Energy Efficient Routing in Ad Hoc Disaster Recovery Networks

Transcription

1 Energy Effcent Routng n Ad Hoc Dsaster Recovery Networks Gl Zussman and Adran Segall Department of Electrcal Engneerng Technon Israel Insttute of Technology Hafa 32000, Israel {glz@tx, segall@ee}.technon.ac.l segall} Abstract. The terrorst attacks on September 11, 2001 have drawn attenton to the use of wreless technology n order to locate survvors of structural collapse. We propose to construct an ad hoc network of wreless smart badges n order to acqure nformaton from trapped survvors. We nvestgate the energy effcent routng problem that arses n such a network and show that snce smart badges have very lmted power sources and very low data rates, whch may be nadequate n an emergency stuaton, the soluton of the routng problem requres new protocols. The problem s formulated as an anycast routng problem n whch the objectve s to maxmze the tme untl the frst battery drans-out. We present teratve algorthms for obtanng the optmal soluton of the problem. Then, we derve an upper bound on the network lfetme for specfc topologes and descrbe a polynomal algorthm for obtanng the optmal soluton n such topologes. Fnally, numercal results regardng the upper bound and the algorthms are presented. Keywords: Routng, Energy effcent, Energy conservng, Power aware, Dsaster recovery networks, Ad hoc networks, Smart badges CCIT Report 392, Technon Dept. of Electrcal Engneerng, July

2 1 Introducton The terrorst attacks on the World Trade Center and the Pentagon on September 11, 2001 have drawn ever-ncreasng attenton to mprovng rescue efforts followng a dsaster. One of the technologes that can be effectvely deployed durng dsaster recovery s wreless ad hoc networkng [23]. For example, rescue forces can use a Moble Ad Hoc Network (MANET) n the lack of fxed communcaton systems. Furthermore, a wreless sensor network can be quckly deployed followng a chemcal or bologcal attack n order to dentfy areas affected by the chemcal/bologcal agents [2]. We propose another applcaton of an ad hoc network, whch can be used n order to gather nformaton from trapped survvors of structural collapse. There are varous possble reasons for structural collapse. The most frequent reasons are earthquakes, terror attacks, structural problems, and mssle attacks. Regardless of the reason, the consequences of a collapse are usually very tragc. For example, n 1995 alone, the Kobe earthquake resulted n the death of nearly 5,500 people, 168 people were klled n the Oklahoma Cty bombng, and more than 500 people were klled n the collapse of the Sampoong department store n Seoul. Thus, the mportance of mprovng rescue technques requres no explanaton. There are a few technques for locatng survvors of structural collapse trapped n the rubble: fber optc scopes, senstve lstenng devces, sesmc sensors, search-and-rescue dogs, etc. [10]. Moreover, durng the rescue attempts n the World Trade Center dsaster ste, the Wreless Emergency Response Team (WERT) attempted to locate survvors through sgnals from ther moble phones [27]. We propose to extend these capabltes and to enable the locaton of survvors by acqurng nformaton from ther smart badges. Smart badges (a.k.a. RFID badges [28]) wll gan ncreased popularty n the near future and wll apparently be used n any modern offce buldng [25]. Snce the transmsson range of a badge s very short and snce rescue equpment can usually be deployed at the perphery of the dsaster scene, there s a need to construct an ad hoc network connectng vctms trapped n the debrs to the rescuers. In such a network, the nformaton acqured from the badges (such as last known locaton, body temperature, etc.) wll be repeatedly routed through other badges to wreless recevers deployed n the dsaster scene. The recevers wll forward the nformaton through wred or wreless lnks to a central unt. In the comng years, smart badges wll use a propretary technology (e.g. [26]) or the new IEEE standard for Low-Rate Wreless Personal Area Networks (LR- WPAN) [15], [17]. Ether way they wll be smple devces wth very low data rates and very lmted power sources. These data rates and power sources are expected to be adequate for regular use. For example, the data rate of an IEEE devce wll be 20 Kb/s or 250 Kb/s [17]. A smart badge based on ths standard s expected to establsh about 20 connectons per day [25]. Thus, the average data rates are expected to be much lower than the possble data rate. Moreover, the duty cycle of such a devce s expected to be less than 1%, thereby enablng a long battery lfe. However, n an emergency network constructed after a collapse, whch may connect thousands of nodes and may route crtcal nformaton, the requred data rates and the consumed energy may be much hgher than n daly use. Thus, the low data rates and the lmted power sources are a major constrant on the performance of an emergency net- 2

3 work. Furthermore, n such a network depletng the battery of a node may have tragc results. Ths report focuses on energy effcent routng protocols for emergency networks of badges. We note that snce wreless devces usually have a fnte power supply, there s an ncreasng nterest n research regardng energy-conservng protocols (see Secton 2). Thus, our network model s based on the model for energy conservng routng n a wreless sensor network presented by Chang and Tassulas [7]. However, unlke a wreless sensor network n whch the avalable bandwdth s usually suffcent, n the emergency network there s a strct bandwdth restrcton along wth a strct energy restrcton. Hence, the soluton of the problem calls for the development of new protocols. We assume that snce the proposed network wll be composed of trapped survvors badges, the network topology and the requrements wll be rather statc. Therefore, our major nterest s n dstrbuted algorthms for quas-statc anycast routng n a statc network wth statonary requrements and unchangng topology. The objectve of the algorthms s to maxmze the tme untl the frst battery drans-out (.e. to solve a max-mn optmzaton problem). Ths objectve functon has been defned by Chang and Tassulas [7] and although t s controversal when appled to MANETs or sensor networks, t s approprate for an emergency network n whch every node s crtcal. In ths report we formulate the problem and present teratve algorthms for obtanng ts optmal soluton. These algorthms are based on the formulaton of the problem as a concurrent max flow problem [19] and the complexty of one of them s logarthmc n the network lfetme. We derve an upper bound on the network lfetme for specfc topologes that s based on the new noton of non-max capacty cut. Then, a polynomal algorthm for obtanng the optmal soluton n specfc topologes s descrbed. Fnally, numercal results regardng the upper bound and the algorthms are presented. The man contrbuton of ths report s the extenson of the energy conservng routng model presented by Chang and Tassulas [7] to a network n whch some of the nodes have a very low data rate as well as a lmted battery. Another contrbuton s the dervaton of bounds on the network lfetme and the development of optmal algorthms, whch can be mplemented n a dstrbuted manner (as requred n a dsaster recovery network). Ths report s organzed as follows. In Secton 2, we dscuss related work and n Secton 3, we present the model and formulate the routng problem. Algorthms for obtanng the soluton of the problem and an upper bound on the network lfetme are ntroduced n Secton 4. In Secton 5, we present numercal results and n Secton 6, we summarze the man results and dscuss future research drectons. 2 Related Work In 1998, Bambos [4] revewed developments n power control n wreless networks and dentfed the need for mnmum-power routng protocols. Snce then, the ssue of energyconservaton n ad hoc and sensor networks has attracted a vast amount of research (see for example, [18], [24], and references theren). Ths research deals wth all layers of the protocol stack and s mostly motvated by the fact that wreless devces usually have a very lmted power supply. In partcular, there s an ncreasng nterest n algorthms for the network layer, namely n energy effcent routng algorthms. 3

