Routing Definition 4.1

Transcription

1 4 Routing So far, we have only looked at network without dealing with the iue of how to end information in them from one node to another The problem of ending information in a network i known a routing Routing involve two baic activitie: The determination of routing path, and the tranport of information group (typically called packet) along the path The firt item i uually referred to a path election, and the econd item i uually called packet witching In thi ection we only concentrate on the problem of electing good path Strategie for ending packet along the path will be dicued in the next ection Certainly, if there i only one ource-detination pair in the network that want to exchange information (and all the edge in the network have the ame capacity), then the bet olution would be to connect them via a hortet path But what about multiple ource-detination pair? Chooing a hortet path for each of them may lead to a high congetion and therefore a poor routing performance For example, conider a complete binary tree in which the leave are connected o that they form an n n-meh Then for mot pair of node the hortet path would lead via the root of the binary tree, but if many node want to exchange information, it i much better to ue the meh-edge intead becaue otherwie the root would become a highly congeted point So it appear that a certain degree of coordination i neceary among the node to arrive at good path A naive trategy would be to imply collect information about all the meage that the node want to end out, and then to compute a bet poible collection of path for them But thi i certainly not practical in a large network Ideally, we would like to have a path election trategy that allow the node to decide locally, ie without conulting other node, along which path (rep edge) to forward a packet There are baically two approache to that: obliviou routing and adaptive routing In obliviou routing, a fixed ytem of optional path i computed in advance for every ource-detination pair, and every packet for that pair mut travel along one of thee optional path (ee Figure 1) Thu, the path a packet take only depend on it ource-detination pair (and maybe a random choice to elect one of the option) Formally, thi can be expreed a follow: Definition 41 An obliviou routing trategy i pecified by a path ytem P and a weight function w : P IR + with the property that for every ource-detination pair (, t), the ytem of flow path P,t for (, t) in P fulfill q P,t w(q) = 1 In the cae of a flow problem, the weight indicate how a flow from to t ha to be plit among the path in P,t, and in the cae of a packet routing problem, the weight indicate the probability that a packet from a ource to a detination t chooe ome particular path p P,t In adaptive routing, the path taken by a packet may alo depend on other packet or event taking place in the network during it travel However, in thi ection we will only concentrate on obliviou routing We tart with an example of how to elect a good path ytem in a meh, followed by a general lower bound on the congetion if every ource-detination pair i jut given a ingle path Afterward, we how how to get around thi lower bound for the hypercube At the end we refine the path election problem for the meh to be more competitive with bet poible olution than the path election rule in the following ubection, which will demontrate that depite the retrictive nature of obliviou routing it i a quite powerful concept 1

2 1/8 1/4 1/8 1/2 t Figure 1: A ytem of optional path for the pair (, t) A can be eaily checked, q P,t w(q) = 1, ie the weight condition in Definition 41 i atified 41 Routing in a meh Conider the two-dimenional n n-meh Every node in thi meh ha a number (x, y) [n] 2 where x repreent it number in the x-dimenion and y repreent it number in the y-dimenion The x y routing trategy work a follow: Given a packet with ource-detination pair ((x 1, y 1 ), (x 2, y 2 )), firt route the packet along the x- dimenion from (x 1, y 1 ) to (x 2, y 1 ) and then along the y-dimenion from (x 2, y 1 ) to (x 2, y 2 ) Thi i certainly an obliviou routing trategy, ince the path of a packet only depend on it ource and detination How well can thi trategy now route arbitrary permutation routing problem? A permutation routing problem i a problem in which every node i the ource of exactly one ourcedetination pair and the detination of exactly one ource-detination pair and all demand are equal to 1 Thu, a permutation routing problem can be pecified by a permutation π : V V on the et of node V Theorem 42 The x y routing trategy can route arbitrary permutation in an n n-meh of unitcapacity edge with congetion at mot 2d and dilation at mot d, where d i the maximum ditance of a ource-detination pair in the permutation Proof We only prove the theorem for the wort cae, namely, that path can have a length of up to 2n The general cae will be an aignment Recall that in a permutation routing problem every node i the ource and detination of a demand of exactly 1 Thu, every x-dimenional line in the meh inject a total demand of at mot n, and every y-dimenional line in the meh ha to aborb a total demand of at mot n When uing the x y routing trategy, a total demand of at mot n can therefore overlap at an edge in x-direction, and a total demand of at mot n can overlap at an edge in y-direction Hence, the maximum fraction of each demand that can be atified o that we obtain a feaible flow i at leat 1/n, and therefore the congetion i at mot n Since the x y routing trategy ue hortet path and the diameter of the n n-meh i equal to 2(n 1), the dilation of the x y routing trategy can be at mot 2n 2

