Topology Design in Time-Evolving Delay-Tolerant Networks with Unreliable Links

Transcription

1 Topology Design in Time-Eoling Dely-Tolernt Networks with Unrelile Links Minsu Hung Siyun Chen Fn Li Yu Wng Deprtment of Computer Science, Uniersity of North Crolin t Chrlotte, Chrlotte, NC 83, USA. School of Computer Science, Beijing Institute of Technology, Beijing, 8, Chin. Astrct With possile prticiption of lrge numer of wireless deices in dely tolernt networks (DTNs), how to mintin efficient nd dynmic topology ecomes crucil. In this pper, we study the topology design prolem in predictle DTN where the time-eoling network topology is known priori or cn e predicted. We model such network s weighted spce-time grph with oth spcil nd temporl informtion. Links inside the spce-time grph re unrelile due to either the dynmic nture of wireless communictions or the rough prediction of underlying humn/deice moility. The im of our topology design prolem is to uild sprse spce-time structure such tht () for ny pir of deices, there is spce-time pth connecting them with the reliility lrger thn required threshold; () the totl cost of the structure is minimized. We first show tht this prolem is NP-hrd, nd then propose seerl heuristics which cn significntly reduce the totl cost of the topology while mintin the relile connectiity oer time. I. INTRODUCTION In dely tolernt networks (DTNs), the lck of continuous connectiity, network prtitioning, long delys, unrelile time-rying links, nd dynmic topology pose new chllenges in design of DTN protocols. Recently, seerl DTN routing schemes [] [3] he een proposed to tke the intermittent connectiity nd time-rying topology into considertion. In ddition, different moility study [], [5] nd grph modeling [6], [7] he een conducted for DTNs to understnd the underlying socil nd temporl chrcteristics of the network prticipnts. Howeer, there is little reserch on how to mintin cost-efficient nd relile topology of DTNs. Network topology is lwys criticl issue in design of ny networking systems. For different network pplictions, network topology cn e designed or controlled under different ojecties (such s power efficiency, fult tolernce, nd throughput mximiztion). Topology control protocols he een well studied in wireless networks [8]. The focus of preious reserch is minly on how to construct cost-efficient structure from sttic nd connected topology. Howeer, in DTNs, the underlying topology is lck of continuous connectiity which mkes existing topology control lgorithms useless. Therefore, how to efficiently mintin dynmic The work of M. Hung, S. Chen, nd Y. Wng is supported in prt y the US Ntionl Science Foundtion (NSF) under Grnt No. CNS-9533 nd CNS The work of F. Li is prtilly supported y the Ntionl Nturl Science Foundtion of Chin under Grnt 6935, Beijing Nturl Science Foundtion under Grnt 7 nd Scientific Reserch Foundtion for the Returned Oerses Chinese Scholrs, Stte Eduction Ministry. Y. Wng (yu.wng@uncc.edu) is the corresponding uthor. topology in DTNs ecomes crucil, especilly with the prticiption of lrge numer of wireless deices. In this pper, we study the topology design prolem in time-eoling nd predictle DTN. DTNs often eole oer time: chnges of network topology cn occur if nodes moe round. For certin type of DTNs, the temporl chrcteristics of topology could e known priori or cn e predicted from historicl dt. For exmple, it is esy to discoery the temporl pttern of topology for DTN formed y either pulic uses [9] or stellites [] which he fixed tours nd schedules, or moile socil network [5] consisting of students who shre fixed clss schedules. A recent study [] on humn moility lso shows tht 93% potentil predictility cn e chieed in humn moility prediction. For this kind of time-eoling nd predictle DTNs, the spce-time grph model [], insted of the sttic grph model, cn e used to cpture oth the spce nd time dimensions of the dynmic network topology nd to enle the emultion of ny storend-forwrd DTN routing methods. Gien such spcetime grph including ll possile temporl nd spcil links, the topology design prolem ims to find sugrph which mintins the connectiity oer time etween ny two deices while minimizing the cost. In our recent study [3], [], we he proposed