4 A poneerng work regardng energy effcent routng was presented by Sngh et al. [30] who studed va smulaton the ssue of ncreasng node and network lfe by usng power-aware metrcs for routng. In [34] other power-aware metrcs are presented and ther performance s studed va smulaton. Some of the prevous work regardng energy effcent routng n moble ad-hoc networks (MANETs) focused on performance comparson of exstng ad hoc routng protocols (such as DSR, AODV, TORA, and DSDV [23]) wth respect to energy consumpton (e.g. [11]). Recently, new power-aware routng protocols for MANETs have been proposed. For example, n [13] a technque (named PARO) desgned as a power-aware enhancement for MANET routng protocols has been ntroduced. In addton, n [36] an algorthm (named GAF) that s desgned to reduce the energy consumpton n the network by turnng off unnecessary nodes and whch s ndependent of the underlyng ad hoc routng protocol s ntroduced. We note that Weselther et al. have publshed numerous papers on energy-aware broadcastng and multcastng (see for example, [35] and references theren) and that ther work s closely related to the ssue of energy effcent routng. For example, Mchal and Ephremdes [21] study the problem of energy effcent routng of connecton orented traffc. Ths problem has some relatonshp to the problem studed n ths report snce unlke other authors they take nto consderaton the fact that the nodes have fnte capacty. However, they provde heurstc algorthms whereas we attempt to develop optmal algorthms. The specal characterstcs of wreless sensor networks and energy conservng technques for such networks are descrbed n [2], [31], and n a number of papers related to the MIT s µamps project [22]. For nstance, n [16], an energy effcent routng algorthm based on clusterng s descrbed and n [5], a methodology for computng upper bounds on the lfetme of a sensor network s presented. Chang and Tassulas [6] ntroduced one of the frst models of energy conservng routng n sensor networks. They defned the energy conservng routng problem as an optmzaton problem n whch the performance objectve s to maxmze the lfetme of the network (.e. to maxmze the tme untl the frst battery drans-out). They proposed heurstc routng protocols for the soluton of the problem and evaluated ther performance by smulaton. The work of Chang and Tassulas [6] has been extended n several dfferent drectons. In [7], they have extended ther model for the case of multcommodty flow and n [8], they have proposed algorthms for obtanng an approxmate soluton of the routng problem. Moreover, n [20], approxmate onlne algorthms for the case n whch the message sequence s not known have been proposed. The problem of fndng a flow control strategy that maxmzes the sources utltes subject to a constrant on the network lfetme has been addressed n [32]. In [9], technques to maxmze the network lfetme n the case of cluster-based networks have been devsed. A scheme for energy aware routng n a network of pconodes [26] that chooses among possble paths based on a probablstc fashon has been ntroduced n [29]. Fnally, n ths report, we ntroduce an extenson to the model of Chang and Tassulas for the case n whch the nodes have a very lmted bandwdth as well as a lmted battery. 4

5 3 Formulaton of the Problem 3.1 Model and Prelmnares Consder the connected drected network graph G = (N,L). N wll denote the collecton of nodes {1,2,,n}. A node could be a badge, a recever (the collecton of recevers s denoted by R), or the central unt (referred to as the destnaton and denoted by d). Recall that recevers are deployed at the perphery of the dsaster scene (ther role s to connect the badges network to the central unt). The collecton of the drectonal lnks wll be denoted by L. We assume that snce smart badges are ntended to be very smple and cheap devces they wll usually transmt at a constant power level. Thus, a unt j that s wthn the transmsson range of node s connected to by a drectonal lnk, denoted by (,j). For each node, Z() wll denote the collecton of ts neghborng nodes (nodes connected to node by a drectonal lnk). Let F j be the average flow on lnk (,j) (F j 0 (,j) L). We defne f j as the rato between F j and the maxmal possble flow on a lnk connectng smart badges 1 (0 f j 1). f j wll be referred to as the flow on lnk (,j). The rato between the rate n whch nformaton s generated at badge node and the maxmal possble flow on a lnk connectng smart badges, s denoted by r (0 r < 1). The transmsson energy requred by node to transmt an nformaton unt s denoted by e. Let each node have an ntal energy level E (we assume that E > 0 N). If a node s a recever or the destnaton, ts energy source s much larger than the energy source of a badge, and therefore, for such a node E =. For low-power devces operatng n the 2.4 GHz ISM band, the transmtter and recever currents are often smlar [15]. Thus, we assume that energy s consumed only when a node transmts nformaton (alternatvely, the energy consumed when t receves nformaton can be ncluded n e ). Moreover, snce the energy requred n order to receve a message s not neglgble, we assume that although a few nodes are able to receve the same message, only the node to whch t s ntended wll receve the full message and forward t. The other nodes wll be n sleep mode or communcate wth ther other neghbors. We have mentoned that the objectve of our energy conservng routng protocols s to obtan lnk flows such that the tme untl the frst battery drans-out wll be maxmzed. Thus, followng the formulaton of [7] and usng the above assumptons, we defne the lfetme of a node and of the network as follows. Defnton 1. (Chang and Tassulas [7]) The lfetme of node under a gven flow s denoted by T and s gven by: T = E e fj. (1) j Z() 1 For example, n IEEE the maxmal data rate (.e. the maxmal possble flow) s 20Kb/s or 250Kb/s. 5

