M. T. Goodrich, Y. Matias, and U. Vishkin. Optimal parallel approximation algorithms for prefix sums and integer sorting. In Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 241--250, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 [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 ....

....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 ....

M. T. Goodrich, Y. Matias, and U. Vishkin. Optimal parallel approximation algorithms for prefix sums and integer sorting. In Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 241--250, 1994.

....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 Delta ; y n , y i 2 [n] such that for i 2 [n] x i y i Gamma y i Gamma1 bx i . As we mentioned in Fact 8.1, the n processor C DMM is essentially equivalent to the n processor Arbitrary CRCW PRAM with O(n) shared memory. Hence we can use the ....

.... and Berkman and Vishkin (1993) 3) O(log n) time randomized algorithm for load balancing was independently presented by Gil et al. 1991) and by Bast and Hagerup (1991) Bast and Hagerup (1993) 4) Approximate parallel prefix sums can be solved within desired bound using an algorithm of Goodrich et al. 1994). 2 8.3 Reduction from CRCW Simulations to EREW Simulations We begin with a very fast transformation from simulating the CRCW PRAM to simulating a weaker model, the EREW PRAM. The input to the CRCW PRAM computation is a sequence ( 1 ; 1 ) n ; n ) of n pairs of requests. In the ....

Goodrich, M. T., Matias, Y., and Vishkin, U. (1994), "Optimal parallel approximation algorithms for prefix sums and integer sorting," In Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 241--250.

....also [14, p. 275] 2) The all nearest one problem can be solved in O(ff(n) O(log n) time deterministically on a CRCW PRAM by an algorithm due to Ragde [26] and Berkman and Vishkin [4] 3) Approximate prefix sums can be solved within the desired bound using an algorithm of Goodrich et al. [13]. 4. The Access Graph. In this section we show how PRAM simulation can be modeled as the access game on the access graph. We also discuss basic properties of the access graph. 4.1. Definition and Properties of the Access Graph. We start with the simulation of an EREW PRAM. The memory of the PRAM ....

M. T. Goodrich, Y. Matias, and U. Vishkin, Optimal parallel approximation algorithms for prefix sums and integer sorting, in Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM Press, New York, NY, 1994, pp. 241--250.

....and Hagerup. See also [16, p. 275] Chaining can be accomplished in O(ff(n) time deterministically on a CRCW PRAM by an algorithm due to Ragde [28] and Berkman and Vishkin [4] Approximate parallel prefix sums can be solved within the desired bound using an algorithm of Goodrich et al. [15]. 2 4 The Access Graph 4.1 Definition and Properties We start with the simulation of an EREW PRAM. The memory of the PRAM is hashed using three hash functions h 1 ; h 2 , and h 3 . That means, each memory cell u 2 U of the PRAM is stored in the modules M h1 (u) M h2 (u) and M h 3 (u) of the ....

M. T. Goodrich, Y. Matias, and U. Vishkin, Optimal parallel approximation algorithms for prefix sums and integer sorting, in Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, 1994, pp. 241--250.

....in O(h lg p) time and linear work with high probability, as follows. The elements are first placed in an array of size O(hn) sorted by the index of their destination processor; this can be done in O(lg (nh) time and O(nh) work w.h.p. using an algorithm for approximate integer sorting [27]. Then, for each element a pointer to its nearest element on its right in the array is found; this can be done in O(ff(nh) time (i.e. o(lg (nh) and O(nh) work, using a nearest zero algorithm [11] The last step creates a list of elements, consisting of sub lists of elements with the same ....

M.T. Goodrich, Y. Matias, and U. Vishkin. Optimal parallel approximation algorithms for prefix sums. In Proc. 5th ACM-SIAM Symp. on Discrete Algorithms, pages 241--250, 1994.

....require Omega Gammaeq n= lg lg n) time to be solved by a polynomial number of processors, as implied by the lower bound of Beame and Hastad [4] This lower bound holds even for randomized algorithms. More recent results have found other, more involved, ways to circumvent these barriers; cf. [38, 3, 26, 30]. We circumvent the obstacle of learning buckets sizes for the purpose of appropriate memory allocation by a technique of oblivious execution, sketched by Figure 1. 1. Partition the input set into buckets by a random polynomial of constant degree. 2. For t : 1 to O(lg lg n) do (a) Allocate M t ....

M. T. Goodrich, Y. Matias, and U. Vishkin. Optimal parallel approximation algorithms for prefix sums and integer sorting. In Proc. 5th ACM-SIAM Symp. on Discrete Algorithms, pages 241--250, Jan. 1994.

....pram, by replacing each prefix sums computation by either an approximate prefix sums computations or by a chaining computation. Algorithms for approximate prefix sums and for chaining are known to take O(taps ) where taps = lg lg p in the worst case and taps = lg p with high probability [BV93, GMV94, GZ95, Rag93] In order to use the approximate version of the prefix sums computation, we must allow for a small fraction of null cells in arrays P Ready and Active, and allow for a little less than p to be scheduled at each step even if p are available (as was already allowed to handle large ....

M.T. Goodrich, Y. Matias, and U. Vishkin. Optimal parallel approximation algorithms for prefix sums and integer sorting. In Proc. 5th ACM-SIAM Symp. on Discrete Algorithms, pages 241--250, January 1994.

....which require Omega Gammaeq n= lg lg n) time to be solved by polynomial number of processors, as implied by the lower bound of Beame and Hastad [4] This lower bound holds even for randomized algorithms. More recent results have found other, more involved, ways to circumvent these barriers; cf. [38, 3, 26, 30]. We circumvent the obstacle of learning buckets sizes for the purpose of appropriate memory allocation by a technique of oblivious execution, sketched by Figure 1. 1. Partition the input set into buckets by a random polynomial of constant degree. 2. For t : 1 to O(lg lg n) do (a) Allocate M t ....

M. T. Goodrich, Y. Matias, and U. Vishkin. Optimal parallel approximation algorithms for prefix sums and integer sorting. In Proc. 5th ACM-SIAM Symp. on Discrete Algorithms, pages 241--250, Jan. 1994.

....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 efficient simulation of erew pram on an n processor collision rmesh with O(lg lg lg n) delay. As a result, the self simulation of an N rmesh on a ....

M.T. Goodrich, Y. Matias, and U. Vishkin. Optimal parallel approximation algorithms for prefix sums and integer sorting. In Proc. 5th ACM-SIAM Symp. on Discrete Algorithms, pages 241--250, January 1994.

.... 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 precisely which step it will awake. Thus the ....

M.T. Goodrich, Y. Matias, and U. Vishkin. Optimal parallel approximation algorithms for prefix sums and integer sorting. In Proc. 5th ACMSIAM Symp. on Discrete Algorithms, pages 241--250, January 1994.

....in O(h lg p) time and linear work with high probability, as follows. The elements are first placed in an array of size O(hn) sorted by the index of their destination processor; this can be done in O(lg (nh) time and O(nh) work w.h.p. using an algorithm for approximate integer sorting [26]. Then, for each element a pointer to its nearest element on its right in the array is found; this can be done in O(ff(nh) time (i.e. o(lg (nh) and O(nh) work, using a nearest zero algorithm [11] The last step creates a list of elements, consisting of sub lists of elements with the same ....

M.T. Goodrich, Y. Matias, and U. Vishkin. Optimal parallel approximation algorithms for prefix sums. In Proc. 5th ACM-SIAM Symp. on Discrete Algorithms, pages 241--250, 1994.

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