Shortest Path Routing in Arbitrary Networks

Transcription

1 Journal of Algorithm, Vol 31(1), 1999 Shortet Path Routing in Arbitrary Network Friedhelm Meyer auf der Heide and Berthold Vöcking Department of Mathematic and Computer Science and Heinz Nixdorf Intitute, Univerity of Paderborn Paderborn, Germany Abtract We introduce an on-line protocol which route any et of N packet along hortet path with congetion C and dilation D through an arbitrary network in O(C +D +log N) tep, with high probability. Thi time bound i optimal up to the additive log N, and it ha previouly only been reached for bounded-degree leveled network. Further, we how that the above bound hold alo for random routing problem with C denoting the maximum expected congetion over all link. Baed on thi reult, we give application for random routing in Cayley network, general node ymmetric network, edge ymmetric network, and de Bruijn network. Finally, we examine the problem ariing when our approach i applied to routing along non-hortet path, determinitic routing, or routing with bounded buffer. 1 Introduction Communication among the proceor of a parallel computer uually require a large portion of runtime of a parallel algorithm. Thee computer are often realized a relatively pare network of a large number of proceor uch that each proceor can directly communicate with a few neighbor only. Thu, mot of the communication mut proceed through intermediate proceor. One of the baic problem in thi context i to route imultaneouly many meage packet through the network. Wherea mot previou theoretical reearch on packet routing concentrate on pecial clae of network a, e.g., leveled network, we are intereted in univeral routing algorithm that can be ued in any network. Aume that we are given an arbitrary proceor network. A packet routing problem of ize N on thi network i defined by a et of N packet each of which ha a ource and a detination node. The goal i to route each packet from it ource to it detination. A routing problem in which every node i the ource of h packet and the detination of h packet i called an h-to-h-routing problem, and a routing problem in which every node end h packet to random detination choen independently and uniformly from the et of node i called a random h-routing problem. A preliminary verion wa preented at the 12th STACS, 1995, ee [9]. {fmadh,voecking}@uni-paderborn.de. Supported in part by DFG-Sonderforchungbereich 376 Maive Parallelität: Algorithmen, Entwurfmethoden, Anwendungen, by EU ESPRIT Long Term Reearch Project (ALCOM-IT), and by DFG Leibniz Grant Me872/6-1. 1

2 Our invetigation are baed on the tore-and-forward model. In thi model, the packet are viewed a atomic object, and it i aumed that the routing proceed in ynchronized tep uch that a packet can cro at mot one link in a tep. In the multi-port model, each link can forward at mot one packet a tep, wherea in the ingle-port model, each proceor can forward at mot one packet a tep. We aume the multi-port model, ince it ha become mot common for tore-and-forward routing in recent year. But note that all our technique and reult can be eaily adapted to the ingle port model. At the beginning of the firt tep, each packet i tored in an initial buffer at it ource node. During the routing, it move forward tep by tep, and at each link on it path, it i tored in a link buffer at the end of the link until it i allowed to move forward along the next link. Upon travering the lat link on it path, the packet i removed from the link buffer and placed in a final buffer at it detination. In the following, any bound on the buffer ize required by a routing protocol refer only to the link buffer, becaue the ize of the initial and final buffer are determined by the particular routing problem. The path travered by a packet from it ource to it detination i called the routing path of the packet. A routing protocol decribe the rule for moving the packet to their detination. We aim to contruct routing protocol that minimize the total number of tep required to deliver all packet. We break thi problem into two part: the problem of electing the routing path and the problem of cheduling the movement of the packet along thee path. The path election problem i defined a follow. We are given the ource and the detination of the packet, and we have to determine the routing path. Thi can be done by a path ytem W which i a et of path through the network. It include a path w(u, v) = (u v) for every pair u and v of node. If all path in W are hortet path, then we call W a hortet path ytem. For every packet with ource u and detination v, we chooe the path w(u, v) a it routing path. Intead of determining the routing path by a path ytem beforehand, the routing path can be elected during the routing, i.e., each node chooe the link for tranmitting a elected packet jut before the packet i paed on. For intance, the next link on a packet routing path can be choen uniformly and randomly from the et of outgoing link which belong to a hortet path to the detination of the packet. But note that we alway aume that the path election proce i completely independent from the cheduling proce. Thu, the routing path can be viewed a input for the cheduling proce. An obliviou routing problem i defined by a et of N packet with preet routing path. The term obliviou mean that the packet are not allowed to leave their preet routing path. An obliviou routing problem i called hortet path routing problem if all routing path are hortet path. Since the path have already been pecified, a routing protocol for obliviou routing problem, which we call obliviou routing protocol, ha only to determine which packet are allowed to move forward in a tep and which have to wait. If we allow a global controller to precompute thi chedule, we talk about off-line routing. If the chedule i produced while the packet are routed through the network, thi i called on-line routing. We are intereted in the contruction of on-line routing protocol. The following parameter greatly influence the routing time for obliviou routing problem: the congetion C, i.e., the maximum number of routing path that pa through the ame link, and the dilation D, i.e., the maximum length of the routing path in the problem. Clearly, max{c, D} = Ω(C + D) i a lower bound on the routing time for any protocol on any obliviou routing problem with congetion C and dilation D, becaue at leat one link 2

