Shortest-Path Routing in Arbitrary Networks

Transcription

1 Ž. Journal of Algorithm 31, Article ID jagm , available online at on Shortet-Path Routing in Arbitrary Network Friedhelm Meyer auf der Heide and Berthold Vocking Department of Mathematic and Computer Science and Heinze Nixdorf Intitute, Unierity of Paderborn, Paderborn, Germany Received July 27, 1996; revied September 1, 1998 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 OC Ž 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 preceding 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 Academic Pre Open acce under CC BY-NC-ND licene. 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 * A preliminary verion wa preented at the 12th STACS, 1995; ee 9. Supported in part by DFG-Sonderforchungbereich 376 Maive Parallelitat: Algorithmen, Entwurfmethoden, Anwendungen, by EU ESPRIT Long Term Reearch Project Ž ALCOM-IT., and by DFG Leibniz Grant Me fmadh@uni-paderborn.de. voecking@uni-paderborn.de Copyright 1999 by Academic Pre Open acce under CC BY-NC-ND licene.

2 106 MEYER AUF DER HEIDE AND VOCKING pecial clae of network a, e.g., leveled network, we are intereted in unieral 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. 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 wu, Ž. Ž u. for every pair u and 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, we chooe the path wu, Ž. 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 tranmit-

3 SHORTEST-PATH ROUTING IN ARBITRARY NETWORKS 107 ting 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 obliiou 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 a 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 an obliiou 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, the dilation D, i.e., the maximum length of the routing path in the problem. Clearly, maxc, D4 Ž 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 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 nonpredictie 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.

4 108 MEYER AUF DER HEIDE AND VOCKING In the following, we repreent the underlying proceor network by a digraph G Ž 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 diamž G 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 et al. 6 how that any obliviou routing problem can be routed in time OC Ž D. with contant-ize link buffer, thereby achieving the naive lower bound. Their proof i baed on the Lovaz local lemma and how only the exitence of the optimal chedule. In 8, Leighton et al. preent an algorithm for computing thi chedule. But ince the runtime of thi algorithm i polynomial in the number of packet and link, it cannot 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 et al. uch that they can replace the Lovaz local lemma by a Chernoff bound in the analyi. Thi modification allow u to calculate the chedule in a ditributed way and on-line. The protocol guarantee routing time OC Ž. Ž. OŽlog N. Ž. 1 log N D polylog N, w.h.p. Recently, Otrovky and Rabani 10 have improved on thi reult. They achieve routing time OC Ž D 1 log N., w.h.p., for arbitrary 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 et al. preent in 6 a imple on-line chedule protocol which complete the routing in time OC Ž D logž DN.., w.h.p., and require buffer of ize logž DN.. A imilar routing time but with much maller buffer ize i achieved in 3 : a protocol i given requiring routing time OC Ž 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 1 Ž. Throughout thi paper, w.h.p. with high probability mean with probability at leat 1 N for any fixed contant with N denoting the number of packet.

5 SHORTEST-PATH ROUTING IN ARBITRARY NETWORKS 109 by Aleliuna 1 and Upfal 14. The protocol can be eaily extended to the cla of bounded-degree leveled network 4, 5. In a leeled 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 packet are routed only from level 0 to level L. Ranade protocol complete the routing in time OC Ž 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 i 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 OeriewNew Reult In Section 2, we introduce a new probabilitic on-line routing protocol which we call the growing-rank protocol. We how that the growing-rank protocol route any hortet-path routing problem of ize N with congetion C and dilation D in OC Ž 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 preceding 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 Ž V, E. i aid to be hortcut-free, if there i a ubnetwork Ž 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 preceding time bound hold even if C denote the maximum expected congetion over all link. Thi

6 110 MEYER AUF DER HEIDE AND VOCKING 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 by 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 diamž G.. Hence, the growing-rank protocol route random h-routing problem on any Cayley network G of ize n in time Oh Ž diamž G. log n., w.h.p. Further, we invetigate node and edge ymmetric network. Intuitively, a network i node Ž edge. ymmetric, if it look the ame viewed from any node Ž 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 diamž G.. Thi implie routing time Oh Ž diamž G. log n., w.h.p. Given any edge-ymmetric network G of ize n and degree, we how that the maximum expected congetion for random h-routing problem i at mot hdiamž G.. Thi implie routing time OŽŽ h 1. diamž 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 Oh Ž 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 Oh Ž log n., w.h.p. By applying Valiant paradigm firt routing to a random detination 15, all application reult for random-h routing problem alo hold 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

