M.T. Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, San Juan, Puerto Rico, pages 711-722, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 processors. The execution ....

....b = log m (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 ....

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M.T. Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, San Juan, Puerto Rico, pages 711--722, 1991.

....counterpart. We first compute T m j=1 I j in O(log n) time using O(n) processors. If the intersection is empty, we construct L T and U B , and determine if there is any point that lies above U B and below L T . To construct the envelopes in parallel, we employ the algorithm by Goodrich [12]. Given a collection F = ff 1 ; f 2 ; f n g of functions, where f i : R R; 1 i n, such that every pair of functions has at most k intersections, Goodrich s algorithm constructs the upper envelope of the functions in F in O(log n) time using O( k (n) processors in Valiant s parallel 5 ....

M. T. Goodrich, Using approximation algorithms to design parallel algorithms that may ignore processor allocation, Proc. 32nd Annu. Symp. on Foundations of Computer Science (1991), 711-- 722.

.... and Azar [1] In the PRAM setting, the lower bound of Beame and Hastad does not apply directly to approximate selection, although it can be used to place bounds on the accuracy obtainable with a given amount of resources (i.e. processors and time) Hagerup [13] extending a result of Goodrich [12], showed that approximate selection problems of size n can be solved in constant time with high probability on an n processor CRCW PRAM for = 1= log n) O(1) which is the best possible accuracy for the stated time and processor bounds. On the other hand, no deterministic PRAM algorithms for ....

M.T. Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. In Proc. 32nd IEEE FOCS (1991), pp. 711--722.

.... simulations of strong CRCW PRAM models on weaker CRCW PRAM models, in MacKenzie and Stout [37] for padded sorting, finding nearest 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 ....

....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 ## log # n) time lower bounds for the ....

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<F3.785e+05> M. T.<F3.82e+05> Goodrich,<F3.749e+05> Using approximation algorithms to design parallel algorithms that may ignore processor<F3.82e+05> allocation, in Proc. 32nd IEEE Symp. on Found. of Comput. Sci., 1991, San Juan, PR, IEEE Computer Society Technical Committee on Mathematical Foundations of Computing, pp. 711--722.

....be implemented to run in time O(n p) w.h.p. on a p processor qrqw pram when p = O(n (t t # # log n) The spawning model can be further generalized to include a start operation in which one task may spawn n new tasks to begin in the next time step. This extended model is called v pram in [35], where it was suggested. It was shown in [35] that the work preserving scheme for the spawning model can be extended to the v pram model as well, with the same overhead. Accordingly, Theorem 4.8 and Corollary 4.9 apply to the v pram model. A more general type of spawning algorithm, the ....

....on a p processor qrqw pram when p = O(n (t t # # log n) The spawning model can be further generalized to include a start operation in which one task may spawn n new tasks to begin in the next time step. This extended model is called v pram in [35] where it was suggested. It was shown in [35] that the work preserving scheme for the spawning model can be extended to the v pram model as well, with the same overhead. Accordingly, Theorem 4.8 and Corollary 4.9 apply to the v pram model. A more general type of spawning algorithm, the L spawning algorithm, is studied in [29] In the ....

<F3.748e+05> M.<F3.851e+05> Goodrich,<F3.971e+05> Using approximation algorithms to design parallel algorithms that may ignore processor<F3.851e+05> allocation, in Proc. 32nd IEEE Symp. on Foundations of Computer Science, San Juan, Puerto Rico, 1991, pp. 711--722.

....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 ....

....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 of linear ....

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M.T. Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, San Juan, Puerto Rico, pages 711-- 722, 1991.

....and prefix sums, have been studied in connection to the CRCW PRAM model. MacKenzie [23] proved that padded sorting and linear approximate compaction require Omega (log n) time on n processor randomized CRCW PRAMs. Matching upper bound for linear approximate compaction were obtained in [12, 13, 17]. A deterministic O(log log n) 3 algorithm was obtained in [18] Approximate prefix sum (hence also approximate compaction) can be computed on a randomized CRCW PRAM in log n steps [4, 14, 15, 20] A deterministic COMMON PRAM algorithm running in O(log log n) time was presented in [16] It ....

