M.J. Atallah, S.R. Kosaraju, L.L. Larmore, G.L. Miller, S-H Teng, "Constructing Trees in Parallel", In Proc. of the 2nd IEEE Symposium on Parallel Algorithms and Architectures, 1989.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... or achieving a nearly optimal one, with a smaller computational effort. A through treatment is given in [22, x6.2. 2] Further work, especially on balanced BSTs, is surveyed in [27] Parallel algorithms have also been considered for construction of optimal and nearly optimal BSTs (see, e.g. [2, 20]) The results in [4] refer to the case where the elements of the RPV are known only up to a permutation. A general account of the properties of BSTs created under the same assumptions we use here is given by Mahmoud in his book [25] In this paper we introduce and analyze policies that achieve ....

M.J. Atallah, S.R. Kosaraju, L.L. Larmore, G.L. Miller, S-H Teng, "Constructing Trees in Parallel", In Proc. of the 2nd IEEE Symposium on Parallel Algorithms and Architectures, 1989.

....the problem. A matrix M[0: n; 0: m] is called concave if the following hold: M [i 1 ; j 1 ] M [i 2 ; j 2 ] M [i 2 ; j 1 ] M [i 1 ; j 2 ] for 0 i 1 i 2 n and 0 j 1 j 2 m (1) Concave matrices were first discussed in [12] and have been very successfully used in solving various problems (see [2, 3, 4, 6, 7, 8, 9, 11, 12, 13] and the references cited within) Given two matrices A[0: n; 0: m] and B[0: m; 0: n] the product matrix W [0: n; 0: n] A Theta B is defined by: W [i; j] min 0km (A[i; k] B[k; j] 2) For the SPBD problem, we require that G contains no negative cycles since otherwise the shortest path of ....

M. J. Atallah, S. R. Kosaraju, L. L. Larmore, G. L. Miller, and S-H Teng, Constructing Trees in Parallel, in Proc. ACM SPAA, 1989, pp. 421-431.

....O(n 6 ) processors on the CRCW PRAM model. The codeword for each source character can be generated in O(log n) time using O(n= log n) processors by tree contraction [MR85] Tree contraction is useful in parallel tree manipulation and is the basis of the approach taken to improve this result [AKLMT89]. Miller and Reif define RAKE and COMPRESS operations on trees [MR85] Let RAKE be an operation that removes all leaves from a tree and let COMPRESS be an operation that halves each chain of nodes (from leaf to root) by pointer doubling. By considering a restricted form of the RAKE operation ....

....leaf to root) by pointer doubling. By considering a restricted form of the RAKE operation where a leaf is removed only if its siblings are leaves, any left justified 6 tree can be reduced to a single chain of vertices along the leftmost path of the tree in at most dlog ne applications of RAKE [AKLMT89]. Also, each iteration of Step 2 in Teng s sequential algorithm simulates the RAKE operation and can be done in O(log n) time using n 3 = log n processors on the CREW PRAM model of computation. However, the algorithm requires O(n) iterations and therefore yields an O(n log n) total time bound. ....

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Atallah, M. J., Kosaraju, S. R., Larmore, L. L., Miller, G. L., and Teng, S.-H. Constructing trees in parallel. In Proceedings 1989 ACM Symposium on Parallel Algorithms and Architectures, Sante Fe, New Mex., 1989, pp. 283--290.

....search trees this can be reduced to O(n 2 ) using additional tabulated values of the cuts in table CUT (see [10] Once the table cost(i; j) is computed then the construction of an optimal tree can be done very efficiently in parallel. The following (easy to see) result was also observed in [2]: Lemma 2. If the table of costs is computed then an optimal tree can be constructed in O(logn) time with n 2 = log(n) processors. The structure of our algorithm is to mimic the sequential computation, however instead of computing one diagonal after the other we advance in larger steps. Let ....

....working in n 1 Gammaffl time, for some ffl 0, whose total work is quadratic. Another important problem is the design of an efficient NC algorithm for the general OBST problem. Even an improvement O(n 6 Gammaffl ) in the number of processors would be of considerable interest (cf. also, [2]) ....

M. J. Atallah, S. ~ R. Kosaraju, L. L. Larmore, G. L. Miller, and S--H. Teng, Constructing trees in parallel, Proceedings of the 1 st ACM Symposium on Parallel Algorithms and Architectures (1989), pp. 499--533.

....tree must support efficient insertion, deletion, and searching. Due to the sorted nature of a search tree, all insertions and searches invariably start at the root. The hot spot at the root makes all of these algorithms inappropriate for efficient cachecoherence protocols. The theory community [MoIy85, DePI86, AKLM89, KiPr90] constructs various optimal and near optimal search trees for various applications, such as Huffman encoding. However, these theoretical algorithms make unrealistic assumptions about communication through shared memory, with at least two [MoIy85, DePI86] assuming that the data structure is stored ....

