u E u E X IX >u k )D t u k ) + E u E X Iu k X>u k 1 )D r X) E u F X u k )D t u k ) + E X [D r X). 3) When the distribution of X is known in advance, a natural objective is to deterine strategies that iniize u subject to soe constraint on DX u. In general, such coputations are nuerical and the optial solutions can be deterined by standard constrained optiization techniques [11 [12. In Section IV-B, we will derive the optial strategy for a particular distribution of X and delay constraint under which the optial strategy has a very interesting structure. On the other hand, when the distribution of X is not known in advance, as is often the case, then we need a different approach. In this study we adopt a worst-case perforance easure. Consider an oniscient observer who knows the object location in advance and will use a TTL of X, incurring an expected cost of. We can then easure the perforance of a strategy u by the following: ρ u, 4) where denotes the set of all probability distributions for X such that <. Theterρ u is an upperbound, or worst-case easure, on the ratio between the cost of strategy u and the oniscient observer, over all X. We will refer to ρ u as the copetitive ratio, or worst-case cost ratio, of u. This type of worst-case easure is coonly used in any online decision and coputation probles [13. It was introduced in [7 as a ethod of analyzing flooding strategies, and generalized in [8 to study randoized strategies.

4 We apply a siilar worst-case analysis to delay. The iniu expected delay is E[D r X), obtainable by either an oniscient observer or a strategy that uses the highest TTL u [L as L ). Hence the worst-case delay ratio is defined as: τ u D u X E[D r X), 5) where we note in this case is the set of all location distributions such that E[D r X) <. Note that the worstcase cost and delay ratios are always strictly greater than 1 for any adissible strategy as it is ipossible to equal or do better than the oniscient observer. We define the following set: { U d u U : D u X E[D r X) d }, 6) for soe constant d > 1. This is the set of all strategies whose delay is always within a factor d of the delay of the oniscient observer, regardless of X. We will call d the delay constraint. Note that as d, the delay constraint becoes less restrictive and the set U d approaches U. We seek a strategy that satisfies this delay constraint d and has the sallest worst-case cost ratio, i.e. achieves the iniu worst-case cost ratio aong all u U d : ρ d u U d. 7) This essentially constitutes our constrained optiization proble P), rewritten as follows: u s.t. 8) D u X E[D r X) d 9) Note that the two reus in P), one in the objective function and the other in the constraint, are in general not achieved under the sae distribution p X x). The intention for adopting such a worst-case forulation, which ay be viewed as soewhat conservative, is to place an upper bound on both the delay and the cost over all possible locations. The above definitions also hold analogously for continuous strategies, by siply replacing U with V, and replacing the set with {f X x)}, which is the set of density functions such that <, ore[d r X) < depending on whether we consider worst-case cost or delay. We will thus denote ρ v, τ v and V d as the continuous versions of 4), 5), 6), respectively. We will use the sae notation ρ d to denote the iniu worst-case cost ratio achieved by continuous strategies satisfying a delay constraint d; the distinction should be clear fro the context. V d is defined as follows for any d>1: { V d v V : {f X x)} D v X E[D r X) d }. 10) We now show that there is no loss in generality in assuing that D t ) D r ). Let D X u denote expected delay of strategy u for object location X when these two functions are equal. Then note the following: DX u E[D r X) E[D tx) E F u X u k )D t u k ) +1 E[D r X) E[D t X) E[D tx) E[D r X) D u X E[D t X) 1 ) +1 11) Hence, the delay ratio when D t D r is siply a rescaling of the ratio when D t and D r are the sae functions. Specifically, astrategyu satisfies DX u /E[D rx) d if and only if: D X u E[D t X) E[D rx) d 1) ) E[D t X) Therefore, the set U d that we defined for the case of D t D r can easily be redefined if D t D r, by siply rescaling the delay constraint d. Note that this result holds in both the discrete and continuous cases. Therefore for the rest of the analysis, we will assue these two functions are equal while noting that the results apply to the unequal case by scaling the constant d. WeletDu) D t u) D r u) for all u. It follows that using a TTL value u for object location X will incur a delay of D in {X, u}). B. Main Results In this section we present our ain results to be proven and discussed later in this paper. We begin by exaining optial