Chnging the routing protool without trnsient loops Nny Rhkiy, Alexnre Guitton To ite this version: Nny Rhkiy, Alexnre Guitton. Chnging the routing protool without trnsient loops. Computer Communitions, Elsevier, 2016, 82, pp.49-58. <10.1016/j.omom.2016.02.010>. <hl- 01448127> HAL I: hl-01448127 https://hl-lermont-univ.rhives-ouvertes.fr/hl-01448127 Sumitte on 27 Jn 2017 HAL is multi-isiplinry open ess rhive for the eposit n issemintion of sientifi reserh ouments, whether they re pulishe or not. The ouments my ome from tehing n reserh institutions in Frne or ro, or from puli or privte reserh enters. L rhive ouverte pluriisiplinire HAL, est estinée u épôt et à l iffusion e ouments sientifiques e niveu reherhe, puliés ou non, émnnt es étlissements enseignement et e reherhe frnçis ou étrngers, es lortoires pulis ou privés.
Chnging the Routing Protool without Trnsient Loops Nny El Rhkiy, Alexnre Guitton Clermont Université, Université Blise Psl, LIMOS, BP 10448, F-63000 Clermont-Ferrn, Frne CNRS, UMR 6158, LIMOS, F-63173 Auière, Frne Astrt Computer networks generlly operte using single routing protool. However, there re situtions where the routing protool hs to e hnge (e.g., euse n upte of the routing protool is ville, or euse n externl event hs triggere trffi with ifferent qulity of servie requirements). In this pper, we show tht n unontrolle hnge of the routing protool might yiel to trnsient routing loops (even if the involve routing protools re loop-free). We show tht it is possile to hieve loop-free hnge for multiple estintions using strongly onnete omponent pproh prouing suessive steps, where eh step ontins noes tht n hnge the routing protool in prllel. Our im is to reue the numer of steps in orer to reue the time require for the network to hnge from one routing protool to nother. Simultion results show tht our strongly onnete omponent pproh gretly reues the numer of steps ompre to the stte of the rt, n thus it gretly reues the time for the hnge. Keywors: Routing protools, routing protool hnge, trnsient routing loops. 1. Introution Computer networks generlly operte single routing protool whih etermines the route pkets hve to follow in orer to reh estintion. However, some situtions require to hnge the urrent routing protool. For exmple, this hnge might e triggere y the vilility of mjor upte of the protool or the orretion of seurity issue [1]. Another exmple onerns monitoring pplitions in wireless sensor networks, where the etetion of ritil event might trigger the hnge from n energy-effiient routing protool to ely-sensitive routing protool [2]. Another exmple fouses on the hnges in routing eisions use y mjor moifitions of the topology(ue to link or noe filures, or to signifint hnges in routing metris) [3, 4, 5]. If noes re urtely synhronize, they n perform the hnge simultneously from the urrent routing protool R 1 to the new routing protool R 2. However, this solution is often iffiult to implement in prtie, espeilly in lrge networks. Inee, the ost of n urte synhroniztion might e prohiitive, or noes might e operte y ifferent network ministrtors, leing to ifferent plnnings for the hnge. We ssume in the following tht noes nnot e synhronize in suh preise mnner. If noes re not synhronize n if noes perform the hnge ritrrily, trnsient routing loops might our, even if the routing protools re loop-free when onsiere inepenently. Figure 1 shows suh n exmple. Initilly, ll pkets towrs re route oring to routing protool R 1, n R 1 is loop-free (see Figure 1()). If noes re requeste to hnge to nother protool R 2 in ritrry orer (R 2 is shown on Figure 1()), it is possile tht hnges first, resulting into the routing epite on Figure 1(). In this se, trnsient routing loop ours etween noes n. This loop will eventully ispper when noe hnges to R 2 too, ut the impt on the network performne is not negligile. Corresponing uthor. Emil Aress: lexnre.guitton@univ-plermont.fr (Alexnre Guitton) Phone: +33 473405229, Fx: +33 473407639 Preprint sumitte to Elsevier Ferury 18, 2016
R 2 R 1 R 1 () () () Figure 1: Unontrolle hnges might yiel to routing loops. () The initil routing protool R 1 is loop-free for estintion. () The new routing protool R 2 is lso loop-free for estintion. () A loop ours etween noes n, if n route oring to R 1 n routes oring to R 2. R 2 Routing loops reue network performne s they n use noe inessiility issues or overlo the network. Even if the routing loops use y the hnges re trnsient (euse ll noes will eventully perform the hnge to the new, loop-free routing protool), our im is to ompletely voi them, s they might hve signifint impt for the pplitions. In [6], we lssifie routing protools into three tegories: (i) omptile routing protools, whih o not yiel to routing loops when they re use together, (ii) elyle routing protools, where noes might voi loops se on the knowlege of the istne funtions of the two protools, n (iii) omine routing protools, when the istne funtions of the two protools re not known or hr to ompute lolly y the noes. In [7], we use proilisti pproh to voi loops. Inee, some noes hoose rnomly whether to forwr pkets or to hol them. However, the ommon ssumption of [6, 7] is tht routing protools lternte. In this pper, we mke more generl ssumption, where the hnge is finl: one noe hs hnge to R 2, it oes not hnge k to R 1 nymore. This new ssumption mkes the previous solutions inpplile. Moreover, we show tht the hnge to R 2 n e performe in suessive steps, where ll noes of the sme step n perform the hnge ritrrily without using loops. Our ontriution is three-fol. First, we show tht the proility of trnsient loops is high for oth rnom n rel topologies, n for severl types of routing protools pirs. Seon, we improve the min heuristi propose y [8] for networks with severl estintions, y proposing greey mehnism to el with troulesome estintions, n y omputing sequene of steps rther thn sequene of noes. Thir, we propose our entrlize heuristi se on the omputtion of strongly onnete omponents in orer to lssify noes into steps. It is ime t reuing the overll numer of steps n thus reuing the hnge urtion, whih is our min metri. The