M. J. Atallah, S. R. Kosaraju, L. L. Larmore, G. L. Miller, and S-H. Teng, "Constructing Trees in Parallel," Proceedings of the 1989 ACM Symposium on Parallel Algorithms and Architectures (SPAA '89), June 1989, 421-431.

....the optimal tree or achieving a nearly optimal one, with a smaller computational effort. An early outline is given in [15] A survey of more recent work on balanced BSTs appears in [18] Recently, parallel algorithms have been considered for construction of optimal and nearly optimal BSTs ([2, 13]) We devise and analyze policies which achieve a close approximation to the optimal BST, with lower organization costs than any of the previously studied heuristics. Section 2 defines some additional notation, and presents the dynamic programming algorithm that constructs the optimal tree for a ....

M.J. Atallah, S.R. Kosaraju, L.L. Larmore, G.L. Miller, S-H Teng, "Constructing Trees in Parallel", In Proc. of the 2nd IEEE Symposium on Parallel Algorithms and Architectures, (1989).

....problem constraints. 1 Introduction Recently, much research has gone into designing efficient parallel algorithms for problems with elementary serial dynamic programming solutions. These problems include string editing [1, 3] context free grammar recognition [22, 21] and optimal tree building [2, 19]. Polylog time parallel algorithms for solving these problems use new approaches since straightforward parallelization of sequential dynamic programming algorithms produces very slow (linear time) parallel algorithms. Many efficient parallel algorithms designed to date rely on monotonicity ....

M. J. Atallah, S. R. Kosaraju, L. L. Larmore, G. L. Miller, and S.-H. Teng: "Constructing Trees in Parallel," Proc. 1st Symp. on Parallel Algorithms and Architectures, 499-533, 1989

....in this area, hopefully there will be many more. Directions of Further Research and Conclusions 123 9.1. 1 Optimal Binary Search Trees The development of algorithms for building optimal binary search trees in polylog parallel time with low processor complexity is cited as an open problem in (Atallah et al. 1989) and (Larmore, Przytycka, and Rytter, 1993) By low processor complexity we mean n k for some constant k 6, The best sequential algorithm for building optimal binary search trees takes Theta(n 2 ) time (Knuth, 1973) There are efficient parallel algorithms for building Huffman trees, ....

.... we mean n k for some constant k 6, The best sequential algorithm for building optimal binary search trees takes Theta(n 2 ) time (Knuth, 1973) There are efficient parallel algorithms for building Huffman trees, alphabetic trees and approximate binary search trees, see (Teng, 1987; Atallah et al. 1989; Larmore and Przytycka, 1992; Larmore, Przytycka, and Rytter, 1993) But, to date, there have been no efficient parallel algorithms for the construction of optimal binary search trees. The optimal binary search tree problem (OBST) is: Given n search keys K i , for 1 i n each with an associated ....

[Article contains additional citation context not shown here]

M. J. Atallah, S. R. Kosaraju, L. L. Larmore, G. L. Miller, and S.-H. Teng: "Constructing Trees in Parallel," Proc. 1st Symp. on Parallel Algorithms and Architectures (SPAA), ACM Press, 499--513, 1989.

....the optimal tree or achieving a nearly optimal one, with a smaller computational effort. An early outline is given in [15] A survey of more recent work on balanced BSTs appears in [18] Recently, parallel algorithms have been considered for construction of optimal and nearly optimal BSTs ([2, 13]) In this paper we devise and analyze policies which achieve a close approximation to the optimal BST, with lower organization costs than any of the previously studied heuristics. Section 2 defines some additional notation, and presents the dynamic programming algorithm that constructs the ....

M.J. Atallah, S.R. Kosaraju, L.L. Larmore, G.L. Miller, S-H Teng, "Constructing Trees in Parallel", In Proc. of the 2nd IEEE Symposium on Parallel Algorithms and Architectures, (1989).

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M. J. Atallah, S. R. Kosaraju, L. L. Larmore, G. L. Miller, and S.-H. Teng. Constructing trees in parallel. In ACM-SIGACT; ACM-SIGARCH, editor, Proceedings of the 1st Annual ACM Symposium on Parallel Algorithms and Architectures, page 421, Santa Fe, NM, June 1989. ACM Press.

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M. J. Atallah, S. R. Kosaraju, L. L. Larmore, G. L. Miller, and S.-H. Teng. Constructing trees in parallel. In ACM-SIGACT; ACM-SIGARCH, editor, Proceedings of the 1st Annual ACM Symposium on Parallel Algorithms and Architectures, page 421, Santa Fe, NM, June 1989. ACM Press. 12

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M.J. Atallah, S.R. Kosajaru, L.L. Larmore, G.L. Miller, S-H. Teng, Constructing trees in parallel, SPAA 1989, pp. 421--431.

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