seerl heuristics for the sic topology design prolem in timeeoling nd predictle DTNs. Those methods ll ssume tht the prediction of future links re perfect nd links re relile for communictions, i.e., the pcket deliery oer these spcil or temporl links is gurnteed nd without errors. This is clerly too optimistic since in prctice, the wireless communictions re unrelile due to the lossy nture of wireless chnnels. In ddition, een though certin type of network moility could e predicted sed on user ehior or historicl dt, the link prediction could e inccurte sometime. Therefore, in this pper, we remoe the strong ssumptions on relile links nd perfect prediction. Insted, for ech spcil or temporl link in the spce-time grph, we ssume it hs proility to reflect its reliility, either sed on link qulity estimtion or moility prediction. Lrger the relile proility of link, higher chnce the DTN trnsmissions oer tht link to e successful. Therefore, gien the weighted spce-time grph, the new relile topology design prolem (RTDP) ims to find sugrph in which the reliility of DTN routing etween ny two deices is gurnteed lrger thn required threshold nd the totl cost

2 of topology is minimized. In this pper, we show tht this new topology design prolem is NP-hrd nd propose fie heuristics which cn significntly reduce the totl cost of network topology while mintining the DTN reliility oer time. For our est knowledge, this pper is the first ttempt to study topology design for time-eoling networks with unrelile links. spce t= t= t=3 t= () 5 time t= t= t=3 t= () Fig.. A time-eoling DTN nd its corresponding spce-time grph: () the time-eoling topology of DTN ( sequence of snpshots); () the corresponding spce-time grph G, where spce-time DTN pth from the source to the destintion 5 is highlighted in lue. II. MODELS, ASSUMPTIONS, AND THE PROBLEM A. Modeling Time-eoling Networks: Spce-time Grphs Assume tht V = {,, n } e the set of ll indiidul nodes (wireless deices) in the network oer period of time T. Here, time is diided into discrete nd equl time slots, e.g. {,, T }. Since positions of indiidul nodes nd the topology co-eole oer time, sequence of sttic grphs cn e defined oer V to model the interctions mong nodes in the time-eoling DTN during certin time slot. Fig. () illustrtes such n exmple. Let G t = (V t, E t ) e directed grph representing the snpshot of the network t time slot t nd link i tt j Et represents tht node i cn communicte to j t time t. Then, the dynmic network is descried y the union of ll snpshots {G t t = T }. Howeer, such sequence of sttic grphs is not directly usele in the nlysis of DTN routing protocols. Some of the snpshots my not e connected grphs t ll (e.g., the first three snpshots in Fig. ()), which mkes routing tsk oer them chllenging. In some existing DTN protocols, n ggregted grph oer {G t } is used, where n edge etween pir of nodes represents these two nodes he intercted t certin point during the time period T. The weight of such n edge could e the frction of the occurrence oer the historicl period of tht interction, i.e., contct proility. Seerl DTN routing protocols [], [3] estimte this contct proility sed on pst contct history nd use it to select rely nodes. Howeer, such ggregted iew discrds temporl informtion out the timing nd order of interctions, which my cuse filure of deliery or d performnces. In this pper, we dopt the spce-time grph [] to model the time-eoling DTNs. We conert the sequence of sttic grphs {G t } into spce-time grph G = (V, E), which is directed grph defined in oth 3 5 spcil nd temporl spces. Fig. () illustrtes the spcetime grph of the sme network in Fig. (). In the spcetime grph G, T + lyers of nodes re defined nd ech lyer hs n nodes, thus the whole ertex set V = {j t j =,, n nd t =,, T } nd there re n(t + ) nodes in G. Two kinds of links (sptil links nd temporl links) re dded etween consecutie lyers in E. A temporl link t j j t (those horizontl links in Fig. ()) connects the sme node j cross consecutie (t )th nd tth lyers, which represents the node crrying the messge in the tth time slot. A sptil link t j k t represents forwrding messge from one node j to its neighor k in the tth time slot (i.e., j k E t ). By defining the spce-time grph