continuous strategies, i.e., finding the strategy in V d that achieves iniu worst-case cost ratio. We define the following class of continuous strategies: Definition 1: Assue that the cost function C ) C. Let v[r, F v1 ) denote a jointly defined sequence v [v 1,v 2,... with a configurable paraeter r, generated as follows: J.1) The first TTL value v 1 is a continuous rando variable taking values in the interval [ 1,C 1 r ) ),with its cdf given by soe nondecreasing, right-continuous function F v1 x) Prv 1 x). Note that this eans F v1 1) 0 and F v1 C 1 r ) ) 1. J.2) The k-th TTL value v k is defined by v k C 1 r k 1 Cv 1 ) ) for all positive integers k. Fro J.1) and J.2), it can be seen that r and F v1 ) uniquely define the TTL strategy, and that given the selection of v 1,the cost of successive TTL values essentially for a geoetric sequence of base r, i.e., Cv k )r k 1 Cv 1 ). More discussion on this structure is given in Section V. Our ain theore regarding the class of continuous strategies V is as follows. Theore 1: When C ) C and C ) βd ) for soe, β > 0, wehave: 1) For any fixed 1 <d<+1, v V d {f X x)} d 1) e d 1. 13)

5 Moreover, this iniu worst-case ratio is achieved by using 1 C ) the strategy v[r, with r e d 1. 2) For d +1,wehave: v V d {f X x)} v e. 14) Moreover, this iniu worst-case ratio is achieved by using 1 C ) the strategy v[r, with r e. Note the optial strategy of Theore 1 can be adjusted for different delay constraints by varying the paraeter r. These optial continuous strategies will be used to derive discrete strategies which perfor well in the worst-case and are optial for a subset of C. In particular, we have the following. Theore 2: When C ) C and C ) βd ) for soe, β > 0, wehave: 1) For 1 <d<+1, u U d 2) For d +1, d 1) e d 1. 15) e. 16) u U d Whether the upper bounds in Theore 2 becoe equalities appears to depend on the specific cost function C ). By restricting our attention to cost functions C ) C 1 we have the following result. For any strategy v, let v denote the strategy [ v 1, v 2,, where v k denotes the greatest integer less than or equal to v k. Theore 3: Consider C ) C 1 and C ) βd ) for soe, β > 0. 1) For 1 <d<+1, u U d d 1) e d 1, 17) where this iniu worst-case ratio can be achieved by the discrete strategy u v, where v denotes strategy v [e d 1, d 1 C ) given by Definition 1. 2) For d +1,wehave u U d u e. 18) Moreover, this iniu worst-case cost ratio is achieved by strategy u v, where v denotes strategy v [e, C ). This result shows that we can take the floor of the optial continuous strategy to obtain a discrete strategy which is optial when the cost is a subclass of C. C. Discussion of Main Results The ain results described in the previous subsection are derived under a worst-case perforance easure. This iplies that for any object location, the optial for 1 <d<+1) strategy v of Theore 1 has an expected search cost within d 1 e d 1 ties the expected cost of the oniscient observer. Siilarly, its expected delay is always within factor d of the delay incurred by an oniscient observer. The differentiation between the two cases, 1 <d<+ 1 vs. d +1, in all three theores is due to the fact that the optiization proble P) has an active/binding constraint in the forer, and an inactive/non-binding constraint in the latter, as we show in the next section. The ain results rely on the relationship C ) βd ) for soe, β > 0, where the factor essentially describes the relative rate at which the cost and delay functions grow with respect to TTL. First note that the constant positive factor β cancels out in the cost or delay ratio calculated in 4) and 5). Hence we can assue β 1without loss of generality. Secondly, the relationship C ) D ) holds, for exaple, in a very representative case of a searching in a 2-diensional network with search cost proportional to the nuber of transissions incurred. In this case Cv) is well approxiated by a quadratic function see e.g., [7, [8) and Dv) can be chosen to be a linear function of v iplying 2), or quadratic iplying 1). Another scenario described by this relationship is when 1, where the cost and delay scale in the sae fashion. This could be a good odel in a linear network with constant node density where both cost and delay increase proportionally linearly) to the nuber of transissions. IV. OPTIMAL STRATEGIES WITH DELAY CONSTRAINTS In this section we prove the results shown in the previous section, i.e., the solution to proble P). The solution approach we take is outlined as follows. We first in Section IV-B) consider the continuous version of proble P) and derive a tight lower-bound to the iniu worst-case cost under the delay constraint. This is accoplished by interchanging the and in 7), and introducing a constrained optiization