F.E. Fich, F. Meyer auf der Heide, and A. Wigderson, Lower bounds for Parallel random-access machines with unbounded shared memory, Advances in Computing Research 4 (1986), 1--15.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....two models, as algorithm designers prefer the power of the Priority model and for computer architects the Common model is easier to implement in hardware. The element distinctness problem is solvable in constant time on a Priority[n] using n processors) Fish, Meyer auf der Heide and Wigderson [109] showed that a Common[n] machine (using n processors) solving Element Distinctness[n] with n integers) requires##qui log log n) time. Ragde, Steiger, Szemeredi and Wigderson [210] improved the lower bound to ## # log n) Boppana improved further these results to the best possible lower bound, ....

F.E. Fich, F. Meyer auf der Heide, and A. Wigderson, Lower bounds for Parallel random-access machines with unbounded shared memory, Advances in Computing Research 4 (1986), 1--15.

....PRAM, are: 1. Sorting requires Omega Gamma n=p log d(p=n) log n 1e n) time [9, 19] 2. Element distinctness requires Omega Gamma n=p log d(p=n) log n 1e n) time if the memory size is bounded [9, 14, 21] 3. Finding the maximum and merging require Omega Gamma n=p log log dp=n 1e n) time [17, 22]. 4. String matching and some related problems on strings require, assuming bounded memory size, Omega Gamma n=p log log dp=n 1e n) time. 2 5. Finding an approximate maximum, namely, an element whose rank belongs in the top ffln ranks, requires Omega Gamma n=p log log dp=n 1e (1=ffl) ....

....dp=n 1e n) lower bounds for the PRAM model. Lemma 5.2 The Omega Gammae 1 log dp=n 1e n) lower bounds are resilient. The comparison model lower bounds for finding the maximum and merging hold under the distinctness assumption. Hence, these lower bounds can be transformed into PRAM lower bounds [17, 22]. The comparison model lower bounds for string matching and the related problems mentioned above do not hold under the distinctness assumption, and therefore, these lower bounds translate only to lower bounds in the PRAM with bounded memory. 8 5.3 Finding an approximate maximum Alon and Azar ....

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F.E. Fich, F. Meyer auf der Heide, and A. Wigderson. Lower bounds for parallel random-access machines with unbounded shared memory. Advances in Computing Research, 4:1--15, 1987.

....BOUND FOR ROUTING IN OPTICAL NETWORKS 1089 2. The (t, f) knowledge set of each processor p has size at most one. 3. Each processor q is in the (t, f) knowledge set of at most k t processors. Condition 2 captures a crucial idea, which can be traced to Fich, Meyer auf der Heide, and Wigderson [8], and may be expressed informally as follows. Suppose that A is run on input g, where g is a 2 relation that refines f . Then the entire state of the n OCPC at time t depends in a particularly simple way on the restriction of g to the processors p with f(p) #. 3.4. Refining partial ....

....3 sets AFFECTS(p) Thus each p # W ## j is an endpoint of at most 6k t edges in E and therefore E # 3k t W ## j . We conclude that W j # W ## j 7k t . A construction similar to the one used in the proof of Claim 3. 11 was used by Fich, Meyer auf der Heide, and Wigderson [8]. Corollary 3.12. If f is t good then for each j # J we have W j # w t . Proof. Since f is t good, S j = s t . Then the corollary follows from Claims 3.9, 3.10, and 3.11. Claim 3.13. If f is t good then the number of groups used by algorithm CONSTRUCT is w 4 7 t . Proof. ....

F. E. Fich, F. Meyer auf der Heide, and A. Wigderson, Lower bounds for parallel randomaccess machines with unbounded shared memory, Advances in Computing Research 4, F. Preparata, Ed., JAI Press, Greenwich, CT, 1987, pp. 1--15.

....minimizing the time required to multicast when internode latency can vary from node pair to node pair. Other work on sorting in the PRAM(m) model includes [2] Sorting in the BSP, another limited bandwidth model, has been studied in [17] and [18] Other work in the PRAM(m) model includes [6] 8] [15], and [23] Variants of the P PRAM(m) are studied in [4] and [16] and a related model is studied in [3] One other paper that uses related ideas from Ramsey theory for compression is [11] where Chandra, Furst and Lipton introduce multi party parallel computation in the number on the forehead ....

