F. E. Fich. The complexity of computation on the parallel random access machine. In J. H. Reif, editor, Synthesis of Parallel Algorithms, chapter 20, pages 843-900. Morgan Kaufmann Publishers, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Construction of a switching network of polylogarithmic depth generating random permutations may be used to design a fast EREWPRAM algorithm. EREW PRAM (Exclusive Read Exclusive Write Parallel Random Access Machine) is one of the major theoretical models for a parallel computer with a shared memory [17]. It consists of a number of independent processors using a shared memory consisting of a number of memory cells. In one step of a PRAM, each processor may read an arbitrary memory cell, perform an arbitrary computation using its local memory, and, nally, write a value into a chosen cell of the ....

F. Fich, The Complexity of Computation on the Parallel Random Access Machine, in Synthesis of Parallel Algorithms, Morgan Kaufmann, San Mateo, CA, 1993, 843-900.

....location. One frequently used arbitration scheme is the concurrent write model in which processors are assigned a write priority. When more than one processor attempts to write to a given location, the processor with the highest priority succeeds. This model is known as the PRIORITY CRCW P RAM [18]. We adopt this model and simply refer to it as the P RAM. In the P RAM model any processor can access any global memory location in one time step; the model allows unlimited parallelism. For this reason the P RAM serves as a convenient model for designing and analyzing parallel algorithms, for ....

....takes O(n) time. The parallel solution outlined previously shows a P RAM can solve this problem in O(log n) time using n=2 processors. Thus, the parity problem is in the class NC and a qualitative speed up is achieved in the parallel setting. On the other hand, parity is not in AC 0 , see [18] for example. We will use the terminology that problems in NC (and thus AC 0 ) are efficiently solved in parallel, since we obtain a qualitative speed up solving these problems in parallel. On the other hand, problems that are in P but likely not in NC are called inherently sequential. The ....

F. E. Fich. The complexity of computation on the parallel random access machine. In J. H. Reif, editor, Synthesis of Parallel Algorithms, chapter 20, pages 843--899. Morgan Kaufman, San Mateo, CA, 1993.

....) max u#Bn # u (#) and the average sensitivity of # is s(#) 2 n X u#Bn n X i=1 #(u) #(u (i) 2 n X u#Bn # u (#) Clearly, s(#) # #(#) # n for any #. Sensitivity can be used to obtain lower bounds for the CREW PRAM complexity of Boolean functions (see [25, 18, 19, 27, 32]) The average sensitivity of a function # can be defined equivalently as the sum of the influences of all variables on #, where the influence of u i on #, denoted I i (#) is the probability that flipping the i th variable of a random Boolean input will flip the output. In other words, I i ....

F. E. Fich, `The complexity of computation on the parallel random access machine', Handbook of Theoretical Comp. Sci., Vol. A, Elsevier, Amsterdam (1990), 757--804.

....by (2) we get property (1) We use the property (1) to estimate time complexity of Boolean function f . Recall that certificate complexity of a Boolean function g is defined as follows: cf(g) max u2f0;1g n min n jSj : 8 v 2 f0; 1g n (8i 2 S u i = v i ) g(u) g(v) o (see, e.g. [9]. For a Boolean function g, let T g denote the time required for computing g on CREW PRAMs. Nisan [20] showed that T g 1 4 log(cf(g) for every Boolean function g. Now we show that cf(f) n Gamma bnc dk=2e . Let us consider w = 0; 0) as an input for f . Let S be a certificate for ....

F.E. Fich, The complexity of computation on the Parallel Random Access Machine, in Synthesis of Parallel Algorithms, J.H. Reif (ed.) (Morgan Kaufmann, San Mateo, 1993) 843-- 899.

.... the maximum, over all binary vectors x = x 1 , x r ) # 0, 1 r , of the number of points y # 0, 1 r on the unit Hamming sphere around x with B(y) #= B(x) This parameter is of interest because it can be used to obtain lower bounds for the CREW PRAM complexity of B , see [7, 8, 9, 21, 27]. That is. the complexity on a parallel random access machine with an unlimited number of all powerful processors such that simultaneous reads of a single memory cell by several processors are permitted, but simultaneous writes are not. Now, let us select an r bit square free integer x with x # ....

F. E. Fich, `The complexity of computation on the parallel random access machine', Synthesis of parallel algorithms , Morgan Kaufmann Publ., San Mateo, CA, 1993, 843--899.