proble whose objective is to iniize the average search cost subject to a delay constraint. Then in Section IV-C we derive a class of randoized continuous strategies whose worst-case cost ratio atches this lower bound for all d, proving that they are optial. These continuous strategies are then used in Section IV-D to derive good discrete strategies whose perforance is at least as good in the worst case. We will also prove that they are optial for a subclass of C. Unless otherwise stated, all proofs can be found in the Appendix. A. Preliinaries The following leas are critical in our subsequent analysis. We will let Jx u and Dx u denote the expected search cost and delay, respectively, of using strategy u when PrX x) 1. Lea 1: For any search strategy v V and u U, {f X x)} Jx v x [1, ) Cx), 19) Jx u x Z Cx), + 20) where Z + denotes the set of natural nubers. Proof of this lea can be found in [9. In words, this lea states that the cost ratio is axiized when the object

6 location is a single point. We also have an analogous lea for search delay. Lea 2: For any search strategy v V and u U, {f X x)} D v X E[DX) D u X E[DX) Dx v x [1, ) Dx), 21) Dx u x Z Dx). + 22) Proof of Lea 2 can be found in [14. These two leas reduce the space over which the worst-case cost or delay can occur, and thus are very useful in subsequent analysis. B. A Tight Lower Bound Consider any d>1. To establish a tight lower bound to the iniu worst-case cost ratio, we can interchange iu and reu [11 to obtain the following: {f X x)} v V d v v V d {f X x)}. 23) Any lower bound of the left-hand side of 23) can be found by fixing soe object location distribution f X and finding the strategy within V d that iniizes the expected cost. Note that the strategy in V d that iniizes the cost ay be randoized, which akes the iniization very difficult. Therefore, we further lower-bound the left hand side by considering a larger set of strategies than V d. In particular, let V d Y ) denote the following set of strategies for soe object location Y such that E[DY ) < : V d Y ) { v V : D v Y E[DY ) d }. 24) Clearly, V d Y ) V d for any Y because any v V d has a delay ratio upper bounded by d for all object locations. Therefore, {f X x)} v V d Y ) {f X x)} v V d, 25) because for any object location X, the iu on the right hand side is over a saller set. A valid lower bound of the left hand side of 25) can be obtained by choosing particular distributions for X and Y, and finding the strategy within V d Y ) that iniizes the expected cost. To obtain a tight lower bound, we need to find a cobination of X and Y such that the optial average-cost strategy under X satisfying the delay constraint induced by Y has a high expected cost ratio. It is iportant to note that it is not necessary that Y and X have the sae distribution; this property allows us to obtain tight lower-bounds. We consider the following proble P1), whose solution not only provides a tight lower bound to 23) but also serves as an exaple for deriving optial average-cost strategies subject to a delay constraint. Proble 1: Suppose C ) βd ). Let FX x) P X >x)cx)/), and F Y x) P Y > x) Cx)/) +1 1,forsoe>1and for all x 1. Consider the following constrained optiization proble: v DY v s. t. v E[DY ) d 26) We solve the above proble for the following choice of : 1) If 1 <d<+1, choose to be such that 1 <<1+ +1 d d 1). 27) 2) If d +1, choose any >1. As explained earlier, the distinction between the two cases is that Proble 1 under the forer 1 <d<+1) has a binding constraint, while it has a non-binding constraint under the latter d +1), which also eans in this case Proble 1 reduces to an unconstrained optiization proble. Solution: The optial strategy v for this proble satisfies Cv j )/ γ j for all j. Thevalueofγ depends on d as follows details can be found in the Appendix). If 1 <d<+1, then γ is: ) 1 1 1) γ 1+ [ 1) + 1d 1). 28) The optial cost ratio for this case is given by: v [ ) d d γ. 29) If d +1, then γ and the optial cost ratio is Using this solution, we see that as approaches 1 fro above, the optial cost ratio for the case 1 <d<+1 has the following liit: li 1 + d 1) e d 1, 30) where the liit is reached fro below. When d +1, then the optial cost ratio satisfies: li e. Hence, the highest iniu cost ratio is lower bounded as follows: Theore 4: When C ) βd ) for β, > 0, for any 1 <d<+1the best worst-case cost ratio is lower bounded by the following: v V d {f X x)} d 1) e d 1. Therefore, any strategy in V d which achieves a worst-case cost ratio of d 1) e d 1 ust be optial. Siilarly, when d +1,wehave: v V d {f X x)} v e. 31) Therefore any strategy in V d which achieves a worst-case cost ratio of e ust be optial. We next derive strategies achieving the lower bounds established above.

