J. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms. In Proc. 32nd Symp. on Found. of Comp. Sci., pages 698--710, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....approximate prefix summation. Their algorithm runs in constant time with probability 1 Gamma 2 Gamman ffi for some positive constant ffi . Assuming the slightly weaker ARBITRARY CRCW PRAM, in which an arbitrary processor succeeds in writing, would entail using an O(log n) time algorithm [10] for processor allocation. This change multiplies our time bounds by O(log n) 2.2 Parallel breadth first search The fastest known parallel algorithm for breadth first search involves repeatedly squaring a matrix, where the element wise operations are in the min plus semiring. In fact, this ....

....belonging to previous levels) and so forth. Suppose there are p processors, To go from level i to level i 1, the processors first divide up the edges outgoing from the level i nodes. This step can be done with high probability in O(log p) time using the parallel load balancing algorithm of [10]. Next, for each such edge, if the other endpoint has not yet been traversed, it is assigned to level i 1. This step can be done in O(e i =p) time, where e i is the number of such edges. Hence the time required to traverse k levels is O(e=p k log p) with high probability. 2.3 Limited ....

J. Gil, Y. Matias, and U. Vishkin, "Towards a theory of nearly constant time parallel algorithms," Proceedings, 32nd Symposium on on Foundations of Computer Science (1991), pp. 698-710.

....in a padded consecutive array, such that all variables with the same value occur in a padded consecutive subarray. Given n bits x 1 ; x 2 ; xn , the chaining problem [R93, BV93] is to find for each x i , the nearest 1 s both to its left and to its right. The processor allocation problem [GMV91] is to redistribute m tasks among n processors, so that each processor gets O(1 m=n) tasks. For a given sequence of integers, x 1 ; Delta Delta Delta ; xn ; x i 2 [n] the approximate parallel prefix sums problem [GMV94] is to find a sequence y 0 = 0; y 1 ; Delta Delta Delta ; yn , y i 2 ....

....tasks. For a given sequence of integers, x 1 ; Delta Delta Delta ; xn ; x i 2 [n] the approximate parallel prefix sums problem [GMV94] is to find a sequence y 0 = 0; y 1 ; Delta Delta Delta ; yn , y i 2 [n] such that for i 2 [n] x i y i Gamma y i Gamma1 bx i . We combine results of [R93, BV93, GMV91, GMV94, BH93] in the following lemma. Lemma 3. The strong semisorting problem, the chaining problem, the processor allocation problem, and the approximate parallel prefix sums can be solved on an Arbitrary CRCW PRAM in O(log n) time with linear total work and linear space, with probability at least 1 Gamma ....

J. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms. In Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science, pages 698--710, 1991.

....stage of our algorithm we use a simple load balancing scheme. As this scheme may also be useful in other situations, we describe this scheme here in a fairly general setting. Other load balancing schemes were obtained by Cole and Vishkin [CV88] for the EREW PRAM, and by Gil, Matias and Vishkin [GMV91] Goodrich [Goo91] and Hagerup [Hag92] Hag93] for the CRCW PRAM. These balancing schemes are much more sophisticated than our balancing scheme. Yet, neither one of them suits our purposes. Let al.G be a (possibly randomised) loosely specified 1 parallel algorithm that uses m virtual ....

....(we assume for simplicity that b is an integer dividing P ) Each such group is initially assigned with mb=P of the virtual processors. To record the allocation of the virtual processors to the actual processors, we use a P=b Theta mb=P matrix 1 The term loosely specified is used in [Goo91] and [GMV91] in a broader sense than that used here. Our loosely specified algorithms are similar to the task decaying algorithms of [Goo91] and [GMV91] 4 MAT. The virtual processors allocated to the i th group are listed in the i th row of this matrix. Before we start the simulation of the loosely ....

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J. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, San Juan, Puerto Rico, pages 698--710, 1991.