M. T. Goodrich, Using approximation algorithms to design parallel algorithms that may ignore processor allocation, in Proc. 32th Symposium on Foundations of Computer Science (IEEE, Los Alamitos, 1991) 711--722.

....to the boundary of a cell in a cutting of the diagram. This problem is basically the computation of the lower envelope (the minimum) of a set of functions of one variable. There is a very simple sequential and deterministic solution for this problem [45] and to a certain extent, it parallelizes [23]. However, its parallelization is not satisfactory, it cannot be made output sensitive, and it is a bottleneck in geometric optimization (diameter problem) Thus, the objective of this work is to present an alternative algorithm for this problem and applications where it outperforms the other ....

.... each stage takes time O(log n) using approximate counting and compaction techniques (but the final output is not padded) In order to achieve the n exponential probability, we make use of the failure sweeping technique introduced in [21] The CRCW algorithm matches the time of an algorithm in [23], but it is considerably simpler. Although the EREW is not work optimal, we will use it to derive a work optimal algorithm for Voronoi diagrams as there fi(n) O(1) Deterministic algorithms Theorem 3.8: LF can be computed deterministically in EREW PRAM using time O(log 2 n) and in CRCW using ....

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M.T. Goodrich. Using approximations algorithms to design parallel algorithms that may ignore processor allocation. In Proc. 32th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS 91), 711--722, 1991.

.... simulations of strong CRCW PRAM models on weaker CRCW PRAM models, in MacKenzie and Stout [37] for Padded Sorting, finding Nearest 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 Theta(log n) expected time randomized load balancing algorithms for worst case inputs, so the lower bound ....

....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 Theta(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 Omega Gamma165 n) time lower ....

[Article contains additional citation context not shown here]

M. T. Goodrich, Using approximation algorithms to design parallel algorithms that may ignore processor allocation, in Proc. 32nd Symp. on Found. of Comp. Sci., 1991, pp. 711--722.

....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. Suppose we are given a (possibly randomized) loosely specified parallel algorithm ALG that uses m virtual processors. The ....

M.T. Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, San Juan, Puerto Rico, pages 711--722, 1991.

....achieve nearly constant time on a CRCW PRAM. The problems include hashing [31, 94, 96, 97, 183, 184] dictionary (insert, delete, query operations) 96] integer sorting [38, 115, 183, 184, 218, 219] integer chain sorting [96, 115, 116] space allocation [96, 116] linear approximate compaction [96, 105, 183], estimation [96, 116] load balancing [93, 96, 116] leaders election [95, 96, 116, 184] generation of random permutations [183] 2 ruling set [55, 96, 105, 183] all nearest smaller values [35] approximate sum [96] efficient simulation of Maximum PRAM model on Tolerant PRAM model [95, 96, ....

.... sorting [38, 115, 183, 184, 218, 219] integer chain sorting [96, 115, 116] space allocation [96, 116] linear approximate compaction [96, 105, 183] estimation [96, 116] load balancing [93, 96, 116] leaders election [95, 96, 116, 184] generation of random permutations [183] 2 ruling set [55, 96, 105, 183], all nearest smaller values [35] approximate sum [96] efficient simulation of Maximum PRAM model on Tolerant PRAM model [95, 96, 117, 184] These new algorithmic techniques provide a theoretical framework for the future development of tools which would automate a number of tedious aspects of ....

[Article contains additional citation context not shown here]

M T Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. In Proc. 32nd Annual IEEE Symposium on Foundations of Computer Science, pages 711--722, 1991.