7 1 <d<+1 d +1 Fig. 1. Continuous Th 1 1) ρ d d 1 e d 1 optial strategy: v [e d 1, d 1 C ) Th 1 2) ρ d e optial strategy: v [e, C ) Discrete Th 2 1) ρ d d 1 e d 1 Th 3 1), C ) C 1 ρ d d 1 e d 1 optial: u v Th 2 2) ρ d e Th 3 2), C ) C 1 ρ d e optial: u v Suary of results on optial worst-case strategies under P). C. Optial Delay-Constrained Strategies For convenience, we suarize the ain results given in Section III in Figure 1. In this and the next subsection we will prove these results. We proceed to find strategies that atch the lower bounds established in the previous subsection. To do so, we will consider strategies of the for v[r, F v1 x) given by Definition 1. Lea 3: Assue C ) C and C ) βd ) for soe β, > 0. Then for any strategy v[r, F v1 ), its worstcase delay ratio is given by: { } 1 gr 1 )+r 1 1)gz) g z) +1, r 1 1 zd1) D1) 1 z<r 1 where gz) is defined as follows for 1 z r 1 : gz) D1) + z D1) D1) F v1 D 1 y)) dy, 32) and g z) denotes the derivative of g with respect to z. Due to space liitations, the proof of this lea is not included in the Appendix, but can be found in [14. Consider the faily of strategies of the for v[r, 1 C ),wehave: gz) D1) [ z z z + z, 33) and g z) D1)1 z ) for all z. We have the following results regarding this faily of strategies: 1) The worst-case delay ratio of these strategies is +1. This is easily verified by using Lea 3. r 2) The worst-case cost ratio of these strategies is. This result was proven in [9. We consider two special cases of this faily of strategies. The first case is when r e d 1 for soe 1 <d<+ 1. With the above results, the worst-case delay ratio of this strategy is exactly d. Hence this specific strategy belongs to V d. Meanwhile, its worst-case cost ratio is d 1) e d 1 plugging r e d 1 into r ), achieving the lower bound established in Theore 4. The second case is when r e. In this case we achieve a worst-case delay ratio of +1 and worst-case cost ratio of e. Hence when d +1, this strategy belongs to V d and is optial since it atches the lower-bound established in Theore 4. If d>+1, the delay constraint becoes non-binding under this strategy. Thus for d>+1 this is also the solution to the unconstrained proble. This result was proven separately in [9 within the context of an unconstrained optiization proble, which we have now shown to be a special case of the ore general result in this paper. Cobining these two cases together, we obtain Theore 1. Therefore, we have obtained the optial worst-case continuous strategies for any delay constraint d>1. D. Optial Discrete Strategies As stated earlier, we are interested in deriving robust integervalued strategies for the TTL-based controlled flooding search, i.e. finding u U d achieving the iniu worst-case cost ratio. We will use our optial continuous strategies of the previous subsection to derive discrete strategies that perfor well in the worst-case. We begin with the following lea: Lea 4: For any v V, we have Dx v Dx v and Jx v Jx v for all x Z +. That is, we can take the floor of any continuous strategy to find a discrete strategy that perfors just as well if the object location is restricted to integers. Using this result, we can prove Theore 2. The proof is given in the Appendix. This theore gives an upper bound on the best worst-case discrete strategy, for all C ) C. It appears that the actual value of the iniu worst-case cost will depend on the specific function C ). A general result is currently not available, but if we restrict ourselves to cost functions C ) C 1, then we can obtain Theore 3 presented earlier. Recall that C 1 contains all polynoial cost functions. Therefore, Theore 3 still holds for a very general class of cost and delay functions. Proof of this theore is rather lengthy and can be found in [14. A sketch is provided in the Appendix. V. APPLICATIONS, EXAMPLES AND DISCUSSION A. Cost-Delay Tradeoff Having derived optial strategies for any delay constraint, it is worth exaining how the delay constraint affects the iniu achievable worst-case cost ratio. Figure 2 depicts the tradeoff between optial worst-case cost ratio as given by Theore 1 and the delay constraint d when C ) βd ). The dotted portion of each curve indicates when the delay constraint is not binding, i.e., for d , 2, 3, respectively. In these cases the optial unconstrained strategy using r e) has a iniu worst-case cost ratio of e. Note that the plot is logarithic. As d approaches 1 fro above, the best worst-case cost ratio approaches for all. Hence, as