....concurrents writes into the communication windows. On the other hand the specific rule how to resolve write conflicts is not of major importance, because several authors have shown efficient simulations among parallel machines with different write conflict resolutions [5] 6] 10] 11] 13] [14]. These simulations are described for PRAMs, but can be transfered to DMMs using Lemma 2.1. 4 For example the TOLERANT rule suffices: If several processors want to write to the same communication window then its contents remains unchanged. This can be done without increase in the time processor ....

J. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms. In FOCS '91, pages 698 -- 710, Oct. 1991.

....de Catalunya.Pau Gargallo 5, 08028 Barcelona, Spain.Contact e mail: conrado lsi.upc.es 1 time O(n=p) These results were improved using parallel dynamic hashing in real time [5] Also, J. Gil, Y. Matias and U. Vishkin implemented a nearly constant time optimal random NC parallel dictionary [10]. Finally H. Bast, M. Dietzfelbinger, T. Hagerup construct what they call a perfect dictionary, a scheme that allows p processors implement a set M in space proportional to jM j to process batches of p insert, delete, and lookup instructions on M in constant time with high probability [2] ....

J. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms. In Proc. 32nd Foundations of Computer Science, pages 698--710, 1991.

.... hashing algorithm in O(logn) expected time was given in [21] followed by an O(log log n) expected time algorithm presented in [14] An O(log 3 n log log 3 n) expected time 8 algorithm was given in [20] its time complexity was later shown to actually be O(log 3 n) with high probability [15]. A similar improvement (from O(log 3 n log log 3 n) to O(log 3 n) was also given by [6] A realtime parallel dynamic hashing algorithm was given by [15] the algorithm supports a batch of n membership query or delete instructions in constant time, and a batch of n insert instructions in ....

.... 3 n) expected time 8 algorithm was given in [20] its time complexity was later shown to actually be O(log 3 n) with high probability [15] A similar improvement (from O(log 3 n log log 3 n) to O(log 3 n) was also given by [6] A realtime parallel dynamic hashing algorithm was given by [15]; the algorithm supports a batch of n membership query or delete instructions in constant time, and a batch of n insert instructions in O(log 3 n) time with high probability, using an optimal number of processors. Applications As noted in [21] hashing can replace sorting based naming ....

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J. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms. In FOCS '91, pages 698--710, Oct. 1991.

....on the processor with the most di#cult subproblem. Unfortunately, the time for load balancing itself can also increase the running time of an algorithm, and that is what we will analyze in this paper. We consider the following load balancing problem, as defined by Gil, Matias, and Vishkin [21]. Load balancing. Given h objects distributed among n processors, redistribute the objects so that each processor gets O(1 h n) objects. Note that under any definition of load balancing, at least one processor will end with##t h n) objects. Consequently, by solving the load balancing problem ....

....neighbors, and constructing the Voronoi diagram, in Gil and Matias [20] to perform parallel hashing, in Hagerup [27] to perform integer chain sorting, and in Goodrich [24] to construct upper envelopes, answer range queries, and perform hidden surface elimination. Recently, Gil, Matias, and Vishkin [21], Hagerup [27] and Goodrich [24] have all developed #(log # n) expected time randomized load balancing algorithms for worst case inputs, so the lower bound given in this paper is tight. The algorithm in Goodrich [24] is for a slightly less general load balancing problem. We will also show ## ....

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<F3.785e+05> J. Gil, Y. Matias, and U.<F3.82e+05> Vishkin,<F3.749e+05> Towards a theory of nearly constant time parallel algorithms,<F3.82e+05> in Proc. 32nd IEEE Symp. on Found. of Comput. Sci., San Juan, PR, IEEE Computer Society Technical Committee on Mathematical Foundations of Computing, 1991, pp. 698--710.

....as subproblems in parallel algorithms. Examples are integer sorting, merging, lowest common ancestor and compaction (see [4, 22, 6, 18, 25] Hence these problems have received considerable attention, and, in recent years, there has emerged a body of literature on very fast parallel algorithms [8, 17, 19]. For the chaining problem, Berkman and Vishkin [8] and, independently, Ragde [25] gave ingenious parallel algorithms that run in O(ff(n) time. For a restricted class of algorithms called oblivious algorithms, Chaudhuri [11] proved that ordered chaining requires Omega Gamma ff(n) time. However, ....

