L.A. Goldberg, M. Jerrum, and P. MacKenzie, An p log log n) lower bound for routing in optical networks, SIAM J. on Computing 27 (1998) 1083-1098.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the underlying network is said to be either ad hoc or unknown or of unknown topology. We assume that multi hop radio networks are ad hoc, when considering distributed communication protocols for them, unless stated otherwise. A model related to radio networks is that of optical communication (see [44, 47, 48]) To explain the relationship, notice that a radio network can be interpreted as operating under just one wave frequency used by all the nodes. In an optical communication parallel computer (OCPC) each node v is assigned its own frequency F (v) Node v can simultaneously receive any message ....

....was shown in [47] a direct randomized al 32 gorithm that can realize any 2 relation with the success probability of at least 1=2 needs time me n) on some 2 relation. An h p log log n) lower bound, for routing h relations in the OCPC model, was shown by Goldberg, Jerrum and MacKenzie [48], with no restriction on algorithms, in particular it covers indirect algorithms. 5.2 DYNAMIC ARRIVALS Let the number of stations be denoted by n. The average arrival rate at station i is i , and = P 1 i n i is the total arrival rate. 5.2.1 Aloha. If arrivals are diversi ed among the ....

L.A. Goldberg, M. Jerrum, and P. MacKenzie, An p log log n) lower bound for routing in optical networks, SIAM J. on Computing 27 (1998) 1083-1098.

....for the multi hop radio networks are often required to be robust enough to run in ad hoc networks. We assume that the multi hop radio networks are ad hoc, when considering distributed protocols, unless stated otherwise. A model related to radio networks is that of optical communication (see [38, 41, 42]) To explain the relationship, notice that a radio network can be interpreted as operating under just one wave frequency used by all the nodes. The term optical communication is usually employed when Randomized Communication in Radio Networks 5 many frequencies are used simultaneously, up to as ....

....[41] a direct randomized algorithm that can realize any 2 relation with the success probability of at least 1=2 needs time Omega Gammame n) on some 2 relation. An Omega Gamma h p log log n) lower bound, for routing h relations in the OCPC model, was shown by Goldberg, Jerrum and MacKenzie [42], with no restriction on algorithms, in particular it covered the indirect algorithms. 5.2 DYNAMIC ARRIVALS Let the number of stations be denoted by n. The average arrival rate at station i is i , and = P 1in i is the total arrival rate. 5.2.1 Aloha. With arrivals diversified among the ....

L.A. Goldberg, M. Jerrum, and P. MacKenzie, An \Omega\Gamma p log log n) lower bound for routing in optical networks, SIAM J. Computing 27 (1998) 1083--1098.

....clients. This idea is central to many MAC protocols, including the Ethernet protocol [92] and the slotted ALOHA 20 protocol [1] and routing protocols for the optical computer [10, 116, 51] For more work in this direction, see [64, 65] Lower bounds for routing in optical computers appear in [62, 88]. 2.2 The 1 out of Protocol Consider the 1 out of problem where 1. Let the hash functions be labeled h i , 0 i , and the shared memory request of node j be for cell x j . Node j needs to successfully access one of the memory locations h i (x j ) 0 i . To solve the 1 out of ....

L. A. Goldberg, M. Jerrum, and P. D. Mackenzie. An \Omega\Gamma p log log n) lower bound for routing on optical networks. In Proceedings of the 6th Annual ACM Symposium on Parallel Algorithms and Architectures, pages 147--156, June 1994.

....to the same wavelength. One may view the abstract pos model as a generalization of the Optical Communication Parallel Computer ocpc. This model, introduced by Anderson and Miller [AM88] has recently attracted increased attention of the theoretical community [GT92, GV94, Rao92, GJLR93, GMR94, GJM94] Whereas the pos model is described in terms of light moving in fibers, the ocpc model is better described in terms of light beams moving in free space. Similar to the pos, an n processor ocpc is composed of n processors, each having a transmitter and a receiver that serve as its opto electronic ....

L.A. Goldberg, M. Jerrum, and P.D. MacKenzie. An \Omega\Gamma p log log n) lower bound for routing in optical networks. In ACM Symposium on Parallel Algorithms and Architectures, pages 147--156, Cape May, NJ, USA, June 1994. SIGACT and SIGARCH, ACM Press.

....respect to the number of processors) is a very well studied and important problem. Routing by itself is not an interesting computational problem in the PRAM context. However, for bounded resources models, routing (and in general information dispersal) is a basic building block for many algorithms ([7, 33, 17, 29]) In particular, as we shall see, routing and integer sorting are quite related in the PPRAM model. Our best algorithm for routing will based on an algorithm for sorting which in turn will be based on a special case of routing. We are interested in the routing problem that is known as the (r ....