7 SHORTEST-PATH ROUTING IN ARBITRARY NETWORKS 111 decribe an obliviou routing problem of ize N with congetion C log Nlog log N and dilation D log N for which the expected routing time of the growing-rank protocol i Ž 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 Ž C D.. The reult hold for any C and D. ŽA imilar example with time bound Ž C Dlog C. can be found in. 7. Interetingly, the underlying network i the butterfly network. Note that Ranade protocol, Leighton randomrank 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,...,em1 with full packet buffer, and every link ei hold only packet that wait for moving forward along e Ži1.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 Qe 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 Qe i allowed to move forward and which packet have to wait. The protocol forward packet whenever poible; i.e., if Qe i not empty, then one of the packet in Qe i moved forward along e. The prioritie among the packet are determined by random rank. Suppoe R and m RD are uitably large integer. ŽThe exact value of R will be pecified later.. Initially, each packet i aigned an integer rank choen randomly, independently, and uniformly from the et

8 112 MEYER AUF DER HEIDE AND VOCKING 0, 1,..., R 14. 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 with minimum rank i choen. Thu, for each outgoing link e with Q 1, a tep look like thi: e 1. chooe a packet p Qe with minimum rank, 2. increae the rank of p by m, and 3. move p forward along e. To break time among packet with the ame rank, we aume that each packet p ha a unique ident-number denoted by idž p.. If there are everal packet with the ame minimum rank, then the one with the 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 Ž p.. Further, we define the ident-rank of p at e a id-rank e Ž p. rank e Ž p. idž p. ŽmaxŽ id 1. with maxž 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 Ž p. id-rank e Ž p 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 OC Ž D log N. tep, w.h.p. Our analyi i baed on a delay equence argument imilar to that in 4, 5, 12. DEFINITION 2.1 ŽŽ, l, r. -delay equence.. An Ž, l, r. -delay equence conit of delay packet p,..., p ; 1 not necearily ditinct link e,...,e uch that e i the lat link 1 1 on the routing path of p, and, for 2 i, e i a link on the routing 1 i path of pi1 and p i; 1 1 integer l 1,...,l1 0 with Ýi1 li l uch that for 1 i i i i1 to ei i1 and including e i ; and 1, l i the number of link on the routing path of p from e Ž excluding e. integer r 1,...,r with 0 r r1 r1 r 1. We call the length of the delay equence, and we ay a delay equence i e actie, if rank iž p. r for 1 i. i i LEMMA 2.2. Suppoe the routing take T Rm D or more tep. Then a Ž T Rm D, Rm D, R D m. -delay equence i actie.

9 SHORTEST-PATH ROUTING IN ARBITRARY NETWORKS 113 Proof. We give a contruction cheme for a delay equence. Let p1 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 e2 and the packet that caued the delay p 2. We now follow the path of p2 backward until we reach a link e3 at which p2 wa forced to wait, becaue the packet p3 wa preferred. We change the packet again and follow the path of p3 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. e For 1 i, we et r rank iž p.. Since the growing-rank protocol i 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 r1 r R. 1 The path from the ource of p to the detination of p recorded by the 1 preceding proce in revered order i called a delay path. It conit of contiguou part of the delay packet routing path. We define the l to i 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 that l m R. Conequently, we have Ý 1 l l R i1 i m Rm 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 Rm D. Thu, if we top the previou contruction at packet p TR md, then we have built an active Ž T Rm D, Rm 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 actie 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 pi p j. Thu, the routing path of p croe the delay path at the colliion link ej and ei in that order. Let denote the number of link on the routing path of p from ej to e i. Then the rank of p i increaed time by m on thi part of the routing path, and, conequently, i id-rank e i p id-rank e j Ž. Ž p. m. Ž 1. On the other hand, each packet pk with i 1 k j delay the packet e kž. e pk1 at link e k. Thu, id-rank pk1 id-rank kž p k.. Further, the rank are increaed by m on every link on the delay path between ej and e i. The