J. Gil, Y. Matias and U. Vishkin, "Towards a theory of nearly constant time parallel algorithms ", Proc. of 32nd IEEE FOCS, 1991, 698--710.

....sciences and humanities. y E mail address: zwick math.tau.ac.il. tremendous progress in the development of parallel algorithms with extremely fast running times. Running times of O(log log n) and O(log n) are now typical (see the works of Gil, Goodrich, Hagerup, Matias, Raman, Vishkin [13] [20], 21] 23] 24] 25] 28] 38] 42] It is clear that prefix sums cannot be used by such fast algorithms. To obtain these extremely fast algorithms, new primitives that replace parallel prefix sums had to be invented. One such primitive, used in many of the previously cited works, is linear ....

....is linear approximate compaction: given an array A of size n that contains k non zero elements, create an array B of size (1 ffl)k that contains the non zero elements of A, where ffl = o(1) with respect to n) Unused cells in B should contain a special null value. Gil, Matias and Vishkin [20], Goodrich [21] and Hagerup [24] have shown that linear approximate compaction can be solved, with high probability, in O(log n) time using O(n= log n) processors in the randomized CRCW PRAM model. This result is work time optimal as shown by a lower bound of MacKenzie [36] Using the tool ....

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J. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, San Juan, Puerto Rico, pages 698--710, 1991.

....array, such that all variables with the same value occur in a padded consecutive subarray. Given n bits x 1 ; x 2 ; x n , the chaining problem (Berkman and Vishkin, 1993; Ragde, 1993) is to find for each x i , the nearest 1 s both to its left and to its right. The load balancing problem (Gil et al. 1991) is to redistribute m tasks among n processors, so that each processor gets O(1 m=n) tasks. For a given sequence of integers, x 1 ; Delta Delta Delta ; x n ; x i 2 [n] the approximate parallel prefix sums problem (Goodrich et al. 1994) is to find a sequence y 0 = 0; y 1 ; Delta Delta ....

.... (4) approximate parallel prefix sums Proof: 1) Bast and Hagerup (1993) showed how to solve string semisorting in O(log n) time with the desired probability on the Tolerant CRCWPRAM, provided the input is from [n] For a general input, O(log n) time hashing (Bast and Hagerup, 1991; Gil et al. 1991) reduces the problem to the solution of Bast and Hagerup. See also (Hagerup, 1992) 2) Chaining can be accomplished in O(ff(n) time deterministically on the Arbitrary CRCW PRAM by an algorithm due to Ragde (1993) and Berkman and Vishkin (1993) 3) O(log n) time randomized algorithm for ....

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Gil, J., Matias, Y., and Vishkin, U. (1991), "Towards a theory of nearly constant time parallel algorithms," In Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science, pages 698--710.

....Lotteries with these properties have many applications in computer science, specially in cases where symmetry breaking is required. Examples include randomized mutual exclusion algorithms [7, 5] broadcast in radio networks [1] elections in anonymous networks [6] and various CRCWPRAM algorithms [2]. 1 Research of Eyal Kushilevitz and Michael Rabin supported by research contracts ONR N0001491 J 1981 and NSF CCR 90 07677 at Harvard University. y Research was done while the author was at Aiken Computation Lab. Harvard University. Current address: Dept. of Computer Science, Technion. ....

Gil J., Y. Matias, and U. Vishkin, "Toward a Theory of Nearly Constant Time Parallel Algorithms", 32nd IEEE Symp. on Foundations of Computer Science, 1991, pp. 698--710.

....where the total number of tasks is at most n. The goal is to redistribute the tasks so that no processor has more than C. This approximate load balancing is a critical step in many randomized ultrafast algorithms, and it had been shown that it can be accomplished in Theta(log n) expected time [4]. log n is a function which grows extremely slowly, being 5 or less for all numbers less than 2 65;536 ) Note that achieving exact load balance takes Omega Gamma 45 n= log log n) expected time, but the extra measure of balance is not needed in many situations. This is especially true in ....