8 Best Worst Case Cost Ratio Delay Constraint d) Fig. 2. When C ) βd ), a logarithic plot of the iniu worstcase cost ratio as a function of the delay constraint d. Dotted portions indicate when the delay constraint is not binding and hence the unconstrained strategy of Theore 1, part 2) is optial. For d 3, the best worst-case cost ratio is e for all three curves. Cost Ratio Delay Ratio Object Location Object Location d1.5 Fig. 3. Cost and delay ratios of optial strategies under different delay constraints, when cost is quadratic and delay is linear, i.e. C ) D ) 2. d2 d3 the constraint on delay becoes tighter, the iniu worstcase cost increases unboundedly. For any fixed d, as increases the iniu worst-case cost also increases. This can be understood by fixing soe delay function D ). As increases, the cost function C ) βd ) increases faster. For any given delay constraint, it then becoes ore difficult to achieve a low cost ratio. B. Exaples We present an exaple scenario where the search delay grows linearly in the TTL value used, while the search cost grows quadratically. Specifically, consider Dx) βx and Cx) ξx 2 for soe β,ξ > 0 so 2. As entioned earlier, this could be a good representation of a two-diensional network, where transissions are on the order of x 2, and the delay is proportional to nuber of hops. Fro Theore 1, the optial strategy is v[e 2 d 1, d 1) x whenever 1 <d<3. When d 3, the optial strategy is v[e, 2 x. Figure 3 depicts the cost and delay ratio curves, with respect to object location, of the corresponding optial strategies when d 1.5, 2, and 3. Note that both the delay and cost ratio curves approach their axiu values very rapidly. Hence, the worst-case value of cost and delay under asyptotic network size as L ) can approxiate the perforance when the network size is finite. At the sae tie, the worst-case is approached asyptotically. Hence the cost ratio at any finite object location is less than the worst-case cost ratio. Also note that the cost and delay ratio curves are sooth and nearly flat with respect to object location. Thus the actual object location does not significantly change the perforance of these strategies. One can view this as a built-in robustness for both the cost and delay criteria. Siilar results hold for other values of d and, and other functional fors of C ) and D ). They are not repeated here. C. Coparison It was shown in [8 and [9 that when adopting a worst-case cost easure, randoized strategies outperfor deterinistic ones. The results of the previous sections show that randoized strategies also perfor better when delay constraints are added. Here we illustrate this in ore detail. Note that both the optial deterinistic strategy for Proble 1 and the optial randoized strategies of Section IV-C share the property that the costs of the TTL values grow geoetrically. That is, for any realization, Cv k )r k 1 Cv 1 ) for all k. It was shown in [7 that the unconstrained optial deterinistic strategy under linear cost C ) is also a geoetric sequence: u k 2 k 1 for all k. Below we copare deterinistic and randoized geoetric strategies when both cost and delay are linear, i.e. when Dv) Cv) v for all v. For deterinistic geoetric strategies with paraeter r, Cv k )r k 1 Cv 1 ) for all k 1. It can be shown [14 that such strategies have a worst-case delay ratio of 2r 1)/r 1) and worst-case cost ratio of r 2 r 1. Now consider the randoized strategies v[r, 1 C ), shown to be optial in Theore 1. For any r>1and 1, r it was shown that the worst-case cost ratio of v is and the 1 worst-case delay ratio is +1. In Figure 4 we plot the worst-case cost and delay ratios, as functions of r, for the aforeentioned geoetric deterinistic and randoized strategies. Note that for any r, the randoized strategy achieves a lower worst-case cost and a lower worstcase delay than its deterinistic counterpart. Hence, randoization has the effect of decreasing worst-case cost and delay at the sae tie. In addition, note that the worst-case delay ratio of the randoized strategies approaches 1 as r, but for the fixed strategies this liit is 2. It can be seen that even by arbitrarily increasing the value of r for deterinistic geoetric

9 Worst Cost Ratio Worst Delay Ratio randoized deterinistic Fig. 4. Coparison of deterinistic and randoized strategies discussed in Section V-C, as a function of r. Note that for any r, the randoization achieves lower worst-case cost ratio and delay ratio. strategies, it is not possible to atch the worst-case delay ratio of the optial randoized geoetric strategies of Theore 1 which always achieve a worst-case delay ratio below 2). By varying the cost/delay functions and, the curves in Figure 4 ay change, but the general relationship between randoized and deterinistic strategies will still hold. D. Generalization of the Proble Abstraction The proble introduced in this paper is priarily otivated by the flooding search application in networks. However, further consideration