H. Attiya, M. Snir and M. Warmuth, Computing on an Anonymous Ring, Journal of the ACM, 35 (1988), 845-875. 21  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

....author s visit at INRIA Sophia Antipolis. 1 Introduction 1.1 The problem Designing network algorithms without complete information about the network is an important problem whose many variations have been extensively studied. Of particular interest are computations in anonymous networks (cf. [2, 3, 4, 8, 10, 12, 13, 16, 17, 18, 19]) in which processors do not have distinct identities and execute identical algorithms. The impossibility of distinguishing processors yields symmetry in computations and restricts the computational power of the network. The situation is even more drastic if the topology of the network and even ....

H. Attiya, M. Snir and M. Warmuth, Computing on an Anonymous Ring, Journal of the ACM 35, (1988), 845-875.

....attention to computing in anonymous networks. In such networks, nodes are unlabeled. However, the links incident to each node are given distinct local labels between 1 and d where d is the degree of the node. The argument supporting the need for such an edgelabeling is often stated as (see, e.g. [2, 3, 8, 13, 16, 22]) if links would be unlabeled, then one could not distinguish them . This armation is correct. However, distinctness is one thing, and comparability is another. For instance, the streets out going from the A o o (Akropolis) are labeled with labels from a totally ordered set, but this total ....

H. Attiya, M. Snir, and M. Warmuth, Computing on an anonymous ring. Journal of the ACM 35:845-875, 1988.

....symmetry in computations and restricts the power of the network both in terms of the class of functions that can be computed and the time and cost of computations. The study of anonymous networks was initiated in [1] and then pursued by many authors, both for specific networks, such as rings [2, 3, 10] and hypercubes [8] and for arbitrary networks [4, 9, 13] Among the problems studied in this context are the following: Which functions a given anonymous network can compute What is the message and bit complexity of such computation Is it possible to perform leader election in the network An ....

H. Attiya, M. Snir and M. Warmuth, Computing on an Anonymous Ring, Journal of the ACM, 35 (1988), 845-875.

....theory is used in the proof of our upper bound, rather than its traditional application to the proof of lower bounds. 1 Introduction There have been several studies in the literature concerning the construction of communication efficient algorithms for computing functions on anonymous rings [2, 4, 5, 6, 1, 10], as well as on more general anonymous networks, like tori [3] hypercubes [7] Cayley networks [8] etc. In general, studies of the bit complexity of computing boolean functions of the inputs mainly resort to input collection before determining the output of the boolean function. In particular, ....

....anonymous networks, like tori [3] hypercubes [7] Cayley networks [8] etc. In general, studies of the bit complexity of computing boolean functions of the inputs mainly resort to input collection before determining the output of the boolean function. In particular, Attiya, Snir and Warmuth [2] showed that all the functions that are computable on an anonymous asynchronous ring of processors, can be computed using O(n ) messages (one bit message if the functions are boolean) using input collection. They also proved that every algorithm that computes the minimum of all inputs (the OR ....

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H. Attiya, M. Snir and M. Warmuth, "Computing on an Anonymous Ring", Journal of the ACM, 35 (4), 845-875, (1988).

....for Future Technologies) 1 Introduction Due to the simplicity of its topology (small number of links and low branching) the ring has been the focus of investigation in several papers on distributed computing. Issues studied include message and bit complexity of computing boolean functions [3, 4, 10], computations like extrema finding [7, 8] leader election [1, 2, 9, 11] orientation [3] symmetry breaking [12] etc. The ring model (unidirectional and bidirectional) currently used in the literature is hardware based. Each processor has physical links to both its neighbors. Within this ....

....of links and low branching) the ring has been the focus of investigation in several papers on distributed computing. Issues studied include message and bit complexity of computing boolean functions [3, 4, 10] computations like extrema finding [7, 8] leader election [1, 2, 9, 11] orientation [3], symmetry breaking [12] etc. The ring model (unidirectional and bidirectional) currently used in the literature is hardware based. Each processor has physical links to both its neighbors. Within this framework there are already several variants of the model and hence of the problems previously ....

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H. Attiya, M. Snir and M. Warmuth, "Computing on an Anonymous Ring", Journal of the ACM, 35 (4), 1988.