reminer of the pper is orgnize s follows. Setion 2 esries relevnt reserh works of the literture. Setion 3 first presents two improvements for the min protool of the literture [8]: greey mehnism for the per-estintion orering, n the omputtion of sequene of steps rther thn sequene of noes. Then, it presents our entrl theorem se on strongly onnete omponents. Finlly, it presents our heuristi se on this theorem. Setion 4 evlutes the performne of these heuristis on severl topologies n for severl routing protools pirs. Finlly, Setion 5 onlues this work. 2. Relte work In this setion, we first esrie rhitetures where routing loops might our. Then, we present the solutions from the literture tht voi routing loops ourring uring the hnge from one protool to nother. Then, we esrie in etils the Routing Tree Heuristi (RTH) presente in [8], s it is the min heuristi to hnge routing protools. Finlly, we esrie the min ifferenes etween RTH [8] n this pper. 2.1. Networks n protools with risks of loop ourrene Severl routing protools tht omine ifferent routing eisions hve een propose in the literture. In[9], the uthors propose to omine retive routing protool with greey geogrphil routing protool. When pket hs to e forwre, the retive protool estlishes the whole route to the estintion. The 2
geogrphil protool is use when the next-hop oring to the retive protool eomes unrehle. Routing loops n our if the geogrphil protool forwrs pkets to noe tht uses the retive protool. In [10], routing protool R tht reues ely is omine with routing protool R e tht reues energy. R n R e re use epening on the trffi proue y the pplition: urgent pkets re forwre oring to R, while perioi pkets re forwre oring to R e. Routing loops n our if n urgent pket rehes noe tht hs limite energy n uses R e. In [11], pkets re given priority se on trffi type. The next-hop of pket is ompute oring to severl prmeters, inluing pket priority, numer of hops to the estintion, link qulity for the next-hop, resiul energy for the next-hop, lo of the next-hop, et. Routing loops n our if the prmeters use y noe re ifferent from the prmeters use y nother noe on the pth to the estintion. Multi-purpose Wireless Sensor Networks (WSNs) [12] hve een propose to enle single WSN eployment to support severl pplitions. The min vntge of multi-purpose WSNs is tht the ost of eployment is shre y ll the pplitions. Severl reserhers hve propose protools for multi-purpose WSNs [13, 14, 15, 16]. In suh networks, severl routing protools n e use simultneously, euse the lrge mount of pplitions yiel to ifferent requirements tht nnot e met y single routing protool. However, eling with severl routing protools might use routing loops when the hoie of the routing protool is me lolly y eh noe [6, 7]. 2.2. Heuristis tht voi routing loops The prolem of voiing trnsient loops uring hnge of routing protools is reent issue. We summrize here the min relte works. In [6, 7], noes re synhronize n forwr pkets oring to sheule ompose of two perios p 1 n p 2. During p 1, noes forwr pkets oring to R 1, n uring p 2, noes forwr pkets oring to R 2. Routing loops n our if noes eome esynhronize or if the eision to forwr pket oring to R 1 or R 2 is me lolly y eh noe, inepenently of the perio. In [6], properties of pirs of routing protools re stuie, n three tegories re ientifie: (i) omptile routing protools, whih o not yiel to routing loops, (ii) elyle routing protools, where noes might voi loops se on the knowlege of the istne funtions of the two protools, n (iii) omine routing protools, where the istne funtions of the two protools re not known or hr to ompute lolly y the noes. The lst two tegories require R 1 n R 2 to lternte, s some noes hol pkets inefinitely for one routing protool. They re not suitle for finl hnge of routing protool, s we onsier in this pper. In [7], two heuristis re propose to voi loops or reue their ourrenes. In the first heuristi, ll loops re voie y foriing some noes to forwr pkets oring to one routing protool R i. In the seon heuristi, proilisti pproh is use to reue the risk of loops: noes tht oul potentilly e involve in loops hoose rnomly whether to forwr or to hol pkets for one routing protool. These two heuristis nnot e pplie here euse the hnge from R 1 to R 2 is finl: one noe hs performe the hnge to R 2, it oes not hnge k to R 1. In[4], the uthors show tht most routing protools n proue trnsient routing loops fter topologil hnge. They show tht suh loops n e voie y hving routers proess routing uptes in speifi orer. Their mehnism is le to el with link filures, new links, or uptes on link metris. The ifferenes with this pper re the following: (i) we onsier ritrry protools for R 1 n R 2, while [4] onsiers single routing protool on two similr topologies, (ii) we reue the numer of steps require for the hnge, while [4] provies n orering of uptes tht oes not use loops, n (iii) in our proposition, eh noe is upte extly one for eh estintion, while the lgorithm of [4] might upte the sme noe severl times. In [3, 5], the uthors show tht orering the routing uptes yiels to itionl messge overhe n inreses the hnge ely. They voi trnsient routing loops y exploiting the existene of one forwring tle per interfe. Messges rriving through unexpete interfes re isre, euse they inite isrepny etween the view of the router n of its neighors. One ll routers hve the sme view of the topology, the protool onverges n proues loop-free routes. The min ifferene with this pper is tht we o not rop pkets uring the hnge to reue the impt of loops, ut we voi hnging the routing protool of noes if it uses loop. 3