G, ny communiction opertion in the time-eoling network cn e simulted on this directed grph. As the lue pth shown in Fig. (), spce-time DTN pth from to 5 shows prticulr DTN routing strtegy to delier the pcket from to 5 in the network: holds the pcket for st time slot, then psses it to 3 t t =, etc. We cn define the connectiity of spce-time grph which is different from tht of sttic grph. Definition : A spce-time grph H is connected oer time period T if nd only if there exists t lest one directed pth for ech pir of nodes (i, T j ) (i nd j in [, n]). This gurntees tht ny pcket cn e deliered etween ny two nodes in the network oer the period of T. Note tht connected spce-time grph does not require the connectiity in eery snpshot. Herefter, we ssume tht the originl spce-time grph G is connected oer time period T. For ech direct link e E, we further ssume tht there is cost c(e), representing the cost ssocited with trnsiting messge oer tht link (trnsiting from one node to the other node oer spcil link or holding the messge within one node oer temporl link). The totl cost of spce-time grph c(h) is defined s the summtion of costs of ll links in H, i.e., c(h) = c(e). () e H The lest cost pth Pc H (u, ) is defined s the pth from u to in H with the minimum totl cost. Its totl cost is denoted y c H (u, ) = e Pc H (u,) c(e). B. Reliility of DTN Topology oer Unrelile Links To consider the reliility of lossy wireless links or unperfected link predictions, we lso define relile proility r(e) for ech link e E, which represents the proility of successful dt trnsmission oer link e. The unreliility of the link could come from either link filure due to lossy wireless signl or poor prediction of future links. Here, we ssume tht the relile proility of ech link cn e otined through link estimtion techniques t the link nd physicl lyers [5] or moility prediction techniques [6]. Fig. () illustrtes simple spce-time grph with just two deices. The lel of ech link e is the pir of cost nd reliility, i.e., (c(e), r(e)). For exmple, in the first time slot, the cost of spcil link is 3 nd its reliility is.6. Gien

3 (,.5) (,.5) (,.5) (,.5) (,.5) (,.5) (3,.6) (,.8) (3,.6) (,.8) (3,.6) (,.8) (3,.6) (,.8) (3,.6) (,.8) (3,.6) (,.8) (5,.) (5,.) (5,.) (5,.) (5,.) (5,.) () G () H (c) H Fig.. Exmples of relile topology design: () the originl spce-time grph G with cost nd reliility.36; () H with cost 6 nd reliility.5; nd (c) H with cost nd reliility.36. Here, links re leled y (cost, reliility), nd green links re remoed links from G. the reliility of ech link, we cn then define the reliility of pth P or structure H. In this pper, we only consider the reliility for single-copy DTN routing where only one copy of ech messge is propgted in the network. Thus, the resulting propgtion pth of messge is siclly single spce-time pth in G. Gien pth P (u, ) connecting nodes u nd, the reliility of P (u, ) is the production of reliility of ll links in tht pth. For exmple, the pth in Fig. () hs reliility of.5.8 =.. For gien routing topology H, we cn define the most relile pth Pr H (u, ) s the pth from u to in H with the highest reliility. Let r H (u, ) = Π e P H r (u,)r(e) e the reliility of pth Pr H (u, ). For exmple, etween nd in Fig. (), is the most relile pth with reliility of.6. =.6. Then the reliility of the topology H is defined s follows r(h) = min i,j n rh (i, j T ). () The reliility of structures shown in Fig. (), () nd (c) re.36,.5 nd.36 respectiely. Notice tht it is esy to clculte r(h) gien H y using ny shortest pth lgorithms. C. Relile Topology Design Prolem We now cn define the relile topology design prolem (RTDP) on weighted spce-time grphs s follows. Definition : Gien connected spce-time grph G, the im of relile topology design prolem (RTDP) is to construct sprse spce-time grph H, which is sugrph of the originl spce-time grph G, such tht () H is still connected oer the time period T ; () the reliility is lrger thn or equl to predefined threshold γ; nd (3) the totl cost of H is minimized. Notice tht generlly speking denser spce-time topology leds to etter connectiity nd reliility ut more expensie in term of totl cost. Therefore, the topology design ims to find topology with certin reliility gurntee while