A. Czumaj, F. Meyer auf der Heide, and V. Stemann. Shared memory simulations with triple-logarithmic delay. In Proc. of the 3rd European Symposium on Algorithms, pages 46--59, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....The time required by a parallel access to an arbitrary n tuple of variables is often referred to as the slowdown of the scheme. Numerous randomized and deterministic schemes have been developed in the literature for a number of specific processor networks. Randomized schemes (see e.g. CMS95,Ran91] usually distribute the variables randomly among the memory modules local to the processors. As a consequence of such a scattering, a simple routing strategy is sufficient to access any n tuple of variables efficiently, with high probability. Following this line, we can give a simple, ....

A. Czumaj, F. Meyer auf der Heide, and V. Stemann. Shared memory simulations with triple-logarithmic delay. In Proc. of the 3rd European Symposium on Algorithms, pages 46--59, 1995.

....randomly from smaller families of hash functions; see [KLM96] 1.1. 2 Shared memory emulations on DMMs One of the earliest applications of the two choice paradigm is in the study of algorithms to emulate shared memory machines (as, for example, PRAMs) on distributed memory machines (DMMs) CMS95, KLM96, MSS96] In such emulations, the processors and the memory cells of the shared memory machine are distributed to the processors and memory modules of the DMM using appropriately chosen (universal) hash functions. Typically, the goal of the emulation algorithm is to minimize slowdown, or ....

....Section 4. The rst rigorous analytical demonstration of the power of two choices is due to Karp, Luby, and Meyer auf der Heide [KLM92, KLM96] who considered the possibility of using two hash functions in the context of PRAM emulation by DMMs. Subsequent work on shared memory emulations on DMMs [CMS95, MSS96] has given rise to a powerful technique for analysis called the witness tree method. See Section 3 for more details on this technique. The balls and bins problem has proven to be a fertile ground for investigating the power of two choices. The classical balls and bins problem, where ....

[Article contains additional citation context not shown here]

A. Czumaj, F. Meyer auf der Heide, and V. Stemann. Shared memory simulations with triple-logarithmic delay. Lecture Notes in Computer Science, 979:46-59, 1995.

....randomly from smaller families of hash functions; see [KLM96] 1.1. 2 Shared memory emulations on DMMs One of the earliest applications of the two choice paradigm is in the study of algorithms to emulate shared memory machines (as, for example, PRAMs) on distributed memory machines (DMMs) CMS95, KLM96, MSS96] In such emulations, the processors and the memory cells of the shared memory machine are distributed to the processors and memory modules of the DMM using appropriately chosen (universal) hash functions. Typically, the goal of the emulation algorithm is to minimize slowdown, or ....

....4. The first rigorous analytical demonstration of the power of two choices is due to Karp, Luby, and Meyer auf der Heide [KLM92, KLM96] who considered the possibility of using two hash functions in the context of PRAM emulation by DMMs. Subsequent work on shared memory emulations on DMMs [CMS95, MSS96] has given rise to a powerful technique for analysis called the witness tree method. See Section 3 for more details on this technique. The balls and bins problem has proven to be a fertile ground for investigating the power of two choices. The classical balls and bins problem, where ....

[Article contains additional citation context not shown here]

A. Czumaj, F. Meyer auf der Heide, and V. Stemann. Shared memory simulations with triple-logarithmic delay. Lecture Notes in Computer Science, 979:46--59, 1995.

....shared variables are distributed among the memory modules via one (or more) hash functions randomly drawn from a suitable universal class. Among the most relevant results, we recall that a PRAM step can be simulated, with high probability, in O (log log log n log n) time on the complete network [CMS95], in O (log n) time on the butterfly [Ran91] and in O ( p n) time on the mesh [LMRR94] In contrast, the development of efficient deterministic schemes, that is, schemes that guarantee a fast worst case simulation time for any PRAM step, appears to be much harder. A simple argument shows that in ....

A. Czumaj, F. Meyer auf der Heide, and V. Stemann, Shared memory simulations with triple-logarithmic delay, in Proc. of the 3rd European Symposium on Algorithms, Corfu, Greece, 1995, pp. 46--59.

....the variables to the modules. The distribution properties of these functions yield very efficient simulations in the probabilistic 3 sense. For instance, using a few copies per variable, work efficient simulations exhibiting triply logarithmic slowdown can be achieved with high probability [3]. In contrast, the development of fast deterministic PRAM simulations appears to be much harder. The pioneering work of Mehlhorn and Vishkin [10] introduced the idea of representing each variable by several copies, so that a read operation needs to access only one (the most convenient) copy. For m ....