6 Defnton 2. (Chang and Tassulas [7]): The lfetme of the network under a gven flow s the tme untl the frst battery drans-out, namely the mnmum lfetme over all nodes. It s denoted by T and s gven by: T = mnt = mn N N e E fj. (2) j Z() 3.2 Problem Formulaton As mentoned before, badges wll generate nformaton that wll be routed through any of the recevers to the destnaton. Thus, the resultng problem s an anycast routng problem. Accordngly, the energy effcent routng problem can be formulated as follows. Problem EER: Gven: Topology and requrements (r ) Objectve: Maxmze the network lfetme: E max T = max mn N e f (3) j j Z() Subject to: f 0 (, j) L (4) j f + r = f N { R, d} k j k Z() j Z() f = f R (6) k j k Z() j Z() f + fj 1 N { R, d} k k Z() j Z() Constrants (4) - (6) are the usual flow conservaton constrants. The meanng of (7) s that the total flow through a node cannot exceed the maxmal badge node capacty (.e. the maxmal data rate of a badge). (5) (7) 3.3 Numercal Example Fg. 1 llustrates a smple network composed of fve badges, two recevers, and a sngle destnaton. In the optmal soluton, the network lfetme s 7.69 tme unts (the batteres of nodes 1,2, and 4 are depleted after 7.69 tme unts). It can be seen that node 5, whose battery has remanng power at tme 7.69, utlzes ts full capacty throughout the operaton of the network. 6

7 r = E = 5 f = 0.35 r = 0.3 E = 5 f = f = r = 0.4 E = 10 f = 0.65 f = 0 f = 0.15 r = 0.4 E = 10 f = f = 0.33 r = 0.5 E = 5 f = 0.32 f = 0.98 f = Badge Recever Destnaton Fg. 1. The requred transmsson rates (r ), the ntal energy values (E ), and the optmal flows (f j ) n a network of badges (assumng that e = 1 ) 4 Algorthms and Bounds In ths secton, we present an equvalent formulaton of Problem EER. Ths formulaton s requred n order to develop dstrbuted algorthms. Iteratve algorthms for obtanng the optmal soluton of the problem are descrbed. Then, we derve an upper bound on the network lfetme for specfc topologes. Fnally, we descrbe a polynomal algorthm for obtanng the optmal soluton n these topologes. 4.1 Lnear Programmng Formulaton The frst step towards obtanng a soluton to Problem EER s convertng t to a lnear programmng problem (Problem EER-LP). Followng the approach n [7], we frst defne f as the amount of nformaton transmtted from node to node j untl tme T j ( f j = f j T ). Then, we formulate Problem EER-LP as follows. Problem EER-LP: Gven: Topology and requrements (r ) Objectve: Maxmze the network lfetme: maxt Subject to: f 0 (, j) L (9) j f + r T = f N { R, d} (10) k j k Z() j Z() fk = fj R (11) k Z() j Z() e f E N { R, d } (12) j j Z() f + fj T N { R, d } (13) k k Z() j Z() Problem EER-LP s a lnear programmng problem, and therefore, t can be solved by well-known algorthms (e.g. Smplex [1, p. 810]). However, these algorthms cannot be easly modfed n order to allow dstrbuted mplementaton, whch s requred n an ad (8) 7

8 hoc network. Thus, analyzng the characterstcs of the problem s requred n order to develop dstrbuted algorthms. If the last set of constrants (13) s gnored, Problem EER-LP becomes a concurrent max-flow problem wth constrants on the flows at the nodes. A concurrent max-flow problem s a multcommodty flow problem n whch a demand s assocated wth each commodty and the objectve s to maxmze a common fracton of each demand wthout exceedng the capacty constrants [3], [19]. Accordngly, we defne Problem CMF as follows (T s the common fracton of each demand). Problem CMF (Concurrent Max Flow): Gven: Topology and requrements (r ) Objectve: max T (as n (8)) Subject to: Flow conservaton constrants (such as (9)-(11)) Capacty constrants (such as (12)) In the followng sectons, we shall defne dfferent nstances of Problem CMF by alterng ether the flow conservaton constrants or the capacty constrants. Addng (13) to Problem CMF means that T has to be maxmzed subject to the addtonal constrant that the flow through a node cannot exceed some percentage of the flow n the network. Recall that (13) results from the fact that the data rate of a badge mght be lower than the requred bandwdth n an emergency stuaton. In sectons , we shall dscuss two dfferent methods for dealng wth the complextes mposed by (13). 4.2 Iteratve Algorthms An algorthm for obtanng the optmal soluton of Problem EER-LP can be based on repeated solutons of dfferent nstances of Problem CMF. Followng the soluton of an nstance of Problem CMF, the node capactes, whch depend both on the energy (12) and the value of the network lfetme (13), have to be recomputed accordng to the obtaned lfetme (T). Then, another nstance of Problem CMF (wth the new capactes) has to be solved. Ths process should be repeated untl the optmal soluton to Problem EER-LP s obtaned. In ths secton, we descrbe an algorthm (referred to as the Iteratve Algorthm) based on the above methodology. Then, we present an mproved verson of the algorthm whch utlzes bnary search (we shall refer to t as the Bnary Iteratve Algorthm). We note that both algorthms obtan an optmal soluton to Problem EER and that the complexty of the Bnary Iteratve Algorthm s logarthmc n the network lfetme. Snce there s a sngle destnaton node, an nstance of Problem CMF can be solved by usng bnary search wth a max flow algorthm (e.g. the preflow-push algorthm [12]). 1 Specfcally, f for a gven set of demands (.e. for a gven T) there exsts a feasble flow (e.g. flow satsfyng (8)-(12)), t can be found by a max flow algorthm. Thus, n order to check the feasblty of a gven T as a soluton to an nstance Problem CMF, the network graph should be converted such that every badge node s connected to a super orgn by a lnk whose capacty s r T. The max flow algorthm should be used n order to maxmze 1 Problem CMF can also be solved by usng bnary search wth a verson of the approxmate algorthm presented n [3]. 8