3 Thu, when uing the objective function behind the flow number, ie to minimize max{c(s), D(S)} over all feaible olution S, then the x y routing trategy i optimal up to a factor of 2 becaue the congetion never exceed the dilation by more than a factor of 2 42 The Borodin-Hopcroft lower bound The nice property of the x y routing trategy i that it jut ha to pecify one path for each ourcedetination pair Doe thi uffice to obtain good obliviou routing trategie for arbitrary network? The next theorem how that there i a limit to thi Theorem 43 ([1]) For every graph G of ize n and degree d and every obliviou routing trategy uing only a ingle path for every ource-detination pair, there i a permutation π in which a node i travered by at leat n/d path Proof Let [n] = {0,, n 1} repreent the et of all node in G and let P = {p i,j : i, j [n]} be any path ytem with exactly one path for every ource-detination pair A node i called a ource for node i wrt t if p,t move through i In Figure 2, for example, 3 i a ource for i wrt t G 2 1 i t 3 4 Figure 2: Illutration of the path to k In the following, we will contruct a permutation π with a high congetion Firt we how that for every node t there are many node that have many ource wrt t Let A(t, z) = {i [n] : i ha wrt t at leat z ource} be the et of all node that are contained in at leat z different path of P that lead to t Then the following lemma hold Lemma 44 For every t [n], A(t, z) n d z Proof For any fixed t [n], let L = {p,t : [n] and A(t, z)}, or in word, the number of path that tart outide of A(t, z), and let B L be the et of all direct neighbor of node in A(t, z) that are not in A(t, z) Obviouly, L = n A(t, z) Since the maximum degree of G i d, it further hold that B A(t, z) d Becaue B A(t, z) =, every node in B ha at mot z 1 path that lead to t Hence, 3

4 G B t A( t, z ) Figure 3: Illutration of A(t, z) and B B (z 1) L and therefore A(t, z) d (z 1) B (z 1) L A(t, z) d (z 1) n A(t, z) A(t, z) (d (z 1) + 1) n n A(t, z) d (z 1) + 1 n d z Now, let X(z) = {(i, t) : i, t [n] and i A(t, z)} = t [n](a(t, z) {t}) Then it hold X(z) = A(t, z) t [n] Lemma 44 n n d z = n2 d z For every node i let T i = {t : (i, t) X(z)} be the et of all detination for which at leat z path move through i Since T i = X(z) n2 d z i [n] but on the other hand there are only n et T i, there mut exit a node i with T i n Chooe z o d z that z and T i are of the ame ize Thi i the cae for z = n/d n or z = d z Thu, there mut be a node i for which there are at leat n/d detination that have at leat n/d path through i Simply chooing for all of thee detination one after the other any ource that ha not been choen by a previou detination reult in a partial permutation with an overlap of at leat n/d path at i Thu, for contant degree network with unit-capacity edge, the theorem implie that the congetion for routing a permutation can be a high a Θ( n) Wherea thi i fine for the 2-dimenional meh, for network with flow number O(log n) uch a the butterfly thi i unacceptably high, ince we know from Section 3 that every BMFP and therefore alo every permutation routing problem can be olved in the butterfly with congetion and dilation at mot O(log n) 4