In [17, 18], the uthors show tht trnsient routing loops tht our fter topologil hnges n e voie y pplying sequene of topology uptes. Between two topology uptes, the sme routing protool is use, ut some link vlues re moifie, whih result into hnges in routing eisions. They propose protool tht minimizes the numer of topology uptes. The ifferene with this pper re the following: (i) we onsier two ifferent routing protools on single topology, while [17, 18] onsier single routing protool pplie on sequene of upte topologies, (ii) we onsier ritrry routing protools, while [17] onsiers hnges on only one link n [18] onsiers hnges onerning the links of single router, n (iii) we reue the numer of steps to perform the hnge, where eh step is set of noes n eh noe ppers extly one per estintion, while [17, 18] ttempts to minimize the numer of topology uptes to gurntee the onvergene of single routing protool without trnsient loops. In [8], the uthorsesrie heuristi whih is se onsimilr ssumptionssin this pper. We esrie this heuristi in etils in the next susetion. 2.3. Routing Tree Heuristi [8] In [8], uthors propose orering lgorithms in orer to voi loops uring the hnge from one routing protool R 1 to nother routing protool R 2. They propose heuristi lle Routing Tree Heuristi (RTH). RTH onsists of the following. For eh estintion, the orering onstrints re ompute seprtely. First, greey lgorithm is use to ompute the set S of noes tht o not yiel to loops in the network. A noe is e to S if n only if its next-hops using R 1 n using R 2 re lrey in S. Seon, the set V of noes is uilt y ing eh noe tht oes not hve the sme next-hops using R 1 n R 2. Thir, RTH uils set of onstrints C in the following wy: for eh pth on R 2 from soure noe to the estintion, onstrint is generte for the lst pir of noes (u,v) on this pth suh tht u V n u / S. In this wy, the hnge from R 1 to R 2 strts from the estintion kwrs (oring to R 2 ) to the soures, whih gurntees tht no loop our on the pth. This mens tht noe n oes not hnge efore ll its suessors on R 2 hve lrey hnge. Finlly, RTH retes n yli irete grph G C from the set of onstrints C, n the orering is ompute s topologil sort of G C. 2.4. Differenes etween RTH n our proposition The ifferenes with this pper re the following: RTH hnges the routing protool of noes one y one, so the urtion of the hnge is proportionl to the numer of noes in the network. However, we show here tht it is possile to hnge the routing protool of severl noes in prllel, without ny loop ourrene. We use this prllelism to reue the time require for the routing protool hnge. RTH hnges the routing protool of noes y following the rs of R 2 from the estintion kwrs to the soures. However, we show tht it is possile to hnge the routing protool for noes inepenently of their position on the pths from soures to estintions oring to R 2. We use strongly onnete omponent pproh to ientify those noes. The uthors of RTH ssume tht there re few troulesome estintions, so RTH tries to ompute the orering for ll estintions together when possile. However, we show tht troulesome estintions re reltively frequent, so we eie to onsier estintions inepenently. 3. Fst hnges of routing protools The min riteri when performing hnges of routing protools without trnsient loops is the overll time requirefor the hnge. We onsierin this pper tht it is possile toperform the hngessequene of steps, where noes of eh steps n hnge their routing protool in prllel. Thus, the time require for the hnge is proportionl to the numer of steps, rther thn to the numer of noes in the network. In this setion, we first esrie improvements to RTH [8]: the first improvement is greey mehnism tht els with troulesome estintions, n the seon improvement enles the support of steps. Seon, we give mthemtil kgroun for prllel steps se on strongly onnete omponents. Thir, we esrie our heuristi se on this kgroun. 4
3.1. Improvements of RTH We present here two improvements of RTH. The first improvement esries the per-estintion orering, when per-router orering oes not exist ue to troulesome estintions. The seon improvement esries how to enle prllel hnges in RTH. 3.1.1. Improvement 1: Greey mehnism of per-estintion orering in RTH The first improvement is greey mehnism of per-estintion orering. Note tht this orering n e pplie when per-router orering exists (in this se, ll estintions n e regroupe, n the result is the sme s with the originl RTH) s well s when per-router orering oes not exist (in this se, severl estintions re regroupe in greey mnner, n troulesome estintions re proesse inepenently). Authors of [8] minly esrie the per-router orering (whih ssumes tht there is no troulesome estintion), n expline tht troulesome estintions n e proesse inepenently from the other estintions. However, they i not propose n lgorithm tht ientifies troulesome estintions, or tht ggregtes non-troulesome estintions(this ws left s future work ue to the ft tht troulesome estintions were limite in their experimentl evlution, f Susetion V.C of [8]). The RTH heuristi (see Figure 8 of [8]) is moifie in the following wy. Initilly, the set of troulesome estintions is initilize to. In the min loop tht onsiers eh estintion, the onstrint grph G C is ompute using V n the set of onstrints C. If the new grph G C, inluing the onstrints generte y ll previous non-troulesome estintions n the new estintion, hs no loop, then the new estintion is onsiere s non-troulesome estintion. However, if the new grph G C hs loop, estintion is e to the list of troulesome estintions, n the onstrints generte from this estintion re remove from C. In this se, moifitions to G C onerning re isre. After ll estintions hve een onsiere, the orering is me first for ll non-troulesome estintions together using G C, n then for ll troulesome estintions one y one, using the per-estintion orering. 3.1.2. Improvement 2: Prllel hnges in RTH The seon improvement onsists of enling prllel hnges in RTH, in orer to otin sequene of steps rther thn sequene of noes. This moifition n e performe effiiently from the onstrint grph G C ompute for ll non-troulesome estintions (s well s for eh troulesome estintion inepenently). This improvement onsists of hnging the lst step of RTH, y repling the topologil sort of G C with Algorithm 1. In this lgorithm, new set S is e to sequene T t eh itertion. At eh itertion, ll noes tht hve no inoming eges in G C (tht is, no onstrints), re e to the new set S n ll eges leving from these noes re remove from G C. Sine G C hs no loop (ue to the onstrution of ll non-troulesome estintions, n to the ft tht troulesome estintions re onsiere inepenently), the proess ens with ll noes eing in T. In our implementtion, sequene T ontins first the steps for ll non-troulesome estintions, then the steps for eh troulesome estintion suessively. In the following, we enote y RTH-p this heuristi. 