3 mut be travered by C packet, and at leat one packet ha to travere D link. An obliviou protocol i aid to be greedy if packet only have to wait at a link becaue they are delayed by another packet which move along thi link or becaue the link buffer at the end of the link i full. C D i an upper bound on the routing time of greedy protocol on network with unbounded buffer, becaue each packet ha to wait at mot C 1 tep on every link of it routing path. An obliviou routing protocol i aid to be nonpredictive if contention i reolved by a determinitic algorithm that i baed only on the hitory of the contending packet travel through the network and on information carried with the packet that i independent of their detination [4]. For intance, the firt-in firt-out protocol i nonpredictive. Finally, we call an obliviou routing protocol trongly on-line if we aume that the proceor do not ue any information about the parameter of a routing problem, i.e., the congetion, the dilation, or the ize of the problem. In the following, we repreent the underlying proceor network by a digraph G = (V, E), where V i the et of node or proceor, and E V V i the et of directed edge or link. Of coure, any network decription which i baed on undirected graph can be repreented in the digraph model jut by replacing each undirected edge by two directed edge in oppoite direction. We denote the diameter of the network by diam(g). 1.1 Known Reult All reult in thi ection relate the routing time and buffer ize required by obliviou routing protocol to the ize, the congetion, and the dilation of the underlying obliviou routing problem. The ize i denoted by N, the congetion by C, and the dilation by D. Leighton, Magg, and Rao [6] how that any obliviou routing problem can be routed in time O(C + D) with contant-ize link buffer, thereby achieving the naive lower bound. Their proof i baed on the Lováz Local Lemma and how only the exitence of the optimal chedule. In [5], Leigthon, Magg, and Richa preent an algorithm for computing thi chedule. But ince the runtime of thi algorithm i polynomial in the number of packet and link, it can not be applied to turn the above off-line protocol into an efficient on-line protocol. Rabani and Tardo [11] modify the off-line protocol of Leighton, Magg, and Rao uch that they can replace the Lováz Local Lemma by a Chernoff bound in the analyi. Thi modification allow to calculate the chedule in a ditributed way and on-line. The protocol guarantee routing time O(C)+(log N) O(log N) D+polylog(N), w.h.p. 1. Recently, Otrovky and Rabani [10] have improved on thi reult. They achieve routing time O(C + D + log 1+ɛ N), w.h.p., for arbitrary ɛ > 0. Both protocol are uitable for arbitrary imple routing path, that i path without cycle. They require buffer of ize C. Beide their off-line reult, Leighton, Magg, and Rao preent in [6] a imple on-line cheduling protocol which complete the routing in time O(C+D log(dn)), w.h.p., and require buffer of ize log(dn). A imilar routing time but with much maller buffer ize i achieved in [3]: a protocol i given requiring routing time O(C + D log N), w.h.p., uing buffer of ize log D. Better reult are known for pecial clae of network. For intance, Ranade [12] propoe a probabilitic on-line routing protocol for butterfly network. The proof i baed on the delay equence technique developed by Aleliuna [1] and Upfal [14]. The protocol can be eaily extended to the cla of bounded-degree leveled network ([4], [8]). In a leveled network, the node can be partitioned into level 0,..., L uch that each link in the network lead from ome node on level i to ome node on level i + 1, for 0 i L 1. Motly, it i aumed that 1 Throughout thi paper, w.h.p. (with high probability) mean: with probability at leat 1 N α for any fixed contant α with N denoting the number of packet. 3

4 packet are routed only from level 0 to level L. Ranade protocol complete the routing in time O(C + L + log N), w.h.p., uing buffer of contant ize. Note that all of the preceding protocol delay packet even if the next edge on their path i free. Thu, none of them are greedy. Leighton [4] introduce a imple probabilitic greedy protocol for butterfly network. It i called the random-rank protocol. Thi protocol i a implified verion of Ranade protocol. Initially, each packet i aigned a random rank. The rank are ued to determine which packet move forward and which have to wait in a tep. Applied to leveled network, the protocol achieve aymptotically the ame performance a Ranade protocol, but it require buffer of ize C. A detailed urvey about all thee routing protocol, including alo mot of the reult preented in thi work, i given in a book of Scheideler [13]. 1.2 Overview New Reult In Section 2, we introduce a new probabilitic on-line routing protocol which we call growingrank protocol. We how that the growing-rank protocol route any hortet path routing problem of ize N with congetion C and dilation D in O(C + D + log N) tep, w.h.p. Thu, we obtain the ame bound for arbitrary network a previouly known only for bounded-degree leveled network. Our protocol i greedy and very imple. The main difference to Leighton random-rank protocol i that the packet rank are increaed whenever the packet move forward. We preent three verion of the growing-rank protocol. The firt require that etimation of C, D, and N are ditributed among the proceor, the econd require only that an upper bound on N i known by all proceor, and the third make no ue of any of thee parameter. Therefore, the above reult i trongly on-line. The drawback of the growing-rank protocol i that it require hortet path. Thi condition can be lightly weakened: A collection of path P on a network G = (V, E) i aid to be hortcutfree, if there i a ubnetwork G = (V, E ) with E E uch that the path in P are hortet path in G. Of coure, every et of hortet path i hortcut-free. Further, we invetigate the behavior of the growing-rank protocol on random routing problem. We how that the above time bound hold even if C denote the maximum expected congetion over all link. Thi value can be calculated very eaily and exactly for many randomized path election trategie. Thi i illutrated by everal application in Section 3. We tart with calculating the maximum expected congetion for random routing problem on Cayley network. Thi cla include many important tandard network, e.g., all tori, the cube-connected cycle, and the butterfly network. We give a imple cheme for the contruction of ymmetric hortet path ytem in thee network. If the packet of a random h-routing problem on a network G are ent along the path in thi ytem, then the maximum expected congetion i at mot h diam(g). Hence, the growing-rank protocol route random h-routing problem on any Cayley network G of ize n in time O(h diam(g) + log n), w.h.p. Further, we invetigate node and edge ymmetric network. Intuitively, a network i node (edge) ymmetric, if it look the ame viewed from any node (edge) of the network. For intance, every Cayley network i node ymmetric, and all equal-ided tori are edge ymmetric. We give a very imple randomized path election trategy which generate the routing path to the detination during the routing. Thi trategy achieve optimal maximum expected congetion for random routing problem on network in both clae. Given any node ymmetric network G of ize n, we how that the maximum expected congetion for random h-routing problem i at mot h diam(g). Thi implie routing time O(h diam(g) + log n), w.h.p. Given any 4

