Providing Protection in Multi-Hop Wireless Networks

Transcription

1 Technicl Report, My 03 Proviing Protection in Multi-Hop Wirele Network Greg Kupermn MIT LIDS Cmrige, MA 039 Eytn Moino MIT LIDS Cmrige, MA 039 Atrct We conier the prolem of proviing protection gint filure in wirele network uject to interference contrint. Typiclly, protection in wire network i provie through the proviioning of ckup pth. Thi pproch h not een previouly coniere in the wirele etting ue to the prohiitive cot of ckup cpcity. However, we how tht in the preence of interference, protection cn often e provie with no lo in throughput. Thi i ue to the fct tht fter filure, link tht previouly interfere with the file link cn e ctivte, thu leing to recpturing of ome of the lot cpcity. We provie oth n ILP formultion for the optiml olution, well lgorithm tht perform cloe to optiml. More importntly, we how tht proviing protection in wirele network ue much 7% le protection reource compre to imilr protection cheme eigne for wire network, n tht in mny ce, no itionl reource for protection re neee. I. INTRODUCTION Multi-hop wirele meh network hve ecome increingly uiquitou, with wie-rnging ppliction from militry to enor network. A thee network continue gining in prominence, there i n increing nee to provie protection gint noe n link filure. In prticulr, wirele meh network hve recently emerge promiing olution for proviing Internet cce. Since thee network will e tightly couple with the wire Internet to provie Internet ervice to en-uer, they mut e eqully relile. Wire network hve long provie pre-plnne ckup pth, which offer rpi n gurntee recovery from filure. Thee protection technique cnnot e irectly pplie to wirele network ue to interference contrint. A oppoe to wire network, two wirele noe in cloe proximity will interfere with one nother if they trnmit imultneouly in the me frequency chnnel. So, in ition to fining ckup route, cheule of link trnmiion nee to e pecifie. In thi work, we conier the prolem of proviing gurntee protection in wirele network with interference contrint vi pre-plnne ckup route, well their correponing link trnmiion cheule. Gurntee protection cheme for wire network hve een tuie extenively [ 5], with the mot common cheme eing Thi work w upporte y NSF grnt CNS-609 n CNS , y DTRA grnt HDTRA , n y the Deprtment of the Air Force uner Air Force contrct #FA87-05-C-000. Opinion, interprettion, concluion n recommention re thoe of the uthor n re not necerily enore y the Unite Stte Government. + gurntee pth protection [5]. The + protection cheme provie n ege-ijoint ckup pth for ech working pth, n gurntee the full emn to e ville t ll time fter ny ingle link filure. Protection cheme optimize for wirele network with interference contrint hve not yet een coniere. Typiclly, n pproch for reiliency in wirele network (in prticulr enor network) i to enure tht there exit coverge for ll noe given ome et of link filure [6, 7]. Thi pproch to reiliency oe not conier routing n cheuling with repect to interference contrint, n ume tht there exit ome mechnim to fin route n cheule t ny given point in time. Furthermore, there i no gurntee tht ufficient cpcity will e ville to protect gint filure. The ie of pplying + protection in wirele network i riefly mentione in [8]. However, [8] oe not tuy the pecific technicl etil of uch n pproch to wirele protection. The gol of thi chpter i to tuy protection mechnim for wirele network with prticulr focu on the impct of wirele interference n the nee for cheuling. The ition of interference contrint mke the protection prolem in wirele etting funmentlly ifferent from the one foun in wire context. After filure in wirele network, link tht coul not hve een ue ue to interference with the file link ecome ville, n cn e ue to recover from the filure. In fct, it i often poile to protection in wirele etting without uing ny itionl reource. Conier llocting protection route for the following exmple, hown in Fig.. The wirele network operte in time-lotte fhion, with equl length time lot ville for trnmiion. Any two noe within trnmiion rnge hve link etween them, n ech link time lot ignment i hown in the figure. We ume -hop interference moel where ny two link tht hve noe in common cnnot e ctive t the me time. Aitionlly, we ume unit cpcity link. Before ny filure, the mximum flow from to i, n cn e chieve uing two time lot cheule, hown in Fig.. At ny given point in time, only one outgoing link from cn e ctive, n imilrly, only one incoming link to cn e ctive. Wirele link {, c}, n {c, } cnnot e ue prior to the filure of {, }, ut ecome ville fter {, } fil. After the filure of {, }, flow cn e route from to c uring

2 c () Before filure c () After {, } fil Fig. : Time lot ignment for protection in wirele network time lot, n from c to uring lot, hown in Fig.. Similr cheule cn e foun for filure of the other link. The mximum flow from to i for oth efore n fter filure; i.e., there i no reuction in mximum throughput when llocting reource for protection route on {, c} n {c, }: protection cn e igne for free. Thi i in contrt to wire network where the mximum throughput without protection from to i 3, n the mximum throughput when igning protection route on {, c} n {c, } i, which mount to 3 lo in throughput ue to protection. The novel contriution of thi chpter i introucing the Wirele Gurntee Protection (WGP) prolem in multihop network with interference contrint. In Section II, the moel for WGP i preente. In Section III, propertie of n optiml olution re exmine for ingle emn with - hop interference contrint, which re then ue to motivte the evelopment of time-efficient lgorithm. In Section IV, n optiml olution i evelope vi mixe integer liner progrm for generl interference contrint. In Section V, time-efficient lgorithm re evelope tht perform within 4.5% of the optiml olution. II. MODEL AND PROBLEM DESCRIPTION In thi chpter, olution to the gurntee protection prolem for multi-hop wirele network uject to interference contrint re evelope n nlyze. Our gol i to provie protection in mnner imilr to wht h een one in the wire etting. Nmely, fter the filure of ome network element, ll connection mut mintin the me level of flow tht they h efore the filure. In orer to o o, reource re llocte n cheule in vnce on lternte (ckup) route to protect gint filure. In wire network, two jcent noe cn trnmit imultneouly ecue they o not interfere with one nother; if cpcity exit on et of link, pth cn e route uing tht cpcity. Wirele network re ifferent; interference contrint mut e coniere. A et of link in cloe proximity cnnot trnmit imultneouly on the me frequency chnnel; only one link from tht et cn e ctive t time, or ele they will interfere with one nother. Not only mut pth etween the ource n etintion e foun with ville cpcity, ut lo cheule of link trnmiion nee to e etermine. Thi i known the routing n cheuling prolem [8 6], which i known to e NP-Hr [9]. The ition of interference contrint complexity to the tritionl wire protection prolem, ut lo preent n opportunity to gin protection from filure with miniml lo of throughput. After filure in wirele network, link tht coul not hve een ue ue to interference with the file link ecome ville, n cn e ue to recover from the filure. In fct, it i often poile to protection in wirele etting without ny lo in throughput. The following network moel i ue for the reminer of the chpter. A grph G h et of vertice V n ege E. An interference mtrix I i given, where Iij kl I i if link {i, j} n {k, l} cn e ctivte imultneouly (o not interfere with ech other), n 0 otherwie. The interference mtrix i gnotic to the interference moel ue (i.e., it cn e ue to repreent nerly ny type of link interference). For the reminer of thi work, we focu on the -hop interference moel (ny two link tht hre noe cnnot e ctivte imultneouly), ut our cheme cn e pte to the K-hop [7] interference moel well. Our gol in thi chpter i to evelop frmework for routing n cheuling with protection uner interference contrint. We ume noe re fixe, link re iirectionl, n tht the network ue ynchronou time lotte ytem, with equl length time lot; the et of time lot ue i T. Only link filure re coniere, n inglelink filure moel i ume; it i trightforwr to pply the olution evelope in thi chpter to noe filure well. For now, we ume centrlize control; the lgorithm preente cn e moifie to work in itriute fhion, one in [8]. Aitionlly, we only conier ingle frequency chnnel. III. EFFICIENT ALGORITHM FOR A SINGLE DEMAND In thi ection, we im to chieve inight into proviing protection for wirele network with interference contrint y exmining the olution for ingle emn uner ic et of network prmeter: -hop interference contrint n unit cpcity link. In Section III-A, propertie of n optiml olution re exmine for routing n cheuling with n without protection. In Section III-B, time efficient lgorithm i evelope tht fin mximum throughput gurntee to e within.5 of the optiml olution. In Appenix B, polynomil time lgorithm i preente tht tighten thi oun n fin olution gurntee to e within 6 5 of the optiml olution. A. Solution Propertie In thi ection, propertie of n optiml olution for WGP for ingle emn re exmine. Firt, we look t routing n cheuling without protection, n then thoe reult re extene to the protection etting. Lemm. The mximum flow tht cn e route n cheule etween the ource n etintion uner -hop interference contrint without protection i. Proof: Uner -hop interference contrint, only one link exiting the ource noe cn e ctive t time. Since ech link