....several other problems which can be similarly solved. We also hope that this lower bound proof technique can be extended to other problems which are known to be solvable in Theta(log n) time, including hashing, generation of random permutations, integer chain sorting, and some PRAM simulations [4]. Unfortunately we still do not know if our padded sort algorithm is optimal, and we are trying to find a lower bound for it. Similarly we are looking at other problems with ultrafast solutions, such as pattern matching [15] or the geometry problems mentioned above, trying to establish lower ....

J. Gil, Y. Matias, and U. Vishkin, "Towards a theory of nearly constant time parallel algorithms ", Proc. 32 Symp. on Theory of Comput. (1990), pp. 244-253.

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Y. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms. In Proc. of the 32nd IEEE Annual Symp. on Foundation of Computer Science, pages 698--710, 1991.

....memory location and all pre x sums obtained in the (virtual) serial process are recorded. 17 memory is managed through a dictionary algorithm: we use a parallel dictionary algorithm which with linear space supports each instruction in O(log m) time and O(m) operations, with high probability [18, 10]. Theorem 7 The expected cost for generating m random variates according to the current weights is N) using m processors on a crcw pram, where N is the number of elements. Updating the weight of m elements can be done in O(log N) amortized expected time and in O(2 time in the worst ....

J. Gil, Y. Matias, and U. Vishkin. Towards a Theory of Nearly Constant Time Parallel Algorithms. Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, 698-710, October 1991.

.... time, O(m) space, and O(m) operations, with high probability [9] As in the sequential case, the memory is managed through a dictionary algorithm: we use a parallel dictionary algorithm which with linear space supports each instruction in O(log m) time and O(m) operations, with high probability [18, 10]. Theorem 7 The expected cost for generating m random variates according to the current weights is N) using m processors on a crcw pram, where N is the number of elements. Updating the weight of m elements can be done in O(log N) amortized expected time and in O(2 time in the worst case, ....

J. Gil, Y. Matias, and U. Vishkin. Towards a Theory of Nearly Constant Time Parallel Algorithms. Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, 698--710, October 1991.

.... time, O(m) space, and O(m) operations, with high probability [9] As in the sequential case, the memory is managed through a dictionary algorithm: we use a parallel dictionary algorithm which with linear space supports each instruction in O(log # m) time and O(m) operations, with high probability [17, 10]. Theorem 7 The expected cost for generating m random variates according to the current weights is O(log # N) using m processors on a crcw pram, where N is the number of elements. Updating the weight of m elements can be done in O(log # N) amortized expected time and in O(2 log # N ) expected ....

J. Gil, Y. Matias, and U. Vishkin. Towards a Theory of Nearly Constant Time Parallel Algorithms. Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, 698--710, October 1991.

....Given k nonempty cells at unknown positions in an array of size n, with k known, move the contents of the nonempty cells to an output array of O(k) cells. The linear compaction problem can be solved by a randomized crcw pram algorithm in time T # lc (n) O(log # n) time and linear work w.h.p. [33]. In section 7 (Theorem 7.4) we show that the linear compaction problem can be solved by a randomized simd qrqw pram algorithm in time T lc (n) O( # log n) and linear work w.h.p. Sometimes the linear compaction algorithm is used under the assumption that the number of nonempty cells is at most ....

<F3.748e+05> J. Gil, Y. Matias, and U.<F3.851e+05> Vishkin,<F3.971e+05> Towards a theory of nearly constant time parallel algorithms,<F3.851e+05> in Proc. 32nd IEEE Symp. on Foundations of Computer Science, San Juan, Puerto Rico, 1991, pp. 698--710. <F4.491e+05>768<F3.851e+05> P. B. GIBBONS, Y. MATIAS, AND V. RAMACHANDRAN

....Approximation 3 as a subroutine, in spite of its widely recognized usefulness in polylogarithmic parallel algorithms. Several problems were suggested instead, which may be viewed as much relaxed versions of the prefix sums problem, and for which nearly constant time algorithms were developed [9, 11, 12, 15, 22, 23]. These problems include the linear approximate compaction [23] load balancing [9] interval allocation [15] and density partitioning [12] See also [22] While these problems can be used, often in concert, to replace parallel prefix for some applications, their use is not always as natural as ....