.... computing the minimum among (2) 3) and (4) 2 Theorem 1 implies that for any nonconstant p and any n = o(m) it is not possible to devise a deterministic algorithm that simulates an (n; m) PRAM step on a p DMM work efficiently in O(n=p) time, which is the case, instead, for randomized simulations [3]. However, the theorem has been proved under the somewhat restrictive assumptions listed at the beginning of this section, so it is not inconceivable that deterministic work efficient simulations might be attainable in a more general setting. In fact, the following subsection shows that this is ....

A. Czumaj, F. Meyer auf der Heide, and V. Stemann. Shared memory simulations with triple-logarithmic delay. In Proc. of the 3rd European Symposium on Algorithms, pages 46--59, 1995.

....shared variables are distributed among the memory modules via one (or more) hash functions randomly drawn from a suitable universal class. Among the most relevant results, we recall that a PRAM step can be simulated, with high probability, in O (log log log n log n) time on the complete network [CMS95], in O (log n) time on the butterfly [Ran91] and in O ( p n) time on the mesh [LMRR94] In contrast, the development of efficient deterministic schemes, that is, schemes that guarantee a fast worst case simulation time for any PRAM step, appears to be much harder. A simple argument shows that ....

A. Czumaj, F. Meyer auf der Heide, and V. Stemann, Shared memory simulations with triple-logarithmic delay, in Proc. of the 3rd European Symposium on Algorithms, Corfu, Greece, 1995, pp. 46--59.

....proc greedy for all processors pardo while processor has packets do choose an unsent packet at random and try to send it Figure 1: Greedy routing algorithm. an h relation for h 2 1 5 p) where p is the number of processors. Other algorithms with even lower latency were proposed by [6, 7, 2, 3, 11]. Contrary to these theoretically strong algorithms, the algorithm of Ger b Graus and Tsantilas [5] in Figure 1 has the advantage of being direct, i.e. the packets are sent to their targets directly, without intermediate nodes. For other results related with direct or indirect routing, see also ....

A. Czumaj and F. Meyer auf der Heide, and V. Stemann. Shared memory simulations with triple-logarithmic delay. Proc. ESA95, 4659, 1995.

....reduces to realizing an h relation. In an h relation, each processor is the source as well as the destination of at most h messages. We assume that each processor knows the value of h, when the routing begins. Algorithms for solving the h relation problem on the OCPC model have been provided by [2, 3, 4, 5, 6, 8, 9, 11]. A goal in routing h relations is to route messages within (h) steps. Algorithms should be optimal for as small h as possible. In Section 2 we present two algorithms for routing h relations, and in Section 3 we present some experimental results. Simulations are done with a routing simulator. 2 ....

A. Czumaj, F. Meyer auf der Heide, and V. Stemann. Shared memory simulations with triple-logarithmic delay. Lecture Notes in Computer Science, 979:4659, 1995.

.... which eliminates the assumption of random inputs, is to distribute the data according to some hash function which is chosen uniformly from a universal class of hash functions [267] Even sharper upper bounds can be obtained if in addition to some clever distribution technique data are copied [57, 64, 87, 136, 209, 217]. In these papers, it is assumed that the network is completely connected (by an optical crossbar switch) Other authors have dealt with bounded degree networks [105, 244] or even with simulations on meshes and mesh like networks [106, 187, 245, 246] More recently the focus appears to be no ....

.... is completely connected (by an optical crossbar switch) Other authors have dealt with bounded degree networks [105, 244] or even with simulations on meshes and mesh like networks [106, 187, 245, 246] More recently the focus appears to be no longer on achieving even faster simulations (in [57], a step of a PRAM with N processors is simulated with a delay of O(log log log N Delta log N) only) but on more diverse aspects such as fault tolerant simulations [25] and simulations on reconfigurable networks [58] 3 If in a given round of a PRAM simulation on a DMM with N nodes, k is ....

Czumaj, A., Meyer auf der Heide, F., and Stemann, V. Shared memory simulations with triplelogarithmic delay. Proc. 3rd European Symp. Alg., Lect. Notes Comput. Sci. 979, 1995, pp. 46--59.