5 edge ymmetric network G of ize n and degree, we how that the maximum expected congetion for random h-routing problem i at mot h/ diam(g). Thi implie routing time O((h/ + 1) diam(g) + log n), w.h.p. Our lat application i a imple routing cheme for de Bruijn network. We how that the maximum expected congetion i at mot O(h log n) if the packet of a random h-routing problem are routed along hortet path in the n-node de Bruijn network. Thi give optimal routing time O(h log n), w.h.p. By applying Valiant paradigm firt routing to a random detination [15], all application reult for random h-routing problem hold alo for arbitrary h-to-h-routing problem. In Section 4, we examine the limit of our approach to deign efficient obliviou routing protocol and ak: I the retriction to hortet routing path neceary? Do we need randomization? What happen if the buffer ize i bounded? We anwer thee quetion by three example. The firt example how that the growing-rank protocol perform poorly on ome routing problem with non-hortet path. For intance, we decribe an obliviou routing problem of ize N with congetion C = log N/ log log N and dilation D = log N for which the expected routing time of the growing-rank protocol i Ω(C D). The econd example illutrate that randomization i neceary. We how that given any nonpredictive protocol, there i a hortet path routing problem with congetion C and dilation D that take time Θ(C D). The reult hold for any C and D. (A imilar example with time bound Ω(C D/ log C) can be found in [7].) Interetingly, the underlying network i the butterfly network. Note that Ranade protocol, Leighton random-rank protocol, and the growing-rank protocol are nonpredictive for any fixed choice of the initial rank. A a conequence, all of the three protocol perform poorly for routing on the butterfly in a determinitic etting. The lat example illutrate that routing with bounded buffer i a much more challenging tak than routing with unbounded buffer, i.e., buffer of ize C. In particular, the example how that, in cae of bounded buffer, a packet p can be delayed by packet whoe routing path do not overlap with the routing path of p. Thi can lead to a routing time much wore than C D tep which i the upper bound for greedy routing with bounded buffer. Ranade protocol [12] ue ghot packet to deal with thi problem. But thi technique i uitable only for leveled network. Another difficulty arie if we conider non-leveled network: the deadlock problem. Suppoe there are m link e 0,..., e m 1 with full packet buffer, and every link e i hold only packet that wait for moving forward along e (i+1) mod m, for 0 i m 1. Then all link are blocked, i.e., a deadlock occur. We believe that avoiding deadlock i the major problem to be olved in order to generalize our reult to network with bounded buffer. 2 The Growing-Rank Protocol Now we introduce the growing-rank protocol. Suppoe we are given a hortet path routing problem of dilation D, congetion C, and ize N on an arbitrary network G. Let Q e denote the et of packet that wait for moving forward over an outgoing link e in a tep. Becaue the routing path have already been determined, the protocol only ha to pecify which of the packet in Q e i allowed to move forward and which packet have to wait. The protocol 5

6 forward packet whenever poible, i.e., if Q e i not empty, then one of the packet in Q e i moved forward along e. The prioritie among the packet are determined by random rank. Suppoe R and m := R/D are uitably large integer. (The exact value of R will be pecified later.) Initially, each packet i aigned an integer rank choen randomly, independently, and uniformly from the et {0, 1,..., R 1}. Whenever a packet travere a link, it rank i increaed by m. If two or more packet are contending to move forward along a link, then one of thoe with minimum rank i choen. Thu, for each outgoing link e with Q e 1, a tep look like thi: 1. chooe a packet p Q e with minimum rank, 2. increae the rank of p by m, and 3. move p forward along e. In order to break tie among packet with ame rank, we aume that each packet p ha a unique ident-number denoted by id(p). If there are everal packet with the ame minimum rank, then one with mallet ident-number i choen. Thee ident-number can be eaily generated. For example, the ith packet tarting at the jth proceor get the ident-number i n + j with n denoting the total number of proceor. In the following, we denote the rank of p while waiting for moving forward along link e by rank e (p). Further, we define the ident-rank of p at e a id-rank e (p) := rank e (p) + id(p)/(max(id) + 1) with max(id) denoting the maximum ident-number. Note that, at each link, the ident-rank of all packet are ditinct. The protocol enure that, whenever a packet p delay a packet p at a link e, then id-rank e (p) < id-rank e (p ). 2.1 Analyi of the Protocol We will how that the growing-rank protocol complete the routing of any hortet path routing problem of ize N with congetion C and dilation D in O(C + D + log N) tep, w.h.p. Our analyi i baed on a delay equence argument imilar to that in [4], [8], and [12]. Definition 2.1 ((, l, r)-delay equence) An (, l, r)-delay equence conit of delay packet p 1,... p ; not necearily ditinct link e 1,..., e uch that e 1 i the lat link on the routing path of p 1, and for 2 i, e i i a link on the routing path of p i 1 and p i ; 1 integer l 1,..., l 1 0 with 1 i=1 l i l uch that for 1 i 1, l i i the number of link on the routing path of p i from e i+1 to e i (excluding e i+1, and including e i ); and integer r 1,..., r with 0 r r 1... r 1 r 1. We call the length of the delay equence, and we ay a delay equence i active, if rank ei (p i ) = r i for 1 i. Lemma 2.2 Suppoe the routing take T R/m + D or more tep. Then a (T R/m D, R/m + D, R + D m)-delay equence i active. 6

7 Proof. We give a contruction cheme for a delay equence. Let p 1 be a packet that move forward in tep T or later along the lat link on it routing path. Call thi link e 1. We follow p 1 routing path backward to the lat link on thi path where it wa delayed. Call thi link e 2 and the packet that caued the delay p 2. We now follow the path of p 2 backward until we reach a link e 3 at which p 2 wa forced to wait, becaue the packet p 3 wa preferred. We change the packet again and follow the path of p 3 backward. We can continue thi contruction until we reach a packet p which wa not delayed in a tep before. Thu, we have determined the delay packet and the link of a delay equence of length. For 1 i, we et r i := rank ei (p i ). Since the growing-rank protocol prefer packet with maller rank and ince the maximum rank occurring during the routing i maller than R := R + D m, we have 0 r r 1... r 1 R. The path from the ource of p to the detination of p 1 recorded by the above proce in revered order i called delay path. It conit of contiguou part of the delay packet routing path. We define the l i to be the length of thee part a decribed in the definition of the delay equence. Let l denote the number of link on the delay path. Since the rank in our equence are increaed by m at each of thee link, it follow l m R. Conequently, we have 1 i=1 l i l R /m R/m + D. Our contruction cover up at leat T tep and conit of l move and delay. Conequently, we have T l T R /m = T R/m D. Thu, if we top the above contruction at packet p T R/m D, then we have built an active (T R/m D, R/m + D, R + D m)-delay equence. Lemma 2.3 If the routing path of the packet are hortet path, then the delay packet in an active delay equence are pairwie ditinct. Proof. Suppoe, in contrat to our claim, that there i ome packet p appearing twice in the delay equence. Then there are i and j with 1 i < j and p = p i = p j. Thu, the routing path of p croe the delay path at the colliion link e j and e i in that order. Let κ denote the number of link on the routing path of p from e j to e i. Then the rank of p i increaed κ time by m on thi part of the routing path, and conequently, id-rank ei (p) = id-rank ej (p) + κ m. (1) On the other hand, each packet p k with i + 1 k j delay the packet p k 1 at link e k. Thu, id-rank e k (p k 1 ) > id-rank e k (p k ). Further, the rank are increaed by m on every link on the delay path between e j and e i. The number of thee link i j 1 k=i l k. Thi give j 1 id-rank ei (p) > id-rank ej (p) + l k m k=i id-rank ej (p) + κ m. (2) Note that j 1 k=i l k κ, becaue the routing path of p i a hortet path. Clearly, (2) contradict (1). Conequently, there i no packet that appear twice in the delay equence. Lemma 2.4 The number of different (, l, r)-delay equence i at mot ( ) 2eC ( + r) d(, l, r) := N 2 l. 7