reveals soe quite general and appealing features about the abstraction of this proble that can potentially be applied to a variety of probles involving constrained resource allocation. Here we restate the sae proble in a ore general context. Consider an individual who seeks to coplete a task e.g., a coputing job). There is a iniu level of resources/effort X required to accoplish the task e.g., an updating step size in the coputing job). X is a rando variable whose distribution ay be unknown to the individual. Its realization is not known in advance. The individual ay choose fro a range of resources/effort levels she is willing to put in the job, and the outcoe e.g., the precision of the coputing result obtained) depends on the effort level. If she chooses alevelv X, then the task returns successfully and the process terinates. Otherwise the task returns failure and the individual increases her resources/effort level and tries again. When a level v is chosen, the individual coits to paying a cost of Cv) e.g., eory and processing needed in the coputing job), regardless of whether she succeeds or not. At the sae tie, with a level v the job takes a certain aount of tie to return either with a success or a failure), and this delay is given by D in {v, X}). r r The successive resource levels v [v 1,v 2, chosen by the individual for a strategy, which deterines the total cost paid and tie expended by the individual in accoplishing the job. As the cost is coitted when a level is chosen, the individual ust balance between selecting too low a level ore likely to be unsuccessful) and too high a level ore costly or wasteful). When one wishes to find a low cost strategy subject to a delay constraint, a constrained optiization proble is obtained. If furtherore the objective and the constraint are in the for of worst-case cost/delay easure, then a forulation akin to the one presented in this paper arises. VI. CONCLUSION In this paper we studied the class of TTL-based controlled flooding search and presented a constrained optiization fraework in order to derive strategies that iniize a worstcase search cost easure subject to a worst-case search delay constraint. Optial strategies were obtained in the continuous as well as discrete cases and their perforance was studied. These results were used to discuss the trade-off between cost and delay using this type of search ethod. We also showed there the abstraction underlying the search application has a broad generalization that can be applied to solve a range of constrained resource allocation probles. REFERENCES [1 D. Johnson and D. Maltz, Dynaic source routing in ad hoc wireless networks, Mobile Coputing, pp , [2 J. Xie, R. Talpade, T. McAuley, and M. Liu, AMRoute: Ad Hoc Multicast Routing Protocol, ACM Mobile Networks and Applications MONET) Special Issue on Mobility of Systes, Users, Data and Coputing, vol. 7, no. 6, pp , Deceber [3 C. Carter, S. Yi, P. Ratanchandani, and R. Kravets., Manycast: Exploring the space between anycast and ulticast in ad hoc networks, Proceedings of the Ninth Annual International Conference on Mobile Coputing and Networks MobiCOM 03), Septeber 2003, San Diego, California. [4 D. Braginsky and D.Estrin, Ruor routing algorith for sensor networks, Proc. International Conference on Distributed Coputing Systes ICDCS-22), [5 S. Shakkottai, Asyptotics of query strategies over a sensor network, Proceedings of IEEE Infoco, March 2004, Hong Kong. [6 Z. Cheng and W. Heinzelan, Flooding strategy for target discovery in wireless networks, Proceedings of the Sixth ACM International Workshop on Modeling, Analysis and Siulation of Wireless and Mobile Systes MSWiM 2003), Sept [7 Y. Baryshinikov, E. Coffan, P. Jelenkovic, P. Mocilovic, and D. Rubenstein, Flood search under the california split rule, Operations Research Letters, vol. 32, no. 3, pp , May [8 N. Chang and M. Liu, Revisiting the TTL-based controlled flooding search: Optiality and randoization, Proceedings of the Tenth Annual International Conference on Mobile Coputing and Networks MobiCo 04), pp , Septeber 2004, Philadelphia, PA. [9 N.B.Chang and M. Liu, Optial controlled flooding search in a large wireless network, Third International Syposiu on Modeling and Optiization in Mobile, Ad Hoc, and Wireless Networks WiOpt 05), pp , April 2005, Riva Del Garda, Italy. [10 P. Billingsley, Probability and Measure, John Wiley & Sons, 1995, New York, USA. [11 S. Nash and A. Sofer, Linear and Nonlinear Prograing, McGraw- Hill, 1996, New York, USA. [12 E. Altan, Constrained Markov Decision Processes, Chapan & Hall/CRC, 1999, Boca Raton, Florida. [13 A. Borodin and R. El-Yaniv, Online Coptutation and Copetetive Analysis, Cabridge University Press, 1998, Cabridge, UK.