....n rmesh, for r 3. More specifically, lower bounds for crcw pram translate to similar lower bounds on the 2 Theta n rmesh, up to a factor of ff(n) Thus, we obtain as a corollary several new lower bounds for the 2 Theta n rmesh. Postscript Recently, Czumaj, Meyer auf der Heide, and Stemann [13] provided improved simulation algorithms of the erew pram on the dmm. They showed that an (n lg lg lg n lg n) processor erew pram can be simulated on an n processor dmm with O(n lg lg lg n lg n) delay. Combined with Lemma 3.1, this implies a similar simulation result of the erew pram on an ....

....6.1 (simulating pram on rmesh) We have, ffl One step of an (n lg lg lg n lg n) processor crcw pram can be simulated in O(lg lg lg n lg n) time w.h.p. on a collision Linear n rmesh. ffl One step of an n processor crcw pram can be simulated in O(1) time w.h.p. on a arbitrary Linear n rmesh [13]. Theorem 6.2 (self simulating rmesh) The self simulation of a General N rmesh on a Linear p rmesh can be performed w.h.p. in ffl O(lg N (N=p) 2 ) steps on the arbitrary model; and in ffl O(lg N lg lg lg p lg p (N=p) 2 lg p) steps on the collision model. Acknowledgments We thank ....

A. Czumaj, F. Meyer auf der Heide, and V. Stemann. Shared memory simulations with triple-logarithmic delay. In Proc. European Symp. on Algorithms, Corfu, September 1995. (To appear).

....OCPC are equivalent. If these machines may have memory faults, then the DMM is weaker because the processors cannot exchange messages directly, the only way to communicate is via the memory. Related research. Simulations of the PRAM on a DMM were given by Czumaj, Meyer auf der Heide and Stemann [4], Dietzfelbinger and Meyer auf der Heide [5] Karp, Luby and Meyer auf der Heide [8] Mehlhorn and Vishkin [13] Meyer auf der Heide, Scheideler and Stemann [15] see also the survey article [14] and the references therein. All these simulations assumed a fully operational DMM. There has been a ....

A. Czumaj, F. Meyer auf der Heide, and V. Stemann, Shared Memory Simulations with Triple-Logarithmic Delay, in Proceedings of the 3rd Annual European Symposium on Algorithms, 1995, Springer LNCS 979, pp. 46--59.

....email Ville.Leppanen cs.utu. Another routing algorithm was proposed by Anderson and Miller [1] and it was improved by Valiant [15] They realize work optimally an h relation for h 2 1 4 p) where p is the number of processors. Other algorithms with even lower latency were provided by [6, 7, 2, 3, 11]. Contrary to these, the h relation algorithm of Ger b Graus and Tsantilas [5] has the advantage of being direct, i.e. the packets go to their target directly, without intermediate nodes. For other results related with direct or indirect routing, see also [4, 14, 8, 9, 12, 13] The algorithm of ....

A. Czumaj and F. Meyer auf der Heide, and V. Stemann. Shared memory simulations with triple-logarithmic delay. Proc. ESA95, 4659, 1995.

....n rmesh, for r 3. More specifically, lower bounds for crcw pram translate to similar lower bounds on the 2 Theta n rmesh, up to a factor of ff(n) Thus, we obtain as a corollary several new lower bounds for the 2 Theta n rmesh. Postscript. Recently, Czumaj, Meyer auf der Heide, and Stemann [13] provided improved simulation algorithms of the erew pram on the dmm. They showed that an (n lg lg lg n lg n) processor erew pram can be simulated on an n processor dmm with O(n lg lg lg n lg n) delay. Combined with Lemma 3.1, this implies a similar simulation result of the erew pram on an ....

....pram on an n rmesh. Specifically, we obtain that an (n lg lg lg n lg n) processor erew pram can be simulated on an n processor collision rmesh with O(lg lg lg n lg n) delay. Similarly to our approach in Section 3. 2, we observe that the factor of (lg n) in the erew pram simulation of [13] is due to the usage of crcw pram O(lg n) time algorithms for problems such as approximate prefix sums and approximate integer sorting [22] The above statement should be verified (by looking at their paper) We replace these algorithms by constant time rmesh algorithms to derive an ....

[Article contains additional citation context not shown here]

A. Czumaj, F. Meyer auf der Heide, and V. Stemann. Shared memory simulations with triplelogarithmic delay. Manuscript, January 1995.