....33095 Paderborn, Germany (gathen uni paderborn.de) School of MPCE, Macquarie University, Sydney, NSW 2109, Australia (igor mpce.mq.edu.au) 1 2 J. VON ZUR GATHEN AND i.e. SHPARLINSKI By [22] 0. 5 log 2 (#(f) 3) is a lower bound on the parallel time for computing f on such machines, see also [6, 7, 8, 31]. This yields immediately the lower bound #(log n) for the OR and the AND of n input bits. It should be contrasted with the common CRCW PRAM, where write conflicts are allowed, provided every processor writes the same result, and where all Boolean functions can be computed in constant time (with a ....

Faith E. Fich, `The complexity of computation on the parallel random access machine', Synthesis of parallel algorithms, Morgan Kaufmann Publ., San Mateo, CA, 1993, 843--899.

....We also show that improving time bound W( p log n) for 2 compaction on EREW PRAM requires novel and more sophisticated techniques. 1 Introduction Parallel Random Access Machines (PRAMs) are widely used for designing parallel algorithms (for a detailed definition of PRAMs see for instance [6]) There are many fast and work efficient parallel PRAM algorithms. However, many of them can be hardly implemented in the real world, due to communication problems. Even if the messages between processors and the shared memory are routed efficiently (say by optical channels) it remains a problem ....

F.E. Fich, The complexity of computation on the Parallel Random Access Machine, in Synthesis of Parallel Algorithms, J.H. Reif (ed.) (Morgan Kaufmann, San Mateo, 1993) 843--899.

....Here we adopt the concurrent read, exclusive write or CREW P RAM model in which only one processor is allowed to write to a given memory location at a time. In this model if two processors attempt to write to the same memory cell, the program fails. Write arbitration schemes are discussed in [13]. In the P RAM model any processor can access any global memory location in one time step; the model allows unlimited parallelism. The P RAM is an idealized model which allows easy expression of algorithms and provides a convenient framework for addressing new problems. The P RAM is also useful ....

F. E. Fich. The complexity of computation on the parallel random access machine. In J. H. Reif, editor, Synthesis of Parallel Algorithms, chapter 20, pages 843--899. Morgan Kaufman, San Mateo, CA, 1993.

....is an evidence that improving time bound Omega ( p log n) for 2 compaction on EREW PRAM requires novel and more sophisticated techniques. 1 Introduction Parallel Random Access Machines (PRAMs) are widely used for designing parallel algorithms (for a detailed definition of PRAMs see for instance [6]) PRAM consists of a number of processors (each possessing a local memory for local computations) and a number of shared memory cells. The execution is divided into steps, each partially supported by KBN grant 8 S503 002 07 and DFG Sonderforschungsbereich 376 Massive Parallelitat step ....

F.E. Fich, The complexity of computation on the Parallel Random Access Machine, in Synthesis of Parallel Algorithms, J.H. Reif (ed.) (Morgan Kaufmann, San Mateo, 1993) 843--899.

....cell, whereas in concurrent access we allow any number of processors to access any given cell at the same time. The three most popular models are the CRCW (concurrent read, concurrent write) CREW (concurrent read, exclusive write) and EREW (exclusive read, exclusive write) PRAMs (see [J aJ92, Fic93] In the case of concurrent write, we need some 58 way to arbitrate write conflicts. There are several variations; we will describe two of them. In a priority CRCW PRAM, processors are assigned distinct priorities and in the case of more than one processor trying to write into the same cell, ....

Faith E. Fich. The complexity of computation on the parallel random access machine. In John H. Reif, editor, Synthesis of Parallel Algorithms, chapter 20, pages 843--899. Morgan Kaufmann, 1993.

....We also show that improving time bound Omega ( p log n) for 2 compaction on EREW PRAM requires novel and more sophisticated techniques. 1 Introduction Parallel Random Access Machines (PRAMs) are widely used for designing parallel algorithms (for a detailed definition of PRAMs see for instance [6]) There are many fast and work efficient parallel PRAM algorithms. However, many of them can be hardly implemented in the real world, due to communication problems. Even if the messages between processors and the shared memory are routed efficiently (say by optical channels) it remains a problem ....

F.E. Fich, The complexity of computation on the Parallel Random Access Machine, in Synthesis of Parallel Algorithms, J.H. Reif (ed.) (Morgan Kaufmann, San Mateo, 1993) 843--899.