3 3 h unit cpcity, the mximum flow tht cn leve the ource (or enter the etintion) i. While Lemm inicte tht flow of i poile, it oe not necerily men tht flow of cn e chieve. We note tht if the ource n etintion re jcent, then mximum flow of cn lwy e chieve y uing one ege etween the two. We ume for the reminer of thi ection tht n re t let two hop prt. We now give the propertie of mximum flow for ingle emn in unitcpcity wirele network uner -hop interference contrint. Lemm. To chieve the mximum flow of, there mut exit t let two noe-ijoint pth from to. Proof: Aume otherwie: there re no noe-ijoint pth, n there i mximum flow of poile from to. If there re no noe-ijoint pth etween n, then y Menger theorem there exit ingle noe j whoe removl will eprte n [9]; hence, ll pth from to mut p through j. In orer for flow of to exit etween n, noe j mut hve totl of unit of flow coming in, n unit of flow going out. Thi i only poile if noe j i oth receiving n trnmitting the entire time, which i not poile uner the -hop interference moel ince noe j cnnot e oth receiving n trnmitting imultneouly. Corollry. If two noe-ijoint pth from to o not exit, then the mximum flow i. Proof: In the proof for Lemm, it w hown tht if no noe-ijoint pth exit etween n, ll pth mut cro ome noe j. Noe j cnnot e receiving n trnmitting t the me time uner the -hop interference moel. The mximum flow tht j cn upport i to e recieving hlf of the time, n trnmitting the other hlf. Increing the mount of time j i trnmitting will reuce the mount of time it cn trnmit, which reuce the overll flow. The me i een if the mount of time j i receiving i incree. Hence, j will hve incoming flow hlf of the time, n outgoing flow for the other hlf. Since ech ege h unit cpcity, the mximum flow poile through noe j i. Any loop-free pth from to (when n re greter thn one hop prt from one nother) cn hve n interferencefree cheule y lternting etween time lot n for ech ege of the pth; hence, ny ege of the pth will only e ctive for hlf of the time, n the pth will upport flow of. If two or more noe-ijoint pth o exit, then mximum flow i epenent on the totl numer of ege in the ijoint pth. Lemm 3. If there exit two noe-ijoint pth etween n with n even totl numer of ege over oth pth, then the mximum flow of i chievle. Proof: When oth pth hve n even totl numer of ege, n interference-free cheule uing two time lot i poile y lternting time lot ignment on ech pth. If ech pth h n even numer of ege, then pth will egin with time lot n en with time lot, n pth will egin with time lot n en with time lot. Ech unit-cpcity ege will e ctive for hlf of the time; hence, ech pth crrie totl of unit of flow, giving the mximum flow of uing oth pth. A imilr cheule cn e hown for the ce when ech pth h n o numer of ege. To help ee Lemm 3, two exmple re hown in Fig., with the time lot ignment for the link lele in the figure. In Fig. n, there re two noe-ijoint pth from the ource to etintion tht hve n even totl numer of ege. In Fig., ech pth h n even numer of ege, n in Fig., ech pth h n o numer of ege. An interference-free cheule for the two pth cn e foun uing two time lot. Ech link i ctive for of the time; hence, ech pth cn upport flow of, giving totl flow of from to. () Mx flow of c e () Mx flow of Fig. : Noe-ijoint pth with n even totl numer of ege Corollry. If there exit more thn two noe-ijoint pth etween n, mximum flow of i lwy chievle. Proof: If there re more thn two noe-ijoint pth, then there lwy exit pir of noe-ijoint pth tht h n even totl numer of ege. If the totl numer of ege in the two noe-ijoint pth i o, then the two-time lot cheule ue in Fig. to chieve the mximum flow of over the two pth i not poile; itionl time lot re neee. While mximum flow of my not e poile, minimum flow of 3 i lwy feile over two noe-ijoint pth if thir time lot i ue. Lemm 4. For pir of noe-ijoint pth tht hve n o totl numer of ege, flow of 3 i lwy poile etween n uing three time lot. Proof: Remove one of the ege from one of the pth; there i now n even numer of ege in the two pth. Scheule the two pth uing two time lot if they were pir of noeijoint pth with n even numer of ege. After thi tep, ll of the cheule ege cn operte without interference. Now reintrouce the remove ege. Thi reintrouce ege i jcent to two ege tht ech hve time lot ignment of n, repectively. Clerly, time lot or cnnot e igne to the reintrouce ege, o ign it time lot 3. With three time lot, ech link i ctive for 3 of the time. Since ech link h unit cpcity, ech pth crrie 3 flow, n the totl flow over oth pth i 3.

4 4 An exmple emontrting Lemm 4 i hown in Fig. 3. The two noe-ijoint pth hve n o totl numer of ege n it i not poile to cheule the two pth uing only two time lot. A thir time lot i e, n feile cheule i now poile. Uing thee three time lot, ech link i ctive for 3 of the time, n ech pth cn upport flow of 3, which give totl flow of 3. c Fig. 3: Noe-ijoint pth with n o totl numer of ege upporting flow of 3 We note tht i in fct poile to contruct cheule uing more thn three time lot to chieve higher throughput on noe-ijoint pth tht hve n o numer of ege. Thee reult re given in Appenix B. Thee reult cn e extene to the ce where protection i require. For protection gint ny ingle link filure in grph G = (V, E), conier ech ugrph fter link filure: G e = (V, E \ e), e E; ll the previou reult till pply to ech of thee new ugrph. To fin the mximum poile protecte flow, the mximum flow i foun fter ech ege i iniviully remove (ech poile ege filure). The minimum of thee flow i the mximum protecte flow. B. Time Efficient Algorithm Uing the ifferent propertie of olution for ingle emn uner -hop interference contrint, we evelop n lgorithm to olve the prolem efficiently. If there exit two noe-ijoint pth with n even totl numer of ege, then the mximum flow i etween the ource n etintion. If there re no noe-ijoint pth, then the mximum flow i. If there exit only pir of ijoint pth tht h n o totl numer of ege, then flow of 3 cn e gurntee. To fin the mximum protecte flow etween noe n in grph G = (V, E), the mximum flow i foun for ech link filure y uing ugrph with ech link e remove: G e = (V, E \ e), e E. The minimum of thee mximum flow i the mximum protecte flow poile for the emn. The key to fining the mximum protecte flow i to e le to ientify noe-ijoint pth etween n with either n even or o totl numer of ege. If there re t mot two noe-ijoint pth, then the mximum flow cn only e foun if it i poile to fin pir of pth with n even totl numer of ege. Hence, we focu on trying to fin pir of noe-ijoint pth tht hve n even totl numer of ege over oth of the pth. There h een limite work on trying to ientify hortet pth with n even numer of ege [0], ut no work looking t uch n lgorithm for ijoint pth. 3 Development of the optiml lgorithm i follow: we firt fin the hortet pir of ege-ijoint pth with n even numer of totl ege, n then we exten thi lgorithm to fin the hortet pir of noe-ijoint pth with n even numer of totl ege. ) Shortet pir of ege-ijoint pth with n even numer of totl ege: To fin the hortet pir of ege-ijoint pth with n even numer of ege, we egin y coniering the more generl ce without the even-ege retriction (the pth cn hve ny numer of ege), which w previouly coniere in []. We ue ifferent formultion for the prolem y uing minimum-cot flow, which re efine fining flow of minimum cot etween ource n etintion in network tht h oth ege cot n ege cpcitie [9]. Minimum-cot flow hve the property tht when given ll integer input (for ege cot n cpcitie), they will hve ll integer olution (integer flow). We olve the hortet ijoint pir of pth prolem y olving the following optimiztion prolem: fin flow of minimum cot to route two unit from to in grph with unit cpcity n unit cot ege. Thi will fin the hortet pir of ijoint pth ince two unit of flow nee to e route, no ege cn hve more thn ingle unit of flow, n with integer input, the olution will e integer, which will e two ege-ijoint pth of unit flow n minimum cot. One lgorithm to olve the minimum-cot flow prolem i the ucceive hortet pth (SSP) lgorithm [9]. SSP fin the hortet pth, n route the mximum flow poile onto tht pth. Thi repet until the eire flow etween the ource n etintion i route. SSP run in polynomil time to olve the minimum-cot flow formultion for the hortet pir of ijoint pth; further etil of SSP cn e foun in [9]. Uing SSP to olve for minimum-cot flow require the ue of ome hortet pth lgorithm. Aume there exit hortet pth lgorithm tht i cple of fining pth with n even or o numer of ege; lel thee lgorithm Even n O Shortet Pth (ESP n OSP, repectively). Uing SSP to olve for the hortet pir of ijoint pth with either ESP or OSP the hortet pth function will lwy yiel pir of ijoint pth with n even totl numer of ege (if they exit). We cll thi the Even Shortet Pir of Ege-Dijoint Pth lgorithm. Lemm 5. The Even Shortet Pir of Ege-Dijoint Pth lgorithm will fin, if it exit, the hortet pir of ijoint pth with n even totl numer of ege. Proof: Firt, we provie more etil for the hortet ucceive pth (SSP) lgorithm. For ech itertion of SSP, flow i route on the reiul grph, which llow new flow to cncel exiting flow; flow in reiul grph re known ugmenting pth. The reiul grph i efine follow: if ege {i, j} h cpcity n cot of (u ij, c ij ) with flow of f ij u ij on it, the reiul grph will hve two ege {i, j} n {j, i} with repective cot n cpcitie (u ij f ij, c ij ), n (f ij, c ij ) [9]. Fining ugmenting pth on the reiul grph mintin noe conervtion contrint; fter ech itertion of SSP, the reiul grph i upte. To fin the hortet pir of ijoint pth, two itertion of