minimizing the cost. If the threshold γ is constnt, the reliility requirement itself coers the sic connectiity, since it is oiously much stronger requirement thn sic connectiity. Once gin, we ssume tht the originl spcetime grph G is connected nd hs reliility lrger thn γ. Fig. () nd (c) show two su-topologies of Fig. () with different costs nd reliility. Note tht these topologies re still connected oer time, i.e., eery node cn find spce-time pth to ny other node. Hrdness of RTDP: The topology design prolem for spce-time grph een without reliility requirement is much hrder thn the one for sttic grph. For sttic grph without time domin, minimum spnning tree cn chiee the gol of keeping connectiity with miniml cost. Howeer, simply pplying the spnning tree in ech snpshot or oer the entire spce-time grph is not direct solution. First, the grph in ech snpshot my not e connected t ll. On the other hnd, spnning tree or forest of the whole spce-time grph connects ech node in eery snpshot, which is not necessry nd wste of mny links. Plus the reliility requirement, the RTDP ecomes more chllenging. In [3], we he proed tht the topology design prolem without reliility requirement is NP-hrd. Since such topology design prolem is specil cse of RTDP when γ =, RTDP is lso NP-hrd. The RTDP is different with the stndrd spce-time routing [], [7], which only ims to find the most cost-efficient spce-time pth for pir of source nd destintion. The topology design prolem ims to mintin oth cost-efficient nd connected spce-time routing topology for supporting relile DTN trnsmissions etween ll pirs of nodes. III. HEURISTICS OF RELIABLE TOPOLOGY DESIGN Since RTDP oer spce-time grphs is NP-hrd, we now propose fie different heuristics to construct sprse structure tht fulfills the connectiity nd reliility requirements oer time. The first three heuristics re sed on minimum cost relile routing, while the other two re sed on greedy lgorithms. For ll lgorithms, the inputs re the originl grph G nd reliility requirement γ, nd the output is sugrph H of G. A. Heuristics Bsed on Min Cost Relile Pth One nturl ide to construct low-cost relile sugrph, which gurntees the route reliility etween ny two nodes oer the time period T, is keeping ll the minimum cost relile pths from i to T j for i, j =,, n in G. Here the minimum cost relile pth connecting u nd, denoted y P G cr(u, ), is the pth with the lest cost mong ll pths etween u nd which he reliility lrger thn or equl to γ. Oiously, if we cn find such pths for eery pir of source nd destintion, the union of ll of them stisfies the reliility requirement of our topology design prolem. Howeer, to find the shortest pth with dditionl constrins in grph itself is lso well known NP-hrd prolem, clled restricted shortest pth prolem [8]. Therefore, in this pper, we use one of

4 the existing heuristics for restricted shortest pth, Bckwrd- Forwrd method (BFM) y Reees nd Slm [9]. The sic ide of Bckwrd-Forwrd method is quite simple. Assume we wnt to find relile pth from s to t. BFM first determines the lest-cost pth (LCP) nd the most relile pth (MRP) from eery node u to t. It then strts from s nd explores the grph y conctenting two segments: () the sofr explored pth from s to n intermedite node u, nd () the LCP or the MRP from node u to t. BFM simply uses Dijkstrs lgorithm with the following modifiction in the relxtion procedure: link u is relxed if it reduces the totl cost from s to while its pproximted end-to-end reliility oeys the reliility constrint. The computtionl complexity of the BFM is siclly three times tht of Dijkstr s lgorithm. Let P G BF M (u, ) e the resulting pth from u to sed on BFM. Notice tht P G BF M (u, ) is not optiml for restricted shortest pth. Howeer, sed on simultion study y Kuipers et l. [], BFM is one of the most efficient methods mong ll existing methods, it hs smll execution time nd often genertes good qulity pth compred with the optiml. Algorithm Union of Min Cost Relile Pth () : H φ; X = {(i, T j )} for ll integer i, j n. : for ll pirs (i, T j ) X do 3: Find the minimum cost relile pth P G BF M ( i, T j ) in G using Bckwrd-Forwrd method. : if e P G BF M ( i, T j ) nd e / H then 5: H = H {e}. 