8 Proof. We count the number of poible choice for each component: There are N poibilitie to chooe p 1. Of coure, thi fixe e 1 a well. Further, there are ( ) ( ( 1)+l 1 +l ) way to chooe the li, becaue 1 i=1 l i l. Now uppoe p i 1, e i 1, and l i 1 for 2 i are fixed. Then e i i that link on the routing path of p i 1 which ha ditance l i 1 to e i 1. Thu, e i i fixed a well, and hence, we have at mot C poibilitie to chooe p i. Therefore, the number of poibilitie to fix p 2,..., p and e 2,..., e i at mot C 1 C. Finally, there are ( +r ) poibilitie to chooe the ri uch that 0 r... r 1 r 1. Altogether, we find that the number of (, l, r)-delay equence i at mot ( ) ( ) + l + r N C. Applying the inequalitie ( ) a b 2 a and ( ( a b) ea ) b b complete the proof. Theorem 2.5 Suppoe we are given a hortet path routing problem of ize N with congetion C and dilation D on an arbitrary network G. Then the growing-rank protocol complete the routing in O(C + D + log N) tep, w.h.p. Proof. Lemma 2.2 and Lemma 2.3 how that the probability prob(t ) that the routing take T = + R/m + D or more tep i bounded by the probability that an (, R/m + D, R + D m)- delay equence with ditinct delay packet i active. The probability that a fixed delay equence with ditinct packet i active i R becaue the rank of all packet have to match the rank in the equence. Combining thi with the bound on the number of delay equence in Lemma 2.4 give prob(t ) d (, Rm ) + D, R + D m R ( ) 2eC ( + R + D m) N 2 R/m+D R. Auming R give ( ) 2eC (2R + D m) prob(t ) N 2 R/m+D R ( ) 2eC (2 + D m/r) N 2 R/m+D which i at mot N α for = max{4ec (2 + D m/r), R/m + D + (α + 1) log N}. Hence, the routing take T = + R m + D = max{4ec (2 + D m R ), R m + D + (α + 1) log N} + R m + D (3) or more tep with probability N α. Finally, applying m = R/D yield that the routing i completed in O(C + D + log N) tep, w.h.p. 8

9 2.2 Becoming Strongly On-Line The drawback of the protocol preented above i that each proceor ha to know etimation of the congetion C, the dilation D, and the ize of the routing problem N. Thi i becaue we have aumed that the range of the rank i ufficiently large, i.e., R max{4ec (2 + D m/r), 2D + (α + 1) log N}, and that the packet rank are increaed by m = R/D whenever the packet move forward. It i eay to check that the reult on the routing time hold for every choice of R and m that atify R = Ω(C + D + log N), m = Ω(R/(C + D + log N)), and m = O(R/D). In particular, it eem to be difficult to compute the congetion of the routing problem. Fortunately, we need only an upper bound on thi value, e.g., N D or N diam(g). Of coure, D and N can be computed and ditributed among the proceor in O(diam(G)) tep. Alternatively, we can ue upper bound on D and N intead of exact value. But note that wherea the quality of the upper bound on N influence only the range of the rank, the quality of the dilation bound influence the routing time. I.e., if we chooe m = R/D with D D then we get routing time O(C + D + log N). For intance, bounding the dilation by the diameter diam(g) give routing time O(C + diam(g) + log N), even if D << diam(g). The following variation of the protocol achieve routing time O(C + D + log N) without auming that the dilation or the congetion are known by the proceor. The proceor only have to know upper bound on the diameter of the network and the ize of the routing problem. For k 0, define time interval k to begin at tep 2 k and to end at tep 2 k+1 1. Hence, each interval k ha length 2 k. Define k to be the mallet integer atifying 2 k +1 1 N diam(g). Then the routing i completed at the end of interval k urely. Chooe R = 2 k 2. The value of m varie during the routing, i.e., the rank of forwarded packet are increaed by m k := R/2 k 2 in interval k. Note that m k i an integer for every k k. Now let k denote the mallet integer atifying 2 k 1 max{12ec, D +(α +1) log N}+D, and uppoe ome packet have not reached their detination at the beginning of interval k. We want to etimate the probability that the routing i not completed during interval k. Analogouly to the argument in the proof of Theorem 2.5, the probability that the routing i not completed during the next T (3) = max{4ec (2 + D mk R ), R m k m k =R/2 k 2 + D + (α + 1) log N} + R m k max{4ec (2 + D/2 k 2 ), D + (α + 1) log N} + D + 2 k 1 max{12ec, D + (α + 1) log N} + D + 2 k 1 2 k k 1 = 2 k + D tep i at mot N α. Conequently, all packet reach their detination in interval k, w.h.p., and thu, the routing take at mot 2 k +1 1 = O(C + D + log N) tep, w.h.p. Now we aume that the packet do not know any information about the routing problem, neither the congetion, the dilation, nor the ize of the routing problem, nor any etimation of thee value. Then the following variation of the growing-rank protocol achieve routing time O(C + D + log N), w.h.p. A above, define interval k to begin at tep 2 k and to end at tep 2 k+1 1, for k 0. At the beginning of interval k, each packet i aigned a new random rank from the interval R k := 2 k. (Note that it i ufficient to append a new random bit at the leat ignificant poition to each rank intead of aigning completely new rank which implifie the protocol lightly.) The rank of the packet are increaed by m := 4 when they are forwarded. 9