5 5 SSP on unit cpcity grph re neee. The firt p will fin pth with m numer of ege, which epening on if ESP or OSP w ue, will e even or o. The econ p, which i one on the reiul grph, will fin pth with m ege. If the econ pth ue ny reiul flow from the firt pth, it flow will completely cncel the firt pth flow, effectively cnceling the uge of the ege in the finl et of ijoint pth. Cll the numer of ege tht re cncelle m x. Since ech pth ue cncelle ege, when tht ege i remove, oth pth will no longer trvere the cncelle ege. The totl numer of ege in the finl et of ijoint pth i m + m m x, which i lwy even. In orer to ue SSP to fin the hortet pir of ijoint pth with n even numer of ege, hortet pth lgorithm i neee tht cn fin pth with n even or o numer of ege. The lgorithm in [0] tht fin the hortet pth with n even numer of ege cnnot e eily extene to fin the hortet pir of ijoint pth with n even numer of ege. Hence, we firt focu on eveloping n lgorithm to fin the Even Shortet Pth (ESP), n then exten ESP to fin the O Shortet Pth (OSP). We moify the tnr Bellmn-For recurion [9] to erch for only pth with n even numer of ege, which i hown in Eqution. We lel S z (, k) to e the minimumcot pth from noe to k uing t mot z ege. The cot of ege {i, j} i c ij ; in our ce c ij =, {i, j} E. Inte of checking if pth from to j plu ege {j, k} i of lower cot thn the exiting pth from to k, we check to ee if the pth from to i plu two ege {i, j} n {j, k} re of lower cot thn the exiting pth from to k. S z (, k) = min[ min {i,j} E {j,k} E {i,j} ={j,k} (S z (, j)+c ij +c jk ), S z (, k)], z =.. V, k V () To fin the hortet pth from the ource with n o numer of ege, we run ESP from ll neighoring noe of (noe tht re one hop from ). The lowet cot pth leing ck to the ource i the olution to OSP. ) Shortet pir of noe-ijoint pth with n even numer of totl ege: In orer to optimlly olve for routing n cheuling uner -hop interference contrint, pir of noeijoint pth with n even numer of ege mut e foun. The Even Shortet Pir of Ege-Dijoint Pth lgorithm fin the hortet pir of ege-ijoint pth with n even numer of ege. To ue the ege-ijoint lgorithm to olve the noeijoint ce, ech noe i trnforme into two eprte noe with n ege of zero cot etween them: one noe h ll incoming ege, n the other ll outgoing ( hown in Fig. 4). If there exite multiple ege-ijoint pth tht interecte t noe v, they woul no longer e le to e ege-ijoint in the trnforme network, ecue then they woul ll hve to hre the ege {v in, v out }. Running the Even Shortet Pir of Ege-Dijoint Pth lgorithm on the trnforme network will fin noe-ijoint v v in v out Fig. 4: Noe plitting to fin noe-ijoint pth pth, ut not necerily chieve the eire reult of pir of ijoint pth with n even numer of ege. With the ition of zero-cot ege to the trnforme network, fining pir of ijoint pth with n even numer of ege in the trnforme network my not yiel pth with n even numer of ege in the originl network. A moifiction to the lgorithm mut e me to ccount for the new ege: in the trnforme network, when chooing etween n exiting pth from to k, or ome new pth to i plu egment i to k, conier only egment tht hve n even numer of originl ege. Thi will enure tht finl pth in the originl network will hve n even numer of ege. The lgorithm now egin to more cloely reemle the Floy-Wrhll lgorithm [9], which conier joining egment to fin hortet pth. Thi new lgorithm i clle Even Shortet Pir of Noe-Dijoint Pth. Thee reult cn e extene to olve Wirele Gurntee Protection prolem with ingle emn uner -hop interference contrint. The mximum flow i foun fter every poile ege filure for ech ugrph G e = (V, E \ e), e E. The minimum of thee mximum flow i the mximum protecte flow. For ech intnce, we firt ee if there exit pir of noe-ijoint pth with n even totl numer of ege. If thi exit, then mximum flow of i poile. If not, we check to ee if there exit noe-ijoint pth with n o totl numer of ege (y running the tnr egeijoint pth routing lgorithm on the trnforme grph). If thi exit, then flow of 3 i poile uing three time lot. If no noe-ijoint pth exit, then fin ome pth from to, which cn upport flow of. IV. AN OPTIMAL FORMULATION FOR WIRELESS GUARANTEED PROTECTION In the previou ection, n optiml olution for routing n cheuling with protection for ingle emn w preente. While thi provie inight, typicl network will nee to imultneouly hnle multiple connection. Aitionlly, mny network hve interference contrint other thn the -hop moel. Thi ection provie mthemticl formultion to the optiml olution for the Wirele Gurntee Protection (WGP) prolem with generl interference contrint. In prticulr, for et of emn, route n cheule nee to e foun uch tht fter ny link filure, ll en-to-en connection mintin their me level of flow. For generl interference contrint, the routing n cheuling prolem w emontrte to e NP-Hr [9]. We conjecture tht ing protection contrint preerve NP-hrne; hence, mixe integer liner progrm (MILP) i formulte to fin n optiml olution to WGP. In wire network, typicl ojective function for protection i to minimize the totl llocte cpcity neee to tify ll emn. A imilr ojective cnnot e clerly efine for