6: return H Algorithm Greedy with Min Cost Relile Pth () : H φ; X = {(i, T j )} for ll i nd j in [, n]. : while X φ do 3: Find the minimum cost relile pths for eery pir nodes in X using BFM in G, nd ssume P BF M (i, T j ) hs the lest cost mong these pths. : if e P BF M (i, T j ) then 5: H e; c(e). 6: X X (i, T j ). 7: return H Algorithm 3 Lest Cost Pth or Min Cost Relile Pth () : H φ; X = {(i, T j )} for ll integer i, j n. : for ll pirs (i, T j ) X do 3: Find the lest cost pth Pc G (i, T j ) in G. : if e Pc G (i, T j ) nd e / H then 5: H = H {e}. 6: for ll pirs (i, T j ) X do 7: if r H (i, T j ) < γ then 8: Add ll links e P G BF M ( i, T j ) into H if e / H. 9: return H With Bckwrd-Fowrd method, our first heuristic pproch is to find minimum cost relile pth for ech pir of source nd destintion nd tke the union of them to form the sugrph which gurntee the oerll reliility. Algorithm shows the detiled lgorithm. Its time complexity is O(n T (log(nt ) + n)) since we only need to compute minimum cost relile pths of G (with O(nT ) nodes nd t most O(n T ) links). This cn e esily chieed y running 3n times of Dijkstr s lgorithm whose complexity is O(nT (log(nt ) + n)). Herefter, we refer this method s union of minimum cost relile pth lgorithm (). Howeer, the structure uilt y the oe method my contin more links thn necessry. Therefore, we propose new greedy lgorithm (s shown in Algorithm ) to further improe the performnce. The sic ide is s follows. We im to connect n pir of nodes in X. In ech round we pick the minimum cost relile pth etween pir nodes in X which is the minimum mong ll minimum cost relile pths connecting ny pir of nodes in X. Then we dd ll links in this pth into H, cler the costs of these links to zeros, nd remoe this pir from X. This procedure is repeted until ll n pir nodes (i, T j ) re gurnteed to e connected y relile pths in H. It is oious tht the output of this method is much sprser thn the one of method. We refer this method s greedy method sed on minimum cost relile pth (). The time complexity is O(T n 5 + T n log(t n)) since 3n times of Dijkstr s lgorithm re running on the spce-time grph in ech round for n rounds. The third method siclly comines the lest cost pth nd the minimum cost relile pth. It includes two steps. In the first step, we connect n pir of nodes in X = {(i, T j )} for ll integer i, j n y dding ll of the lest cost pths etween them. After this step, we he structure H which connects ll pirs of (i, T j ) ut my not stisfy the reliility requirements yet. In the second step, for ech pir (i, T j ), we clculte its reliility in H. If the cost of r H (i, T j ) is less thn γ, we directly dd the links in the minimum cost relile pth etween this two nodes in G into H. See Algorithm 3 for detil. Herefter, we denote this method s. Both steps of need n O(n T (log(nt )+n)) time since they oth run computtion of shortest pths for O(n ) rounds. Therefore, the totl time complexity is O(n T (log(nt )+n)). B. Greedy-sed Heuristics Both the fourth nd fifth lgorithms re sed on the sme greedy principle, deleting or dding edges in order nd checking whether the reliility is chieed or oid. The sic ide of the fourth method is deleting edges from input grph (either the originl grph G or the output grph from the first two lgorithms) in decresing order sed on link cost. Link i tt+ j is remoed if the sugrph H without this link cn still chiee reliility of γ. Algorithm shows the detiled lgorithm, nd we denote it s herefter. The time complexity nlysis is strightforwrd. The sorting could e done in O(n T log(n T )) with O(n T ) links. The for-loop includes n T rounds of computtions of shortest pths. Thus,