....use heavily the assumption that input is taken uniformly at random from the unit square. Hagerup [14] defined the integer chain sorting problem, and gave an optimal randomized algorithm in the doubly logarithmic level. An algorithm in the O(lg n) time level was subsequently given by Gil et al. [11]. While this is also an interesting approximate version of parallel integer sorting, its applications are limited in that it amounts to a reduction of sorting to the well known list ranking problem [17] for which the near logarithmic lower bound still holds [3] This lack of applicability ....

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Joseph Gil, Yossi Matias, and Uzi Vishkin. Towards a theory of nearly constant time parallel algorithms. In Proc. of the 32nd IEEE Annual Symp. on Foundation of Computer Science, pages 698--710, October 1991.

....preprocessing time and query processing time. Prefix maxima with restricted input. 6] gave a prefix maxima algorithm for input restricted to integers in the range [1: s] The algorithm runs in O(log log log s) time and O(n) operations, using O(ns ffl ) space. Using a parallel hashing algorithm [23, 17], this prefix maxima algorithm can be implemented using only linear space (the time then increases by a factor of O(log n) with high probability) For the case s = n, 6] give an algorithm that takes O(log n) time and O(n) operations, using O(n) space. Hagerup [19] includes an O(log n) ....

....confidence in probabilistic analysis into polynomial and exponential. Precise degrees of polynomials and exponent parameters are 6 Berkman, Matias Vishkin of lesser significance. This somewhat coarse notation is convenient for a clearer presentation and analysis as explained below, and in [17]. Given a procedure that succeeds with n polynomial probability, where n is the size of the problem, we will always be able to repeat it a constant number of times so that its success probability increases to 1 Gamma n Gammac for any constant c, with the only requirement being that a failure ....

J. Gil, Y. Matias, and U. Vishkin, Towards a theory of nearly constant time parallel algorithms, in Proc. of the 32nd IEEE Annual Symp. on Foundation of Computer Science, 1991, 698--710.

....hashing algorithm. The algorithm that realizes Lemma 3.5 is the basis for the algorithm that realizes the crcw pram simulation of Lemma 3. 2; the (lg n) overhead and the (lg n) factor in the delay of the latter simulation algorithm are due to the application of a crcw pram hashing algorithm [20], adapted to the dmm. As a result we have: Corollary 3.6 One step of an (n lg lg n) processor arbitrary crcw pram can be simulated on a collision n rmesh by O(lg lg n) calls to a hashing algorithm. Proof. By Lemma 3.5 and Lemma 3.1. A hashing algorithm takes Theta(lg n) steps on the dmm. A ....

....algorithm on the rmesh was known. A parallel dictionary handles one batch of operations at a time. Each batch consists of an array of elements and an instruction: insert, delete, or lookup. The parallel dictionary processes a batch using several processors. We use the following theorem: Lemma 3. 9 ([20]) There exists a dictionary implementation on an n processor arbitrary crcw pram, with the following features, all w.h.p. a) At all times, the total space used by the dictionary is linear in the number of elements currently stored in the dictionary. b) Any batch of n lg n elements is ....

J. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms. In Proc. 32nd IEEE Symp. on Foundations of Computer Science, pages 698--710, October 1991.

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J. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms. In Proc. 32nd Symp. on Found. of Comp. Sci., pages 698--710, 1991.

No context found.

J. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, San Juan, Puerto Rico, pages 698-710, 1991.

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J. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms. Proc. of the 32nd IEEE Annual Symp. on Foundation of Computer Science, pages 698--710, 1991.

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J. Gil, Y. Matias, and U. Vishkin. Towards a theory of nearly constant time parallel algorithms, in: "Proc. 32nd Foundations of Computer Science," (1991).

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J. Gil, Y. Matias, U. Vishkin, "Towards a theory of nearly constant time parallel algorithms, Proc. 36st Annual IEEE Symp. on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA,1991, pp. 698-710.

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