F. Fich, M. Li, P. Ragde, and Y. Yesha. Lower bounds for parallel random access machines with read only memory. Information and Computation, 83(2):234-244, November 1989.

....steps. Also, the same authors show in [ACS89] that sorting requires time Omega Gamma n log n p l log p) in a model where reading or writing a block of size b from memory takes time l b. The PRAM(m) model was introduced in [VW85] and has been studied subsequently in [LY86] FRW88] [FLRY89], Aza92] MNV94] Adl96] and [BFS97] The case where n AE p was first examined in [MNV94] where Mansour, Nisan and Vishkin prove a lower bound of Omega Gamma n p mp ) for several problems, including sorting, in a concurrent read version of the PRAM(m) which implies the same bound in the ....

F. Fich, M. Li, P. Ragde, and Y. Yesha, Lower Bounds for Parallel Random Access Machines with Read Only Memory,, Information and Computation, vol. 83, no. 2, November 1989, pages 234-244.

....can be considered as a PRAM with a single shared memory cell. They also point out that in [GGKMRS] K] and [V] it is implied that the size of the shared memory may determine the hardware feasibility of the parallel machine. More lower bounds for PRAM with small shared memory appear in [B] [FMW], FRW] and others. Li and Yesha made more progress on analyzing the complexity of threshold languages in this model. In [LY2] they considered the threshold languages L g for g = o(n) They noticed that the Omega Gamma p g) lower bound (m = 1) can be matched using Gamma n g 0:5 Delta ....

....we cannot recognize L g in less than g=k steps which completes the proof of the main theorem. 9 4. The Complexity of Computing Symmetric Functions Comparing the relative power of models is known to be an important question. It is known that PRIORITY is strictly stronger than ARBITRARY (see [FMW], FRW] for models without a ROM and [LY] LY1] FLRY] for models with a ROM) The question is whether this is true for symmetric functions. Note that there is a strong connection between the threshold decision problem and computing symmetric functions. In [LY1] it is shown that PRIORITY(1) and ....

F. Fich, F. Meyer auf der Heide, and A. Wigderson, Lower bounds for parallel random access machines with unbounded shared memory, Advances in Computing Research, 1987 pp. 1-15.

....log p) time on the two dimensional sub bus mesh. From the work of Hao, MacKenzie and Stout [12] a lower bound of Omega# log log p) is obtained for computing MINIMUM on a two dimensional sub bus mesh. Their proof is based on a PRAM simulation of the mesh model, and applies a result of Fich et al. [11] in which an equivalent lower bound is proved for the CRCW PRAM. However, this proof requires that the inputs be very large. Another lower bound of Omega# log log p) for computing MINIMUM on the two dimensional sub bus mesh follows from the general lower bound for the parallel comparison model of ....

F. E. Fich, F. Meyer auf der Heide, P. Ragde and A. Wigderson. Lower bounds for parallel random access machines with unbounded shared memory. Advances in Computing Research, Vol. 4, pp. 1-15, 1987.

....algorithm. They include: 1) OR and AND functions, 2) maximum finding for small integers [FRW 88] Small integers are integers in the range [1, n c ] for some constant c . 3) finding whether n integers drawn from the domain [1. M ] are pairwise distinct given a memory of size M [FMW 87], and (4) logn coloring of a cycle [CV86 ] Until recently only two other highly parallelizable problems were known: general) maximum finding and merging of sorted arrays. In both cases, Valiant [Va 75] first described an O (loglogn ) algorithm in the parallel comparison model. His algorithms ....

F.E. Fich, F. Meyer auf der Heide and A. Wigderson, "Lower bounds for parallel random access machines with unbounded shared memory", Advances in Computer Research, 4 (1987), 1-15.

....m j ; Note that at Line 2, exactly one processor satisfies the condition m j = i and there can be no write conflict. By definition, the maximum of n numbers can be computed in O(1) time with n processors on a Maximum CRCW PRAM. But on a Priority CRCW PRAM, Fich, Meyer auf der Heide, and Wigderson [16] showed that the computation requires Omega Gammaqui log n) time with n processors. We can now conclude the following. Theorem 4 A Maximum (or equivalently, Minimum) CRCW PRAM can simulate a Priority CRCW PRAM within the same time and processor bounds, but not vice versa. A CRCW PRAM is called ....