9 the flow from the super orgn to the destnaton. If n the obtaned soluton the flow outgong from the super orgn s then T s feasble. rt, (14) N { R, d} Accordngly, we defne the process of bnary search for the soluton of Problem CMF as follows. Defnton 3. Algorthm BMF (Bnary Max Flow) A bnary search algorthm for the soluton of Problem CMF (.e. for obtanng T). At each teraton of the bnary search (.e. for a gven T), a max flow algorthm s executed. It s executed n a network wth a super orgn node whch s connected to every badge node by a lnk whose capacty s r T. Snce the complexty of a max flow algorthm s O(n 3 ) [1, p.240], [12], the number of steps requred to fnd a soluton to Problem CMF by Algorthm BMF s 3 ( log max ) O n T, (15) where T max s the maxmal possble value of network lfetme (T). It can be shown that for a network of badges and the resultng Problem CMF, the value of T max s bounded by n tmes the maxmal lfetme of a sngle node (recall that n s the number of nodes). Consequently, assumng that the requred tolerance of the soluton s n seconds and that regardless of the traffc, the lfetme of a battery s bounded by about 10 years, O(log T max ) s actually O(log n). As mentoned before, the soluton of Problem EER-LP can be based on solutons of dfferent nstances of Problem CMF. Problem EER-LP ncludes constrants on the node flows. Thus, n order to enable the executon a max flow algorthm (requred for the soluton of Problem CMF), the network graph should be converted as descrbed n the followng. Snce the capactes mposed by (12) and (13) are node capactes, each badge node should be dvded nto two subnodes ( and o ) connected by an nternal lnk. If node generates nformaton, we assume t s generated at. Accordngly, for a gven T, the capacty of the nternal lnk (, o ) s defned as: c o E T(1 + r) = mn, N { R, d}. (16) e 2 Followng the dvson of the nodes, every drectonal lnk (,j) connectng badge nodes should be replaced by a drectonal lnk ( o,j ). The capactes of all these lnks should be set to. Notce that drectonal lnks connectng a badge node to recever node j should be replaced by a drectonal lnk ( o,j) wth nfnte capacty. In the frst teraton of the Iteratve Algorthm, the value of T at all the nternal lnks should be set to. A soluton to the resultng Problem CMF n the converted network should be obtaned by Algorthm BMF. 1 We shall denote the value of T obtaned at ths stage by T 1. Then, the nternal lnk capactes should be updated accordng to (16) and the value of T 1. A soluton to the resultng Problem CMF n the converted network should be obtaned by Algorthm BMF. Ths process s repeated untl the flow values computed by 1 Notce that snce T s set to n (16), Problem CMF s equvalent to the problem defned by (8)-(12). 9

10 Algorthm BMF satsfy all the nternal lnk capactes (16) computed accordng to the obtaned T. The complexty of the Iteratve Algorthm s not necessarly polynomal n the number of nodes or lnks. Specfcally, the number of executons of Algorthm BMF, whose complexty s gven n (15), s not necessarly polynomal n the number of nodes or lnks. A straghtforward mprovement s the use of bnary search n order to obtan the value of the optmal T. Namely, after obtanng T 1, a max flow algorthm should be used n order to check whether the lfetme of T 1 /2 s feasble when the nternal lnk capactes are computed accordng to (16) and T = T 1 /2. If t s, the feasblty of 3T 1 /4 should be checked n a smlar manner. Otherwse, the feasblty of T 1 /4 should be checked. The algorthm termnates when the dfference between the feasble and non-feasble T s wthn the requred tolerance. We refer to the algorthm based on the above methodology as the Bnary Iteratve Algorthm and descrbe t n Fg. 2. Recall that checkng the feasblty of a gven network lfetme requres O(n 3 ) steps. Thus, the complexty of ths algorthm s gven by (15). Although max flow algorthms (as the preflow-push algorthm) can be used n envronments where decsons have to be made locally, the dstrbuted mplementaton of the Bnary Iteratve Algorthm, descrbed n Fg. 2, requres some coordnaton mechansm. Ths mechansm s requred snce the nodes should be aware of the bnary search and the value of T, whch determnes the nternal lnk capactes (see steps 6 and 10 n Fg. 2). The defnton of the exact procedure n whch a dstrbuted teratve algorthm has to be executed s subject for further research. 1 transform the node-capactated network to a lnk capactated network 2 set c = E / e N { R, d} o T 1 = T max (.e. n max lfetme of a battery) 3 execute bnary search untl feasble T 1 non-feasble T 1 < tolerance 4 check feasbly of T 1 (by a max flow algorthm) 5 update T 1 (accordng to the bnary search) 6 set ( ) c = mn E / e, T (1 + r) / 2 N { R, d} o 1 T = T 1 7 execute bnary search untl feasble T non-feasble T < tolerance 8 check feasbly of T (by a max flow algorthm) 9 update T (accordng to the bnary search) 10 set c = mn E / e, T(1 + r) / 2 N { R, d} ( ) o 11 obtan the flow values ( f (, j) L) 12 set f = f / T (, j) L j j j Fg. 2. The Bnary Iteratve Algorthm for obtanng the optmal soluton of Problem EER 4.3 Upper Bound on the Network Lfetme In ths secton, we derve an upper bound on the network lfetme. It s based on a few observatons regardng the relatonshp between the optmal network lfetme and the capactes of dfferent cuts n the network. The bound can be computed usng a max flow algorthm (e.g. the preflow-push algorthm [12]). In the next secton, we shall show that 10

11 n a network wth a sngle orgn node, the bound s equal to the optmal soluton and outlne an O(n 4 ) algorthm for obtanng the optmal soluton. In ths secton, we focus on the case n whch only a subset of the badges generates nformaton (we shall refer to these badges as the orgn nodes and denote the collecton of orgn nodes by A). Moreover, we assume that these nodes do not forward nformaton generated by other nodes. 1 The resultng network graph s descrbed n Fg. 3. In the future, we ntend to extend the bound for the case n whch all the badges may generate nformaton and forward nformaton of other badges. Moreover, we conjecture that n some cases the bound s tght when there s more than a sngle orgn node. Badges Recevers (R) Orgn nodes (A) Destnaton (d) Fg. 3. A network graph n whch nformaton s generated only by orgn nodes. These nodes are not able to forward nformaton generated by other nodes We shall now redefne the transformaton of a node-capactated network to a lnkcapactated network and restate well-known defntons of a cut and related notons [1, p. 177]. In order to ncorporate node capactes, the network graph s transformed n a smlar manner to the transformaton descrbed n the prevous secton. However, there are two major dfferences. Frst, n some cases, whch wll be descrbed below, there s no need to separate orgn nodes nto subnodes. Second, the nternal lnk capactes do not take nto account the value of T. Accordngly, we defne the nternal lnk capactes of badges that are not orgns as: E c = N { A, R, d }. (17) o e Smlarly, n case orgn nodes have to be dvded nto subnodes, ther nternal lnk capactes are defned as: c o E e = A. (18) Notce that accordng to the context, n the rest of ths secton, N, whch orgnally denotes the collecton of nodes, sometmes denotes all the subnodes ( and o ). 1 We note that n [33], a smlar network topology s studed n the context of routng and schedulng n packet rado networks. 11