10 Let k denote the mallet integer atifying 2 k 1 max{12ec, D + (α + 1) log N} + D, and uppoe the routing i not completed at the beginning of interval k. Then the probability that ome packet have not reached their detination in T (3) = max{4ec (2 + D R k /m=2 k 2 m R k ), R k m + D + (α + 1) log N} + R k m + D max{4ec (2 + D/2 k 2 ), D + (α + 1) log N} + D + 2 k 1 max{12ec, D + (α + 1) log N} + D + 2 k 1 2 k k 1 = 2 k tep i at mot N α. Therefore, the routing take at mot 2 k +1 1 = O(C + D + log N) tep, w.h.p, which give the following corollary. Corollary 2.6 Any hortet path routing problem with congetion C, dilation D, and ize N can be routed trongly on-line in time O(C + D + log N), w.h.p. 2.3 Analyi for Random Routing Problem Suppoe N packet hould be routed along randomly choen hortet routing path from their ource node to a random detination in a network G = (V, E). Let P denote the et of packet, and M the et of all hortet path in G. We model the election of the random detination for a packet p P and the election of the routing path to thi detination together, i.e., by a random choice of a path m from M. For m M and p P, we denote the probability that m i the randomly choen routing path for p by prob(p, m). Note that we do not demand that the routing path are choen uniformly from the et of all path tarting at the ource of p. Further, we do not demand that all node have the ame probability to become the random detination of p. However, in all of our application they have. The only retriction we place on the path election proce i that the routing path for a packet i choen independently from the routing path of other packet and from the cheduling proce. For any N-tuple of path M M p we ay M decribe the reult of the random path election, if M = p P m p with m p denoting the randomly choen routing path for packet p. Finally, the probability that M decribe the reult of the random path election i denoted by prob(m). Then for any M = p P m p M p, we have prob(m) = p P prob(p, m p) becaue the routing path are choen independently from each other. The following example how how random routing problem in which the routing path are determined by a hortet path ytem W, which include exactly one path w(u, v) = (u v) for every pair u and v of node, can be repreented in the above model. We aume that the random detination for each packet p P i choen randomly, independently, and uniformly from the et V of node. Let ource(p) denote the ource node of p and det(p) the randomly elected detination of p. Then we chooe the path w(ource(p), det(p)) W a p routing path. Thi trategy can be eaily expreed in term of the above model by imply pecifying the probabilitie that a path m M i the routing path of a packet p P, i.e., we et prob(p, m) := { 1 V if m W ource(p) 0 otherwie with W v W denoting the et of path tarting at proceor v V. 10

11 The following theorem bound the routing time of the growing-rank protocol on random routing problem a decribed above. We aume that the packet rank are increaed by m := R/diam(G) when the packet move forward. The given routing time depend on the total number of packet N, the diameter diam(g), and the maximum expected congetion C exp := max{e(c e ) e E} with E(C e ) denoting the expected number of packet travering link e. We will ee later that C exp can be calculated very eaily and exactly for random h-routing problem on everal clae of network. Theorem 2.7 Suppoe we are given an arbitrary network G in which N packet hould be routed along random routing path. Suppoe all routing path are hortet path which are choen independently from each other. Let C exp denote the maximum expected congetion. Then the growing-rank protocol complete the routing in O(C exp + diam(g) + log N) tep, w.h.p. Proof. Becaue of Lemma 2.2 and 2.3, we can bound the probability prob(t ) that the routing take T = + R/m + diam(g) or more tep by the probability that an (, R/m + diam(g), R + diam(g) m)-delay equence with ditinct packet i active. For any N-tuple of path M M P let d(, l, r, M) denote the number of poible (, l, r)-delay equence with ditinct delay packet under the aumption that M decribe the reult of the random path election. Then prob(t ) prob(m) d (, Rm ) + diam(g), R + diam(g) m, M R. (4) M M p Note that d(, l, r) := M M p prob(m) d(, l, r, M) i equal to the expected number of poible (, l, r)-delay equence for randomly choen M. We can count thi number a follow. There are at mot N (+l ) ( +r ) way to chooe p1, e 1, the l i, and the r i. Now uppoe p i 1, e i 1, and l i 1 are fixed, for 2 i. Then e i i that link on the routing path of p i 1 which ha ditance l i 1 to e i 1. Thu, e i i fixed a well. What i the expected number of candidate for p i under the aumption that e i and p 1,..., p i 1 are already fixed? - The routing path of p i mut travere e i, and p i mut be ditinct from p 1,..., p i 1. Let M ei M be the et of all path in M that cro e i. Then, the expected number of poibilitie to chooe p i i at mot prob(p, m) prob(p, m) = E(C ei ) C exp. m M ei p P m M ei p P\{p 1,...,p i 1} Becaue thi bound i independent from the choice for the delay packet p 1,... p i 1, the expected number of choice for p 2,..., p and e 2,..., e i at mot Cexp 1. Putting all the piece together, we get ( ) ( ) + l + r d(, l, r) N Cexp 1 ( ) N 2 l 2eCexp ( + r). Applying thi to equation (4) yield ( ) prob(t ) N 2 R/m+diam(G) 2eCexp ( + R + diam(g) m) R 11