....and cannot be encoded, concatenated, or take part in any computation. This operation was studied in the context of the implementation of certain parallel algorithms, as well as in general simulations of shared memory models such as the PRAM on distributed memory machines, see 5 for example [7, 33, 17, 29]. In particular, Valiant [33] showed that an efficient, distributed implementation of this operation implies efficient realization of universal shared memory models for parallel programming, such as the BSP [33] We are going to use the following fixed r relation: Definition 2.2 Consider a r ....

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L.A. Goldberg, M. Jerrum, and P.D. MacKenzie. An \Omega\Gamma p log log n) Lower Bound for Routing in Optical Networks. In Proc. ACM Symp. on Parallel Alg's and Arch's, June 1994.

....by a receiver to stop receiving on a given wavelength when it is tuning to another one. Clearly, ffi ae r . When ffi = 0, we say that detuning is instantaneous . 2. 2 Optical Communication Parallel Computer A highly theoretical model known as the OCPC (for Optical Communication Parallel Computer)[17 19] has received considerable attention lately. It can be seen as N elementary processors with a local memory, whose nodes are connected by a complete graph and can communicate with any other node (one to one communications) in a single hop. In this model, the transmitters are tunable, but the ....

....to send and at most h messages to receive. In case one knows the distribution of the messages, this problem can be trivially solved in h steps. Otherwise, several random algorithms have been proposed to solve the problem, whose best known lower bound is Omega Gamma h p log log N) steps[19]. The best solution to date runs in time O(h log log N) 18] This problem can be seen as a self simulation problem: what is the cost of simulating a OCPC with hN processors on a smaller OCPC with N processors This latter problem has been studied for the ORPC leading to better simulation ....

L.A. Goldberg, M. Jerrum, and P.D. MacKenzie. An \Omega\Gamma p log log n) lower bound for routing in optical networks. In ACM Symposium on Parallel Algorithms and Architectures, June 1994.

....comfortable to program. There are many studies about simulation of SMM on DMM in different interconnection networks, such as mesh (Leighton 1992) butterfly (Ranade 1991) hypercube (Valiant 1990) complete network (Anderson and Miller 1988, Ger b Graus and Tsantilas 1992, Goldberg al. 1993, Goldberg al. 1994, Karp al. 1992) and MOB (Goldberg al. 1993) The simulation of DMM has two components: Address map. Mapping the address space of the PRAM onto the N memory modules of the network, so that the PRAM consists of processors P 1 ,P 2 , P p and shared memory cells U = M[1, m] A DMM has ....

....an arbitrary h relation on a n processor OCPC. This is asymptotically the fastest known. If h log n then the failure probability of that algorithm can be made as small as n a for any positive constant a. The algorithm contains four phases and it is complex, maybe too complex to be practical. Goldberg al. 1994) prove a lower bound for the number of communication steps required, when h 1. The case h = 1 is trivial because each processor is the destination of at most one packet. 3.2. Decreasing the simulation cost Slackness is a one method to decrease the simulation cost. The processors are mapped so ....

Goldberg L. A., Jerrum M., MacKenzie P.D. 1994: An Lower Bound for Routing in Optical Networks. Proc. 6th ACM Symp. on Parallel Algorithms and Architectures, pp. 147-156. June 1994.

....one of each being tunable. One may view the abstract pos model as a generalization of the Optical Communication Parallel Computer ocpc. This model, introduced by Anderson and Miller [AM88] has recently attracted increased attention of the theoretical community [GT92, GV94, Rao92, GJLR93, GMR94, GJM94] Whereas the pos model is described in terms of light moving in fibers, the ocpc model is better described in terms of light beams moving in free space. Similar to the pos, an n processors ocpc is composed of n processors, each having a transmitter and a receiver which serve as its ....

L.A. Goldberg, M. Jerrum, and P.D. MacKenzie. An \Omega\Gamma p log log n) lower bound for routing in optical networks. In 6-th ACM Symposium on Parallel Algorithms and Architectures, pages 147--156, June 1994.

....in Dietzfelbinger, Kuty lowski, and Reischuk [13] and in fact, improve the constants on some of those lower bounds) and to prove lower bounds on restricted domain compaction problems on Exclusive Write PRAMs. The Random Adversary technique has also been used in Goldberg, Jerrum and MacKenzie [23] to prove a lower bound for a fundamental routing problem on the Optical Communication Parallel Computer (as defined by Anderson and Miller [2] under the name Local Memory PRAM ) We refer the reader to that paper for a full overview of the results, and a discussion of the model and the specific ....