F. Fich, F. Meyer auf der Heide, and A. Wigderson. Lower bounds for parallel random access machines with unbounded shared memory. In F. Preparata, editor, Advances in Computing Research, volume 4, pages 1--15. JAI Press, 1987.

....Sometimes, when a problem is restricted to a small domain, the time required to solve it strictly decreases. For example, consider the problem of finding the maximum element of a set of n numbers. This problem has time complexity Theta(log log n) on PRIORITY or COMMON PRAM for general inputs [FMW86], but when the elements composing the input are restricted to lie within the range [1: n k ] it can be done in O(k) time [FRW88] The parallel random access machine (PRAM) is a natural model of parallel computation that is used both for algorithm design and for obtaining lower bounds. On this ....

....When both models have p = n processors, FRW88 2] show that the one step of PRIORITY can be simulated in O (log n= log log n) time steps on COMMON. B89] and [R] independently showed how these algorithms could be combined giving a tradeoff between n and p. Fich, Meyer auf der Heide, and Wigderson [FMW86] first separated the models using the Element Distinctness problem, a problem closely related to sorting. An input hx 1 ; x n i 2 [1: d] n is said to be element distinct if each variable x i has a distinct value. In other words, for all i; j 2 [1: n] if i 6= j then x i 6= x j . This ....

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F. E. Fich, F. Meyer Auf Der Heide, and A. Wigderson, "Lower bounds for parallel random-access machines with unbounded shared memory," Advances in Computing Research., vol. 4, pp. 1-15, 1986.

....the input variables are known to be all distinct can be converted into a corresponding lower bound in the priority CRCW PRAM with infinite memory. The proof techniques used are standard multivariable Ramsey theoretic arguments that were developed by several authors for studying specific problems [9,19,21 24]. The main idea is that one can restrict the original input domain in such a way that the processors must communicate in a manner that depends only on the relative order between the input variables. This implies that the PRAM can only determine the relations between input variables that were ....

.... for input of size n on a p processor PRAM, are: 1) Sorting requires f(n p logi(p log 1 n) time [9,21] 2) Element distinctness requires fl(n p 1ogi(v n)log , 1 n) time if the memory size is bounded [9,16] 3) Finding the maximum and merging require fl( n p log Iogfp time [ 19,24]. 4) String matching and some related problems on strings require fl(n p log log I , ln) time, if the memory size is bounded. 5) Finding an approximate maximum, namely, an element whose rank belongs in the top en ranks, requires s2(n p log logi. l(1 e) tog n log (p n) time ....

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F.E. Fich, F. Meyer auf der Heide and A. Wigderson, Lower bounds for parallel random-access machines with unbounded shared memory, Adv. in Cornput. Research 4 (1987) 1-15.

....n Gamma log p) steps to be computed by an EREW PRAM with p processors. Lower bounds for a number of other problems have been obtained using the Ramsey theory technique. The ELEMENT DISTINCTNESS problem 21.4 Lower Bound Techniques 45 was studied by Fich, Meyer auf der Heide, and Wigderson [FMW87] They showed that COMMON with n processors and an infinite amount of shared memory requires Omega Gammaqui log log n) steps to solve this problem for a domain of size 2 Omega Gamma n log n) Ragde, Steiger, Szemeredi, and Wigderson [RSSW88] improved the lower bound on time to ....

F. Fich, F. Meyer auf der Heide, and A. Wigderson. Lower bounds for parallel random access machines with unbounded shared memory. In F. Preparata, editor, Advances in Computing Research, volume 4, pages 1--15. JAI Press Inc., 1987.

....first n 0 shared memory cells at the end of the computation When the number of shared memory cells is too small, the input values can be distributed approximately equally among the processors. Another possibility is to have a separate, read only memory containing the input values [VW85, LY89, FLRY89] In this case, each processor is also allowed to read from one of the n read only memory cells during each computation step. EXERCISE 21.1 Explain why it doesn t matter whether the input is initially located in the shared memory, the processors local memories, or a separate readonly memory, ....