3.1.3. Exmple of RTH-p In the following, we esrie RTH-p in n exmple. This exmple shows our greey mehnism for per-estintion orering, s well s prllel hnges. Figure 2 shows grph of five noes, with three estintions n two routing protools: () estintion with R 1 n R 2, () estintion with R 1 n R 2, n () estintion e with Re 1 n Re 2. When onsiering estintions inepenently, RTH proues the orer (,e,,,) for estintion, (,e,,,) for estintion, n (e,,,, ) for estintion e. When onsiering ll estintions together, our greey mehnism first onsiers estintion, ientifies estintion s troulesome, n ientifies estintion e s non-troulesome. Thus, RTH-p proues sequene ({, e},{},{},{}) for oth non-troulesome estintions n e, n sequene ({,, e},{},{}) for troulesome estintion. The resulting sequene is ({,e},{},{},{},{,,e},{},{}) (to simplify the nottion, we i not inite whih estintion is onerne y the hnge t eh step). Note tht eh noe ppers severl times ut for ifferent estintions. 5
Algorithm 1 Prllel hnges in RTH. Require: G C grph of onstrints Ensure: T is sequene of steps T while some noes re not in T o S while noe n G C o if n / S n n / T then if n hs no inoming eges in G C then S S {n} en if en if en while while noe n S o remove from G C ll eges leving from n en while set S to sequene T en while R 1 R 2 R 2 R e 2 R 1 R e 1 e e e () () () Figure 2: Exmple of routing protools for three estintions: () is the estintion, () is the estintion, n () e is the estintion. 6
3.2. Prolem formultion for the onstrution of sequene of steps Let V e set of noes n V estintion. Let us onsier two routing protools for estintion : R 1 is the routing protool initilly use y noes, n R 2 is the new protool to use. For ll i {1,2} n for ll n V, we enote y R i (n) V the next-hop of n towrs oring to R i. Given prtition (V 1,V 2 ) of V (i.e., V 1 V 2 = V n V 1 V 2 = ), we enote y R V1,V 2 the routing protool efine in the following wy: for ll n V 1, R V1,V 2 (n) = R 1 (n), n for ll n V 2, R V1,V 2 (n) = R 2 (n). In other wors, noes of V 1 route oring to R 1, n noes of V 2 route oring to R 2. The set of routing eisions of oth R 1 n R 2 form the set E of the eges of the grph G = (V,E). Definition 1 (Loop-free step). Let (V 1,V 2 ) e prtition of noes, n S V 1 set of noes tht hnge their routing protool to R 2 in ritrry orer. S is lle step. Step S is si to e loop-free if n only if for ll S S, the routing protool R V1\S,V 2 S is loop-free. Definition 1 sttes tht step S is loop-free if n only if ll the possile intermeite su-steps S S orrespon to loop-free routing protool. This mens tht the noes of S n hnge their routing protool ritrrily without using loops. Definition 2 (Loop-free sequene). Let T = (S 1,...,S m ) e sequene of steps. Sequene T is si to e loop-free if n only if oth onitions pply: (S 1,S 2,...,S m ) prtitions the set V, for ll i [1;m], S i is loop-free step on prtition (V i 1,V i 2), with V i 2 = S 1... S i 1 n V i 1 = V\V i 2. Definition 2 sttes tht sequene T = (S 1,...,S m ) is loop-free if n only if eh step S i is loop-free, n fter step S m, ll noes elong to V2 m S m = V (tht is, they hve hnge to the trget routing protool R 2 ). Note tht the set of noes running R 2, whih is V2 i, inreses s the sequene progresses. Figure 3 shows n exmple of loop-free sequene T = ({,, },{}). Figure 3() shows the initil routing protool R 1. Figure 3() shows oth R 1 n R 2 for noes of the first step {,,}, to inite tht these noes might route either oring to R 1 (if they hve not hnge yet) or R 2 (if they hve lrey hnge). It n e seen tht for ny routing protool use y the noes of the first step, no loop ours for ny ritrry orer of hnge. Figure 3() shows R 2 for noes tht hve hnge their routing protool on the previous step, n shows oth R 1 n R 2 for the noe of the seon step. On sie note, it n e notie tht T is loop-free sequene with minimum numer of steps, s it is not possile to hve oth n in the sme loop-free step. R 2 R 2 R 1 R 1 R 1 () () () Figure 3: An exmple of loop-free sequene T = ({,,},{}). () Initilly, ll noes route oring to R 1. () During the first step, noes from {,,} might route oring to R 1 or R 2 without using loops. () During the seon step, noe routes oring to R 1 or R 2, while ll noes from {,,} route oring to R 2. At the en of the seon step, ll noes route oring to R 2 (not shown). Theorem 1. Let G, V 1, V 2, R 1 n R 2 e efine s previously for estintion. Let G = (V,E ) with E E, suh tht for ll x V 1, (x,r 1 (x)) E n (x,r 2 (x)) E, n for ll x V 2, (x,r 2 (x)) E. Let C = {C i } i e the set of ll strongly onnete omponents of G. For ll i, let us enote y C i C i set of noes tht verifies the following properties: C i V 1, 7
G i = (V,E i ) oes not ontin ny loop, with E i efine s follows: if x C i then (x,r 1(x)) E i n (x,r 2(x)) E i, if x V 1 \C i then (x,r 1(x)) E i, if x V 2 then (x,r 2 (x)) E i. Then, S = i C i is vli step for estintion. Proof. Let us ssume tht {C i } i verifies the properties. We hve to show tht S = i C i is vli step, tht is, for every S S, R V1\S,V 2 S is loop-free n les to estintion. Let S e n ritrry suset of S, n let us uil R V1\S,V. 2 S Let us first show tht R V1\S,V 2 S is loop-free. By ontrition, let us suppose tht there is loop (x 0,x 1,...,x n ) in R V1\S,V, 2 S with x n = x 0. Suppose here tht this loop spns single strongly onnete omponent C i. For ll k, we hve (x k,x k+1 ) E S. (i) If x k S, then x k+1 = R 2 (x k ) y onstrution of E S (sine S S C i ). Beuse of the properties of C i, we hve (x k,r 2 (x k )) E i. (ii) If x k / S n x k V 1, then x k+1 = R 1 (x k ) y onstrution of E S. We hve to onsier the two following su-ses: x k C i n x k / C i. If x k C i, then (x k,r 1 (x k )) E i. If x k / C i, then (x k,r 1 (x k )) E i. (iii) If x k / S n x k V 2, then x k+1 = R 2 (x k ) y onstrution of E S, n x k / C i. Thus, (x k,r 2 (x k )) E i. To summrize these three ses, ll the rs of the loop on E S re lso inlue in the rs of E i, thus G i ontins loop. However, this is