12 m=r/diam(g) N 2 2 diam(g) ( ) 2eCexp ( + 2R) R. We et := max{12ec exp, 2 diam(g)+(α+1) log N}, for contant α. Thu T = +2 diam(g) = O(C exp + diam(g)) + (α + 1) log N. Further, we chooe R. Then prob(t ) N 2 2 diam(g) Thi complete the proof of Theorem Application ( ) (α+1) log N+2 diam(g) 2eCexp 3R = N α. 12eC exp R Now we give everal application for the growing-rank protocol. We invetigate random routing problem on node ymmetric network, edge ymmetric network, and de Bruijn network. All reult in thi ection are conequence of Theorem Node Symmetric Network An automorphim of a network G = (V, E) i a permutation φ : V V with the property that (u, v) E (φ(u), φ(v)) E. The automorphim of G form an algebraic group under the operation of compoition. Thi group i denoted by Aut(G). An automorphim group U Aut(G) i aid to be tranitive on G if, given any two node u and v, there i an automorphim φ U uch that φ(u) = v, and a network G i called node ymmetric if Aut(G) i tranitive on it. Intuitively, a node ymmetric network look the ame, if viewed from any node of the network. The cla of Cayley network i an important ubcla of node ymmetric network. Many tandard network belong to thi cla, e.g., all tori, the cube-connected-cycle, and the wrapped butterfly network. Cayley network are defined a follow. Let Γ be a finite algebraic group with identity 1, and uppoe Σ i a et of generator of Γ with 1 / Σ. Then the Cayley network G Γ,Σ = (V, E) i defined by V = Γ and E = {(a, b) a 1 b Σ}. Figure 1 how an example for a Cayley network. Suppoe W i a path ytem on a network G = (V, E) that include a hortet path w(u, v) = (u v) for every pair u and v of node. We call W ymmetric if given any two node u and v there i a permutation ψ : V V uch that for every path (w 0 w 1 w l ) W with w i = u there i a path (ψ(w 0 ) ψ(w 1 ) ψ(w l )) W with ψ(w i ) = v, for 0 i l. Roughly peaking, a ymmetric path ytem ha the property that it look the ame viewed from any node of the network. Lemma 3.1 For every Cayley network, there i a ymmetric hortet path ytem. Proof. Let G Γ,Σ = (V, E) be a Cayley network. Then there i a tranitive automorphim group U of ize V [2]. We denote by φ v u the automorphim of U which map the node u onto the node v, for u, v V. Thu, U = {φ v u v V }, for any u V. Suppoe w = (w 0 w 1 w l ) i a path in G and φ i an automorphim of G. Then we define φ(w) := (φ(w 0 ) φ(w 1 ) φ(w l )). Since φ i an automorphim, φ(w) i a hortet path in G if and only if w i a hortet path in G. We contruct a ymmetric hortet path ytem in two tep. (For implicity of notation, we aume V = {0, 1..., n 1}.) 12

13 Figure 1: The Cayley network G Γ,Σ with Γ = ( Z 8, +) and Σ = {1, 2}. Note that the identity in Γ i 0 rather than 1 a Γ i an additive group. Step 1: chooe arbitrarily a hortet path w(0, v) from the node 0 to every node v V. Step 2: for every u V \ {0} and every v V, define the path w(u, v) from u to v by w(u, v) := φ u 0 (w(0, φ 0 u(v))). In the firt tep we have choen n prototype path (including the trivial one from 0 to 0). In the econd tep we have made n 1 copie of each prototype path. Thu, every automorphim of U, except for the identity, ha been ued once for copying each prototype path. Let u and v be two node of G. We have to how that there i a permutation ψ which map every path w = (w 0 w 1 w l ) W with u = w i onto a path w = (w 0 w 1 w l ) W with v = w i for 0 i l. For ψ we chooe the automorphim φv u U. Clearly, φv u map u onto v. Thu, it remain only to prove that w = φ v u(w) i a member of the path ytem W. From the contruction cheme, we know that w i a copy of a prototype path w or a prototype path w itelf. We claim Thi can be proved a follow: w = φ v (a) u (w) = φ w 0 w 0 (w) (b) = φ w 0 0 φ 0 w 0 (w) (c) = φ w 0 0 (w ). a) There i exactly one automorphim in U that map w 0 onto w 0. Conequently, φv u = φw 0 w 0. b) Since U i a group, φ w 0 0 φ0 w 0 i an element of U, and ince there i only one automorphim in U that map w 0 onto w 0, it follow φw 0 0 φ 0 w 0 = φ w 0 w 0 c) It i w = φ w0 0 (w ), and conequently, w = φ 0 w 0 (w). Hence, the automorphim φ w 0 0 U map w onto w. If φ w 0 0 i the identity, then w i generated in Step 1. Otherwie, w i generated a a copy of w in Step 2. 13

14 Theorem 3.2 Let G = (V, E) be a Cayley network with ymmetric hortet path ytem W. Suppoe each proceor end h packet to randomly and uniformly choen detination along the path decribed in W. Then the growing-rank protocol complete the routing in time O(h diam(g) + log V ), w.h.p. Proof. For every path w W, the expected number of packet that travere w i h/ V. Further, for ymmetry reaon, the number of path in W paing through a node v i the ame for all node v V, namely at mot diam(g) V. Hence, the expected number of packet that pa through a node i at mot h diam(g) V = h diam(g), V and therefore C exp h diam(g). Finally, applying Theorem 2.7 yield that the routing time of the growing-rank protocol i O(h diam(g) + log(h V )) = O(h diam(g) + log V ), w.h.p. For bounding C exp in the proof of the above theorem, we ued the ymmetry propertie of the path ytem W. A een, ymmetric path ytem can be eaily contructed for Cayley network. For non-cayley node ymmetric network, like for example the Peteren graph [16], the contruction in the proof of Lemma 3.1 fail. Here we have to chooe another path election trategy. Suppoe the detination for the packet are pecified. Then we elect the routing path randomly during the routing intead of beforehand by a path ytem. We aume that each proceor chooe randomly the link for tranmitting a elected packet jut before the packet i paed on. Thi link i choen randomly and uniformly from the et of outgoing link which belong to a hortet path to the detination of the packet. Theorem 3.3 Let G = (V, E) be a node ymmetric network. Suppoe each proceor end h packet to randomly and uniformly choen detination. Further, uppoe that the routing path are elected randomly during the routing a decribed above. Then the growing-rank protocol complete the routing in time O(h diam(g) + log V ), w.h.p. Proof. Let C v denote the number of packet that travere a node v V. E(C v ) i the ame for all node v V for ymmetry reaon. Therefore, V C exp v V E(C v ) h V diam(g) which give C exp h diam(g). Finally, it follow from Theorem 2.7 that the routing time of the growing-rank protocol i O(h diam(g) + log V ), w.h.p. 3.2 Edge Symmetric Network We ay that a network G = (V, E) i edge ymmetric, if given any pair of edge (u, v) and (u, v ) there i an automorphim φ Aut(G) uch that φ(u) = u and φ(v) = v. Thu, each edge in an edge ymmetric network can be mapped by an automorphim onto any other edge. Intuitively, all edge in an edge ymmetric network look the ame. All equal-ided tori, for example, are edge ymmetric. For thee network we ugget the ame path election trategy a for the general node ymmetric network. The following reult improve the one for node ymmetric network lightly. 14