12 Defnton 4. A cut s dentfed by a par [O,D] of complementary subsets of nodes ( D = N O). The capacty of the cut [O,D] s denoted by C[O,D] and s the sum of the capactes of all the lnks whch are drected from O to D. The set of lnks drected from O to D are denoted by (O,D). We shall now defne the new noton of non-max capacty of a cut, whch s requred n order to determne the upper bound on the network lfetme. Defnton 5. The non-max capacty of the cut [O,D] s the sum of the capactes of the lnks drected from O to D not ncludng the lnk wth the hghest capacty. It s denoted by Y[O,D] and t s gven by: YOD [, ] = COD [, ] max ( c). (19) (, j) ( O, D) The next proposton provdes an upper bound on the optmal lfetme of the network (T * ). Proposton 1. If there exsts O N n the transformed network wth the lnk capactes determned by (17) that satsfes: then: The proof appears n the appendx. j d O (20) r > 0.5 (21) O (, j) ( O, D) s an nternal lnk, (22) T * mn O N: O satsfes (20)-(22) 2 YOD [, ]. (23) 2 r 1 The network lfetme s also bounded by the soluton of Problem CMF n the transformed network wth the lnk capactes determned by (17) and (18). It s well known [19] that ths soluton s bounded by the sparsest cut (a.k.a. mn cut), whch shall be denoted by C: C = mn O N: d O O COD [, ] r. (24) We note that n a network wth multple orgns and a sngle destnaton, the soluton to Problem CMF s equal to the sparsest cut (24) [14]. Notce also that the value of C s computed accordng to the transformed network determned by (17) and (18), whereas the bound descrbed n Proposton 1 s computed accordng to the transformed network determned only by (17). The next theorem combnes the results of Proposton 1 and (24). Theorem 1. O = C r 0.5 N * T 2 YOD [, ] C r O N: O satsfes (20)-(22) 2 r 1 N O mn mn, > 0.5 (25) 12

13 The proof appears n the appendx. As mentoned before, the value of C s the soluton to Problem CMF, and therefore, t can be computed by Algorthm BMF, defned n Defnton 3. We shall now show that n addton, the computaton of the bound on T *, descrbed n Theorem 1, requres several teratons of a max flow algorthm. Furthermore, Corollary 1 wll show that n a network wth a sngle orgn node, the optmal network lfetme (T * ) can be computed by O(n) teratons of a max flow algorthm. Consder a subgroup of orgn nodes (denoted wthout loss of generalty by { 1,, k }) such that { 1,, k } A and { 1,, k } r > 0.5. (26) For such a group there may be a few possble cuts [O,D] such that { 1,, k } O and {d,a { 1,, k }} D. Snce the sum of r s equal for all these cuts, obtanng the bound descrbed n Proposton 1 (23), s equvalent to obtanng Y {,, } = mn Y[ O, D] (27) mn 1 k O N: O satsfes (22) { 1,, k} O,{ d, A { 1,, k}} D for every subgroup { 1,, k }. Namely, for every subgroup of orgn nodes satsfyng (26), the mnmum of the non-max capactes of the cuts that separate { 1,, k } and {d,a { 1,, k }}, and whch are composed of nternal lnks should be computed. We shall now defne the noton of an Internal-zero Graph, whch s requred n order to compute Y mn { 1,, k } (27). Defnton 6. The Internal-zero Graph G l (l N {A,R,d}) s dentcal to the Graph G except that the capacty cll of the nternal lnk (l,l o o ) s taken to be 0. Accordng to the followng proposton, Y mn { 1,, k } s equal to the mnmum of the values of mnmal capacty of an [O,D] cut (mn C[O,D]) separatng { 1,, k } and {d,a { 1,, k }} n O(n) dfferent Internal-zero Graphs (G l ). Proposton 2. Y {,, } mn mn C[ O, D] mn 1 The proof appears n the appendx. k = Gl: l N { A, R, d} O N:{ 1,, k} O, { d, A { 1,, k }} D Consequently, for every subgroup { 1,, k } A satsfyng (26), there s a need to obtan the capacty of the mnmum cut n O(n) dfferent graphs 1. Accordng to the Max- Flow Mn-Cut Theorem [1, p. 185], for each of the O(n) graphs the soluton of a max flow problem s equal to the mn cut. The max flow problem should be solved when the subgroup of orgn nodes s connected to a super orgn and the objectve s to maxmze the flow from the super orgn to the destnaton. To conclude, the bound defned n Theorem 1 can be computed by Algorthm BMF and by O(n) executons of a max flow algorthm (e.g. the preflow-push algorthm) for every subgroup of orgn nodes satsfyng (26). Although the complexty of a max flow algorthm s O(n 3 ) [1, p.240], [12], the computaton of the bound becomes mpractcal for (28) 1 The graphs are actually the same. The only dfference s n the lnk capactes. 13