6 6 wirele network ince the concept of cpcity chnge in the preence of interference contrint. Conier ome ctive link {i, j}. An jcent link {j, k} cnnot e ue imultneouly with {i, j} ecue of interference; hence, imply ing itionl link cpcity (in wire ene) will not llow it ue. Another time lot mut e llocte to llow connection to ue {j, k} uch tht it oe not interfere with {i, j}. Aing n itionl time lot will reuce the time tht ech iniviul time lot in the cheule i ctive, which reuce the overll throughput of the network [8, 9, 3]. For exmple, conier network with two time lot n connection tht upport flow of uing thee two time lot. If thir time lot i e to the cheule, then the originl two time lot re only ctive for 3 of the totl time, n tht flow cheule throughput i reuce from to 3. Thu, the ojective we conier i to ue minimum numer of time lot to route n cheule ech emn with protection. Fining protection route n cheule uing the minimum numer of time lot llow for imple comprion to exiting wire n wirele protection cheme. The ifference etween the numer of time lot necery to route n cheule et of emn efore n fter ing protection will e coniere the reuction of the mximum throughput. To e conitent with the wirele protection cheme mentione in [8], wirele flow re retricte to ingle pth (no flow plitting llowe). For ee of expoition, the MILP ign the me throughput to ll emn; ee Appenix Section A for the formultion with ifferent throughput requirement. For the MILP, the following vlue re given: G = (V, E) i the grph with et of vertice n ege D i the et of flow requirement u ij i the cpcity of link {i, j} I i the interference mtrix, where I kl ij I i if link {i, j} n {k, l} cn e ctivte imultneouly, 0 otherwie T i the et of time lot in the ytem, T Z + The MILP olve for the following vrile: x ij i routing vrile n i if primry flow i igne for emn (, ) on link {i, j}, 0 otherwie yij,kl i routing vrile n i if protection flow i igne on link {i, j} for the emn (, ) fter the filure of link {k, l}, 0 otherwie λ,t ij i cheuling vrile n i if link {i, j} cn e ctivte in time lot t for the emn (, ), 0 otherwie δ,t ij,kl i cheuling vrile n i if link {i, j} cn e ctivte in time lot t fter filure of link {k, l} for the emn (, ), 0 otherwie t i if time lot t i ue y ny emn, n 0 otherwie The ojective function i to minimize the numer of time lot (the length of the cheule) neee to route ll emn with protection: Ojective: min t () t T The following contrint re impoe to fin feile routing n cheuling with protection. Before link filure: Flow conervtion contrint for the primry flow: route primry trffic efore filure for ech emn. {i,j} E x ij {j,i} E x ji = if i = if i =, 0 otherwie i V, (, ) D (3) In ny given time lot, for given emn, only link tht o not interfere with one nother cn e ctivte imultneouly. + λ,t ij λ,t kl + I ij {i,j} E, {k,l} E kl, {i,j} ={k,l}, t T (4) Only one emn cn ue given link t time. λ,t ij, {i,j} E t T (5) Enure enough cpcity exit to upport the necery flow for emn (, ) on ege {i, j} for the length of time tht the link i ctive. x ij λ,t ij u ij, t T {i,j} E (6) Mrk if lot t i ue to cheule emn efore filure. λ,t ij t, After link filure: {i,j} E {k,l} ={i,j} y ij,kl {j,i} E {k,l} ={j,i} {i,j} E t T, Flow conervtion contrint for protection flow: route protection trffic fter ech link filure {k, l} E. y ji,kl = if i = if i =, 0 otherwie i V, {k, l} E, (, ) D (7) In ny given time lot fter the filure of link {k, l}, only link tht o not interfere with one nother cn e ctivte imultneouly. δ,t ij,kl + δ,t uv,kl + {i,j} E, {k,l} E Iij uv, {u,v} E, t T {i,j} ={k,l} ={u,v} Only one emn cn ue given link t time fter the filure of link {k, l}. δ,t {i,j} E, {k,l} E ij,kl, (9) t T Enure enough cpcity exit fter the filure of link {k, l} to upport the necery flow on ege {i, j} for the length of time tht the link i ctive. yij,kl δ,t ij,kl u ij, t T (8) {i,j} E, {k,l} E (0) Mrk if time lot t i ue to cheule emn fter the filure of link {k, l}. δ,t ij,kl t, {i,j} E, {k,l} E t T,

7 7 % Reuction Mx Flow Wire WGP Avg. Noe Degree Fig. 5: Reuction of throughput when ing protection To emontrte how protection cn e e to wirele network with miniml reuction of throughput, WGP i compre to oth the wire (without interference) n wirele protection (with interference) cheme. One hunre rnom grph were generte with 5 noe ech. Noe tht re phyiclly within certin trnmiion rnge of one nother re coniere to hve link, n the trnmiion rnge i vrie to give ifferent eire verge noe egree. The noe egree i vrie from.5 to 6.5, n for ech grph, ten ource/etintion pir re rnomly choen to e route concurrently. All link hve unit cpcity; -hop interference contrint were ue for the wirele network. The imultion reult re foun in Fig. 5. For comprion to wire protection, we ue the me network topologie, however, in the wire ce we o not enforce the interference contrint (i.e., ll link cn e ctivte imultneouly). For wire protection, we compute the reuction in throughput the reuction in mximum flow fter protection i e. A compre to the wire protection cheme, WGP h lower reuction in throughput for ll noe egree exmine. For noe egree.5, oth the WGP n the wire protection cheme hve lrger reuction in throughput: 0% for WGP n 37% for wire. Thi i ecue t lower noe egree, there re fewer ville en-to-en pth, n therefore fter filure, there re fewer routing option ville. A the noe egree incree, n there re more ville en-to-en pth, the reuction in throughput ecree when ing protection. In fct, it i often poile for WGP to hve no reuction in the throughput etween the protecte n unprotecte etting. For n verge noe egree of 3.5, WGP only loe out 0% of throughput when ing protection, while the wire cheme loe 3%. For 0% of the imultion t noe egree 3.5, there w no lo in throughput for WGP. When the noe egree goe to 6.5, WGP no longer h ny lo in flow, while the wire etting till h lo of %. We compre WGP to wirele + protection cheme. In prticulr, wirele + protection pplie the wire + protection cheme to wirele network ( mentione in [8]): i.e., fin cheule for the hortet pir of ijoint pth in the network etween the ource n etintion, with the primry flow efore filure route onto one pth, n the ckup flow route onto the other. To compre WGP to wirele +, the numer of time lot neee eyon the non-protection routing Avg. Noe Degree % Reuction of Protection Time Slot TABLE I: WGP v. Wirele + re compre; thee re the time lot neee to meet the protection requirement. Tle I how the percent reuction in numer of time lot neee to provie protection uing WGP over wirele +. When the verge noe egree i.5, WGP h up to 7% reuction of time lot neee to meet protection requirement. The reon for thi i tht wirele + i cheuling two pth for ech emn, primry n ckup, n not trying to recpture ny cpcity fter filure; thi in turn cue ignificnt incree in interference etween connection. A the noe egree incree, there i incree pth iverity n more opportunitie to fin interference-free routing; hence, wirele + h etter performnce. But t ll time, wirele + nee ignificntly more time lot to provie protection for ll of the emn thn WGP oe, which i le to recpture cpcity fter filure. V. ALGORITHMS FOR PROVIDING WIRELESS PROTECTION In the previou ection, n MILP w preente to fin n optiml olution to Wirele Gurntee Protection (WGP), which i not computtionlly efficient metho of fining olution. In thi ection, two time-efficient lgorithm re preente to olve the Wirele Gurntee Protection prolem for et of emn. Similr to the previou ection, primry n ckup flow re retricte to ingle pth, n the ojective i to minimize the length of the cheule to route ll emn with protection. We firt how tht thi prolem i NP- Hr uner -hop interference contrint. Next, lgorithm re evelope uming unit emn, unit cpcity ege, n ingle link filure moel; the lgorithm cn e moifie to reflect other vlue of emn n cpcity. The lgorithm re evelope for ynmic (one-t--time) rrivl: n incoming emn nee to e route n cheule over n exiting et of connection; the exiting et cnnot hve their routing or cheule chnge. A -hop interference moel i ue, ut the lgorithm cn e extene to generic K-hop interference moel, with the extenion etile in the en of Section V-B. We fin tht when compre to the optiml tch ce (ll connection re route n cheule imultneouly), the ynmic routing perform within few percentge point of optiml. Firt, in Section V-A, we emontrte WGP to e NP-Hr uner -hop interference contrint when flow re retricte to ingle pth. Next, in Section V-B, n lgorithm to fin hortet -hop interference-free pth uing miniml numer of time lot i preente. Thi erve the uiling lock for the next two lgorithm tht re evelope. In Section V-C,