15 Theorem 3.4 Let G = (V, E) be a edge ymmetric network of degree. Suppoe each proceor end h packet to randomly and uniformly choen detination. Further, uppoe that the routing path are elected randomly during the routing a decribed above. Then the growingrank protocol complete the routing in time O((h/ + 1) diam(g) + log V ), w.h.p. Proof. Let C e denote the number of packet that travere an edge e E. For ymmetry reaon, E(C e ) i the ame for all e E, namely C exp. Hence, E C exp = e E E(C e ) h V diam(g) which give C exp h V diam(g) E = h diam(g) Now applying Theorem 2.7 yield that the routing time of the growing-rank protocol i O(h/ diam(g) + diam(g) + log V ), w.h.p. 3.3 De Bruijn Network The k-dimenional de Bruijn network ha n = 2 k node. Thee node are repreented by k-bit binary tring, and each node u 1 u 2... u k ha a link to the node u 2... u k 0 and to the node u 2... u k 1. The diameter of the network i k = log n. Figure 2 give an example Figure 2: The 3-dimenional de Bruijn network. For two node u = u 1... u k and v = v 1... v k, we define κ = κ(u, v) k to be the larget integer atifying u k κ+1... u k = v 1... v κ. For intance, κ( , ) = 3. Let W be the path ytem in which the path w(u, v) W from a node u to a node v i defined by w(u, v) = (u = u 1... u k = u 1... u k κ v 1... v κ u 2... u k κ v 1... v κ+1 u 3... u k κ v 1... v κ+2. v 1... v k = v). Obviouly, the length of thi path i k κ, and ince thi i equal the ditance between u and v, the path i a hortet path. 15

16 Theorem 3.5 Suppoe each proceor in the de Bruijn Network of ize n end h packet to randomly and uniformly choen detination along the path in W. Then the growing-rank protocol complete the routing in time Θ(h log n), w.h.p. Proof. Define k := log n. Firt, we how that there are at mot k n path in W that pa through an arbitrary link (u, u ). Define M i to be the et of all node v uch that the ditance from v to u i i, and define M i to be the et of all node v uch that the ditance from u to v i i, for 0 i k. Obviouly, M i 2 i and M i 2i. Suppoe w(v, v ) i a path from W of length l uch that (u, u ) i the ith link on thi path for 1 i l. Then v M i 1 and v M l i. Thu, the number of path that pa through (u, u ) i at mot k l M i 1 M l i k l 2 i 1 2 l i (k 1) 2 k + 1 k n. l=1 i=1 l=1 i=1 A a conequence, the expected number of packet that travere through (u, u ) i at mot (h k n)/n = h k, and hence, C exp h k. Now our theorem follow by applying Theorem Limit of our Approach In thi ection, we try to illutrate which additional problem occur for routing along nonhortet path, for determinitic routing, and for routing with bounded buffer. 4.1 The Growing-Rank Protocol on Non-Shortet Path Here we invetigate the behavior of the growing-rank protocol on non-hortet path. We give an obliviou routing problem with congetion C and dilation D where the protocol behave poorly, e.g., take expected time Θ(C D) for D = log N and C = log N/ log log N. The routing path in thi example are non-hortet but imple, i.e., each node appear at mot once in the path. Theorem 4.1 Suppoe N, D, and C atify log N/ log log N C N ɛ with ɛ < 1 and C D/ log log N. Then there i an obliviou routing problem of ize N, dilation D, and congetion C uch that the expected routing time of the growing-rank protocol on thi problem i Ω(C + D log N/ log log N). Remark 4.2 We aume that the packet rank are increaed by m = R/D when the packet move forward. It i eay to check that the reult hold alo for any m with m = Ω(R/(C + D + log N)) and m = O(R/D) if C atifie C (R/m)/ log log N intead of C D/ log log N. Note that thi give example with routing time Θ(C D) for every m fulfilling the condition decribed in Section

17 Proof. Conider the zip network in Figure 3. For implicity, we aume that C i even and Figure 3: The zip network and the routing path for the packet in A. (Sorry, I lot the picture, ee Journal of Algorithm.) D = 2d 1 for ome d a given in thi picture. Suppoe we are given two et A and B each of C/2 packet with ource node u 1 and v 1 repectively. Thee packet hould be routed with the growing-rank protocol. The routing path of the packet in A i u 1 u 2 v 1 v 2 v 3 v 4 u 3 u 4 u 5 u 6 v 5 v d 3 v d 2 v d 1 v d u d 1 u d a hown in the picture, and the routing path of the packet in B i v 1 v 2 u 1 u 2 u 3 u 4 v 3 v 4 v 5 v 6 u 5 u d 3 u d 2 u d 1 u d v d 1 v d. Define A A and B B to be the et of packet with initial rank maller than 2m. Suppoe A = B = k. Then the rank of the packet in A are bigger than the rank of the packet in B at node v 1, becaue they have been increaed twice by m on their way from u 1 to v 1. Conequently, thee packet are delayed by the packet of B for k 2 tep at thi node. By the ame argument, the packet in B are delayed for k 2 time at node u 1. Further, uppoe all other packet have rank not maller than 4m. Then the packet in A and B are not affected by thee packet, and the above event recur at the node u 3, v 3 ; u 5, v 5 ; and o on. A a conequence, the firt packet reache it detination after (k 2) d/2 + (2d 1) tep, and thu, the routing time i at leat (k 2) d/2 + (2d 1) + (c 1) k D/4 + C/2. Now aume that we have a routing network which include N/C dijoint copie of the zip network each of which with the above decribed routing problem. Thi give an obliviou routing problem of ize (at mot) N, dilation D, and congetion C. We will how that the expected routing time for thi problem i Ω(C + D k) for uitable k = Ω(log N/ log log N). Thi i trivially true for C = Ω(k D) ince any protocol require at leat C tep. Further, it i true for D = O(1) ince C log N/ log log N. Therefore, we aume that C k D/2 and D 8. Then the probability that k C/2 = Ω(log N/ log log N) packet from the et A in a fixed copy have rank maller than 2m and C/2 k packet have rank of at leat 4m i ( C/2 k ) ( ) k ( 2m 1 4m ) C/2 k (a) R R (b) ( ) ( ) k C/2 2m 4m (C/2 k) ( 4 R ) k R ( ) ( ) k C/2 2m 4 k k R ( ) k C m, 4k R where equation a) hold becaue 4m/R 4/D 1/2, and equation b) hold becaue C k D/2 k R/2m. A the ame bound hold for the packet in B, the probability that the event decribed above happen in none of the at leat N/C N/2C N 1 ɛ /2 copie i at 17