10 114 MEYER AUF DER HEIDE AND VOCKING number of thee link i Ý j1 l. Thi give k1 k j1 ei e id-rank p id-rank j Ž. Ž p. Ý l k m k1 id-rank e j Ž p. m. Ž 2. Note that Ýki j1 lk, becaue the routing path of p i a hortet path. Clearly, Ž. 2 contradict Ž. 1. Conequently, there i no packet that appear twice in the delay equence. LEMMA 2.4. The number of different Ž, l, r. -delay equence i at mot 2eC Ž r. l d, l, r N 2. Ž. ž / Proof. We count the number of poible choice for each component: There are N poibilitie to chooe p 1. Of coure, thi fixe e1 a well. Further, there are ž / Ž 1. l l 1 way to chooe the l i becaue Ýi1 1 li l. ž / Now uppoe p, e, and l for 2 i are fixed. Then e i i1 i1 i1 i that link on the routing path of p which ha ditance l to e. Thu, i1 i1 i1 ei 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. r Finally, there are Ž. 0 r r r 1. 1 poibilitie to chooe the r uch that i Altogether, we find that the number of Ž, l, r. -delay equence i at mot Applying the inequalitie complete the proof. ž / ž / l N C r. ž/ ž/ a a ea a 2 and b b ž b / b

11 SHORTEST-PATH ROUTING IN ARBITRARY NETWORKS 115 THEOREM 2.5. Suppoe we are gien 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 OŽ C D log N. tep, w.h. p. Proof. Lemma 2.2 and 2.3 how that the probability probž T. that the routing take T Rm D or more tep i bounded by the probability that an Ž, Rm 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, ince 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 R prob T d, D, R D m R m Auming R give Ž. ž / 2eC Ž R D m. R MD ž / N 2 R. Ž. ž / 2eC Ž 2 D mr. R md ž / 2eC Ž 2 R D m. R md prob T N 2 R N 2, which i at mot N for max4ec Ž 2 D mr., Rm D Ž 1. log N 4. Hence, the routing take R T D m D m R R max 4eC 2, D Ž 1. log N D Ž 3. R M m ½ ž / 5 or more tep with probability N. Finally, applying m RD yield that the routing i completed in OC Ž D log N. tep, w.h.p Becoming Strongly On-Line The drawback of the preceding protocol i that each proceor ha to know etimation of the congetion C, the dilation D, and the ize of the

12 116 MEYER AUF DER HEIDE AND VOCKING routing problem N. Thi i becaue we have aumed that the range of the rank i ufficiently large, i.e., R max4ec Ž 2 D mr.,2d Ž 1. log N 4, and that the packet rank are increaed by m RD 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 ŽC D log N., m ŽRŽ C D log N.., and m OŽ RD.. 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 diamž G.. Of coure, D and N can be computed and ditributed among the proceor in OŽdiamŽ 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 RD with D D, then we get routing time OC Ž D log N.. For intance, bounding the dilation by the diameter diamž G. give routing time OC Ž diamž G. log N., even if D diamž G.. The following variation of the protocol achieve routing time OC Ž D log N. without auming that the dilation or the congetion i 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 interal k to begin at tep 2 k and to end at tep 2 k1 1. Hence, each interval k ha length 2 k. Define k to be the mallet integer k 1 atifying 2 1 N diamž G.. Then the routing i completed at the end of interval k urely. Chooe R 2 k2. The value of m varie during the routing; i.e., the rank of forwarded packet are increaed by m R2 k2 in interval k. Note that m i an integer for every k k k k. k 1 Now let k denote the mallet integer atifying 2 max12ec, D Ž 1. log N4 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 ½ ž / 5 Ž. 3 m k R R T max 4eC 2 D, D Ž 1. log N D R m m m R2 k 2 k k k 2 Ž. 4 Ž. max 4eC 2 D2, D 1 log N D 2 max 12eC, D 1 log N D 2 k 1 Ž. 4 2 k 1 2 k 1 2 k k k 1