8 8 n lgorithm for fining miniml length cheule for WGP i preente, where ckup route n cheule i foun for ech poile filure. Thi pproch h rwck in tht fter ny filure, new route i foun; hence, route n cheule for ech filure event nee to e tore. To overcome thi, n lgorithm i evelope in Section V-D uing ijoint pth uch tht only two pth re neee: primry n ckup. In Section V-E, the performnce of the two lgorithm re compre to the optiml MILP formultion. A. Complexity Reult uner -hop Interference Contrint Without protection, the routing n cheuling prolem i NP-Hr uner generl interference contrint [9]. But if flow for ech emn re llowe to e plit, polynomil time lgorithm i poile for -hop interference contrint []. We emontrte tht when flow cnnot e plit, the routing n cheuling prolem ecome NP-Hr uner -hop interference contrint. Theorem. Fining the minimum length cheule to route et of emn uner -hop interference contrint when flow plitting i not llowe i NP-Hr. We firt conier the following necery n ufficient conition for routing et noe-ijoint pir, (, ),..., ( N, N ), in only two time lot without flow plitting. Lemm 6. Uner -hop interference contrint, et of emn tht re noe-ijoint cn e route n cheule uing two time lot without flow plitting if n only if there exit noe-ijoint pth etween ech of the noe pir. Proof: If there exit noe-ijoint pth etween every noe pir in the et of emn, then cheule uing two lot i poile. A proof cn e ccomplihe y contruction. Any iniviul pth cn e cheule in two time lot y uing lternting time lot. If ll pth re noe-ijoint, then there exit no conflict etween pth uner -hop interference contrint. Therefore, ll pth cn e cheule uing the me two-time lot. For the other irection, ume otherwie: two lot cheule i poile without there eing noe-ijoint pth etween every pir of noe in the et of emn. There exit noe v tht h m pth croing it. There re m pth coming into v n m pth out tht nee to e cheule. Uner -hop interference contrint, thi will require t let m time lot to prouce n interference free cheule. Uing Lemm 6, Theorem cn e quickly emontrte. Proof of Theorem : We reuce the Dijoint Connecting Pth Prolem (DCPP) [] to our. DCPP k the following quetion: given grph G = (V, E) n collection of N noe-ijoint pir (, ),..., ( N, N ), oe G contin N mutully noe-ijoint pth, one connecting i n i for ech i, i N? We cn k n equivlent quetion for our routing n cheuling prolem: cn et of N noe-ijoint pir e route n cheule uing the miniml numer of time lot (two) uner -hop interference contrint without A noe i ource or etintion for t mot one emn. flow plitting? If ye, then y Lemm 6 tht men we hve foun N mutully noe-ijoint pth, one connecting i n i for ech i, i N, which olve DCPP. An nwer of no men olution to DCPP oe not exit. Next, we exten thi complexity reult to the ce when protection i require. Theorem. Fining the minimum length cheule to route et of emn with protection uner -hop interference contrint without flow plitting i NP-Hr. The proof cn e foun in Appenix Section C. B. Minimum Scheule for n Interference Free Pth We egin y eveloping n lgorithm to fin hortet interference-free pth uing the minimum numer of time lot uner the -hop interference moel. Thi lgorithm will e uiling lock for the two protection lgorithm tht will e icue in the upcoming ection. We conier n incoming emn for connection etween noe n. Connection lrey exit in the network, with the et of T time lot lrey in ue. Be on how the current connection re route n cheule, et of ege interference I cn e contructe, where for every ege {i, j}, I ij I i the et of time lot tht cnnot e ue on tht ege ecue either tht time lot i lrey ue y {i, j}, or uing tht time lot on {i, j} will interfere with nother ege uing it t tht time. The et of ege interference I cn e contructe in polynomil time, n will e given n input to the lgorithm. Firt, we wih to etermine the hortet interference free pth without uing ny itionl time lot eyon the et T, n without recheuling or rerouting exiting connection. Ech ege {i, j} h et of free time lot uring which it cn e ue: τ ij = T \ I ij. Let P e the et of ege ue in pth. If ech ege of loop-free pth P h t let two free time lot, then tht pth cn e cheule without interference uing the exiting time lot lloction T. Lemm 7. For -hop interference, loop-free pth P cn e cheule without interference if τ ij, {i, j} P. Proof: If τ ij, {i, j} P, then ech ege in P h n ville time lot tht oe not interfere with it jcent et of ege. Since the pth i loop-free, ny two ege tht ue the me time lot will never e le thn one hop prt from one nother, n therefore never interfere with ech other. Uing the reult from Lemm 7, the following lgorithm i contructe to fin -hop interference-free pth uing only the et of time lot T : remove ll ege in G tht hve τ ij, fin the hortet pth P etween n, n ign time lot to the ege in P uch tht it h n interference-free cheule. An improvement cn e me to the lgorithm y ttempting to mximize the numer of free time lot on ny ege, o tht future connection will e le likely to require itionl time lot to fin n interference-free pth. Currently, ege tht hve mny free time lot re not given ny preference. If n ege h only the miniml numer of free time lot, it my e electe

9 9 for ue in pth. Thi my hurt fining interference-free pth for future connection y limiting the numer of ville time lot on n ege, thu neceitting new time lot. We ign cot for ech ege to e equl to the numer of time lot tht tht ege interfere with: c ij = I ij. With repect to thee new ege cot, minimum-cot interference-free pth i foun. The more time lot n ege i in conflict with, the more expenive tht ege will e, n the le likely it will e ue in route. We refer to thi lgorithm int_free_pth, which will return the ege n cheule of pth etween n. To fin n interference free pth tht trie to minimize future conflict, n uing minimum itionl time lot, we firt fin minimum-cot interference free pth for the current et of time lot igne in the network, T. If uch pth oe not exit, incree the et of ville time lot y, n repet. We note tht the et of time lot will never incree y more thn two ince feile cheule cn e foun for ny pth with two free time lot. We cll thi lgorithm fin_pth. C. Minimum Length Scheule for Wirele Protection In thi ection, n lgorithm i evelope tht trie to fin the minimum length cheule for the Wirele Gurntee Protection prolem, with n pproch tht i imilr to the optiml olution foun y the MILP in Section IV. The prolem i roken up into E + uprolem. Firt, the minimum length cheule i foun to route the et of emn efore filure. Then, for ech poile filure, the minimum length cheule i foun to route the et of emn on filure grph G kl = (V, E \ {k, l}), {k, l} E (i.e., the grph tht remin fter the filure of ege {k, l}). Ech of the olution to thee uprolem repreent the route n cheule necery to meet the protection requirement for the et of emn efore n fter ny link filure. The mximum of ny of thee minimum length cheule will e the length of the cheule neee to protection to et of emn in wirele network. The lgorithm i clle minimum_protect; it will return the et of pth n cheule for ech emn, inicting which pth n cheule to ue fter ny link filure. D. Dijoint Pth Wirele Gurntee Protection In Section V-C, n lgorithm w ecrie to fin the minimum numer of time lot to route n cheule et of emn with protection. After ny filure, new route i foun; hence, mny poile routing configurtion exit, n route n cheule for ech filure event nee to e ve. A more eirle pproch my e to limit the numer of pth neee to only two: primry n ckup. Before continuing with the evelopment of the lgorithm, complexity reult i preente regring uing ijoint pth to provie protection in wirele network with -hop interference contrint. For et of time lot T, imply etermining if ny olution exit to WGP uing ijoint pth i NP-Complete. Theorem 3. For n incoming connection etween n, uing ijoint pth to provie protection in wirele network g e h c f Fig. 6: Dijoint pth routing n cheuling with protection with -hop interference contrint for the et of time lot T i NP-Complete. A reuction i performe from the Dynmic Shre-Pth Protecte Lightpth-Proviioning (DSPLP) []. The proof cn e foun in Appenix Section D. Our pproch for eveloping n lgorithm to olve WGP uing ijoint pth i imilr to the wirele + protection cheme ecrie erlier; however, we tke vntge of the time lot reue tht i poile efore n fter filure, well the opportunity to hre protection reource etween filure ijoint emn. If n ege in primry pth P ue time lot t, then for -hop interference, ll ege jcent to tht ege lo cnnot ue t. After the filure of n ege in the primry pth, the time lot ue to route tht pth re no longer neee (ince they re not eing ue). The time lot on the ege of the primry pth tht i not fil now cn e reue for protection; furthermore, the time lot on the ege tht interfere with the file primry pth lo ecome free to ue for protection. Protection reource hring cn lo llow for time lot reue. If two primry pth re filure ijoint uner ingle link filure moel, only one will fil t time. Hence, time lot t on jcent ege cn e hre for protection etween the two filure ijoint connection, ince the two jcent ege will never e ctivte imultneouly. An exmple i hown in Fig. 6. Two emn nee to e route uner -hop interference contrint: one from to, n nother from g to i. Ech ege i igne time lot, with the time lot leling hown in the figure. The ege ue for primry flow re inicte y oli line, n the ege ue for protection re otte line. After the filure of ege {, }, the entire primry pth etween n i no longer ctive, n it time lot will no longer e in ue; hence, ege {, e} n {f, } cn ue time lot, even though they woul hve conflicte with {, } n {c, } efore the filure. Similrly, {g, e} i igne time lot, even though primry ege {g, h} i igne the me time lot. Since oth primry pth re filure ijoint, time lot on {e, f} i hre etween the two connection for protection. Aitionlly, ecue t mot one ckup pth will e ue t time, protection ege {g, e} n {, e} cn oth e igne time lot ; they will never interfere with one nother. Similrly, {f, i} n {f, } cn e i