....of R 0 is 0 HALT stop execution of this processor Table 1: Sample PRAM Instruction Set. memory is performed. This eliminates read write conflicts to shared memory, but does not eliminate all access conflicts. This is dealt with in a number of ways as described in Table 2 (see [Vishkin, 1983] and [Fich, 1993] for more details) All of these variants of the PRAM are deterministic, except for the ARBITRARY CRCW PRAM, for which it is possible that repeated executions on identical inputs result in different outputs. Any given PRAM computation will use some specific time and hardware resources. The ....

Fich, F. E. 1993. The complexity of computation on the parallel random access machine.

....integer x = x 1 : x n is square free. Thus g 0 is very similar to the function g given by (1) It has been shown in [25] that the bound oe(g 0 ) bn=60c holds. This parameter is of interest because it can be used to obtain lower bounds for the CREW PRAM complexity of Boolean functions (see [10, 11, 21, 26]) that is the complexity on a parallel random access machine with an unlimited number of all powerful processors, such that simultaneous reads of a single memory cell by several processors are permitted, but simultaneous writes are not. In particular, from the above bound on oe(g 0 ) one ....

F. E. Fich, `The complexity of computation on the parallel random access machine', Handbook of Theoretical Comp. Sci., Vol. A, Elsevier, Amsterdam (1990), 757--804.

....the bits accessed. The decision of which bit to read at any time may depend on the previous bits read. The complexity measure D(f) is the minimal height of a decision tree that computes f [10, 14] There is a tight connection between CROW PRAM s and decicion trees: CROW(f) log(D(f) Theta(1) [4, 5]. In this section we prove a corresponding relation for CROW PRAM s with simultaneous reading and writing. Theorem11. CROW(f) d log ff D(f)e Gamma O(1) 1:44 log D(f) where ff = 1 2 (1 p 5) Proof. Here we assume that the state of a processor reflects its whole history including all ....

F. E. Fich. The complexity of computation on the parallel random access machine. In J. H. Reif, editor, Synthesis of Parallel Algorithms, chapter 20, pages 843--900. Morgan Kaufmann Publishers, 1993.

....processors at one step, its contents will be the value written by the highest indexed processor among them. This is a very strong convention of resolving write conflicts and any lower bound for this model holds also for any other (standard) PRAM model, like the Arbitrary or Common (see e.g. F93, KR90] Let A be a set of elements drawn from a linearly ordered set. The outcome of a comparison between two elements a i and a j from A is exactly one of the following: a i a j , a i = a j , or a i a j . A parallel comparison tree with p comparisons in each round is a 3 p ary tree in ....

F. E. Fich. The complexity of computation on the parallel random access machine. In J. H. Reif, editor, Synthesis of Parallel Algorithms, chapter 20, pages 843--899. Morgan Kaufman, 1993.

....this provides some intuitive support for the empirical observation that most known CREW PRAM algorithms are CROW PRAM algorithms. In one context, we know the two models are equivalent. Following the appearance of an extended abstract of this paper [12] Ragde (personal communication; see also Fich [13], Nisan [28] observed that nonuniform CROWPRAMs, i.e. ones having arbitrary instructions, exponentially many processors initially active, and allowing different programs for each value of n, running in time t are equivalent to Boolean decision trees of depth 2 t . Nisan [28] established that ....

F. E. Fich. The complexity of computation on the parallel random access machine. In J. H. Reif, editor, Synthesis of Parallel Algorithms, chapter 20, pages 843--899. Morgan Kaufmann, 1993.

....are resolved in favour of the processor with the lowest number. Listed in decreasing order by power, the PRAM models are Priority CRCW Arbitrary CRCW Common CRCW CREW EREW; an algorithm that runs on one model will run unchanged on any of the more powerful models. See Fich s survey [14] for a detailed account of the relationships between the various PRAM models. There are several measures that can be used to assess the performance of an algorithm. The (worst case) running time of an algorithm, T (n) is defined to be the maximum number of parallel steps performed by the algorithm ....

Faith E. Fich. The complexity of computation on the parallel random access machine. In [36], Chapter 20.

....f(x (i) fi fi fi : In [27] it has been shown that for the function g(x) deciding if an n bit integer x is square free, the bound oe(g) bn=60c holds. This sensitivity is of interest because it can be used to obtain lower bounds for the CREW PRAM complexity of a Boolean function f (see [11, 12, 23, 28]) that is the complexity on a parallel random access machine with an unlimited number of all powerful processors, such that simultaneous reads of a single memory cell by several processors are permitted, but simultaneous writes are not. In particular, from the above bound on oe(g) one immediately ....