5 Algorithm Greedy Algorithm to Delete Links () : H G. : Sort ll links in link set E of G sed on their costs. 3: for ll e E (processed in decresing order of costs) do : if r(h {e}) γ then 5: H = H {e}. 6: return H Algorithm 5 Greedy Algorithm to Add Links () : H φ; X = {(i, T j )} for ll integer i, j n. : for ll pirs (i, T j ) X do 3: Find the lest cost pth Pc G (i, T j ) in G. : if e Pc G (i, T j ) nd e / H then 5: H = H {e}. 6: for ll e G H (in incresing order of costs) do 7: if r(h) < γ then 8: H = H + {e}. 9: return H the time complexity of this prt is O(n T (log(nt ) + n)). Therefore, totl time complexity of is lso in order of O(n T (log(nt ) + n)), which is much higher thn those of methods sed on minimum cost relile pth. Note tht since the outputs of our first two lgorithms re much sprser thn the originl grph ut cn gurntee the reliility, if we use them s the input of it cn se certin computtion cost. The fifth lgorithm is lso greedy lgorithm ut processes links in the reerse order compred with the fourth lgorithm. It strts with uilding connected sprse structure nd then dds more links to stisfy the reliility requirement. It lso includes two steps. In the first step, gin we tke the union of ll lest cost pths etween (i, T j ) for ll integer i, j n s H. In the second step, remining links re processed in the incresing order of cost. Link i tt+ j is dded to H if the current sugrph H cnnot chiee reliility of γ yet. See Algorithm 5 for detil. Herefter, we denote this method s. The complexity of is the sme s the one of (O(n T (log(nt ) + n))) since the second step is lmost the sme with nd the first step cost less time. IV. SIMULATIONS To elute our proposed lgorithms deoted to the RTDP prolem we he conducted extensie simultions oer rndomly generted networks. We implement nd test the following six lgorithms: (Alg ), (Alg. ), (Alg. 3), (Alg. ), (Alg. 5), nd. Here, is the union of lest cost pths from i to j T for i, j =,, n), which mintins the connectiity oer time ut cnnot gurntee the reliility. We use s reference. In ll simultions, we tke three metrics s the performnce mesurement for ny lgorithm deoted to RTDP: Totl Cost: the totl cost of the constructed topology H (output of the lgorithm), i.e., c(h) = e H c(e). Totl Numer of Edges: the totl numer of edges in H, i.e., H. Here, G denotes totl edge numer of G. Reliility: r(h) = min i,j n r H (i, T j ). The ojectie of topology design is to construct topologies with smll totl cost, smll edge numer nd lrger reliility. The underlying time-eoling networks re rndomly generted from rndom grph model. We first generte sequence of sttic rndom grphs {G t } with nodes, spreding oer time slots. For ech time slot t, the sttic grph G t is generted using the clssicl rndom grph genertor. For ech pir of nodes i nd j, we insert edge i j with fixed proility δ into G t. Smll lue of δ leds to sprse DTN, nd δ =. implies tht the topology in ech time slot is complete grph. After generting the sequence of {G t }, we conert it into the corresponding spce-time grph G. The cost nd reliility of ech inserted edge re rndomly chosen from to 5 nd from.8 to, respectiely. For ech setting, we generte rndom time-eoling networks nd report the erge performnce of our lgorithms mong them. For the first set of simultions, we increse the network density y rising δ from. to.8, nd keep reliility requirement γ t.6. Fig. 3() nd 3() show the rtios etween the totl cost/numer-edges of the generted grph H nd tht of the originl grph G with incresing δ. This rtio implies how much sing chieed y the topology design lgorithm, compred with the originl network. From the results, ll lgorithms cn significntly reduce the cost of mintining the connectiity. Een with the lest density (δ =.), ll lgorithms except for cn se more thn 7% cost nd round 65% edges. When the network is denser, the sing of topology design is lrger. For δ =.8, more thn 85% cost is sed. Compring ll methods, we cn find the following osertions. () hs the lest cost since it only gurntee the connectiity while other proposed lgorithms gurntee the reliility; () hs the lrgest cost due to simply dding mny low-cost links; (3) chiees smller cost thn since it reuse some links in preious rounds; () s cost is etween s nd s since it comines oth LCP nd MCRP; (5) Oerll, nd he the est costs mong the proposed fie methods. Fig. 3(c) shows the reliility of ech structure generted y ll methods. Clerly, LCP is the only lgorithm tht cnnot stisfy the reliility requirement γ =.6. It is n lgorithm only for mintining the connectiity oer time. All other fie lgorithms cn lwys chiee the desired reliility, nd they he ery close performnce in term of reliility. But considering their costs,,, re etter choices. When G is sprse (e.g. δ =.), its reliility is just oer.6. In tht cse, our proposed lgorithms remoe round 7% edges ut still keep the reliility t the desired leel. This clerly demonstrtes the power of topology design. In the second set of simultions, we fixed the network density δ =.5 nd run our lgorithms with different reliility requirement γ incresing from.3 to.6. From results shown in Fig., we cn osere tht tighter reliility requirement results in higher cost nd lrger numer of