13 SHORTEST-PATH ROUTING IN ARBITRARY NETWORKS 117 tep i at mot N. Conequently, all packet reach their detination in k 1 interval k, w.h.p., and thu, the routing take at mot 2 1 OC Ž D log N. tep, w.h.p. Now we aume that the packet do not have 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 OC Ž D. k log N, w.h.p. A before, define interval k to begin at tep 2 and to end at tep 2 k1 1, for k 0. At the beginning of interval k, each packet i k aigned a new random rank from the interval R 2. Ž 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. k 1 Let k denote the mallet integer atifying 2 max12ec, D Ž 1. log N4 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 ½ 5 ž / Ž. 3 m R R k k T max 4eC 2 D, D Ž 1. log N D R m m R m2 k 2 k k k 2 Ž. 4 Ž. max 4eC 2 D2, D 1 log N D 2 max 12eC, D 1 log N D 2 k 1 Ž. 4 2 k 1 2 k 1 2 k k 1 tep i at mot N. Therefore, the routing take at mot 2 k1 1 OC Ž 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 OŽC D log N., w.h. p 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 Ž 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

14 118 MEYER AUF DER HEIDE AND VOCKING probž 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 probž M.. Then, for any M p m M, we have probž M. Ł probž p, m. p P p p P 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 wu, Ž. Ž u. for every pair u and of node can be repreented in the preceding 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 ourcež p. denote the ource node of p and detž p. the randomly elected detination of p. Then we chooe the path wžourcež p., detž p.. W a p routing path. Thi trategy can be eaily expreed in term of the preceding model by imply pecifying the probabilitie that a path m M i the routing path of a packet p P; i.e., we et 1 if m W ourcež p., probž p, m. V 0 otherwie, with W W denoting the et of path tarting at proceor V. The following theorem bound the routing time of the growing-rank protocol on random routing problem a decribed previouly. We aume that the packet rank are increaed by m RdiamŽ G. when the packet move forward. The given routing time depend on the total number of packet N, the diameter diamž G., and the maximum expected congetion C max EC Ž. e E4 with EC Ž. exp e e denoting the expected number of packet travering link e. We will ee later that Cexp can be calculated very eaily and exactly for random h-routing problem on everal clae of network. THEOREM 2.7. Suppoe we are gien 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

15 SHORTEST-PATH ROUTING IN ARBITRARY NETWORKS 119 Cexp denote the maximum expected congetion. Then the growing-rank proto- col complete the routing in OŽC diamž G. log N. tep, w.h. p. exp Proof. Becaue of Lemma 2.2 and 2.3, we can bound the probability probž T. that the routing take T Rm diamž G. or more tep by the probability that an Ž, Rm diamž G., R diamž G. m. -delay equence with ditinct packet i active. For any N-tuple of path M M p let dž, l, r, M. denote the number of poible Ž, l, r. -delay equence with ditinct delay packet under the aumption that M decribe the reult of the random path election. Then R probž T. probž M. d, diamž G., R diamž G. m, M p m Note that Ý ž / MM R. Ž 4. Ý dž, l, r. probž M. dž, l, r, M. p MM i equal to the expected number of poible Ž, l, r. -delay equence for randomly choen M. We can count thi number a follow. There are at Ž l. Ž r mot N. way to chooe p 1, e 1, the l i, and the r i. Now uppoe p i1, e i1, and li1 are fixed, for 2 i. Then ei i that link on the routing path of pi1 which ha ditance li1 to e i1. Thu, ei i fixed a well. What i the expected number of candidate for p under the aumption that ei and p 1,..., pi1 are already fixed? The routing path of pi mut travere e i, and pi mut be ditinct from p 1,..., p i1. Let Me i M be the et of all path in M that cro e i. Then, the expected number of poibilitie to chooe p i at mot i Ý Ý Ý Ý probž p, m. probž p, m. ppp 1,..., pi14 mme pp mm i ei Ž. E C C. Becaue thi bound i independent of the choice for the delay packet p 1,..., p i1, 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 1 dž, l, r. N C r exp 2eCexp Ž r. l N 2. ei ž / ž / ž / exp i