F. E. Fich, `The complexity of computation on the parallel random access machine ', Handbook of Theoretical Comp. Sci., Vol. A, Elsevier, Amsterdam (1990), 757--804.

....average (taken with respect to the uniform distribution) of the sensitivity of f on input w over all w of a given length. These definitions are made precise below. The sensitivity is of interest because it can be used to obtain lower bounds for the CREW PRAM complexity of Boolean functions (see [9, 10, 16, 19]) that is the complexity on a parallel random access machine with an unlimited number of allpowerful processors, such that simultaneous reads of a single memory cell by several processors are permitted, but simultaneous writes are not. The average sensitivity is a finer characteristic of Boolean ....

F. E. Fich, `The complexity of computation on the parallel random access machine', Handbook of Theoretical Comp. Sci., Vol. A, Elsevier, Amsterdam (1990), 757--804.

....time complexity of Boolean function f (and therefore of algorithm A, since f can be computed using A. Recall that certificate complexity of a Boolean function g is defined as follows: cf(g) max u2f0;1g n min n jSj : 8 v 2 f0; 1g n (8i 2 S u i = v i ) g(u) g(v) o (see, e.g. [11]. For a Boolean function g, let T g denote the time required for computing g on CREW PRAMs. Nisan [25] showed that T g 1 4 log(cf(g) for every Boolean function g. Now we show that cf(f) n Gamma bnc dk=2e . Let us consider the input sequence w = 0; 0) Let S be a certificate for ....

F.E. Fich, The complexity of computation on the Parallel Random Access Machine, in Synthesis of Parallel Algorithms, J.H. Reif (ed.) (Morgan Kaufmann, San Mateo, 1993) 843--899.

....this provides some intuitive support for the empirical observation that most known CREW PRAM algorithms are CROW PRAM algorithms. In one context, we know the two models are equivalent. Following the appearance of an extended abstract of this paper [13] Ragde (personal communication; see also Fich [14], Nisan [28] observed that nonuniform CROWPRAMs, i.e. ones having arbitrary instructions, exponentially many processors initially active, and allowing different programs for each value of n, running in time t are equivalent to Boolean decision trees of depth 2 t . Nisan [28] established that ....

F. E. Fich. The complexity of computation on the parallel random access machine. In J. H. Reif, editor, Synthesis of Parallel Algorithms, chapter 20, pages 843--899. Morgan Kaufmann, 1993.

No context found.

Faith E. Fich. The complexity of computation on the parallel random access machine. In John H. Reif, ed., Synthesis of Parallel Algorithms, pages 843--899. Morgan Kaufmann, 1993.

No context found.

Faith E. Fich. The complexity of computation on the parallel random access machine. In John H. Reif, editor, Synthesis of Parallel Algorithms, pages 843-- 900. Morgan Kaufman, San Mateo, CA, 1993.

....Let f be any n ary function f : D 1 2 1 1 1 2 D n N, where D 1 ; D n N are finite sets. Then the logarithm of f s decision tree complexity characterizes to within a constant factor the time for a (nonuniform) CREW PRAM with an arbitrarily powerful instruction set to compute f [21, 10]. With normal arithmetic capabilities, a nonuniform CROW PRAM can evaluate any decision tree of height h and size s in dlog 2 he O(1) steps using s processors, by pointer jumping (Ragde, personal communication; see also [21, 10] Preinitialized memory is used to specify the decision tree, naming ....

.... CREW PRAM with an arbitrarily powerful instruction set to compute f [21, 10] With normal arithmetic capabilities, a nonuniform CROW PRAM can evaluate any decision tree of height h and size s in dlog 2 he O(1) steps using s processors, by pointer jumping (Ragde, personal communication; see also [21, 10]) Preinitialized memory is used to specify the decision tree, naming the input variable to be tested at each internal node, the out edges from each, and the function value at each leaf. Addition is used to index into the list of outedges at each internal node in constant time. As in the proof of ....

F. E. Fich. The complexity of computation on the parallel random access machine. In J. H. Reif, editor, Synthesis of Parallel Algorithms. Morgan Kaufman, San Mateo, CA, 1993. To appear.

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F. E. Fich. The complexity of computation on the parallel random access machine. In J. H. Reif, editor, Synthesis of Parallel Algorithms, chapter 20, pages 843-900. Morgan Kaufmann Publishers, 1993.

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