impossile y onstrution of C i. Thus, y ontrition, there is no loop on G S. Suppose now tht this loop spns severl strongly onnete omponents, inluing C i n C j, with i j. For ll noes x m of the loop (with (x m,x m+1 ) E S ), let us show tht (x m,x m+1 ) E. (i) If x m S, then x m+1 = R 2 (x m ), n x m C i, whih mens tht x m V 1. Thus, (x m,r 2 (x m )) E y onstrution of E. (ii) If x m / S n x m V 1, then x m+1 = R 1 (x m ). Thus, (x m,x m+1 ) E. (iii) If x m / S n x m V 2, then x m+1 = R 2 (x m ). Thus, (x m,x m+1 ) E. To summrize these three ses, for ll x m of the loop, (x m,x m+1 ) E, so there exists loop in G etween noe of C i n noe of C j, whih mens tht C i n C j re the sme strongly onnete omponent, whih is impossile. Let us show now tht for ny noe x, is rehle from x in G S. By ontrition, let us suppose tht there is noe x from whih is not rehle. Let us onsier the pth strting from x in G S. Either there exists noe on the pth tht hs no next-hop on G S, or the pth hs loop. We just prove tht there is no loop in G S, so there hs to e noe y without next-hop. By onstrution of G S, we n see tht ll noes y S (V 1 \S ) V 2 = V hve next-hop, so estintion is rehle from ny noe x in G S. This ompletes the proof. Figure4showsnexmpleofgrphofninenoes,whereestintionisnoef. TheroutingprotoolsR 1 n R 2 re shown on Figure 4(). To uil the first vli step, ll routing rs re onsiere. The strongly onneteomponentsoftheresultinggrphrethefollowing: C 1 = {f},c 2 = {,}nc 3 = {,,e,g,h,i}. The following sets verify the property of the theorem: C 1 = {f}, C 2 = {} n C 3 = {,g,h,i}. Inee, it n e verifie tht none of the grphs G i (for i {1,2,3}) hs loop (this n lso e seen on the grph shown on Figure 4()). Thus, the step S 1 = {,,f,g,h,i} is vli first step. To uil the seon vli step, the strongly onnete omponents of the grph shown on Figure 4() re ompute. They re the following: C 1 = {}, C 2 = {},..., C 9 = {i}. The following sets verify the property of the theorem: C 1 = {}, C 3 = {}, C 5 = {e}, n C i = otherwise. The seon step is S 2 = {,,e}. After this step, ll noes route oring to R 2. Thus, T = (S 1,S 2 ) is loop-free sequene. 3.3. Centrlize heuristi for loop-free hnge of routing protools In this susetion, we esrie our entrlize heuristi for loop-free hnge, lle Strongly Connete omponent Heuristi with prllel hnges (SCH-p). We ssume tht entrlize entity knows the whole network topology, s well s R 1 n R 2 for eh estintion. 8
e f e f e f g h i g h i g h i () () () Figure 4: Sequene se on strongly onnete omponents. () The grph hs three strongly onnete omponents C 1 = {f}, C 2 = {,}, C 3 = {,,e,g,h,i}. () The first step is S 1 = {,,f,g,h,i}, n only the noes of S 1 n route oring to R 1 or R 2. () The seon step is S 2 = {,,e}, n only the noes of S 2 n route oring to R 1 or R 2. SCH-p is se on Algorithm 2. Eh estintion is onsiere sequentilly n inepenently. First, the set of strongly onnete omponents of the grph is uilt se on the two routing protools for the urrent estintion. Then, eh strongly onnete omponent C i is onsiere iniviully, n noes re istriute into severl steps oring to Algorithm 3. We use greey pproh: for eh step S j, we noes one y one into set C i,j, until it is not possile to more noes without violting the onstrints of Theorem 1. Then, we move to the next step, until ll noes of C i re in step for this estintion. The worst-se omplexity of SCH-p is O( V 5 ). Inee, there re t most V estintions. For eh estintion, the strongly onnete omponents of grph n e ompute in O( V + E ) with Trjn s lgorithm [19]. Note tht in our grphs, E 2 V s eh noe hs t most two outgoing rs (one with R 1 n one with R 2 ). For eh strongly onnete omponent C i, there re t most C i resulting steps (s there is t lest one noe per step). Computing the set C i,j requires onsiering t most C i 2 omintions of noes, eh requiring to uil G n etermining if it hs loop, whih n e one in O( C i ). As the set of strongly onnete omponents prtitions the grph, we otin the overll omplexity of O( V 5 ). 4. Simultion results In this setion, we esrie our simultion results. We strt y esriing our settings. Then, we quntify the proility of loop ourrene when using ritrry routing protools, with unontrolle hnges (tht is, without ny heuristi to voi trnsient loops). We lso quntify the numer of troulesome estintions for RTH n RTH-p. Then, we ompute the numer of steps require to hnge the routing protools for RTH, RTH-p n our heuristi SCH-p. 4.1. Topologies n prmeter settings We use two types of topologiesin orer to evlute the heuristis overlrgenumer of networks. First, we eie to generte rnom onnete grphs. We generte networks ompose of 50,, 150, 200, 250, n 300 noes, rnomly eploye on n re of m m. Noes tht re istnt of less thn 20 m re onsiere onnete. Seon, we eie to test our heuristi SCH-p on rel networks. We use the Roketfuel network topologies [20, 21], whih re lso use for the evlution of RTH in [8]. The resulting topologies hve 79, 87, 104, 138, 161, n 315 noes. In the following, we use three senrios for the routing protools. They re ll se on shortest pths, using ifferent link metris. In Senrio 1, R 1 uses the hop-ount metri n R 2 uses rnom metri, hosen rnomly within [1;] for eh link. R 2 moels protool se on ely or loss rte. In Senrio 2, oth R 1 n R 2 re se on inepenent rnom metris hosen within [1;] (suh s ely for R 1 n loss rte for R 2 ). In Senrio 3, R 1 is se on rnom metri hosen within [1;] for eh link, n R 2 uses orrelte weight for links. If w [1;] enotes the weight of the link for R 1, the weight of the link for R 2 is hosen rnomly within [mx(1,w 10);min(,w + 10)]. This moels the 9