5 43 Valiant Trick We aw in Section 42 that obliviou routing trategie with only a ingle path for each ourcedetination pair can have an extremely high congetion But what about multiple path? Let S be the bet poible olution for the multicommodity flow problem underlying the definition of F, ie F = max{c(s), D(S)} From Theorem 312 we know that by applying S twice we can olve every BMFP with congetion and dilation at mot 2F Thi work in a way that for every ource-detination pair (, t) we firt branch off the demand from to all other node in the ytem and afterward reunite it at the detination t Uing S twice till give an obliviou path ytem, but now we have many optional path for a flow In the cae of actually ending packet, thi boil down to the following trategy, which i a generalization of a well-known trick by Valiant [4]: For every packet with ource-detination pair (, t), chooe a random intermediate detination v [n] with probability c(v)/c(v ) and end the packet firt along a flow path in S from to v and then along a flow path in S from v to t (If there i more than one path from to v rep v to t in S, then there will be another random experiment for picking one of thee optional path baed on their flow value) For the imple cae that S only ha a ingle path for every ource-detination pair and c(v) i the ame for all node v, thi boil down to: For every packet with ource-detination pair (, t), chooe an intermediate detination v [n] uniformly at random and end the packet firt along the path in S from to v and then along the path in S from v to t To demontrate the effect of thi trick, let u conider routing in the d-dimenional hypercube Suppoe that for every ource-detination pair ( d 1,, 0 ), (t d 1,, t 0 ) {0, 1} d we ue the path that firt adjut 0 to t 0, then 1 to t 1, and o on, until all bit have been et to the detination value Let thee path form our path ytem S Becaue S ha exactly one path for every ource-detination pair, it follow from Theorem 43 that there mut be a permutation with at leat 2 d /d path travering a node and therefore a congetion of at leat ( 2 d /d)/d at an edge when uing S directly However, if we ue Valiant trick, we arrive at the following reult Theorem 45 Uing Valiant trick in the d-dimenional hypercube, any BMFP can be routed with congetion at mot 2d and dilation at mot 2d Proof Since the d-dimenional hypercube ha a diameter of d and S ue hortet path, the dilation of Valiant trick mut certainly be at mot 2d Thu, it remain to bound the congetion Firt, we determine the number of path S croing any edge of the hypercube Conider ome fixed edge e, and uppoe that e fixe dimenion i for ome i {0,, d 1}, ie it connect two node v and w in the hypercube that only differ at dimenion i When uing the bit adjutment trategy, then there are 2 i poible ource that can reach v before croing e in the direction of w, and 2 d i 1 poible detination can be reached after croing e Alo, there are 2 i poible ource that can reach w before croing e in the direction of v, and 2 d i 1 poible detination can be reached after croing e Hence, the number of all poible ource-detination pair whoe path cro e i equal to 2 2 i 2 d i 1 = 2 d (1) 5

6 Now, it follow from the definition of the pecial BMFP B that the demand of every ource-detination pair i equal to d d d 2 d = d 2 d (2) So the total demand croing an edge i equal to (1) (2) = d, but every edge can only upport a flow of 1 Hence, the maximum concurrent flow value f for S i 1/d Thi give a congetion of d for S and therefore a congetion of 2d for Valiant trick, becaue it double the overlap In general, it follow from Theorem 312: Theorem 46 For any network G with flow number F it hold: when uing Valiant trick on an optimal path collection for F, any BMFP can be routed in G with congetion and dilation at mot 2F In the cae of actually ending packet intead of flow, 2F i an upper bound on the expected congetion caued by the packet in the network 44 Obliviou routing for the meh reviited A we aw earlier, it i not really neceary to ue Valiant trick for the meh to be good for all BMFP in a ene that the congetion and dilation i alway cloe to the flow number However, if we are more picky here, then the x y routing trategy i till not really atifying, ince there are routing problem (other than BMFP) where the x y routing trategy would perform very poorly Imagine, for example, that we have a multicommodity flow problem for the n n-meh with ource-detination pair ((i, 0), (m, i)) for all i {0,, m 1}, where each pair i ha a demand of d i = m When uing the x y routing trategy, then all path for the pair would go though the edge {(m 1, 0), (m, 0)}, cauing a congetion of m 2 If, however, all pair would have ued a y x routing trategy, the congetion would have only been m (ee Figure 4) In the firt cae it would take Θ(m 2 ) time tep to end a flow of m for every ource-detination pair, wherea in the econd cae it would only take O(m) tep to do thi Hence, there would be a large difference between what the x y trategy can achieve and what can be achieved in the bet cae A imilar counterexample can alo be found for the y x trategy Alo, Valiant trick doe not help, becaue it would create a dilation of Θ(n), cauing a time of Ω(n) to deliver all flow, wherea for the cae that m = n thi can already be achieved in O( n) time tep So we need a different approach Fortunately, there i a better approach For implicity, we aume that we have an n n-meh of unit-capacity edge where n i a power of 2 For every ource-detination pair (, t), a ytem of flow path from to t i recurively contructed in the following way: Let M,t be the mallet poible 2 k 2 k -meh that ha in a corner and that contain t (if thi i not poible, M,t repreent the whole n n-meh) The flow path are contructed recurively a hown in Figure 5 Initially, all the flow tart at Then, it i evenly ditributed among all node in M 0 uing a mixed x y and y x routing trategy a ketched in Figure 5(b) That i, each node in M 0 receive a quarter of the flow, and the flow for the node at the oppoite corner of in M 0 come in equal part from the other two node in M 0 Afterward, the flow in M 0 i evenly ditributed among all node in M 4 Finally, the flow in M 4 i evenly ditributed among all node in M,t The ame i done from t Thu, the beginning and endpoint of the flow path from and t meet in M,t, reulting in a legal flow from to t 6