16 120 MEYER AUF DER HEIDE AND VOCKING Ž. Applying thi to 4 yield Ž. Ž. ž / 2eCexp R diamž G. m R mdiamž G. prob T N 2 R ž / mrdiamž G. 2eCexp Ž 2 R. 2diamŽ G. N 2 R. We et max12ec,2 diamž G. Ž 1. log N 4 exp, for contant. Thu, T 2 diamž G. OC Ž diamž G.. Ž 1. exp log N. Fur- ther, we chooe R. Then Ž 1. logn2diamž G. 2eCexp 3R 2diamŽ G. prob T N 2 N. 12eC R ž / Ž. exp Thi complete the proof of Theorem APPLICATIONS Now we give everal application for the growing-rank protocol. We invetigate random routing problem on node-ymmetric network, edgeymmetric network, and de Bruijn network. All reult in thi ection are conequence of Theorem Node-Symmetric Network An automorphim of a network G Ž V, E. i a permutation : V V with the property that Ž u,. E ŽŽ u., Ž.. E. The automorphim of G form an algebraic group under the operation of compoition. Thi group i denoted by AutŽ G.. An automorphim group U AutŽ G. i aid to be tranitie on G if, given any two node u and, there i an automorphim U uch that Ž u., and a network G i called node ymmetric if AutŽ 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 be a finite algebraic group with identity 1, and uppoe i a et of generator of with 1. Then the Ž. Ž. 1 Cayley network G, V, E i defined by V and E a, b a b 4. Figure 1 how an example for a Cayley network. Suppoe W i a path ytem on a network G Ž V, E. that include a hortet path wu, Ž. Ž u. for every pair u and of node.

17 SHORTEST-PATH ROUTING IN ARBITRARY NETWORKS 121 FIG. 1. The Cayley network G with Ž,. and 1, 24, 8. Note that the identity in i 0 rather than 1 a i an additive group. We call W ymmetric if, given any two node u and, there i a permutation : V V uch that for every path Ž w w w. 0 1 l W with w u there i a path Ž Ž w. Ž w. Ž w.. i 0 1 l W with Ž w. i 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. ytem. For eery Cayley network, there i a ymmetric hortet-path Proof. Let G Ž V, E., be a Cayley network. Then there i a trani- tive automorphim group U of ize V 2. We denote by u the automor- phim of U which map the node u onto the node, for u, V. Thu, U V 4 u for any u V. Suppoe w Ž w w w. 0 1 l i a path in G and i an automorphim of G. Then we define Ž w. ŽŽ w. Ž w. 0 1 Ž w... Since i an automorphim, Ž w. l 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. ŽFor implicity of notation, we aume V 0, 1,..., n Step 1. Chooe arbitrarily a hortet path wž 0,. from the node 0 to every node V. Step 2. For every u V 04 and every V, define the path wu, Ž. from u to by wu, Ž. u ŽwŽ0, 0 Ž... 0 u

18 122 MEYER AUF DER HEIDE AND VOCKING In the firt tep we have choen n prototype path Žincluding 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 be two node of G. We have to how that there i a permutation which map every path w Ž w w w. 0 1 l W with Ž u w onto a path w w w w. i 0 1 l W with wi for 0 i l. For we chooe the automorphim u U. Clearly, u map u onto. Thu, it remain only to prove that w Ž w. u 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 Ž. a Ž. b Ž. c w w 0 w už. w Ž. 0 w Ž. 0 Ž. 0 0 w w w w w. Thi can be proved a follow: Ž. a There i exactly one automorphim in U that map w0 onto w 0. Conequently, w 0 u w 0. Ž. w 0 0 b Since U i a group, 0 w 0 i an element of U, and ince there i only one automorphim in U that map w onto w 0 0, it follow that w 0 0 w 0 0 w0 w0. Ž. w 0Ž c It i w w., and conequently, w 0 Ž w.. 0 w 0 Hence, the automorphim w 0 U map w onto w. If w i the identity, then w i generated in Step 1. Otherwie, w i generated a a copy of w in Step 2. THEOREM 3.2. Let G Ž 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 OŽh diamž G. logv., w.h. p. Proof. For every path w W, the expected number of packet that travere w i hv. Further, for ymmetry reaon, the number of path in W paing through a node i the ame for all node V, namely at mot diamž G. V. Hence, the expected number of packet that pa through a node i at mot h diamž G. V h diamž G., V