Algorithm 2 Min lgorithm for SCH-p. Require: G = (V,E) grph, D V set of estintions, R 1 n R 2 two routing protools (for eh estintion of D) Ensure: T is vli sequene j 1 for D o mx 1 C strongly onnete omponents of G (with rs resulting of R 1 n R 2 for estintion ) for C i C o if C i = 1 then S j S j C i else ol j while there re noes of C i tht re not in step yet o fin suitle set C i,j C i (see Algorithm 3) S j S j C i,j j j +1 en while if j 1 > mx then mx j 1 en if j ol en if en for j j +mx en for return T = (S 1,...,S j 1 ) Algorithm 3 Computtion of one step for C i in SCH-p. Require: C i is strongly onnete omponent (with t lest two noes), is estintion, the list of previous steps is known Ensure: C i,j stisfies Theorem 1 C i,j en flse while not en o en true for noe n in C i o if n is not lrey in step n n / C i,j then uil G with the noes of G n no eges for eh noe m tht hs een e in previous step, r (m,r 2 (m)) to G for eh noe m C i,j {n}, rs (m,r 1(m)) n (m,r 2 (m)) to G for eh other noe m, r (m,r 1 (m)) to G if G oes not ontin loop then C i,j C i,j {n} en flse en if en if en for return C i,j en while 10
se where protool R 2 uses n upte version of the topology, or inlues n itionl prmeter in the omputtion of link weights. 4.2. Simultion on loop ourrene with unontrolle hnges We onsier tht loop ours if it is possile for pket to enter routing loop when noes on the pth eie ritrrily to route oring to R 1 or R 2. For instne, we onsier tht there is loop in the topology shown on Figure 1, euse it is possile tht routes oring to R 1 while routes oring to R 2. Notie tht even if there is loop ourrene in topology, some noes might e le to sen pkets to the estintion without loops. Thus, our metri refers to the risk of ourrene of t lest one loop. In the following, ll the results on loop ourrenes re verge over simultions per estintion, n over ll possile estintions. 4.2.1. Loop ourrenes on rnom networks In this sususetion, we quntify the proility of loop ourrene for rnom grphs, in the three previous senrios. Figure 5 shows the perentge of loop ourrene s funtion of the numer of noes in the network, for the three senrios. In Senrio 1, the perentge of loop ourrene inreses with the numer of noes. Even when the numer of noes is smll (for instne, 50), the perentge of loop ourrene is out 60%, whih mens tht loops re likely to our. This omes from the ft tht the two routing protools uil pths with low orreltionsue to the rnom weight of R 2. In Senrio2, the perentge of loop ourrene is out % for ll numers of noes in the network. This is ue to the ft tht R 1 n R 2 yiel to ifferent routing eisions, thus pths hve very low orreltion. In Senrio 3, the perentge of loops is low for smll numer of noes. Inee, sine the link metris re orrelte, R 1 n R 2 re likely to e similr, so pkets re likely to follow similr pths. However, when the numer of noes is lrge, the two routing protools exhiit ifferenes, n the resulting shortest pths hve low orreltions (even if the weights re orrelte). % loop ourrene 80 60 40 20 Senrio 1 Senrio 2 Senrio 3 0 50 150 200 250 300 Numer of noes in the network Figure 5: In rnom networks, when R 1 is se on hop-ount metri n R 2 is se on rnom metri, the perentge of loop ourrene is high, ue to the low orreltion of the metris of R 1 n R 2. When R 1 n R 2 re oth se on rnom metri, the perentge of loop ourrene is very high, gin ue to the very low orreltion of the metris of R 1 n R 2. When R 1 is se on rnom metri n R 2 is se on evition of the metri of R 1, the perentge of loop ourrene is high for topologies with lrge numer of noes, espite the high orreltion of the metris. 4.2.2. Loop ourrenes on rel networks In this sususetion, we quntify the numer of loops tht might pper using R 1 n R 2, for rel networks. Figure 6 shows the perentge of loop ourrene s funtion of the numer of noes, for the three senrios. We notie tht the perentge of loop ourrene is muh lrger with Senrio 1 n Senrio 2 11
thn with Senrio 3. This is ue to the ft tht the routing protools R 1 n R 2 of Senrio 3 re highly orrelte. However, the perentge of loop ourrene is more importnt for lrge networks s the pths re longer. For rel networks, the perentge of loop ourrene epens on the unerlying topology. In generl, this perentge inreses with the network size, ut there re some rel topologies (suh s the Roketfuel topology 1221.ity with 104 noes) tht yiel few routing loops (see lso Figure 16 of [8]). % loop ourrene 80 60 40 20 Senrio 1 Senrio 2 Senrio 3 0 50 150 200 250 300 350 Numer of noes in the network Figure 6: In rel networks, for Senrio 1, the perentge of loops is high (etween 50% n 80% for smll networks (less thn 104 noes), n ove 90% for lrge networks). For Senrio 2, the perentge of loops is lwys high inepenently of the network size (etween 90% n %). For Senrio 3, the perentge of loops inreses with the network size (10% for smll networks n up to 90% for lrge networks). 4.3. Averge perentge of troulesome estintions In this susetion, we fous on the perentge of troulesome estintions. Troulesome estintions re estintions tht require to e onsiere inepenently y RTH n RTH-p. These estintions nnot follow the ommon per-router orering use for ll the remining estintions. We ompute the numer of troulesome estintions for rnom n rel networks y verging over simultions, n we normlize it with respet to the numer of noes of the network in orer to otin perentge. All noes t s estintions. 4.3.1. Troulesome estintions for rnom networks Figure 7 shows the verge perentge of troulesome estintions in rnom networks. We notie tht the perentge of troulesome estintions is low for smll networks n inreses onsistently for lrge networks. This is ue to the ft tht the perentge of loop ourrene is smll (f Sususetion 4.2.1). Moreover, the perentge of troulesome estintions epens on the senrios. In Senrio 1, we notie tht the perentge of troulesome estintions is low for smll networks (less thn 26%) ut it inreses quikly for lrge networks (up to 94%). In Senrio 2, the perentge of troulesome estintions is out 80% for network of 50 noes n goes up to out % for network of 300 noes. In Senrio 3, the perentge of troulesome estintions is lmost zero for smll networks when R 1 n R 2 re highly orrelte, ut it inreses up to 98% for lrge networks. 4.3.2. Troulesome estintions for rel networks Figure 8 shows the perentge of troulesome estintions in rel networks. For Senrio 1 n Senrio 2, the perentge of troulesome estintions is importnt (etween 70% n % for Senrio 1, n ove 90% for Senrio 2) for lrge networks. This perentge is low for smll networks s pths to estintions re smller, whih les to smll numer of loops in the network (f Sususetion 4.2.2). For Senrio 3, the perentge of troulesome estintions is low, even for lrge networks. Note tht the low perentge of troulesome estintions for Senrio 3 is the min rgument use y uthors of [8] to justify tht there re few troulesome estintions. However, we elieve tht Senrio 3 represents only speifi se, where the two routing protools re highly orrelte. 12