.... processors and m shared memory cells by COMMON with p processors and m shared memory cells and Omega Gamma 104 p=m) log log(p=m) steps are known to be required to simulate PRIORITY with p processors and m shared memory cells by ARBITRARY with m shared memory cells and any number of processors [FLRY89] In contrast, PRIORITY and ARBITRARY require the same amount of time to compute any symmetric Boolean function using one shared memory cell [LY86, LY87] 21.3 Relationships Between PRAMs and Other Models The PRAM is closely related to other models of computation. This section considers ....

[Article contains additional citation context not shown here]

F. Fich, M. Li, P. Ragde, and Y. Yesha. Lower bounds for parallel random access machines with read only memory. Information and Computation, 83(2):234--244, November 1989.

....the input variables are known to be all distinct can be converted into a corresponding lower bound in the priority CRCWPRAM with infinite memory. The proof techniques used are standard multi variable Ramsey theoretic arguments that were developed by several authors for studying specific problems [9, 19, 21, 22, 23, 24]. The main idea is that one can restrict the original input domain in such a way that the processors must communicate in a manner that depends only on the relative order between the input variables. This implies that the PRAM can only determine the relations between input variables that were ....

....are: 1. Sorting requires Omega Gamma n=p log d(p=n) log n 1e n) time [9, 21] 2. Element distinctness requires Omega Gamma n=p log d(p=n) log n 1e n) time if the memory size is bounded [9, 16] 2 3. Finding the maximum and merging require Omega Gamma n=p log log dp=n 1e n) time [19, 24]. 4. String matching and some related problems on strings require Omega Gamma n=p log log dp=n 1e n) time, if the memory size is bounded. 5. Finding an approximate maximum, namely, an element whose rank belongs in the top ffln ranks, requires Omega Gamma n=p log log dp=n 1e (1=ffl) log ....

[Article contains additional citation context not shown here]

F.E. Fich, F. Meyer auf der Heide, and A. Wigderson. Lower bounds for parallel randomaccess machines with unbounded shared memory. Advances in Computing Research, 4:1--15, 1987.

....i h(N) j and (b) T l G (T ) m . As (b) implies that T = Omega i T l G (T ) m j , and T l G (T ) m = Omega i min t0 n t l G (t) m oj , a) and (b) imply the theorem. ffl ad (a) T = Omega i h(N) j is shown for EREW , CREW PRAMs in [CDR86] and for CRCWPRAMs in [FMW87]. Clearly, these lower bounds for PRAMs also hold for our models. ffl ad (b) The proof for T l G (T ) m is inspired by the lower bounds in [FMW87] It uses an adversary argument. First, we fix a T net V 1 P in G of maximum cardinality l G (T ) and fix the values of input variables x i = 2 ....

....G (t) m oj , a) and (b) imply the theorem. ffl ad (a) T = Omega i h(N) j is shown for EREW , CREW PRAMs in [CDR86] and for CRCWPRAMs in [FMW87] Clearly, these lower bounds for PRAMs also hold for our models. ffl ad (b) The proof for T l G (T ) m is inspired by the lower bounds in [FMW87]. It uses an adversary argument. First, we fix a T net V 1 P in G of maximum cardinality l G (T ) and fix the values of input variables x i = 2 V 1 to 1. Now, intuitively, during the whole algorithm a processor P can learn at most one input variable x i 2 V 1 which is transported to P only via ....

F. Fich, F. Meyer auf der Heide, and A. Wigderson. Lower bounds for parallel random access machines with unbounded shared memory. In F. P. Preparata, editor, Advances in Computing Research, volume 4 of Parallel and Distributed Computing, 1987. 11

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F. E. Fich, F. Meyer auf der Heide, P. Ragde and A. Wigderson. Lower bounds for parallel random access machines with unbounded shared memory, Advances in Computing Research, 4 (1987), pp. 1--15.

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F.E. Fich, F. Meyer auf der Heide, P.L. Ragde, and A. Wigderson, "Lower bounds for parallel random access machines with unbounded shared memory", in Advances in Computing Research, Vol. 4, ed. F.P. Preparata (JAI Press, Greenwich, CT, 1987) pp. 1--15.

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F.E. Fich, F. Meyer auf der Heide, and A. Wigderson. Lower bounds for parallel random-access machines with unbounded shared memory. In Advances in Computing Research. JAI Press, 1986. Proof. The previous theorem showed a lower bound of T

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