....implemented to run in time O(n=p) w.h.p. on a p processor qrqw pram when p = O(n= t t 0 Delta p lg n) The spawning model can be further generalized to include a start operation in which one task may spawn n new tasks to begin in the next time step. This extended model is called v pram in [Goo91] where it was suggested. It was shown in [Goo91] that the work preserving scheme for the spawning model can be extended to the v pram model as well, with the same overhead. Accordingly, Theorem 4.8 and Corollary 4.9 apply to the v pram model. A more general type of spawning algorithm, the ....

....qrqw pram when p = O(n= t t 0 Delta p lg n) The spawning model can be further generalized to include a start operation in which one task may spawn n new tasks to begin in the next time step. This extended model is called v pram in [Goo91] where it was suggested. It was shown in [Goo91] that the work preserving scheme for the spawning model can be extended to the v pram model as well, with the same overhead. Accordingly, Theorem 4.8 and Corollary 4.9 apply to the v pram model. A more general type of spawning algorithm, the L spawning algorithm, is studied in [GMR97] In the ....

M.T. Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. In Proc. 32nd IEEE Symp. on Foundations of Computer Science, pages 711--722, 1991.

.... spawning algorithms, and loosely specified algorithms; the proposed scheduling techniques were typically based on very fast crcw pram algorithms for relaxed versions of the prefixsums problem such as linear compaction, load balancing, interval allocation, and approximate prefix sums [GM91, GMV91, Goo91, Hag91, MV91, Mat92, Hag93, Gil94, GM96, GMV94, GZ95] The techniques that were used are insufficient, however, to cope with the model considered in this paper, even when space considerations are ignored. In particular, previous techniques assumed that whenever a thread goes to sleep, it is known ....

M.T. Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. In Proc. 32nd IEEE Symp. on Foundations of Computer Science, pages 711--722, 1991.

....between processors, can be implemented with unit cost per thread. In the connection of PRAM models, the latter is naturally achieved. Currently, there exists a quite rich theory of dynamically balancing the workload of each PRAM processor; see e.g. Gil 1991, Gil, Matias 1991, Gil et al. 1991, Goodrich 1991, Hagerup 1992, Matias, Vishkin 1991] For example, consider a parallel algorithm A that takes time T (q) using q PRAM processors, and for which nt( nt( 1) for all 2 f1; T (q)g. In [Matias, Vishkin 1991] it is proved that there is a work optimal simulation of A on a PRAM with p ( ....

M.T. Goodrich. Using Approximation Algorithms to Design Parallel Algorithms that May Ignore Processor Allocation. In Proceedings, 32th Annual Symposium on Foundations of Computer Science, pages 711 -- 722, 1991.

....for the entire algorithm that takes O(lg lg n lg n) expected time. Subsequent to a preliminary version of this paper (presented in [7] other policies of effective load balancing for other classes of algorithms were introduced, often using amortization arguments similar to the one used here [17, 9, 10, 5] (see also [16] ....

M.T. Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. In Proc. 32nd IEEE Symp. on Foundations of Computer Science, pages 711--722, 1991.

....; d 2 ; d n and n points a 1 ; a 2 ; a n , determine for every j = 1; 2; n, whether a j 2 Pi j = j i=1 d i . We now give an O(log n) time, O(n) processor algorithm for solving this problem in the parallel comparison model. Our algorithm is based on Goodrich s paradigm [12] for computing the upper envelope of n surfaces in the so called k intersecting class. We first build an n leaf complete binary tree T . The ith leaf of T stores d i and a i . Every internal node v of T is associated with (1) the intersection I(v) of the discs stored at the leaves of the subtree T ....

....stages are used for this pipelined procedure. Every I(v) is maintained as a linked list. At the end of each stage t, a list I t (v) is stored at v. We say v is full when I t (v) I(v) The merge at the children of v is pipe lined to v by maintaining a list which is an approximately uniform (see [12]) subsequence of I t (v) and I t 1 (v) is defined as I t (x) I t (y) The information for the points in A(v) can be maintained and pipe lined in a similar fashion as for I(v) These structures at every node v can be maintained in O(1) time per stage. By following the techniques of [12] it ....