10 0 oth igne time lot. Thi ie of time lot reue fter filure form the i for the the ijoint pth wirele protection lgorithm, which we lel ijoint_protect. We conier n incoming emn requeting connection etween noe n. Connection lrey exit in the network, with the et of T time lot lrey in ue. A interference-free primry pth etween n, P, i foun uing fin_pth. Once primry pth fil, none of the time lot neee for tht pth, or on the ege tht interfere with tht pth, re neee, n they ecome ville to e ue for protection. Next, ckup pth B i foun tht i ijoint to P, n oe not interfere with ny of the other connection tht i not fil. Aitionlly, the ckup pth B will not interfere with the protection routing for the ifferent exiting emn tht woul fil if n ege in P fil (i.e., B will not interfere with the protection pth for emn whoe primry pth re not ijoint with P ). The lgorithm i etile in Algorithm. Algorithm (P, I, T ) = ijoint_protect(g, I, T,, ) Fin route n cheule efore filure: (P, T, I ) = fin_pth(g, I, T,, ) Contruct new network without the ege in pth P : G F = (V, E \ P ) Once n ege for tht emn fil, none of the lot neee to upport it re ue n ecome ville. Contruct filure interference et uing the interference et for the primry route efore tht emn w route, n the filure interference et for ech ege {k, l} in P I F = ( {k,l} P I kl ) I Fin ijoint pth, n cheule it: (P F, T, I ) = fin_pth(g F, I F, T,, ) Upte interference et: Before filure: I = I After ech primry pth filure: I kl = I, {k, l} P I = Set of I n ll I kl P = Set of P n ll P kl Return (P, I, T ) E. WGP Algorithm Simultion The lgorithm minimum_protect n ijoint_protect re compre to the optiml olution foun y the MILP in Section IV. A imilr imultion etup i ue tht in Section IV. One hunre rnom grph were generte with 5 noe ech. The noe egree i vrie from.5 to 6.5, n for ech rnom grph, ten ource/etintion pir re rnomly choen to e route concurrently, ech with unit emn. All link hve unit cpcity, n -hop interference contrint were ue. The lgorithm route n cheule emn one-t--time, while the MILP optimize the route n cheule for ll emn together (in tch). To compre the two, the lgorithm rnomly orer the et of Avg. Min Length Scheule Avg. Noe Degree Fig. 7: Avg. time lot neee for WGP Dijoint Minimum MILP emn, n then olve for ech emn one-t--time. The imultion reult re foun in Fig. 7. Similr to the previou imultion, noe egree incree, the verge minimum length cheule ecree. Thi i ecue of the incree iverity in poile numer of ento-en pth, which le to greter opportunity of fining interference free pth. On verge, minimum_protect neee only 4.5% more time lot to meet ll requirement thn the optiml MILP neee, n ijoint_protect neee 0.% more time lot thn the MILP. VI. CONCLUSION In thi chpter, the prolem of gurntee protection in multi-hop wirele network i introuce. Becue of link interference, reource tht were unville prior to filure cn e ue for protection fter the filure. In fct, protection cn often e provie uing no itionl reource. For the ce of ingle emn with -hop interference contrint, propertie of n optiml olution re preente, n time-efficient lgorithm i evelope tht olve the prolem of wirele routing n cheuling with n without protection, gurnteeing mximum throughput tht i within.5 of optiml. For generl interference contrint n multiple concurrent emn, n optiml olution i evelope for the protection prolem vi mixe integer liner progrm. When compre to uing tritionl wire protection cheme on wirele network, our Wirele Gurntee Protection (WGP) cheme ue much 7% le protection reource to chieve the me level of reiliency. Two low-complexity lgorithm to olve WGP re evelope, n on verge, thee lgorithm perform cloe to the optiml olution. A future irection for our work i to pt the cheme evelope in thi chpter to itriute etting. APPENDIX A. MILP for WGP with Different Throughput Some emn etween noe n h it own throughput requirement f. The ojective of the MILP i to minimize the numer of time lot neee to cheule every emn. Since ech emn h throughput requirement, fining minimum length cheule will e with repect to keeping the rtio of the ifferent emn throughput contnt. We ume f

11 i integer (, ) (V, V ). If necery, the emn n link cpcitie cn e cle y the mllet integer tht mke ll emn vlue integer (hence, f i ume to e t the very let rtionl, (, ) (V, V )). For the MILP, the following vlue re given: G = (V, E) i the grph with et of vertice n ege D i the et of flow requirement f i the flow require etween noe (, ); f Z u ij i the cpcity of link {i, j} I i the interference mtrix, where I kl ij I i if link {i, j} n {k, l} cn e ctivte imultneouly, 0 otherwie T i the et of time lot in the ytem, T Z + The MILP olve for the following vrile: i routing vrile n i if primry flow i igne for emn (, ) on link {i, j}, 0 otherwie yij,kl i routing vrile n i if protection flow i igne on link {i, j} for the emn (, ) fter the filure of link {k, l}, 0 otherwie λ,t ij i cheuling vrile n i if link {i, j} cn e ctivte in time lot t for the emn (, ), 0 otherwie δ,t ij,kl i cheuling vrile n i if link {i, j} cn e ctivte in time lot t fter filure of link {k, l} for the emn (, ), 0 otherwie t i if time lot t i ue y ny emn, n 0 otherwie x ij The ojective function i to minimize the numer of time lot (the length of the cheule) neee to route ll emn with protection: Ojective: min t () t T The following contrint re impoe to fin feile routing n cheuling with protection. Before link filure: Flow conervtion contrint for the primry flow: route primry trffic efore filure for ech emn. {i,j} E x ij {j,i} E x ji = if i = if i =, 0 otherwie i V, (, ) D () In ny given time lot, for given emn, only link tht o not interfere with one nother cn e ctivte imultneouly. + λ,t ij λ,t kl + I ij {i,j} E, {k,l} E kl, {i,j} ={k,l}, t T Only one emn cn ue given link t time. λ,t ij (3), {i,j} E t T (4) Enure enough cpcity exit to upport the necery flow f for emn (, ) on ege {i, j} for the length of time tht the link i ctive. f x ij λ,t ij u ij, t T {i,j} E (5) Mrk if lot t i ue to cheule emn efore filure. After link filure: {i,j} E {k,l} ={i,j} λ,t ij t, y ij,kl {j,i} E {k,l} ={j,i} {i,j} E t T, Flow conervtion contrint for protection flow: route protection trffic fter ech link filure {k, l} E. y ji,kl = if i = if i =, 0 otherwie i V, {k, l} E, (, ) D (6) In ny given time lot fter the filure of link {k, l}, only link tht o not interfere with one nother cn e ctivte imultneouly. δ,t ij,kl + δ,t uv,kl + {i,j} E, {k,l} E Iij uv, {u,v} E, t T {i,j} ={k,l} ={u,v} (7) Only one emn cn ue given link t time fter the filure of link {k, l}. δ,t {i,j} E, {k,l} E ij,kl, (8) t T Enure enough cpcity exit fter the filure of link {k, l} to upport the necery flow f on ege {i, j} for the length of time tht the link i ctive. f yij,kl δ,t ij,kl u ij, t T {i,j} E, {k,l} E (9) Mrk if time lot t i ue to cheule emn fter the filure of link {k, l}. δ,t ij,kl t, {i,j} E, {k,l} E t T, B. Scheule for Higher Throughput on Noe-Dijoint Pth with n O Numer of Ege In Section III-A, for pir of noe-ijoint pth whoe totl numer of ege i o, cheule uing three time lot w ue to chieve flow of 3 etween the ource n the etintion. Ech link i igne one of three time lot, n ince ech link i ctive for only 3 of the time, ech pth upport flow of 3, n the totl en-to-en flow i 3. An exmple i hown on the five ege network in Figure 8. By uing itionl time lot, it i in fct poile to incree the en-to-en throughput. For the me five ege network, mximum flow of 5 6 i poile uing ix time lot. A cheule tht chieve thi flow i hown in Figure 8. On the horter pth, three time lot re igne to ech link. Since there re totl of ix time lot in ue, ech link i ctive for hlf of the totl time, n i upporting flow of. On the longer pth, ech link i igne two time lot. Thee two time lot