19 SHORTEST-PATH ROUTING IN ARBITRARY NETWORKS 123 and therefore C h diamž G. exp. Finally, applying Theorem 2.7 yield that the routing time of the growing-rank protocol i Oh Ž diamž G. logžh V.. Oh Ž diamž G. logv., w.h.p. For bounding Cexp in the proof of Theorem 3.2, 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, 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 Ž 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 preiouly. Then the growing-rank protocol complete the routing in time OŽh diamž G. logv., w.h. p. Proof. Let C denote the number of packet that travere a node V. EC Ž. i the ame for all node V for ymmetry reaon. Therefore, Ý V C EŽ C. h V diamž G., exp V which give C h diamž G. exp. Finally, it follow from Theorem 2.7 that the routing time of the growing-rank protocol i Oh Ž diamž G. logv., w.h.p Edge-Symmetric Network We ay that a network G Ž V, E. i edge ymmetric, if, given any pair of edge Ž u,. and Žu,., there i an automorphim AutŽ G. uch that Ž u. u and Ž.. 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. Ž. THEOREM 3.4. Let G V, E be an edge-ymmetric network of degree. Suppoe each proceor end h packet to randomly and uniformly choen

20 124 MEYER AUF DER HEIDE AND VOCKING detination. Further, uppoe that the routing path are elected randomly during the routing a decribed preiouly. Then the growing-rank protocol complete the routing in time OŽŽ h 1. diamž G. logv., w.h. p. Proof. let Ce denote the number of packet that travere an edge e E. For ymmetry reaon, EC Ž. e i the ame for all e E, namely C. Hence, exp Ý E C EŽ C. h V diamž G., exp ee e which give h V diamž G. h diamž G. Cexp. E Now applying Theorem 2.7 yield that the routing time of the growing-rank Ž Ž. Ž.. protocol i Ohdiam G diam G log V, w.h.p 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 uu 1 2 uk ha a link to the node u2 uk0 and to the node u2 uk1. The diameter of the network i k log n. Figure 2 give an example. For two node u u u and, we define Ž u,. 1 k 1 k k to be the larget integer atifying uk1 uk 1 k. For in- tance Ž , Let W be the path ytem in which the path wu, Ž. W from a node u to a node i defined by wž u,. Ž u u1 uk u1 uk 1 k u2 uk 1 1 u3 uk k.. Obviouly, the length of thi path i k, and ince thi i equal to the ditance between u and, the path i a hortet path. 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 Žh 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 Žu, u.. Define M to be the i

21 SHORTEST-PATH ROUTING IN ARBITRARY NETWORKS 125 FIG. 2. The three-dimenional de Briujn network. et of all node uch that the ditance from to u i i, and define M i to be the et of all node uch that the ditance from u to i i, for i i 0 i k. Obviouly, Mi 2 and Mi 2. Suppoe wž,. i a path from W of length l uch that Žu, u. i the ith link on thi path for 1 i l. Then M and M i1 li. Thu, the number of path that pa through Žu, u. i at mot k l k l i1 li k i1 li ÝÝ Ž. l1 i1 l1 i1 M M 2 2 k k n. A a conequence, the expected number of packet that travere through Žu, u. i at mot Ž h k n. n h k, and, hence, Cexp h k. Now our theorem follow by applying Theorem LIMITS OF OUR APPROACH In thi ection, we try to illutrate which additional problem occur for routing along non-hortet path, for determinitic routing, and for routing with bounded buffer 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 Ž C D. for D log N and C log Nlog log N. The routing path in thi example are non-hortet but imple; i.e., each node appear at mot once in the path.

22 126 MEYER AUF DER HEIDE AND VOCKING THEOREM 4.1. Suppoe N, D, and C atify log Nlog log N C N with 1 and C Dlog log N. Then there i an obliiou 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 Ž C D log Nlog log N.. Remark 4.2. We aume that the packet rank are increaed by m RD when the packet move forward. It i eay to check that the reult hold alo for any m with m ŽRŽ C D log N.. and m OŽ RD. if C atifie C Ž Rm. log log N intead of C Dlog log N. Note that thi give example with routing time Ž C D. for every m fulfilling the condition decribed in Section 2.2. Proof. Conider the zip network in Fig. 3. For implicity, we aume that C i even and D 2 d 1 for ome d a given in thi picture. Suppoe we are given two et A and B each of C2 packet with ource node u1 and 1, repectively. Thee packet hould be routed with the growing-rank protocol. The routing path of the packet in A i u u u u u u u u d3 d2 d1 d d1 d a hown in the figure, and the routing path of the packet in B i u u u u u u u u u. d3 d2 d1 d d1 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 1, becaue they have been increaed twice by m on their way from u1 to 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 FIG. 3. The zip network and the routing path for the packet in A.