10 [14 N. Chang and M. Liu, Controlled flooding search with delay constraints, EECS Technical Report CGR 05-08, 2005, University of Michigan, Ann Arbor, ingyan/pub/cgr pdf. VII. APPENDIX A. Solution to Proble 1 To begin, we calculate the ean of object location cost and delay, noting that X takes values on [1, ): E [CX) PrC X) >x) dx 0 [ CC 1 x)) + dx. 34) 1 [ E [DY ) E CY ) 1/ ) Pr C Y ) 1/ >x dx ) 1) 1/. 35) Using 3) to evaluate the delay ratio for deterinistic) v: DY v E[DY ) F Y v k )Dv k ) +1 36) E[DY ) By rearranging 36) and observing that only the nuerator of the cost ratio depends on v, Proble 1 is equivalent to: J X v s. t. F Y v k )Dv k ) d 1)E[DY ). v Therefore, we define the Lagrangian for λ 0: Gv,λ) F X v k )Cv k+1 ) k0 ) λ d 1)E[DY ) F Y v k )Dv k ) ) Cvk ) Cv k+1) λd 1)E[DY ) k0 ) +1 1/ Cvk ) + λ Cv k ) 1/, where v 0 1for notational convenience. A necessary condition [11 for optiality of v is that the partial derivative of G with respect to v j is 0, for all j 1. In other words, G Cv j) [ Cv j 1 ) Cv j+1 )Cv j ) 1 v j v j +λ1 )Cv j ) ) Because the derivative of the cost function is strictly positive and > 0, then equation 37) is satisfied if and only if the ter inside the brackets is equal to 0. Setting this ter equal to 0, letting λ λ 1 1 for notational convenience, and rearranging yields the following recursion for j 1: Cv j+1 ) Cv j) [ Cvj ) Cv j 1 ) ) + λ1 ), 38) Hence, any optial strategy ust satisfy the recursion given by 38). Let γ j Cvj) Cv j 1) for all j 1. This quantity indicates the relative aount of cost increase after every unsuccessful search. Then 38) reduces to: γ j+1 1 γ j λ 1) ) 39) Note that the value of γ 1 uniquely defines the reaining values γ j for all j 2. At the sae tie, the entire sequence {γ j } uniquely defines the strategy. Hence it reains to deterine values of γ 1 that define optial strategies. Lea 5: Fix λ 0. A necessary condition for optiality is that for all j 1, γ j γ where γ is the unique solution to the following equation: 1 1) γ γ λ. 40) Hence, any optial strategy for Proble 1 will have costs increasing geoetrically by factor γ. Proof: It should be noted for copleteness that equation 40) has a unique solution because the function 1 x x is strictly increasing in x this can be seen by differentiating with respect to x), is equal to 0 when x 1 1, is continuous, and increases to as x.atthesaetie, λ 1) is a nonnegative finite quantity. Note that if γ 1 γ, then γ j γ for all j 2. Hence, it suffices to prove that γ 1 γ is necessary for optiality. We proceed by contradiction. Case 1: γ 1 >γ. Note that if γ j >γfor soe j, then we have the following: γ j+1 1 γ j λ 1) ) 1 γ j γ j + γ j λ 1) > 1 γ γ λ 1) + γ j γ j, 41) where the last inequality follows fro the fact that 1 x x is strictly increasing in x, as noted earlier. Hence we have the following: if γ j >γfor soe j, then γ j+1 >γ j. This eans that because γ 1 >γ, then γ 2 >γ 1 >γ, and so on. Hence by induction, the {γ j } for a strictly increasing sequence, where γ j >γfor each j. So for each j 1 by rearranging the recursion 39): γ j+1 γ 1 1) λ j γj 1 1) λ γ γ1, 42) where the inequality holds because γj > γ, and the last equality holds fro the definition of γ. The inequality becoes strict when λ >0. Note that for any j 1, we have by the definition of γ j that Cv j ) j γ k. Hence, the expected cost of any such strategy is given by: v F X v j )Cv j+1 ) j0 γ 1 j0 j γ k+1 γ k j0 ) Cvj ) Cv j+1), 43)