12 0 0 c () Flow of 3 0,, 3 4, 5 3, 4, 5 0,, c () Flow of 5 6 Fig. 8: Noe-ijoint pth with n o totl numer of ege upporting flow of 3 n 5 6. repreent 3 of the totl time; hence the longer pth upport flow of 3. The totl flow on oth pth i + 3 = 5 6. If the longer of the two noe-ijoint pth h K ege, then it i in fct poile to lwy chieve throughput over the two pth of K K y employing the following cheuling cheme tht ue K time lot. On the horter pth, we ign hlf of the time lot to ech ege: K to (K ) on the firt ege, 0 to (K ) on the econ ege, n lternte etween thoe two ignment for ech uequent ege for the reminer of the pth. Since ech ege of the horter pth ue hlf of the time lot, ech ege i ctive of the time, n the horter pth crrie flow of. On the longer pth (hving K ege), ign (K ) time lot to ech ege in the following fhion: For the j th ege, where ege 0 leve the ource n ege K enter the etintion, ign time lot mo[j(k ), K] through mo[(j + )(K ), K]. The nottion mo[, ] repreent the moulo function whoe vlue i the integer reminer when i ivie y. Ech ege h (K ) time lot igne to it n i ctive for K K of the time, llowing the longer pth to upport flow of K K. The totl flow cro oth pth i K K + K K = K K. Thi cheuling cheme cn lwy chieve throughput of K K, which i emontrte in Lemm 8. Two exmple re hown in Figure 9. In the firt network, hown in Figure 9, the longer pth h five ege (K = 5), n the horter h four. Ten time lot re ue in totl, with the horter pth upporting flow of, n the longer pth upporting flow of 4 5, reulting in n en-to-en flow of 9 0. It i trightforwr to ee tht if the horter pth h two ege inte of four, the me throughput woul hve een chievle uing the me ten time lot. In the econ network, hown in Figure 9, the longer pth h four ege (K = 4), n the horter h three; eight time lot re ue. The horter pth upport flow of, the longer pth upport flow of 3 8, n the totl en-to-en flow i 7 8. Lemm 8. For pir of noe-ijoint pth with n o numer of ege, where the longer pth h K ege, cheule exit tht chieve throughput of K K over the two pth. Proof: We emontrte the cheuling cheme preente 4,5,6,7 8,9,0,,3,4,5 0,,,3 5,6,7,8,9 0,, 0,,,3,4 5,6,7,8,9 () K = 5, n flow of,3,4 5,6,7 0,,,3 6,7,8,9 0,,,3, ,, 4,5,6,7 4,5,6,7 () K = 4, n flow of 7 8 Fig. 9: Noe-ijoint pth with n o numer of ege upporting flow of K K in thi ection lwy chieve the eire rte of K K. We conier two ce: K i o, n K i even. We firt exmine the ce where K i o; n exmple w hown in Figure 9. Since the longer pth h n o numer of ege, the horter pth mut hve n even numer. On the horter pth, hlf the time lot re igne to ech ege, lternting etween time lot K to K on the firt ege, time lot 0 to K on the econ ege, n o forth until the finl ege. Since there i n even numer of ege, the finl ege of the horter pth entering the etintion will e igne time lot 0 to K. On the longer pth, time lot mo[j(k ), K] through mo[(j + )(K ), K] re igne to the j th ege, with ege 0 leving the ource n ege K entering the etintion. Thi reult in K time lot igne to ech ege. By contruction, the ege leving the ource for ech pth o not interfere with one nother. We nee to verify tht the finl ege entering the etintion lo o not interfere. The finl ege of the pth entering the etintion i numere j = K ; the time lot ignment for tht ege i mo[(k )(K ), K] through mo[k(k ), K]. The vlue of mo[, ] i equl to [3], where q i the integer floor of ome vlue q. The finl time lot igne to ege K i: mo[k(k ), K] K(K ) = K(K ) K K K = K(K ) K K

13 3 Since K i o, K i integer; hence, we get: mo[k(k ), K] ( K = K(K ) K + ) K ( ) K = K(K ) K = K(K ) K(K ) + K = K The finl ege (j = K ) of the longer pth i igne time lot: (K + ) through (K ). The finl ege of the horter pth w igne time lot 0 to K. Hence, when K i o, thi cheuling cheme will not cue interference etween jcent ege, n will chieve n en-to-en flow of K K. We next emontrte imilr reult for when K i even; n exmple network w hown in Figure 9. The longer pth h n even numer of ege, n the horter pth h n o numer. Agin, on the horter pth, hlf the time lot re igne to ech ege, lternting etween time lot K to K on the firt ege, time lot 0 to K on the econ ege, n o forth until the finl ege. Since there i n o numer of ege, the finl ege of the horter pth entering the etintion will e igne time lot K to K. We now conier the finl ege entering the etintion of the longer pth, which will e igne time lot mo[(k )(K ), K] through mo[k(k ), K]. The finl time lot for thi ege will hve the vlue mo[k(k ), K] = K(K ) K K Since K i even, K i not integer, ut K ( K = K(K ) K + K. i; hence, we get: mo[k(k ), K] K = K(K ) K K ) K = K(K ) K ( K ) = K K K + K = K The finl ege of the longer pth i igne time lot through (K ). The finl ege of the horter pth w igne time lot K to K. Therefore, when K i even, thi cheuling cheme will not cue interference etween jcent ege t the etintion, n will chieve n en-to-en flow of K K. Uing the cheme ecrie ove, the minimum throughput tht cn e gurntee on pir of noe-ijoint pth with n o numer of ege i 5 6, which i greter thn the 3 flow ecrie in Section III-A. The minimum gurntee throughput of 5 6 i inepenent of K. Lemm 9. For pir of noe-ijoint pth, cheule cn lwy e foun tht gurntee flow of t let 5 6 from the ource to the etintion. Proof: We conier only the ce when the ource n etintion re more thn one hop prt; otherwie, only one ege nee to e ctivte etween the two noe, crrying the mximum flow of without the ue of noe ijoint pth. When there re n even numer of ege over the two noeijoint pth, then the mximum flow of cn e chieve uing two time lot, w hown in Lemm 3. When there re n o numer of ege over the two noeijoint pth, where the longer pth h K ege, then flow of K K cn lwy e chieve, which w emontrte in Lemm 8. We now how tht the minimum K i 3, hence the minimum flow i 5 6. Since the ource n etintion re more thn one hop prt, the minimum numer of ege over oth pth i 4. With 4 totl ege, the two pth hve n even numer of ege, n mximum flow of i chievle uing two time lot. The next mllet numer of ege for oth pth i 5. Since the ource n etintion cnnot e one hop prt, thi men the longer pth h 3 ege, n the horter h. Hence, the mllet vlue of K poile i 3, which give n chievle throughput of K K = 5 6. Any vlue of K tht i greter thn 3 will reult in higher chievle throughput. If only pir of noe-ijoint pth exit with n o numer of ege etween the ource n etintion, flow of K K cn foun uing K time lot. But thi oe preclue the poiility of higher feile throughput exiting tht ue itionl ege. Conier the exmple in Figure 0. 0,,,3 4,5,6,7 6,7 4,5 6,7 0,,,3 4,5 0,,,3 4,5,6,7 Fig. 0: Noe-ijoint pth with itionl ege upporting flow of The pir of noe-ijoint pth etween noe n re hown uing the oli ege, n itionl ege tht connect with one of the noe-ijoint pth re hown uing the otte ege. In thi network, ll poile pir noe-ijoint pth hve n o numer ege. The longer pth h 4 ege; if we cheule ccoring the cheme ecrie erlier, flow of 7 8 cn e chieve etween n. But y uing the otte ege, in ition to the oli ege, cheule cn e foun tht chieve the mximum flow of, hown in Figure 0. Thi how tht the flow K K i trictly lower oun on the mximum flow tht cn e foun when only pir of noeijoint pth exit tht hve n o numer of ege. We o not currently conier the prolem of etermining whether or