23 SHORTEST-PATH ROUTING IN ARBITRARY NETWORKS 127 not maller than 4m. Then the packet in A and B are not affected by thee packet, and the preceding event recur at the node u 3, 3; u 5, 5; and o on. A a conequence, the firt packet reache it detination after Ž k 2. d2 Ž 2 d 1. tep, and, thu, the routing time i at leat Ž k 2. d2 Ž 2 d 1. Ž c 1. k D4 C2. Now aume that we have a routing network which include NC dijoint copie of the zip network each of which with the routing problem decribed previouly. Thi give an obliviou routing problem of ize Žat mot. N, dilation D, and congetion C. We will how that the expected routing time for thi problem i Ž C D k. for uitable k Ž log Nlog log N.. Thi i trivially true for C Ž k D. ince any protocol require at leat C tep. Further, it i true for D OŽ. 1 ince C log Nlog log N. Therefore, we aume that C k D2 and D 8. Then the probability that k C2 Ž log Nlog log N. packet from the et A in a fixed copy have rank maller than 2m and C2 k packet have rank of at leat 4m i ž / ž / ž / ž / ž / Ž. 2m ž k / ž R / k C m ž /, 4k R k C2k k Ž. a C2 2m 4m C2 2m Ž4 mžc2k. R. 1 4 k R R k R b C2 k 4 where Ž. a hold becaue 4mR 4D 12, and Ž. b hold becaue C k D2 k R2m. A the ame bound hold for the packet in B, the probability that the event decribed previouly happen in none of the 1 at leat NC N2C N 2 copie i at mot C m 2 k N 1 2 N 1 C m 2 k ž ž 1 4k R / / ž ž / / exp 2 4k R 1 2 k N ž Ž. c exp 1 ž / 2 4k log log N / Ž d. N 1 Ž 1. log N2loglog N exp Ž log N ž. 2 / ž / Ž1.2 N exp ož 1., 2 k

24 128 MEYER AUF DER HEIDE AND VOCKING where Ž. c hold becaue C Dlog log N RŽ m log log N., and Ž. d hold becaue k Ž 1. log N4 log log N. A a conequence, the expected routing time i at leat Ž1 ož 1.. Ž C D4 k. ŽC D log Nlog log N 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. ŽA imilar example yielding a lower bound of Ž C Dlog C. rather than Ž C D. i preented in. 6. THEOREM 4.3. Suppoe we are gien any determinitic nonpredictie 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 Ž C D.. Proof. Fix an arbitrary output node 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 previouly defined problem. A node on level 1 l D receive it firt packet at time Ž l 1. C2 l or later. ŽFor implicity, we aume that C i even.. Hence, the root receive it firt packet at time Ž D 1. C2 D or later. A a conequence, the routing take at leat C2 Ž D 1. D Ž C 1. Ž C D. tep.

25 SHORTEST-PATH ROUTING IN ARBITRARY NETWORKS Routing with Bounded Buffer Suppoe the packet that are tranmitted along a link are tored in a 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 hown 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 eery C, D, and B there exit a greedy routing protocol that require time ŽŽ C2 B. D 2. for a routing problem with congetion C and dilation D on a bounded-degree leeled network with buffer ize B. Proof. For C2 B our theorem i trivially true. Therefore, we aume C2 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,..., A, each of which include C2 packet. The packet in A, 0 D i for 0 i D, hould be routed from node ui to node i. The cheduling rate are defined by a imple rule: packet in A are preferred againt packet in packet of A0 reache 0? A for 0 i j D. How long doe it take until the firt After D B 1 tep, B packet from each et A with 1 i D i 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 at thi time. In the i following C2 B time tep, the link ed i travered by packet of A D. A a conequence, all packet tored in the buffer of e,..., e are j 1 D1 blocked. In general, the packet in A with 0 i D 1 travere D i FIG. 4. The railway network of depth D.