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Parallel Algorithmic Techniques for Combinatorial Computation
 Ann. Rev. Comput. Sci
, 1988
"... this paper and supplied many helpful comments. This research was supported in part by NSF grants DCR8511713, CCR8605353, and CCR8814977, and by DARPA contract N0003984C0165. ..."
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Cited by 29 (3 self)
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this paper and supplied many helpful comments. This research was supported in part by NSF grants DCR8511713, CCR8605353, and CCR8814977, and by DARPA contract N0003984C0165.
The parallel complexity of element distinctness is \Omega\Gamma p log n
 SIAM J. Disc. Math
, 1988
"... Abstract. We consider the problem of element distinctness. Here n synchronized processors, each given an integer input, must decide whether these integers are pairwise distinct, while communicating via an infinitely large shared memory. If simultaneous write access to a memory cell is forbidden, the ..."
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Abstract. We consider the problem of element distinctness. Here n synchronized processors, each given an integer input, must decide whether these integers are pairwise distinct, while communicating via an infinitely large shared memory. If simultaneous write access to a memory cell is forbidden, then a lower bound of f(log n) on the number of steps easily follows (from S. Cook, C. Dwork, and R. Reischuk, SIAM J. Comput., 15 (1986), pp. 8797.) When several (different) values can be written simultaneously to any cell, then there is an simple algorithm requiring O(1) steps. We consider the intermediate model, in which simultaneous writes to a single cell are allowed only if all values written are equal. We prove a lower bound of f((logn) 1/2) steps, improving the previous lower bound of f(log log log n) steps (F.E. Fich, F. Meyer auf der Heide, and A. Wigderson, Adv. in Comput., 4 (1987), pp. 115). The proof uses Ramseytheoretic and combinatorial arguments. The result implies a separation between the powers of some variants of the PRAM model of parallel computation.
Prefix Graphs and Their Applications
 in "Proceedings, 20th International Workshop on GraphTheoretic Concepts in Computer Science," Lecture Notes in Computer Science
, 1994
"... . The range product problem is, for a given set S equipped with an associative operator ffi, to preprocess a sequence a1 ; : : : ; an of elements from S so as to enable efficient subsequent processing of queries of the form: Given a pair (s; t) of integers with 1 s t n, return as ffi as+1 ffi \De ..."
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. The range product problem is, for a given set S equipped with an associative operator ffi, to preprocess a sequence a1 ; : : : ; an of elements from S so as to enable efficient subsequent processing of queries of the form: Given a pair (s; t) of integers with 1 s t n, return as ffi as+1 ffi \Delta \Delta \Delta ffi a t . The generic range product problem and special cases thereof, usually with ffi computing the maximum of its arguments according to some linear order on S, have been extensively studied. We show that a large number of previous sequential and parallel algorithms for these problems can be unified and simplified by means of prefix graphs. 1 Introduction In 1983 Chandra, Fortune and Lipton introduced a computational paradigm closely related to the Ackermann function and used it to study the computation of semigroup products on unboundedfanin circuits [?, ?]. Since then the paradigm was rediscovered several times, under different names and in different guises, and expl...
Arne Andersson
"... We show that a unitcost RAM with a word length of w bits can sort n integers in the range 0 : : 2 w \Gamma 1 in O(n log log n) time, for arbitrary w log n, a significant improvement over the bound of O(n p log n) achieved by the fusion trees of Fredman and Willard. Provided that w (log n) 2+ ..."
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We show that a unitcost RAM with a word length of w bits can sort n integers in the range 0 : : 2 w \Gamma 1 in O(n log log n) time, for arbitrary w log n, a significant improvement over the bound of O(n p log n) achieved by the fusion trees of Fredman and Willard. Provided that w (log n) 2+ffl for some fixed ffl ? 0, the sorting can even be accomplished in linear expected time with a randomized algorithm. Both of our algorithms parallelize without loss on a unitcost PRAM with a word length of w bits. The first one yields an algorithm that uses O(log n) time and O(n log log n) operations on a deterministic CRCW PRAM. The second one yields an algorithm that uses O(log n) expected time and O(n) expected operations on a randomized EREW PRAM, provided that w (log n) 2+ffl for some fixed ffl ? 0. Our deterministic and randomized sequential and parallel algorithms generalize to the lexicographic sorting problem of sorting multipleprecision integers represented in several words...
Parallel Processing Letters, fc World Scienti c Publishing Company SIMPLE AND WORKEFFICIENT PARALLEL ALGORITHMS FOR THE MINIMUM SPANNING TREE PROBLEM
"... Communicated by Two simple and worke cient parallel algorithms for the minimum spanning tree problem are presented. Both algorithms perform O(m log n) work. The rst algorithm runs in O(log2 n) time on an EREW PRAM, while the second algorithm runs in O(log n) time on a Common CRCW PRAM. ..."
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Communicated by Two simple and worke cient parallel algorithms for the minimum spanning tree problem are presented. Both algorithms perform O(m log n) work. The rst algorithm runs in O(log2 n) time on an EREW PRAM, while the second algorithm runs in O(log n) time on a Common CRCW PRAM.