% troulesome estintions 80 60 40 20 Senrio 1 Senrio 2 Senrio 3 0 50 150 200 250 300 Numer of noes in the network Figure 7: In rnom networks, the perentge of troulesome estintions is more importnt in lrge networks thn in smll networks. Troulesome estintions pper even with orrelte routing protools (suh s in Senrio 3). % troulesome estintions 80 60 40 20 Senrio 1 Senrio 2 Senrio 3 0 50 150 200 250 300 350 Numer of noes in the network Figure 8: In rel networks, the perentge of troulesome estintions is more importnt in lrge networks thn in smll networks. Correlte routing protools (see Senrio 3) re le to gretly reue the numer of troulesome estintions. 13
4.4. Results on verge numer of steps In this susetion, we evlute the numer of steps require for ll the noes to hnge from R 1 to R 2 without ny loop ourrene. This is our min metri, s the hnge urtion is proportionl to the numer of steps proue y the heuristis. We onsier tht ll noes re estintions. Simultion results re verge over repetitions n onfiene intervls of 95% re shown. We ompre our heuristi SCH-p to RTH [8]. Rell tht RTH provies n orering to hnge the noes sequentilly: the numer of steps of RTH is equl to the numer of noes in the network, if there is no troulesome estintion. We lso ompre our heuristi SCH-p to RTH-p. Rell tht RTH-p onsists of hnging severl noes in prllel. 4.4.1. Numer of steps for rnom networks Figure 9 shows the verge numer of steps require to hnge the routing protool, in term of the numer of noes in the network, for Senrio 1. The y-xis is epite using logrithmi sle. We notie tht the numer of step inreses onsistently with the size of the network for ll the heuristis (RTH, RTH-p n SCH-p). We notie lso tht the numer of steps is very lrge for RTH. This is ue to the ft tht only one noe t eh step is le to hnge from R 1 to R 2. More preisely, the numer of steps for RTH is equl to the numer of noes in the network, times α + 1, where α is the numer of troulesome estintions. Both RTH-p n SCH-p outperform RTH with gin tht rehes up to 97% for RTH-p, n up to 99% for SCH-p. SCH-p outperforms RTH-p: for lrge networks (more thn noes), SCH-p shows gin of 63% for network of noes, n gin of 77% for network of 300 noes. The lrge gin of SCH-p n e expline s follows: SCH-p is le to hnge the routing protool of noes inepenently on their position on the pths from the soures to the estintions (oring to R 2 ), while RTH-p only hnges the routing protool for noes strting from the estintions n going kwr to the soures. Sine the numer of troulesome estintions is lrge (see Fig. 7), RTH-p is not le to ggregte mny non-troulesome estintions together. 000 Numer of steps 00 0 RTH RTH-p SCH-p 10 50 150 200 250 300 Numer of noes in the network Figure 9: In rnom networks for Senrio 1, SCH-p is le to perform the hnge in out 600 steps on verge for lrge networks. RTH n RTH-p require respetively 85182 steps n 2602 steps to perform the hnge of ll the noes. Figure 10 shows the verge numer of steps require to hnge the routing protools in term of the numer of noes, for Senrio 2. The y-xis is epite using logrithmi sle. We n see tht the numer of steps inreses onsistently with the size of the network. We notie lso the sme ehvior for RTH s in Fig. 9: the numer of steps require to hnge ll the noes is very lrge, espeilly for lrge networks. RTH proues 2021 steps for networks of 50 noes, n 90000 steps for networks of 300 noes. 14
000 Numer of steps 00 0 RTH RTH-p SCH-p 10 50 150 200 250 300 Numer of noes in the network Figure 10: In rnom networks n for Senrio 2, SCH-p is le to perform the hnge in out 761 steps on verge for lrge networks. RTH n RTH-p require respetively 90000 steps n 3864 steps to perform the hnge of ll the noes. SCH-p outperforms RTH-p (respetively RTH), with gin vrying from 57% (respetively 94%) for smll networks up to 80% (respetively 99%) for lrge networks. Figure11showsthe vergenumerofstepsintermofthe numerofnoesinthe network,forsenrio3. The y-xis is epite using logrithmi sle. RTH-p shows etter ehvior thn SCH-p for smll networks of less thn 150 noes. Inee, RTH-p shows gin of 89% for network of 50 noes n 11% for network of noes ompre to SCH-p. Those gins re ue to the ft tht there re few troulesome estintions in this se, s R 1 n R 2 re highly orrelte. Rell tht SCH-p onsiers estintions inepenently, n thus is not le to enefit from non-troulesome estintions. SCH-p is etter thn RTH-p for lrge networks. Inee, SCH-p shows gin of 42% for network of 150 noes n of 61% for network of 300 noes, ompre to RTH-p. 000 Numer of steps 00 0 RTH 10 RTH-p SCH-p 1 50 150 200 250 300 Numer of noes in the network Figure 11: In rnom networks n for Senrio 3, SCH-p is le to perform the hnge in out 998 steps on verge for lrge networks. RTH n RTH-p require respetively 89361 steps n 2585 steps to perform the hnge of ll the noes. 4.4.2. Numer of steps for rel networks Figure12 showsthe vergenumer ofsteps in term ofthe numer of noes for Senrio1n Roketfuel topologies. The y-xis is epite using logrithmi sle. We notie the sme ehvior s for rnom networks. For smll networks, SCH-p n RTH-p hieve similr performne. For lrge networks, SCH-p shows etter performne thn RTH-p: SCH-p shows gin tht rehes 99% ompre to RTH n 68% ompre to RTH-p, for network of 315 noes. Figure13showsthe vergenumerofstepsintermofthe numerofnoesinthe network,forsenrio2, using logrithmi sle. The gin of SCH-p rehes 96% for the smllest network ompre to RTH, n 15