18 mot ( 1 ( ) ) 2k N 1 ɛ /2 C m 4k R (c) (d) ( exp N 1 ɛ ( ) ) 2k C m 2 4k R ( exp N 1 ɛ ( ) ) 2k 1 2 4k log log N exp ( N 1 ɛ ) (log N) (1 ɛ) log N/2 log log N 2 exp ( N (1 ɛ)/2 ) = o(1), 2 where Equation (c) hold becaue C D/ log log N = R/(m log log N), and Equation (d) hold becaue k (1 ɛ) log N/4 log log N. A a conequence, the expected routing time i at leat (1 o(1)) (C + D/4 k) = Ω(C + D log N/ log log N). 4.2 Determinitic Routing Now we conider determinitic routing. We invetigate the behavior of nonpredictive routing protocol in which all cheduling deciion have to be independent from the future routing path of the packet. Note that the growing-rank protocol i not determinitic, and hence not nonpredictive. However, for any fixed etting of the initial rank it i nonpredictive. The ame hold for Leighton random-rank protocol [4] and for Ranade protocol [12]. The following example how that all thee protocol perform poorly in a determinitic etting even on leveled network. (A imilar example yielding a lower bound of Ω(C D/ log C) rather than Ω(C D) i preented in [6].) Theorem 4.3 Suppoe we are given any determinitic non-predictive routing protocol Q for routing on the D-dimenional butterfly network. Then, for any C, there i a routing problem with congetion C for which Q take time Ω(C D). Proof. Fix an arbitrary output node v on level D of the butterfly. Thi node i the root of a complete binary tree T of height D whoe leave are the 2 D input node on level 0. We aume that each input node want to end out C packet. For the firt edge on the routing path of each packet we chooe the edge to the parent node of the ource node in the tree T. The following edge are pecified inductively uch that each edge in our tree T i paed by C routing path. Suppoe u i a node on level l with 1 l D 1 which belong to the tree T. Then u i croed by 2C routing path. We aume that thee path are determined already up to level l, and we have to continue the path up to the next level. Thi we do depending on the behavior of protocol Q up to level l. We chooe the path of thoe C packet that would arrive firt at node u to leave the tree and the path of the other C packet to tay in the tree, i.e., to cro the parent node of u. Thi define the routing path inide the tree. For the path outide the tree we only demand that they have congetion C. Now we calculate the time which i taken by Q for routing the above defined problem. A node on level 1 l D receive it firt packet at time (l 1) C/2+l or later. (For implicity, we aume that C i even.) Hence, the root receive it firt packet at time (D 1) C/2 + D or later. A a conequence, the routing take at leat C/2 (D 1) + D + (C 1) = Ω(C D) tep. 18

19 4.3 Routing with Bounded Buffer Suppoe the packet that are tranmitted along a link are tored in an link buffer at the end of the link until they are forwarded along the next link on their path. If a link buffer i full, then the repective link cannot tranmit packet until one of the packet leave the buffer. A een in the introduction, C D i an upper bound on the routing time of greedy protocol on network with unbounded buffer. The following example how that thi bound doe not hold for network with bounded buffer. Theorem 4.4 For every C, D, and B there exit a greedy routing protocol that require time Ω((C/2 B) D 2 ) for a routing problem with congetion C and dilation D on a bounded-degree leveled network with buffer ize B. Proof. For C/2 B our theorem i trivially true. Therefore, we aume C/2 > B. Further, we aume for implicity that C i even. Figure 4 define the railway network of depth D. Suppoe we have D + 1 et A 0,..., A D Figure 4: The railway network of depth D. (Sorry, I lot the picture, ee Journal of Algorithm.) each of which including C/2 packet. The packet in A i, for 0 i D, hould be routed from node u i to node v i. The cheduling rule are defined by a imple rule: packet in A j are prefered againt packet in A i, for 0 i < j D. How long doe it take until the firt packet of A 0 reache v 0? After D + B 1 tep, B packet from each et A i with 1 i D have travered link e i. Furthermore, the buffer at the end of each link e i, for 1 i D 1, are filled with B packet from A i at thi time. In the following C/2 B time tep, the link e D i travered by packet of A D. A a conequence, all packet tored in the buffer of e 1,..., e D 1 are blocked. In general, the packet in A D i with 0 i D 1 travere along link e D i from time D + B + i (C/2 B) to D + B + (i + 1) (C/2 B) 1. Hence, the packet of A 0,..., A D i 1 are blocked during thi time. Conequently, the firt packet of A 0 reache v 0 after time tep D + B + D (C/2 B) = B + D (C/2 B + 1). Now uppoe the packet of A 0 together with the A 0 -packet of D 1 other railway network of ame depth, are ued to build a new routing problem of the above decribed tructure on a further railway network of depth D 1. Note that thi doe not increae the depth of the network ince the packet in A 0 have travered only one edge o far. In the added network there will be a et of C packet which take B+(D 1) (C/2 B+1) further tep until the firt of thee packet travere link e 1 in thi network. Again, thi et of packet can be ued a input for a railway network of depth D 2. If we continue thi contruction until the added network have depth 1, then we get a routing problem with congetion C on a leveled network of depth D. The routing problem contructed in thi way take time D i=1 (i (C B + 1) + B) + (C 1) = Ω(C D2 ). 5 Acknowledgement We would like to thank Chritian Scheideler and Rolf Wanka for helpful dicuion. 19