7 x-y routing trategy y-x routing trategy Figure 4: x y routing v y x routing on a meh It i clear that thi trategy i obliviou, but how good i it? For thi we need ome notation For any multicommodity flow problem P in the n n-meh let C P OPT be the bet poible congetion and and D P OPT be the bet poible dilation achievable for P (by poibly different olution) Theorem 47 For any multicommodity flow problem P our recurive routing cheme ha a congetion of O(C P OPT log n) and a dilation of O(D P OPT) Proof Suppoe that M,t i a 2 k 2 k -meh Then and t mut have a ditance of at leat 2 k 1 On the other ide, the longet poible path our routing trategy would contruct from ome node to ome node t in M,t i k 1 2 i=0 2(2 i 1) + 2(2 k 1) 4 2 k k = 6 2 k Thu, the dilation of our routing trategy i at mot a contant time the maximum ditance between a ource-detination pair in P and therefore bounded by O(D P OPT) Hence, it remain to bound the congetion Our aim will be to how that for every k, the congetion caued by all 2 k 2 k -mehe ued by ource-detination pair i at mot O(C P OPT) Since there are only log n different k, thi reult in a total congetion of O(C P OPT log n) So conider ome fixed k Given a ource-detination pair (, t) with demand d, let M be a 2 k 2 k -meh that i ued by (, t) to pread it demand to all node in M a hown in Figure 5(b) In thi cae, M ha a 2 k 1 2 k 1 -meh M in which the demand wa initially evenly ditributed among all of it node That i, every node in M had a demand of d/2 2(k 1) When uing a mixed x y and y x routing trategy for preading it out to M, every edge i croed by a demand of at mot 3 8 d 2 2(k 1) 2k 1 d 2 (3) k 7

8 M 6 M 8 M10 M 11 t M 9 M 7 3/8 1/8 M 2 M 3 1/8 M 4 M 5 1/4 remain 3/8 M 1 M 0 (a) (b) Figure 5: Recurive routing trategy from to t (a) illutrate the recurive decompoition into ubmehe and (b) illutrate the ditribution of flow from the haded ub-meh to the three other ubmehe in it next higher meh Now conider an edge e that i contained in m different 2 k 2 k -mehe M 1,, M m that belong to ource-detination pair ( 1, t 1 ),, ( m, t m ) with demand d 1,, d m Then it follow from (3) that e i croed by a total demand of at mot 2 k m i=1 d i On the other hand, one can draw a 2 k+1 2 k+1 - meh M around e that contain all ub-mehe M i Suppoe that of the total demand d = m i=1 d i a demand of d i routed completely inide of M from ource to detination, and a demand of d i leaving or entering M at ome point Since the ditance between i and t i mut be at leat 2 k 1 for every i, the average amount of the demand d croing an edge in M mut be at leat 2 k 1 d d = 2 22(k+1) 2 k+4 Furthermore, the average amount of demand croing an edge in (M, M) mut be at leat d d = 4 2k+1 2 k+3 Since either d or d mut be at leat d/2, every routing trategy mut therefore have an edge that i croed by a total demand of Ω(d/2 k ), ie COPT P = Ω(d/2 k ) On the other hand, we calculated that edge e i croed by a demand of O(d/2 k ) Hence, the congetion caued by our recurive cheme i O(COPT), P which complete the proof Thi reult i optimal, ince it i known that for every obliviou routing trategy on the n n-meh there i a routing problem P for which the trategy ha a congetion of Ω(C P OPT log n) [2] One may ak whether our upper bound for obliviou routing on a meh can alo be extended to other network Surpriingly, Räcke recently howed that when only looking at the congetion thi i poible: Theorem 48 ([3]) For every network with non-negative capacitie there i an obliviou routing trategy that achieve for every multicommodity flow problem P a congetion of O(C P OPT polylog(n)) 8

9 Hence, obliviou routing i a urpriingly powerful concept Reference [1] A Borodin and J Hopcroft Routing, merging, and orting on parallel model of computation Journal of Computer and Sytem Science, 30: , 1985 [2] B Magg, F M auf der Heide, B Vöcking, and M Wetermann Exploiting locality for network of limited bandwidth In Proc of the 38th IEEE Symp on Foundation of Computer Science (FOCS), page , 1997 [3] H Räcke Minimizing congetion in general network In Proc of the 43rd IEEE Symp on Foundation of Computer Science (FOCS), 2002 [4] L Valiant A cheme for fat parallel communication SIAM Journal on Computing, 2(11): ,