[Article contains additional citation context not shown here]

M.T. Goodrich, Using approximation algorithms to design parallel algorithms that may ignore processor allocation. Proc. 32nd Ann. IEEE Symp. on Foundations of Computer Science, San Juan, Puerto Rico, 711--722, 1991.

....1 ; d 2 ; d n and n points a 1 ; a 2 ; a n , determine for every j = 1; 2; n, whether a j 2 Pi j = j i=1 d i . We now give an O(log n) time, O(n) processor algorithm for solving this problem in the parallel comparison model. Our algorithm is based on Goodrich s paradigm [12] for computing the upper envelope of n surfaces in the k intersecting class. The algorithm is sketched below. We first build an n leaf complete binary tree T . The ith leaf of T stores d i and a i . Every internal node v of T is associated with (1) the intersection I(v) of the discs stored at the ....

....approximatelyuniform subsequence of I t (v) and I t 1 (v) is defined as I t (x) I t (y) The information for the points in A(v) can be maintained and pipe lined in a similar fashion as for I(v) These structures at every node v can be maintained in O(1) time per stage. By using the techniques of [12] we can show that the problem of checking whether a j 2 Pi j for every j is solvable in O(log n) time using n processors in the parallel comparison model. More details our found in the full version of the paper. We then use this parallel algorithm to drive a parametric search algorithm [21] We ....

M.T. Goodrich, Using approximation algorithms to design parallel algorithms that may ignore processor allocation. Proc. 32nd Ann. IEEE Symp. on Foundations of Computer Science, San Juan, Puerto Rico, 711--722, 1991.

....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 ....

.... 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 is the case for parallel prefix computations in polylogarithmictime algorithms. Indeed, this deficiency motivated us to re examine the ....

[Article contains additional citation context not shown here]

Michael T. Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. In Proc. of the 32nd IEEE Annual Symp. on Foundation of Computer Science, pages 711--722, 1991.

....subroutine in favor of new problems for which they could produce constant time or near constant time algorithms. Indeed, several problems were suggested recently, which may be viewed as much relaxed versions of the prefix sums problem, and for which nearly constant time algorithms can be developed [9, 10, 11, 13, 17, 18]. These problems include the linear approximate compaction problem [18] the load balancing problem [9] the interval allocation problem [13] and the density partitioning problem [11] While these problems can be used, often in concert, to replace parallel prefix for some applications, the goal ....

.... of the prefix sums problem, and for which nearly constant time algorithms can be developed [9, 10, 11, 13, 17, 18] These problems include the linear approximate compaction problem [18] the load balancing problem [9] the interval allocation problem [13] and the density partitioning problem [11]. While these problems can be used, often in concert, to replace parallel prefix for some applications, the goal of the present paper is return to the prefix sums problem. We show that one can solve an approximate version of the prefix sums problem in o(lg lg n) parallel time with high ....

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Michael T. Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. In FOCS '91, pages 711--722, 1991.

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M.T. Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, San Juan, Puerto Rico, pages 711-722, 1991.

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M. Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. Proc. of the 32nd IEEE Annual Symp. on Foundation of Computer Science, pages 711--722, 1991.

No context found.

M. Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. Proc. of the 32nd IEEE Annual Symp. on Foundation of Computer Science, pages 711--722, 1991.

No context found.

M.T. Goodrich. Using approximation algorithms to design parallel algorithms that may ignore processor allocation. In Proc. of the 32nd IEEE Annual Symp. on Foundation of Computer Science, pages 711--722, 1991.

No context found.

M. T. Goodrich, "Using approximation algorithms to design parallel algorithms that may ignore processor allocation", Proc. 32nd Annual Symposium on Foundations of Computer Science, 1991, pp. 711--722.

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