....Ville.Leppanen cs.utu. Another routing algorithm was proposed by Anderson and Miller [1] and it was improved by Valiant [16] They realize work optimally an h relation for h 2 Omega Gamma1 4 p) where p is the number of processors. Other algorithms with even lower latency were provided by [6, 7, 2, 3, 11]. Contrary to these, the h relation algorithm of Ger#b Graus and Tsantilas [5] has the advantage of being direct, i.e. the packets go to their target directly, without intermediate nodes. For other results related with direct or indirect routing, see also [4, 15, 8, 9, 12, 14] The algorithm of ....

A. Czumaj and F. Meyer auf der Heide, and V. Stemann. Shared memory simulations with triple-logarithmic delay. Proc. ESA95, 4659, 1995.

.... is ffi = Theta(OE) If the total routing capacity is P ( packets per physical processor per step) two necessary conditions for time processor optimality are that the load = N=P (parallel slackness factor) is = Omega (maxfOE; flg) and = Omega (OE) Many such solutions can be derived [3, 4, 5, 6, 9]. Often when simulation results are reported, the asymptotic complexity of simulation time is highlighted while other aspects are almost ignored. Better results can be obtained by assuming stronger graph theoretical properties (larger degree Delta, smaller OE) and or stronger (shared resource) ....

A. Czumaj, F. Meyer auf der Heide, and V. Stemann. Shared Memory Simulations with Triple-Logarithmic Delay. In Proceedings of ESA'95, 46 -- 59, 1995.

....same processor, they always fail. Another routing algorithm was proposed by Anderson and Miller [1] and it was improved by Valiant [15] They realize work optimally an h relation for h 2 (log p) where p is the number of processors. Other algorithms with even lower latency were provided by [6, 7, 2, 3, 11]. Contrary to these, the h relation algorithm of Ger b Graus and Tsantilas [5] GGT for short, has the advantage of being direct, i.e. the packets go to their targets directly, without intermediate nodes. For other results related with direct or indirect routing, see also [4, 14, 8, 9, 12, 13] ....

A. Czumaj and F. Meyer auf der Heide, and V. Stemann. Shared memory simulations with triple-logarithmic delay. Proc. ESA95, 4659, 1995.

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Czumaj, A., Meyer auf der Heide, F., and Stemann, V. (1995c), "Shared memory simulations with triple-logarithmic delay," In Proceedings of the 3rd Annual European Symposium on Algorithms, pages 46--59.

....cost of a triangle is the product of the weights at each vertex of the triangle (see also Figure 4. 1) Transformation from one problem to another can be done in a linear sequential time (Hu and Shing, 1980; Hu and Shing, 1982) and also in O(log n) parallel time with linear work on the CREW PRAM (Czumaj, 50 50 100 1 10 1 100 10 20 20 100.000 1.000 1.000 5.000 20.000 200 Figure 4.1: Geometric representation of the evaluation of a matrix chain. The polygons above correspond to the chain M 1 Theta M 2 Theta M 3 Theta M 4 , where dimensions are as follows 50 Theta 20 Theta 10 Theta 100 Theta 1. The ....

Czumaj, A., Meyer auf der Heide, F., and Stemann, V. (1995c), "Shared memory simulations with triple-logarithmic delay," In Proceedings of the 3rd Annual European Symposium on Algorithms, volume 979 of Lecture Notes in Computer Science, pages 46--59, Springer-Verlag.

....size, each being almost a tree. Thus, the above computation of the schedule in fact consists of independent computations on the small trees, which can be shown to only need O(log(maximum size of the tree) O(log log n) phases. More dedicated analyses of the access graph can be found in [6,5,7]. Using very complicated algorithmic tricks, strategies are developed that need approximately log log log n phases to compute a schedule with constant contention. 3 Scenario 2: Low Bandwidth Systems In this section, we focus on data management in parallel processor systems in which the processors ....

A. Czumaj, F. Meyer auf der Heide, and V. Stemann. Shared memory simulations with triple-logarithmic delay. In Proc. of the 3rd European Symposium on Algorithms (ESA), pages 46--59, 1995.

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A. Czumaj, F. Meyer auf der Heide, and V. Stemann, Shared memory simulations with triple-logarithmic delay, in Proceedings of the 3rd Annual European Symposium on Algorithms, vol. 979 of Lecture Notes in Computer Science, Springer-Verlag, 1995, pp. 46--59.

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