14 4 not mximum flow of exit in thi ce, n we leve it future work. C. Proof for Theorem Theorem : Fining the minimum length cheule to route et of emn with protection uner -hop interference contrint without flow plitting i NP-Hr. Proof: We prove the protection verion of the prolem to e NP-hr y reucing the non-protection verion to it. We egin y tking grph G = (V, E) n trnforming it to ome other grph G = (V, E ). Grph G will e contructe uch tht ny feile olution in G uing two time lot for noeijoint pth for et of noe-ijoint emn (i.e., olution for the non-protection prolem in G) will hve n immeite olution tht inclue protection uing two time lot for the me et of noe-ijoint emn. Conier n ege {i, j} of ome pth etween n in G, which i noe-ijoint from ny other pth n i cheule to ue time lot. Three ege, {i, k}, {k, l}, n {l, j}, cn e e to protect {i, j} without neeing ny itionl time lot, with time lot ignment hown in Fig.. Ege {i, k} n {l, j} re igne time lot, n ince they will only e ctivte fter the filure of ege {i, j}, they o not conflict with the time lot ignment for {i, j}. Furthermore, the ege on the pth in G irectly preceing n irectly following ege {i, j} will e time lot (ecue the cheule only conite of two time lot); o, fter {i, j} fil, the protection routing of {i, k}, {k, l}, n {l, j} will not interfere with the exiting cheule ege of the pth. Aitionlly, ince we re currently coniering feile olution of noeijoint pth for the et of noe-ijoint emn, no noe will hve more thn ingle pth croing it, o the protection pth of {i, k}, {k, l}, n {l, j} will not interfere with ny other emn. i j k i l j Fig. : Ege trnformtion for NP-hrne proof To egin, it i cler tht ny olution for non-protection routing in two time lot for the et of noe-ijoint emn in G will immeitely give protection routing n cheule uing two time lot in G for the me et of emn. We now conier the other irection: for et of noe-ijoint emn, oe olution tht ue two time lot for the protection prolem on G give olution for the me et of emn uing two time lot for the non-protection prolem in G? If the protection prolem return olution for routing n cheuling uing two lot, then tht men tht efore ny filure, n fter ny filure, the et of noe-ijoint emn cn e route in two time lot. So, if routing n cheule i foun, then we tke the route n cheule from efore filure (which i in two time lot), n trnfer ny flow tht my hve een route onto {i, k}, {k, l}, n {l, j} to ege {i, j}, with {i, j} hving the me time lot ignment {i, k}. Becue of how our trnformtion w performe, thi will lwy yiel feile olution for the et of noe-ijoint emn in G. In generl, it i not necerily the ce tht no olution to the protection prolem inicte no olution to the non-protection prolem. But for our prticulr grph trnformtion, thi i the ce; we know tht if two time lot olution exit in G, then protection routing mut exit tht ue two time lot. Hence, if the minimum cheule to the protection prolem for the et of noe-ijoint emn in G ue more thn two time lot, then the cheule for the et of emn in G mut ue more thn two time lot. We complete the proof y noting tht our prolem i clerly in NP, n tht the grph trnformtion of G to G cn e ccomplihe in polynomil time. D. Proof for Theorem 3 Theorem 3: For n incoming connection etween n, uing ijoint pth to provie protection in wirele network with -hop interference contrint for the et of time lot T i NP-Complete. Proof: We reuce the Dynmic Shre-Pth Protecte Lightpth-Proviioning (DSPLP) [] to our prolem. We firt egin y giving etil of DSPLP. DSPLP h the following et of prmeter: W i the et of poile wvelength on ny link. L pth re route, where (w i, i ) i the i th working n ckup pth, repectively. The quetion DSPLP k i: oe there exit (w L, L ) from to tht tifie hre pth protection contrint? Thoe contrint eing: () w L n L re link ijoint; () w L n w i, i < L, o not utilize me wvelength on ny common link; (3) w L n i, i < L, o not utilize me wvelength on ny common link; (4) L n i, i < L, cn hre wvelength on common link if w L n w i re link ijoint. The following i provie y DSPLP: grph G with vertice n ege (V, E); W i the et of poile wvelength on ny link. λ ij i the et of wvelength ue on ege {i, j} for primry pth; λ kl ij i the et of wvelength ue on ege {i, j} to protect gint the filure of {k, l}. T = W. More imply put, ome new incoming emn tht nee ijoint primry n protection pth, cn hre ckup reource with ome other emn if the two primry pth re filure ijoint. There i cler prllel etween the wvelength multiplexing cheme tht DSPLP i e on n our wirele protection cheme tht ue time lot: time lot ue for routing/cheuling on link re imilr to wvelength ue on link for routing n protection. In [], NP-Completene i hown for network with T = ; hence, it i ufficient for u to emontrte tht if our prolem cn olve n intnce of DSPLP with only one wvelength, our prolem i lo NP- Complete. If wvelength re coniere timelot, they will interfere with one nother. Clerly more thn one time lot mut exit in orer to fin feile routing n cheule in wirele network with -hop inference contrint. So, we exten new link from ech noe, n we incree T (the numer of time

15 5 lot in the wirele network) uch tht there exit ufficient time lot to chnge exiting pth tht ue one wvelength in the originl network into interference free cheule uing the new ege in G W. It i ey to ee tht the numer of new time lot tht nee to e e will e the mximum noe egree of the network. An exmple i hown in Fig.. The noe egree i 4, n ech ege h pth route on it uing the exiting wvelength. We exten new ege out of the noe, n incree the numer of time lot to ny vlue ove 5. Now n interference free cheule cn e igne to llow the ege tht ue wvelength (now time lot ) to continue routing thoe pth uing time lot. () Noe in DSPLP network () Noe with new ege Fig. : Time lot ignment for extene new ege We moify the network G to G W in the following mnner. We exten new link from ech noe, n incree T W (the numer of time lot in the wirele network) y ome lrge enough vlue, uch tht the new ege in G W will never interfere with one nother, or with exiting emn uing wvelenth/time lot. Ol link in G W tht ue wvelength to upport lightpth in G will hve no ville time lot in G W. Since the primry link uing wvelength in G will no hve no free time lot in G W, every future incoming emn in G W will e ege-ijoint from the exiting primry emn tht ue wvelength/time lot, n hence cn hre ckup cpcity with exiting emn. All link in G tht ue wvelength for protection will hve only one free time lot (time lot ) ville for ue in G W, n only the ckup pth for future emn in G W. All other ol link in G W will hve only one time lot ville, time lot, ville for ue for the primry or protection pth. Conier ome new incoming emn in G W tht nee to e route n cheule with ijoint primry n protection pth. The wy the network G W w contructe will enure tht if olution exit for the new emn, then olution exit for DSPLP in G. The new emn in G W will only e le to ue time lot on the ol link, which i wvelength. 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