6 rtio of totl cost rtio of edge numer unicsting reliility G δ δ () c(h)/c(g) () H / G (c) r(h) Fig. 3. Simultion results on rndom networks with different density nd γ =.6. ()-(): The cost (or edge numer) of H is diided y the cost (or edge numer) of G, which shows how much sing chieed y proposed methods. (c): The reliility r(h) shows the reliility of H. δ rtio of totl cost rtio of edge numer unicsting reliility G γ γ () c(h)/c(g) () H / G (c) r(h) Fig.. Simultion results on rndom networks with fixed network density δ =.5 nd rying reliility requirement γ. γ edge-usge of our topology lgorithms. When the reliility requirement ecomes looser, the restriction on remoing links ecomes weker. In the extreme cse with γ =, our RTDP conerges to the sic topology design prolem [3] which only preseres the connectiity. Notice tht performnces of LCP do not ffect y the reliility requirement. In ddition, performs etter when the reliility requirement is smll. V. CONCLUSION We study relile topology design prolem in predictle time-eoling DTN with unrelile links modeled y proilistic spce-time grph. We propose set of new heuristics which cn not only mintin the connectiity nd reliility of pths oer time ut lso significntly reduce the cost of topology. Simultion results from rndom networks demonstrte the efficiency of our methods. We eliee tht this pper presents the first step in exploiting topology design for time-eoling DTNs with unrelile links. REFERENCES [] Q. Yun, I. Crdei, nd J. Wu, Predict nd rely: n efficient routing in disruption-tolernt networks, in ACM MoiHoc 9, 9. [] T. Spyropoulos, et l., Spry nd wit: n efficient routing scheme for intermittently connected moile networks, in ACM WDTN 5, 5. [3] A. Lindgren, A. Dori, O. Schelén, Proilistic routing in intermittently connected networks, Mo. Comp. Comm. Re., 7(3):9, 3. [] J. Leguy, T. Friedmn, nd V. Conn, Eluting moility pttern spce routing for DTNs, in IEEE INFOCOM, 6. [5] V. Srinisn, et l., Anlysis nd implictions of student contct ptterns deried from cmpus schedules, in ACM Moicom, 6. [6] E.M. Dly nd M. Hhr, Socil network nlysis for routing in disconnected dely-tolernt mnets, in ACM MoiHoc, 7. [7] T. Hossmnn, et l., Know thy neighor: towrds optiml mpping of contcts to socil grphs for DTN routing, in IEEE INFOCOM, 9. [8] Y. Wng, Topology control for wireless sensor networks, ook chpter, Sensor Networks & Applictions (eds. Y. Li et l.), Springer, 7. [9] X. Zhng, J. Kurose, B.N. Leine, D. Towsley, nd H. Zhng, Study of us-sed disruption-tolernt network: moility modeling nd impct on routing, in ACM MoiCom 7, 7. [] S. Burleigh nd A. Hooke, Dely-tolernt networking: n pproch to interplnetry internet, IEEE Commun. Mg., (6):8 36, 3. [] C. Song, Z. Qu, N. Blumm, nd A.-L. Brsi, Limits of predictility in humn moility, Science, 37:8,. [] S. Merugu, M. Ammr, nd E. Zegur, Routing in spce nd time in networks with predictle moility, GIT, Tech. Rep., CC--7,. [3] M. Hung, S. Chen, Y. Zhu, et l., Topology control for time-eoling nd predictle dely-tolernt networks, in IEEE MASS,. [] M. Hung, S. Chen, et l., Cost-efficient topology design prolem in time-eoling dely-tolernt networks, in IEEE Gloecom,. [5] N. Bccour, et l., A comprtie simultion study of link qulity estimtors in wireless sensor networks, in IEEE MASCOTS, 9. [6] W. Su, S. Lee, nd M. Gerl, Moility prediction in wireless networks, in IEEE MILCOM,. [7] B.B. Xun, A. Ferreir, nd A. Jrry, Computing shortest, fstest, nd foremost journeys in dynmic networks, Int l J. of Foundtions of Computer Science, ():67 85, 3. [8] F. A. Kuipers, An oeriew of constrint-sed pth selection lgorithms for QoS routing, IEEE Communictions Mgzine, 5 56,. [9] D. S. Reees nd H. F. Slm, A distriuted lgorithm for delyconstrined unicst routing, IEEE/ACM Trns. Netw., 8:39 5,. [] F. Kuipers, T. Korkmz, et l., Performnce elution of constrintsed pth selection lgorithms, IEEE Network, 8:6 3,.