14 very large sets of orgns. However, n the next secton we use the methodology descrbed above for developng an O(n 4 ) optmal algorthm for a network wth a sngle orgn node. 4.4 Non-max Capacty Algorthm We have mentoned that the complexty of the Bnary Iteratve Algorthm, descrbed n Secton 4.2, s not necessarly polynomal n the number of nodes or lnks. Thus, we have developed an optmal energy effcent routng algorthm wth polynomal complexty. The algorthm s based on the upper bound derved n Theorem 1 and on the noton of nonmax capacty, defned n Defnton 5. Therefore, t s referred to as the Non-max Capacty Algorthm. In the followng theorem we shall show that n a network wth a sngle orgn node (.e. only a sngle badge generates nformaton), the upper bound descrbed n Theorem 1 s equal to the optmal soluton. Ths observaton wll be used n order to develop an O(n 4 ) algorthm for obtanng the optmal flow values. We emphasze that a complexty of O(n 4 ) s usually much lower than the complexty of executng a lnear programmng algorthm, such as the Smplex. Theorem 2. If { N, r 0} = 1, the optmal network lfetme (T * ) s equal to the upper bound defned n (25). The proof appears n the appendx. Accordngly, t s obvous that f r 0.5 (where s the orgn node), the optmal soluton to Problem EER-LP s the sparsest cut (C defned n (24)). Snce there s only a sngle orgn, the value of C can be obtaned by a max flow algorthm. Thus, n ths case the Non-max Capacty Algorthm reduces to a sngle executon of the max flow algorthm n the transformed network wth the lnk capactes determned by (17) and (18). On the other hand, f r > 0.5, the Non-max Capacty Algorthm conssts of O(n) executons of the max flow algorthm n order to obtan the optmal soluton to Problem EER-LP. Accordng to Theorems 1 and 2, obtanng the optmal lfetme (T * ) requres computng the values of the sparsest cut (C) and the value of Y mn {} (defned n (27)). We have already mentoned that the value of C can be obtaned usng a max flow algorthm. Moreover, accordng to Proposton 2, Y mn {} can be obtaned by O(n) executons of a max flow algorthm n the transformed network wth the lnk capactes determned by (17). In each of these executons, c = 0 for a dfferent nternal lnk of a badge node. o Once the optmal network lfetme (T * ) s obtaned, the nternal capactes of the nodes should be updated: * E T c = mn, N {,, } o A R d (29) e 2 c o E e = A (30) Then, a max flow algorthm should be executed n the resultng transformed network n order to derve the flow values ( f j ). In the course of the proof of Theorem 2, we have shown that the flow values derved n ths procedure yeld the optmal network lfetme 14

15 (T * ). The optmal flow values ( f j ), correspondng to the orgnal problem (Problem EER), can be easly derved from the values of f j. The Non-max Capacty Algorthm, whch s based on the above methodology, s descrbed n Fg. 4. It can be seen that t requres O(n) executons of a max flow algorthm. Hence, the followng corollary results from the fact that the complexty of a max flow algorthm s O(n 3 ). Corollary 1. If { N, r 0} = 1, the value of the network lfetme can be computed by an O(n 4 ) algorthm. 1 transform the node-capactated network to a lnk capactated network 2 set c = E / e N { R, d} o 3 obtan the max flow values ( f (, j) L) 4 f r k > 0.5 k A (k s the orgn node) 5 set T f / k = kk r o k 6 l N {A,R,d} 7 set c 0 ll = o 8 obtan the max flow value ( f kk ) o 9 set T = 2 f /( 2r 1) c kko k = E / e ll o l l * 10 set T = mn T N { R, d} ( ) * c = mn E / e, T / 2 N { A, R, d} o c = E / e kk o k k 11 obtan the max flow values ( f (, j) L) 12 set f = f r / f (, j) L j j k kko j j Fg. 4. The Non-max Capacty Algorthm for obtanng the optmal soluton of Problem EER n a network wth a sngle orgn node The Non-max Capacty Algorthm requres O(n) executons of a max flow algorthm such as the preflow-push algorthm [12]. As descrbed n steps 6-9 n Fg. 4, most of these executons can be performed n parallel. After the value of T * s obtaned, there s a need to execute a max flow algorthm once more (step 11 n Fg. 4). Snce the preflow-push algorthm can be executed n a dstrbuted manner and snce the Non-max Capacty Algorthm requres runnng O(n) nstances of the preflow-push algorthm n parallel, the Nonmax Capacty Algorthm can be mplemented as a dstrbuted algorthm. Fnally, we note that the scalablty of the Non-max Capacty Algorthm to a network wth multple orgn nodes requres further research. 5 Numercal Results The teratve algorthms (presented n Secton 4.2) and the Non-max Capacty Algorthm (presented n Secton 4.4) were mplemented and tested on several representatve cases. In addton, the upper bound on the network lfetme (presented n Secton 4.3) was computed and compared to the optmal lfetme for a few network topologes. It was 15

16 puted and compared to the optmal lfetme for a few network topologes. It was found that n many cases the optmal lfetme and the upper bound are equal. Moreover, n a network wth a sngle orgn node, the Non-max Capacty Algorthm usually requres the lowest number of teratons. In a network wth multple orgns, the Bnary Iteratve Algorthm usually converges to the optmal soluton faster than the Iteratve algorthm. However, the Iteratve Algorthm requres less teratons n order to converge to a soluton whch s close to the optmal soluton. In ths secton, we brefly descrbe the numercal results obtaned for a few networks and demonstrate these fndngs. 5.1 Optmal Soluton and Upper Bound Fg. 5 llustrates a network wth a few orgn nodes. Table 1 ncludes dfferent values of ntal energy and requred transmsson rates for whch the optmal results and the upper bound on the network lfetme (presented n Theorem 1) were computed. The values of the optmal network lfetme (T * ), the upper bound, and the sparsest cut (C) are descrbed n Table 2. Notce that n some cases, the optmal lfetme and the upper bound are equal. Moreover, n several other network topologes we have checked, the upper bound and the optmal lfetme have been found to be equal for varous values of ntal energy and requred flow. Ths result supports our conjecture that n some cases the upper bound s tght. However, we note that there are networks n whch the optmal lfetme s lower than the upper bound (e.g. Network 5-D descrbed n Table 2). Orgn nodes (A) Badges Recevers Destnaton Fg. 5. A network of badges wth a few orgn nodes Table 1. The requred transmsson rates (r ) and the ntal energy values (E ) correspondng to the network descrbed n Fg. 5 Network r 1 r 2 r 3 r 4 r 5 E 1 E 2 E 3 E 4 E 5 E 6 E 7 E 8 E 9 E 10 E 11 E 12 E 13 E 14 E 15 E 16 5-A B C D E Table 2. The optmal network lfetme (T * ), upper bound on the system lfetme, and sparsest cut (C) n the network descrbed n Fg. 5 for the dfferent values of requred transmsson rates and ntal energy descrbed n Table 1 (assumng that e = 1 ) Network T * Upper Bound C 5-A B C D E