000 Numer of steps 00 0 RTH RTH-p SCH-p 10 50 150 200 250 300 350 Numer of noes in the network Figure 12: In rel networks n for Senrio 1, SCH-p is le to perform the hnge in out 664 steps on verge for the lrgest network (315 noes). RTH n RTH-p require respetively 94890 steps n 2121 steps to perform the hnge of ll the noes. 48% ompre to RTH-p. The lrgest gin rehe for the lrgest network is 99% ompre to RTH, n 69% ompre to RTH-p. Numer of steps 000 00 0 RTH RTH-p SCH-p 50 150 200 250 300 350 Numer of noes in the network Figure 13: In rel networks n for Senrio 2, SCH-p is le to perform the hnge in out 869 steps on verge for the lrgest network. RTH n RTH-p require respetively 99102 steps n 2851 steps to perform the hnge of ll the noes. Figure14showsthe vergenumerofstepsintermofthe numerofnoesinthe network,forsenrio3, using logrithmi sle. RTH-p is etter thn SCH-p ue to the very smll numer of troulesome estintions in this speifi senrio n for these speifi networks. The gin of RTH-p rehes 94% for network of 104 noes. The omputtion time of SCH-p is lrger thn the omputtion time of RTH n RTH-p. With our mono-thre implementtion on i7-2600 CPU t 3.40 GHz (with eight ores), SCH-p is le to ompute ll the steps for the lrgest rel topology in 91 seons on verge (with stnr evition of 3 seons, n for Senrio 2 whih yiels the lrgest numer of loops). RTH n RTH-p ompute ll the steps in 1.5 seons n 2.2 seons respetively. However, we elieve tht the omputtion time of SCH-p is resonle for lrge networks of 300 noes, n tht the gin of the hnge urtion outweights the omputtion time. 5. Conlusion When routing protool hs to e hnge, trnsient routing loops might our in the network. In this pper, we quntifie the perentge of loop ourrene on severl senrios. We show tht loops n e ompletely voie y hving noes hnge their routing protool oring to sequene whih epens 16
000 Numer of steps 00 0 RTH 10 RTH-p SCH-p 1 50 150 200 250 300 350 Numer of noes in the network Figure 14: In rel networks n for Senrio 3, RTH-p is generlly etter thn SCH-p ue to the very low numer of troulesome estintions. SCH-p performs the hnge in out 666 steps for the lrgest network. RTH n RTH-p require respetively 59629 steps n 698 steps to perform the hnge of ll the noes. on the topology n on the routing protools. This sequene is list of steps, where eh step is set of noes tht n hnge their routing protool in ritrry orer. We propose RTH-p, n enhnement of the RTH heuristi [8], tht is le to hnge severl noes in prllel. Then, we propose SCH-p, entrlize heuristi whih uils sequenes with smll numer of steps. Reuing the numer of steps is ruil, s the urtion of hnge is proportionl to the numer of steps. Simultion results for rnom n rel networks show tht SCH-p outperforms oth RTH n RTH-p in most senrios. The gin of SCH-p in term of numer of steps vries etween 80% n 99% when ompre to RTH, n etween 61% n 77% for most lrge networks when ompre to RTH-p. Referenes [1] K. Butler, T. R. Frley, P. MDniel, J. Rexfor, A survey of BGP seurity issues n solutions, Proeeings of the IEEE 98 (1) (2010) 122. [2] P. Trks, T. Zhriis, S. Voliotis, P. Krkzis, T. Velivsski, L. Srkis, Routing metri seletion n esign for multi-purpose WSN, in: IWSSIP (Interntionl Conferene on Systems, Signls n Imge Proessing), 2014, pp. 199 202. [3] Z. Zhong, R. Kerlpur, S. Nelkuiti, Y. Yu, J. Wng, C. N. Chuh, S. Lee, Avoiing trnsient loops through interfe-speifi forwring, in: IEEE/ACM IWQoS (Interntionl Symposium on Qulity of Servie), Vol. 3552 of Leture Notes in Computer Siene, Springer, 2005, pp. 219 232. [4] P. Frnois, O. Bonventure, Avoiing trnsient loops uring the onvergene of link-stte routing protools, IEEE/ACM Trnstions on Networking 15(6)(2007) 1280 1292. oi:10.1109/tnet.2007.902686. [5] S. Nelkuiti, Z. Zhong, J. Wng, R. Kerlpur, C. N. Chuh, Mitigting trnsient loops through interfe-speifi forwring, Computer Networks 52 (3) (2008) 593 609. [6] N. El Rhkiy, A. Guitton, M. Misson, Avoiing routing loops in multi-stk WSN, JCM (Journl of Communitions) 8 (3) (2013) 751 761. [7] N. El Rhkiy, A. Guitton, M. Misson, Improving routing performne when severl routing protools re use sequentilly in WSN, in: ICC (IEEE Interntionl Conferene on Communitions), 2013, pp. 1408 1413. [8] L. Vnever, S. Vissihio, C. Pelsser, P. Frnois, O. Bonventure, Lossless migrtions of link-stte IGPs, IEEE/ACM Trnstions on Networking 20 (6) (2012) 1842 1855. 17
[9] R. Shirni, M. St-Hilire, T. Kunz, Y. Zhou, J. Li, L. Lmont, Comine retive-geogrphi routing for unmnne eronutil ho networks, in: IWCMC (Interntionl Wireless Communitions n Moile Computing Conferene), 2012. [10] S. Mo, M. T. Hnsen, R. Jurk, B. Kusy, N. Boullh, A. Ksentini, On lning etween minimum energy n minimum ely with rio iversity for wireless sensor networks, in: IFIP Wireless Dys, 2012. [11] M. Fonoge, M. Crei, A. Amrose, A QoS se routing protool for wireless sensor networks, in: IPCCC (IEEE Performne Computing n Communitions Conferene), 2010. [12] J. Steffn, L. Fiege, M. Cili, A. P. Buhmnn, Towrs multi-purpose wireless sensor networks, in: Systems Communitions, 2005, pp. 336 341. [13] J. Steffn, M. Voss, Seurity mehnisms for multi-purpose sensor networks, in: GI Jhrestgung, Vol. 2, 2005, pp. 158 160. [14] D. Mnjunth, S. V. Goplih, A multi-purpose wireless sensor network for resientil lyouts, in: BWCCA (Interntionl Conferene on Bron n Wireless Computing, Communition n Applitions), 2007, pp. 49 58. [15] P. Jvier el Ci, Optimizing resoure use in multi-purpose WSNs, in: PerCom Workshops (Pervsive Computing Workshops), 2011, pp. 395 396. [16] L. Srkis, T. Zhriis, H.-C. Leligou, M. Dohler, A frmework for servie provisioning in virtul sensor networks, EURASIP Journl on Wireless Communitions n Networking (1) (2012) 135. [17] F. Cl, P. Merinol, J.-J. Pnsiot, P. Frnois, O. Bonventure, Greful onvergene in link-stte IP networks: A lightweight lgorithm ensuring miniml opertionl impt, IEEE/ACM Trnstions on Networking 22 (1) (2013) 300 312. oi:10.1109/tnet.2013.2255891. [18] F. Cl, P. Merinol, S. Vissihio, J.-J. Pnsiot, P. Frnçois, Greful router uptes for link-stte protools, in: IEEE ICNP (Interntionl Conferene on Network Protools), 2013. [19] R. E. Trjn, Depth-first serh n liner grph lgorithms, SIAM Journl on Computing 1 (2) (1972) 146 160. [20] N. Spring, R. Mhjn, D. Wetherll, Mesuring ISP topologies with roketfuel, in: ACM SIGCOMM, 2002, pp. 133 145. [21] Roketfuel projet, http://reserh.s.wshington.eu/networking/roketfuel/ (2002). 18