....symmetry in computations and restricts the power of the network both in terms of the class of functions that can be computed and the time and cost of computations. The study of anonymous networks was initiated in [1] and then pursued by many authors, both for specific networks, such as rings [2, 3, 11] and hypercubes [9] and for arbitrary networks [4, 10, 15] Among the problems studied in this context are the following: Which functions a given anonymous network can compute What is the message and bit complexity of such computation Is it possible to perform leader election in the network An ....

....An even more severe restriction on the amount of knowledge available to processors is the lack of sense of direction (cf. 12, 5] when there is no consistent labeling of links. For rings, the sense of direction is equivalent to orientation and its computational aspects were studied, e.g. in [2]. Networks in which neither links nor nodes have a priori assigned labels, are obviously both anonymous and devoid of sense of direction. We will call such networks totally unlabeled. In this paper we consider one of the fundamental tasks in distributed computing: broadcasting. See, e.g. the ....

H. Attiya, M. Snir and M. Warmuth, Computing on an Anonymous Ring, Journal of the ACM, 35 (1988), 845-875.

....symmetry in computations and restricts the power of the network both in terms of the class of functions that can be computed and the time and cost of computations. The study of anonymous networks was initiated in [1] and then pursued by many authors, both for specific networks, such as rings [2, 3, 14] and hypercubes [10] and for arbitrary networks [4, 11, 18] Instytut Informatyki, Uniwersytet Warszawski, Banacha 2, 02 097 Warszawa, Poland. E mail: diks mimuw.edu.pl This work was done during the author s visit at the Universit e du Qu ebec a Hull. Institute of Informatics, Faculty of ....

....An even more severe restriction on the amount of knowledge available to processors is the lack of sense of direction (cf. 15, 7] when there is no consistent labeling of links. For rings, the sense of direction is equivalent to orientation and its computational aspects were studied, e.g. in [2]. Totally unlabeled networks, those in which neither links nor nodes have a priori assigned labels, are obviously both anonymous and devoid of sense of direction. One of the fundamental tasks in distributed computing is broadcasting. See, e.g. the survey [8] for references to the literature on ....

H. Attiya, M. Snir and M. Warmuth, Computing on an Anonymous Ring, Journal of the ACM, 35 (1988), 845-875.

....We de ne similarly n ( for the more general case of hypercubes with at most faulty components. 1.3 Results of the paper Previous results on computing Boolean functions on asynchronous, anonymous, labeled networks can be summarized as follows. Network Bit Complexity Paper Rings O(N ) [3] n Tori, n constant O(N ) 4] Hypercubes: 0 O(N log N) 8] Hypercubes: 1 O(N log log N) This paper The result of [3] is valid both for oriented as well as unoriented rings. The result of [4] is valid for n dimensional tori, where n is a constant (independent of the number ....

....on computing Boolean functions on asynchronous, anonymous, labeled networks can be summarized as follows. Network Bit Complexity Paper Rings O(N ) 3] n Tori, n constant O(N ) 4] Hypercubes: 0 O(N log N) 8] Hypercubes: 1 O(N log log N) This paper The result of [3] is valid both for oriented as well as unoriented rings. The result of [4] is valid for n dimensional tori, where n is a constant (independent of the number of nodes) Moreover, the constant implicit in the bit complexity bound O(N ) depends on n [4] Hence this result cannot apply to the ....

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H. Attiya and M. Snir and M. Warmuth, \Computing on an Anonymous Ring", Journal of the ACM, 35 (4),

....algorithm (its time complexity depending exponentially on the ID s actually in use) with only O(n) messages. This algorithm does not seem to be very interesting in itself, but it is interesting because it demonstrates the need for the assumptions in the lower bound. Attiya, Snir and Warmuth, in [14] used similar ideas to those in [58] but took them much further. They considered the case where there are no ID s built in, but (for certain problems) processes may start with input values. The object is for the processes to compute some function (invariant under circular shifts) of the input ....

....it only yields a lower bound of ] nlogn) because of the possible utility of null messages. Now a stronger definition is needed for fooling pair, in which both x and must be very symmetric. Then it can be shown that the algorithm causes many messages to be sent in both and The lemmas used in [14] are slightly different from those used in [58] instead of analyzing chains of mes sages in detail, they are stated in terms of less tailed information about the number of rounds at which some message is sent. As in [58] much effort is devoted here to producing the strong symmetries needed for ....