11 where the product is defined to be equal to 1 if j 0. We have shown that if γ 1 >γ, then γ k+1 γ k >γ 1 for all k. Hence for any such strategy where γ 1 >γ, the expected search cost is lower-bounded by: γ 1 j0 j γ k+1 γ k > γ j0 j γ 1 44) However, note that the right-hand side of the above equation is siply the expected search cost for a strategy such that γ j γ for all j plug γ j γ into 43)). Hence fro 44), any strategy where γ j >γfor all j has expected search cost strictly greater than using γ j γ, and these strategies cannot be optial. Case 2: γ 1 <γ. Note that for any optial strategy, γ j > 1 for all j, because only strictly increasing TTL sequences can be optial. Hence the sequence {γ j } is always lower-bounded by 1. Note that if γ j <γ for soe j, then we have the following: γ j+1 1 γ j λ 1) ) 1 γ j γ j + γ j λ 1) < 1 γ γ λ 1) + γ j γ j 45) Hence, if γ 1 <γ, then γ 2 <γ 1 <γ, and so on. Because the {γ j } are bounded, then the sequence converges since all onotonic bounded sequences converge). Let γ li j γ j. Because the γ j are strictly less than γ and for a decreasing sequence, then γ <γ. On the other hand, { 1 li γ [ j+1 li γ j j j λ 1) } γ 1 [ γ λ 1) 1 γ γ λ 1) We defined γ as the unique nuber satisfying 40). Because we have just shown that γ is bounded and also satisfies the sae equation we have that γ γ. However, this contradicts the fact that γ <γ, which we showed earlier. Hence, it is not possible to have γ 1 <γ if the {γ j } are bounded. Therefore, cobining Case 1 and Case 2 proves that γ j γ for all j is the only possible optial strategy for fixed λ. For any strategy v where γ j γ, we have the following: Cv j ) j γ k γ j. Therefore we have the following geoetric su, which converges since γ > 1 necessary for increasing sequence) iplies 0 <γ 1 < 1: ) 1 F Y v k )Dv k ) 1/ Cvk ) 1/ γ 1 ) k 1/ γ ) Using 46) into 36), we can see that it is possible to achieve a delay ratio arbitrarily close to 1 by choosing a sufficiently high enough value of γ. Therefore, for every d>1, there exists a strategy achieving delay ratio below d. Hence the optial strategy for Proble 1 ust satisfy the Kuhn-Tucker condition see [11 and [12): ) λ F Y v k )Dv k ) d 1)E[DY ) 0 47) Therefore either λ 0or the delay constraint is satisfied with equality. We use this to prove solutions for two cases, depending on whether 1 <d<+1 or d +1. Case 1: 1 <d<+1 When λ 0, then λ 0and we have an unconstrained optiization proble. In this case, γ 1 1 fro Lea 5. The suation in 46) is then equal to 1/ / 1) for this value of γ. Fro 36), we know that dividing this suation by 35) and then adding 1 gives the delay ratio: D v Y E[DY ) F Y v k )Dv k ) +1 E[DY ) 1+ 1) +1. Fro inequality 27) on, this delay ratio is thus strictly greater than d. Hence, this strategy does not eet the delay inequality requireent. Therefore, we seek solutions for which the delay constraint is et with equality, i.e. the ter inside the brackets of 47) is equal to 0. In this case, 46) needs to be equal to d 1) 1 1)+1 1), and solving for γ gives: ) 1 1 1) γ 1+ 48) [ 1) + 1d 1) Fro the earlier equation 40) relating γ and λ, we have that λ can be calculated as: γ λ 1 ) γ 1 γ [ 1) + 1d 1) 1 The cost ratio can be calculated by ultiplying both sides of 38) by F X v j ) and then suing over j 1 to give: F X v j )Cv j+1 ) j1 1 j0 F X v j )Cv j+1 ) λ 1 ) Cv j ) F X v j ) j1 The left-hand su is siply v Cv 1), so rearranging and solving for v gives: v 1 Cv 1) λ 1 Cv j ) F X v j ) j1 J X v Cv 1) λ j1 Cv j) F v j ) 49) To evaluate this ratio, note that: Cv j ) F ) 1 Cvk ) v j ) γ 1 1 j1 ) 1 d 1) 1+ 50) 1)