17 5.2 Optmal Soluton - Sngle Orgn Fg. 6 llustrates a network wth a sngle orgn node whose optmal flow values were obtaned by the Non-max Capacty and Iteratve Algorthms 1. Table 3 ncludes dfferent values of ntal energy and requred transmsson rates for whch the optmal results were computed. The optmal results are descrbed n Table 4 and the number of executons of a max flow algorthm 2 requred by each of the algorthms s descrbed n Table 5. Badges 5 Recevers 2 9 Orgn Destnaton 7 11 Fg. 6. A network of badges wth a sngle orgn node Table 3. The requred transmsson rates (r 1 ) and the ntal energy values (E ) correspondng to the network descrbed n Fg. 6 Network r 1 E 1 E 2 E 3 E 4 E 5 E 6 E 7 E 8 6-A B C Table 4. The optmal flows (f j ) n the network descrbed n Fg. 6 for the dfferent values of requred transmsson rates and ntal energy descrbed n Table 3 (assumng that e = 1 ) Network f 12 f 13 f 14 f 25 f 26 f 36 f 47 f 48 f 59 f 65 f 67 f 68 f 7,11 f 8,10 T * 6-A B C Table 5. The number of executons of a max flow algorthm requred for obtanng the optmal solutons descrbed n Table 4 Network \ Algorthm Iteratve Bnary Iteratve Non-max Capacty 6-A B C In the soluton descrbed n the frst lne of Table 4 (Network 6-A), nodes 4 and 8 utlze ther full capacty whereas nodes 2,5,6, and 7 utlze ther full energy resources. In the soluton descrbed n the second lne of Table 4 (Network 6-B), node 8 utlzes ts full capacty whereas nodes 5 and 7 utlze ther full energy resources. Fnally, n Network 5- C, node 2 utlzes ts full capacty whle nodes 3 and 4 utlze ther full energy resources. In general, n a network wth a sngle orgn node, the Non-max Capacty Algorthm usually requres the lowest number of executons of a max flow algorthm. The Bnary Iteratve Algorthm usually converges faster than the Iteratve Algorthm. However, after 1 The requred tolerance for the Bnary Iteratve Algorthm was Snce the network has a sngle orgn, there s no need for executng a concurrent max flow algorthm. 17

18 a few teratons, the Iteratve Algorthm usually converges to a soluton whch s close to the optmal soluton. On the other hand, the Bnary Iteratve Algorthm usually converges to such a soluton only after most of the teratons. Ths phenomenon affects the performance of the algorthms when the ntal soluton s very close to the optmal soluton. In such cases the Iteratve Algorthm usually outperforms the Bnary Iteratve Algorthm. 5.3 Optmal Soluton - Multple Orgns Fg. 7 llustrates a network wth a few orgn nodes whose optmal flow values were obtaned by the Iteratve Algorthms 1. In ths network all the badges generate nformaton and are able to forward nformaton generated by other badges. Table 6 ncludes dfferent values of ntal energy and requred transmsson rates for whch the optmal results were computed. Fg. 8 presents the convergence of the Iteratve and Bnary Iteratve Algorthms to the optmal soluton. It can be seen that the Bnary Iteratve Algorthm usually converges to the optmal soluton faster than the Iteratve Algorthm. However, as mentoned before, the Iteratve Algorthm requres less teratons n order to converge to a soluton whch s close to the optmal soluton. Ths results from the fact that the Iteratve Algorthm takes nto account the specal characterstcs of the problem and changes the node capactes at each teraton. However, snce Algorthm BMF fnds a feasble flow for gven demands, at each teraton of the Iteratve Algorthm there s a need to execute a bnary search n order to obtan a feasble soluton. Thus, the complexty of an teraton of the Iteratve Algorthm s hgher than the complexty of an teraton of the Bnary Iteratve Algorthm Badge Recever Destnaton Fg. 7. A network of badges n whch all the badges generate nformaton and are able to forward nformaton generated by other badges Table 6. The requred transmsson rates (r ) and the ntal energy values (E ) correspondng to the network descrbed n Fg. 7 Network r 1 r 2 r 3 r 4 r 5 E 1 E 2 E 3 E 4 E 5 7-A B The requred tolerance for the Bnary Iteratve Algorthm was We assumed that e = 1 and that the maxmal battery lfetme s

19 T Iteratons 7-A Iteratve 7-A Bnary Iteratve 7-B Iteratve 7-B Bnary Iteratve 60 Fg. 8. The network lfetme values (T) obtaned by the Iteratve and Bnary Iteratve Algorthms durng ther convergence to the optmal soluton (T * ) 6 Conclusons and Future Research We have proposed to enable the formaton of a network composed of smart badges n order to acqure nformaton from survvors of structural collapse. The two man aspects that affect the performance of such a network are the lmted batteres of the badges and ther very low data rates (relatvely to the requrements n a dsaster scene). Accordngly, an energy effcent routng problem n such a network has been formulated as an anycast routng problem. The problem has been formulated such that the objectve functon s to maxmze the tme untl the frst battery drans-out and the flow through the badges s bounded by ther data rates. We have presented teratve algorthms for obtanng the optmal soluton of the problem. These algorthms are based on the formulaton of the problem as a concurrent max flow problem and the complexty of one of them s logarthmc n the network lfetme. We have derved an upper bound on the network lfetme for specfc topologes. The bound s based on the new noton of non-max capacty of a cut and s the bass for optmal algorthms. Then, an O(n 4 ) dstrbuted algorthm for obtanng the optmal soluton n a network wth a sngle orgn node has been descrbed. Fnally, we have presented a few numercal results and dscussed the tghtness of the upper bound as well as the performance of the algorthms. The work presented here s the frst approach towards an analyss of the routng problem n an emergency network of smart badges. Hence, there are stll many open problems to deal wth. For example, we would lke to nvestgate the tghtness of the upper bound derved n ths paper and to evaluate the scalablty of the Non-max Capacty Algorthm to networks wth several orgn nodes. We also wsh to utlze methods developed for fractonal packng problems n order to develop fast approxmate algorthms. In addton, 19