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Hagit Attiya, Marc Shit, and Manfred K. War- muth. Computing on an anonymous ring. Oc- tober 1988.

....of IDs to processors that exhibits a large amount of replication symmetry around the ting. We give a relatively simple assignment of values if n is a power of 2, and a somewhat more involved assignment for general values of n. More recently, a different assignment of IDs has been given in [2]. Alternatively, if the number of rounds is required to be bounded by some t in the worst case, then there is a (very fast growing) function f(n, t) which has the following very interesting property. If IDs are chosen from any set T having at least f(n, t) elements, then any t bounded algorithm ....

ATTIVA, C., SNIR, M., AND WARMUT}, M. Computing on an anonymous ring. In Proceedings of the 4th Annual ACM Symposium on Principles of Distributed Computing (Minaki, Ontario, Canada, Aug. 5-7). ACM, New York, 1985, pp. 196-203.

.... left to right from i to f(t, b) 2. each internal node is labeled by the largest integer in its left subtree, and 3. the left branches axe labeled and the right branches are labeled . 0 i 2 3 = 15. The resulting [d,3] bs decision tree is shown in Figure 1. E Figure 1: The [4,3] bs decision tree for solving the , 4, 3 game 3.1.3 The algorithm It follows from Theorem 2 and its proof that the optimal solution strategy for determining the unknown number in a , t, b game is just the searching process in a [t, b] bs decision tree. The first question asked in the , ....

....attentive reader might have already observed that, in Sections 2 and 4, knowledge of n can be replaced by knowledge of the diameter (G) of the network; in such a case, G) can replace n in the time bounds stated by Theorems 5 and 6. That some knowledge of 5(G) is needed is implied by a result in [3], stating that in an anonymous ring network, non constant functions (e.g. min) cannot be computed without any knowledge of the ring size. However, exact knowledge of either (G) or n is not actually required; any value rn (G) would do in the proposed algorithm, provided that this value is ....

C. Attiya, M. Snir, M. Warmuth, Computing on an anonymous ring, in: Proc. 4th Ann. ACM Syrup. on Principles of Distributed Computing, Aug. 1985, 196-204.

....that the network is asynchronous and that initially processors have exactly one input bit but are further identical. We study the bit and message complexity of computing non constant functions on this network. These problems have been studied to a large extent for the ring network (see e.g. [1, 2, 4, 5]) As the torus can be seen as a 2 dimensional version of the ring, it is interesting to compare the results. A large part of this research was done while both authors were at the Laboratory for Computer Science of the Massachusetts Institute of Technology. tAuthor s present address: Computer ....

....result with a very similar algorithm. Theorem 3.2 Every one bit input function on the k x I torus can be computed with bit compleit t O(k . I . max(k, Lower bounds The following lower bound results can be obtained in the same way as similar restfits for the ring, by Attiya, Snir and Warmuth [2], or by an easy simulation on the ring. Theorem 3.3 AND, OR, XOR, SUM have message complexitel fi(nv d ) on anontmous v d x v d tori, and message complexitel fi(k. I. max k, l) on anontmous k x I tori. Theorem 3.4 The probabilitt that a random computable Boolean function on the anontmous ....

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' H. Attiya, M. Snir, and M. Warmuth. Computing on an anonymous ring. In Proc. 4th Ann ACM Slmp. Principles of Distributed Computing, pages 196-203, 1985.

....the theory of graph coverings, which was going to provide, in particular with the work of Yamashita and Kameda [YK96] several characterization for problems which are solvable under certain topological constraints. Further investigation led to the classification of computable functions [YK96,YK98,ASW88,Nor96] and allowed to eliminate several restrictions (such as bidirectionality, distinguished links, synchronicity, DKMP95,BCG 96,BV97a] Few years ago, while lecturing at the Weizmann Institute about possibility and impossibility results for function computation [BV97a] we were asked ....

....a bound is not known, a completely different approach is needed, as shown in [BV99] Our results are mainly of theoretical interest, because of the large amount of information exchanged by the processors. Moreover, complexity issues are not taken into consideration, as opposed, for instance, to [ASW88,ANIM96] We rather concentrate on general decidability properties that hold under any assumption of knowledge or of communication primitives (broadcast, point to point, etc. Our networks are just directed graphs coloured on their arcs (information such as processor identity, communication ....