....the Random Adversary technique has been used in MacKenzie [35] to prove a lower bound for Compaction on the QRQW PRAM (see Gibbons et al. 19] for a description of the QRQW PRAM) 2. Definitions. In this paper we will only be concerned with the Parallel Random Access Machine (PRAM) model. See [23] for the definition changes required for the Optical Communication Parallel Computer (OCPC) model. Relevant definitions for other models should be relatively easy to develop. In the PRAM model, processors communicate by reading and writing to a global shared memory. The PRAM model is further ....

L. A. Goldberg, M. Jerrum, and P. D. MacKenzie, An \Omega\Gamma p log log n) lower bound for routing in optical networks, in Proc. 6th ACM Symp. on Para. Alg. and Arch., 1994, pp. 147--156.

.... The OCPC model The ocpc model was first introduced by Anderson and Miller [2] and Eshaghian and Kumar [10] and has been studied by Valiant [40] Ger eb Graus and Tsantilas [12] Gerbessiotis and Valiant [11] Rao [33] Goldberg, Jerrum, Leighton and Rao [17] and Goldberg, Jerrum and MacKenzie [18]. The feasibility of the ocpc from an engineering point of view is discussed in [2, 12] See also the survey paper of McColl [30] and the references therein. Computing h relation on the OCPC A fundamental problem that deals with contention resolution on the ocpc is that of realizing an h relation. ....

....receive. Following Anderson and Miller [2] Valiant [40] and Ger eb Graus and Tsantilas [12] Goldberg et al. 17] solved the problem in time O(h lg lg n) for an n processor ocpc. A lower bound of Omega Gamma p lg lg n) expected time was recently obtained by Goldberg, Jerrum, and MacKenzie [18]. Simulating PRAM on OCPCs Valiant described a simulation of an erew pram on an ocpc in [40] More specifically, Valiant gave a constant delay simulation of a Bulk Synchronous Parallel (bsp) computer on the ocpc (there called the s pram) and also gave an O(lg n) randomized simulation of an n ....

L.A. Goldberg, M. Jerrum and P.D. MacKenzie, An \Omega\Gamma p lg lg n) Lower Bound for Routing in Optical Networks, Proceedings of the ACM Symposium On Parallel Algorithms and Architectures 6 (1994).

.... The OCPC model The ocpc model was first introduced by Anderson and Miller [2] and Eshaghian and Kumar [10] and has been studied by Valiant [39] Ger eb Graus and Tsantilas [12] Gerbessiotis and Valiant [11] Rao [33] Goldberg, Jerrum, Leighton and Rao [17] and Goldberg, Jerrum and MacKenzie [18]. The feasibility of the ocpc from an engineering point of view is discussed in [2, 12] See also the survey paper of McColl [30] and the references therein. Computing h relation on the OCPC A fundamental problem that deals with contention resolution on the ocpc is that of realizing an h relation. ....

....receive. Following Anderson and Miller [2] Valiant [39] and Ger eb Graus and Tsantilas [12] Goldberg et al. 17] solved the problem in time O(h lg lg n) for an n processor ocpc. A lower bound of Omega Gamma p lg lg n) expected time was recently obtained by Goldberg, Jerrum, and MacKenzie [18]. Simulating PRAMs on OCPCs Valiant described a simulation of an erew pram on an ocpc in [39] More specifically, Valiant gave a constant delay simulation of a Bulk Synchronous Parallel (bsp) computer on the ocpc (there called the s pram) and also gave an O(lg n) randomized simulation of an n ....

L.A. Goldberg, M. Jerrum and P.D. MacKenzie, An \Omega\Gamma p lg lg n) Lower Bound for Routing in Optical Networks, To appear in this journal. (A preliminary version appeared in Proceedings of the ACM Symposium On Parallel Algorithms and Architectures 6 (1994) 147--156.)

.... is an extension and generalization of the Random Restriction technique first used in Furst, Saxe, and Sipser [7] The Random Adversary technique was used to prove lower bounds on randomized CRCW PRAM algorithms in MacKenzie [19] and on randomized OCPC algorithms in Goldberg, Jerrum and MacKenzie [12]. However, this is the first application of the technique to Exclusive Write PRAM algorithms. The Random Adversary technique requires a specific proof structure. Assuming there exists a deterministic lower bound for a problem with this structure, a randomized lower bound sometimes can be obtained ....

L. A. Goldberg, M. Jerrum, and P. D. MacKenzie. An \Omega\Gamma p log log n) lower bound for routing in optical networks. In Proc. 6th ACM Symp. on Para. Alg. and Arch., pages 147--156, 1994.

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