20 future study wll focus on the dstrbuted mplementaton of the proposed algorthms. For nstance, we wsh to study how much energy and how many messages are requred n order to obtan the optmal soluton. Fnally, we note that despte the theoretcal mportance of the optmal algorthms and bounds, n an emergency stuaton there s a need for low complexty heurstc algorthms. Thus, a major future research drecton s the development of approxmate and heurstc algorthms that wll deal wth the specal characterstcs of a smart badges network operated n a dsaster ste. Appendx Proof of Proposton 1 Lemma 1. Assume that there exsts a feasble flow n the network (.e. a flow f j satsfyng (9)-(13)). Every cut [O,D] n the transformed network (wth the lnk capactes determned by (17)) that satsfes (20)-(22) must nclude at least 2 nternal lnks. Proof: Accordng to the Max-Flow Mn-Cut Theorem [1, p. 185], the flow through the cut [O,D] s at least T r. (31) O Thus, accordng to (21), the flow through the cut s bgger than T/2. Assume that the cut ncludes only a sngle nternal lnk (, o ). Then, n the orgnal network: fk + fj > T. (32) k Z() j Z() Equaton (32) does not satsfy (13) and contradcts the assumpton that there exsts a feasble flow n the network. Proof of Proposton 1: We assume that there exsts a feasble flow n the network and consder the transformed network (wth the lnk capactes determned by (17)). Accordng to Lemma 1, every cut [O,D] satsfyng (20)-(22) ncludes at least 2 nternal lnks. For a cut [O,D], the flow through the lnk wth the hghest capacty cannot exceed T/2. Namely: f mm o T = 2 2 O A k Z() where (m,m o ) s the nternal lnk wth the maxmum capacty n the cut. Thus, the flow through the rest of the lnks n the cut satsfes: O r 2 ( 1) k 2 1 ( ) f k r f r T (33) O O Ak Z() O fll = (34) o ( l, ) (, ), 2 2 lo O D r ( l, lo) ( m, mo) O 20

21 Hence, (23) results from the fact that the non-max capacty of a cut s at least the flow through the lnks composng the cut, not ncludng the flow through the lnk wth the hghest capacty. Namely for every cut [O,D]: YOD [, ] fll. (35) o ( l, lo) ( O, D), ( l, lo) ( m, mo) Proof of Theorem 1 Lemma 2. T * = C (C defned n (24)), f r 0.5. (36) N Proof: Assumng that the flow s loop free, f (36) holds, the maxmal possble flow through an nternal lnk cannot exceed T. Thus, (13) becomes redundant and Problem EER-LP reduces to Problem CMF, whose optmal soluton s C. Lemma 3. If (36) does not hold: T * 2 YOD [, ] mn mn, C O N: O satsfes (20)-(22) 2 r 1. (37) O Proof: Problem EER-LP ncludes addtonal constrants to Problem CMF. Therefore, ts optmal soluton T * s bounded by the optmal soluton of problem CMF (C). On the other hand, n Proposton 1, another upper bound on T * s derved for the case (36) does not hold. It s obvous that T * s bounded by the mnmum of these bounds. Proof of Theorem 1: If (36) holds the theorem results from Lemma 2. Otherwse, t results from Lemma 3 and Proposton 1. Proof of Proposton 2 Defnton 7. We shall denote by C mn { 1,, k ;G l }: mn [, ] COD cl, l = 0, O N:{ 1,, k} O,{ d, A { 1,, k}} D o Consder the case n whch there s no [O,D] cut separatng { 1,, k } and {d,a { 1,, k }} whch s composed of only nternal lnks (.e. satsfyng (22)). In such a case even f all the capactes of the nternal lnks are taken to be 0, the capacty of the mn cut s. Therefore, snce Y {,, } = mn C {,, ; G} mn 1 k mn 1 l N { A, R, d} s equal or bgger than the capacty of the mn cut when all the nternal lnk capactes are 0, Y mn { 1,, k } =. Ths conforms to the fact that accordng to (27), Y mn { 1,, k } s undefned n such a case. Now, consder the case n whch there exsts a cut [O,D] separatng { 1,, k } and {d,a { 1,, k }} whch s composed of only nternal lnks. Assume that there exsts Y * = Y mn { 1,, k } such that: k l (38) (39) 21

22 * Y > mn C {,, ; G}. (40) l N { A, R, d} Snce C mn { 1,, k ;G l } s the capacty of a cut that does not nclude one nternal lnk, there exsts a cut [O,D] separatng { 1,, k } and {d,a { 1,, k }} such that: mn 1 k l YOD [, ] = mn C {,, ; G}. (41) l N { A, R, d} mn 1 It s obvous that Y[O,D] Y mn { 1,, k } whch s a contradcton to (40). Assume that there exsts Y * = Y mn { 1,, k } such that: k l * Y < mn C {,, ; G}. (42) l N { A, R, d} We wll refer to the nternal lnk wth the maxmal capacty n the cut [O,D] for whch Y * s computed as (m,m o ). Recall that n the Graph G m : c = mm 0. Accordngly: mn 1 k l o * Y C {,, ; G }. (43) mn 1 Notce that Y * cannot be smaller than C mn { 1,, k,g m }, because n such a case C mn { 1,, k,g m } = Y *, whch s a contradcton. Snce (43) s a contradcton to (42): k m Y {,, } = mn C {,, ; G}. (44) mn 1 k mn 1 k l l N { A, R, d} Proof of Theorem 2 Snce n ths proof we deal wth a network wth a sngle orgn node, we shall denote ths node by o. Moreover, the cuts that affect the network lfetme are only the cuts composed of nternal lnk. Therefore, we shall refer to these cuts as nternal cuts. Lemma 4. If (36) does not hold: f o YOD [, ] C E mn mn,, {,, } O N: O satsfes (20)-(22) 2 r 1 2 e N A R d. (45) O Proof: Accordng to (17), the ntal capacty of an nternal lnk s: E c = N { A, R, d }. (46) o e In order to satsfy the capacty constrants of Problem EER-LP, the flow must satsfy: T f N { A, R, d }. (47) o 2 Thus, (45) s obtaned by combnng (46), (47) and (37) (the result of Lemma 4). 22