Hagit Attiya, Marc Snir, and Manfred K. Warmuth. Computing on an anonymous ring. J. Assoc. Comput. Mach., 35(4):845--875, 1988.

....up. They have shown that a synchronous network of N processors can reach unison within 2N synchronous rounds. Gouda and Herman [4] present a solution to the unison problem which is also stabilizing, i.e. the system is guaranteed to reach unison starting from any state. Attiya, Snir and Warmuth [1] consider the processor synchronization problem on anonymous, synchronous rings in order to reduce input collection and orientation algorithms to the case where all processors start simultaneously, and solve this problem in O(N logN) messages. Singh [6] has shown a lower bound on time message ....

Attyia, H. -- Snir, M -- Warmuth, M.: Computing on an Anonymous Ring, Journal of the ACM, 35 (1998), pp. 845--875.

....an opportunity to study a single problem under many di#erent assumptions, but no general principles have yet emerged. Other Problems Other static problems include the computation of functions, such as the median and other order statistics, where the inputs are initially distributed. Attiya et al. [ASW88] and Abrahamson et al. AAHK86] have obtained especially interesting upper and lower bound results many surprisingly tight about the number of messages required to compute functions in a ring. 44 4.3.2 Dynamic Algorithms Distributed Termination In this problem, each process is either active ....

Hagit Attiya, Marc Snir, and Manfred K. Warmuth. Computing on an anonymous ring. Journal of the ACM, 35(4):845--875, October 1988.

....between the number of tokens at any two nodes is at most one. In this paper, we study static load balancing on a ring network. The ring network has been studied extensively in both theory and practice. Several problems arising in distributed computing have been addressed on the ring (see [5, 13, 14, 19] for a variety of examples) From a practical perspective, the ring is an essential component of several parallel and distributed architectures [17, 25] Our main contribution is a tight analysis of a simple algorithm that is based on the local balancing approach. We show that this algorithm, ....

H. Attiya, M. Snir, and M. Warmuth. Computing on an anonymous ring. Journal of the ACM, 35:845--875, 1988.

....all addition on processor indices is done mod m, i.e. processor p m i is identical to p i ) The ring is an important network in both theory and practice. From a theoretical perspective, the ring is a basic network structure and much work has been done developing and analyzing algorithms for it [ASW88, HP89, Lei91, MS90] From a practical perspective, the ring is either the foundation of or a component of many parallel architectures [Hut88, Rot92, Tel94] In this chapter, we assume that there are m machines, n unit size jobs and that the capacity of each network link is one. We begin by ....

H. Attiya, M. Snir, and M. Warmuth. Computing on an anonymous ring. Journal of the ACM, 35(4):845--875, 1988.

....for any useful computation on the network. The aim of this kind of research is to characterize networks by their bit complexity. In [MW] Moran and Warmuth studied the bit complexity of a ring of n processore under the assumption that all the processors in the ring are identical (anonymous [ASW]) i.e. all processors run the same program and the only parameter to the program is the input of the processor. It was shown that for anonymous rings it takes 2(n logn) bits to compute any non constant function. We would like to release the assumption that the processors are anonymous by allowing ....

C. Attiya, M. Snir and M. K. Warmuth, "Computing on an anonymous ring," extented abstract in Proceedings PODC 1985, p. 161-173, 1985, to appear in JACM, October 1988.

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H. Attiya, M. Snir and M. Warmuth, Computing on an Anonymous Ring, Journal of the ACM, 35 (1988), 845-875. 21

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H. Attiya and M. Snir and M. K. Warmuth, Computing on an Anonymous Ring, JACM, 35 (4), 845--875, Oct., 1988.

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H. Attiya, M. Snir, and M. Warmuth. Computing on an anonymous ring. Journal of the ACM, 35(4):845-875, 1988.

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H. Attiya, M. Snir, and M. Warmuth. Computing on an anonymous ring. Journal of the ACM 35:845-875, 1988.

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H. Attiya, M. Snir, and M. K. Warmuth. Computing on an anonymous ring. J. ACM, 35(4):845--875, Oct. 1988.

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Hagit Attiya, Marc Snir, and Manfred K. Warmuth. Computing on an anonymous ring. Journal of the ACM, 35(4), pages 845--875, October 1988.

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H. Attiya, M. Snir, and M. Warmuth, "Computing on an Anonymous Ring", Journal of the ACM, 35(4), Oct. 1988, pp.845-875.

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