U. Vishkin, W. J. Paul, and H. Wagener. Parallel dictionaries on 2-3 trees. In ICALP 1983, pages 597--609.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(resp. the object holding it) in a concurrent data structure can be used as a handle into the data structure and that the operations Insert, Delete, and Search can be executed concurrently. Concurrent data structures have been heavily investigated in the area of shared memory machines (or PRAMs) [14], though in a slightly weaker form than we require here, since in some of these constructions not every data item can be used as a handle. The reason this was often not necessary is that these approaches rather concentrate on parallelizing the way in which the data structure is accessed instead of ....

U. Vishkin, W. J. Paul, and H. Wagener. Parallel dictionaries on 2-3 trees. In ICALP 1983, pages 597--609.

....comes from the exclusive read restriction of the model. A very elegantway around this restriction was given byPaul, 1 Note that due to the congestion problem, even an efficientembedding of the graph G into the network will not lead to an efficientmultisearch algorithm. 3 Vishkin and Wagener [26] for the case where G is a 2 3 tree. However, it should be noted that they assume a linear ordering on the searchkeys. We cannot afford to make this assumption since we consider applications involving multidimensional searchkeys for which no linear ordering can be used. The multisearch problem ....

....the time per advancement of all queries by one step needs to be O( p n log n ) whichisless than the diameter of the network. The techniques we use to solve the multisearch problem for the mesh are very different from those used in [8] and they are also very different from those used in [26]. In very broad terms, our techniques for solving the multisearch problem are a judicious combination of the following ideas. ffl Partition G into pieces, some of which are processed sequentially, while others are processed in parallel. ffl Create multiple copies of those pieces of G for ....

W. Paul, U. Vishkin and H. Wagener. Parallel dictionaries on 2-3 trees. in Proceedings 10th International Colloquium on Automata, Languages, and Programming (ICALP), LNCS 154, Springer-Vergerlag, Berlin, 1983, pp. 597-609.

....are designed such that each task is some number of steps ahead of the task following it. Pipelining has been used to improve the time of many parallel algorithms for shared memory models. Paul et al. described a pipelined algorithm for inserting m new keys into a balanced 2 3 tree with n keys [28]. They first considered a nonpipelined algorithm that has O(lg m) tasks, each of which takes O(lg n) parallel time (steps) for a total time of O(lg n lg m) on an EREW PRAM. Each task works its way up from the bottom of the insertion tree to the top, one level at a time. They then showed how to ....

....require a dynamic pipeline, which varies in depth depending on the input data. As such asynchronous algorithms have not been considered before, we developed a new technique for analyzing their computation depth. The fourth algorithm is a variant of Paul, Vishkin and Wagener s (PVW) 2 3 trees [28]. Because the bottom up insertion used in the PVW algorithm does not map naturally into the use of futures, we describe a top down variant that does. As with the PVW algorithm, the pipelining improves the algorithm complexity for inserting m keys into a tree of size n from O(lg n lg m) to O(lg n ....

[Article contains additional citation context not shown here]

W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2--3 trees. In Proceedings of the 10th Colloquium on Automata, Languages and Programming, pages 597--609. Lecture Notes in Computer Science, vol. 143. Springer-Verlag, Berlin, July 1983.

....the butterfly network. It runs in time O(log n) with high probability, for n = m = p. However, large constants are involved and it performs badly on the BSP model. Further it is not obvious how to generalize the algorithm work optimally for the case n m. The EREW PRAM algorithm of Paul et al. [14] is capable of executing n search queries on a search tree with m nodes in time O(log m log n) Unfortunately, their approach would require too much fine grained communication on the BSP model. Ranade [15] has developed a multisearch algorithm for the p processor butterfly network for n = p log p ....

W. Paul, U. Vishkin and H. Wagener, Parallel dictionaries on 2-3 trees, in: Proc. of the 10th ICALP (1983) 597--609.

....we use the results of all three chapters. Thus the primary purpose of this chapter is as a case study of the use and limitations of the simulation techniques of Chapters 4 and 5. We omit many of the details of the proofs in this chapter, as they can be found in [46, 47] In multisearch problems [44, 12, 72] a large number of queries are simultaneously processed and answered by navigating through a large data structure on a parallel computer. A closely related application is an Internet search engine. As input for the multisearch problem, we consider a set of n queries and a balanced binary search ....

.... for solving a number of data structuring problems on a parallel machine, e.g. trapezoidal decomposition, next element search on hypercubes [44] multiple planar point location, interval trees, hierarchical representations of polyhedras on meshes [12] dictionaries on 2 3 trees on a EREW PRAM [72], maintaining balanced binary search trees on a BSP [17, 16] multiple point location in a class of hierarchical DAGs on a BSP [46] and a BSP [51] The BSP multisearch algorithm due to Baumker, Dittrich, and Meyer auf der Heide [17] with improvements due to Dittrich [46] was the first ....

W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2-3 trees. In Proc. Int. Colloquium Algorithms, Languages and Programming, LNCS 154, pages 597--604, 1983.

....requires O(1) communication rounds. The case of large m is harder especially in a distributed memory setting, due to the difficulty of providing efficient parallel access to the large number of shared data. An optimal EREW PRAM algorithm, running in O(log m) time on n processors, is given in [9]. The algorithm, however, exploits a linear ordering of the queries which is not possible when working with a multidimensional universe. In [10] Ranade presents a randomized algorithm that achieves optimal speed up on a p processor butterfly for p log p queries on a number of segments polynomial ....

W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2-3 trees. In Proc. of the 10th Int. Colloquium on Automata, Languages and Programming, pages 597--609, 1983.

....of an EM algorithm has been the number of I O operations performed, and internal computation time is ignored (see [32] We will call an algorithm I O optimal if the number of I O operations matches the lower bound for the number of I Os required to solve the problem. In multisearch problems [17, 5, 27] a large number of queries are simultaneously processed and satisfied by navigating through a large data structure on a parallel computer. Multisearch has been shown to be very useful for solving a 1 P1 P2 Pp Processors Processors Router Network CPU Memory Disk Processor Comm ....

....Memory number of data structuring problems on a parallel processor, e.g. trapezoidal decomposition, next element search, etc. on hypercubes [17] multi planar point location, interval trees, hierarchical representations of polyhedras on meshes [5] dictionaries on 2 3 trees on a EREW PRAM [27], maintaining binary search trees on a BSP [7, 6] multi point location in a class of hierarchical DAGs on a BSP [19] and a BSP [18] Acheiving blocking for multisearch is especially difficult. Batch filtering was introduced in [21] as a sequential EM paradigm for executing a collection of ....

Paul, W., Vishkin, U., and Wagener, H. Parallel dictionaries on 2-3 trees. In Proc. Int. Colloquium Algorithms, Languages and Programming, LNCS 154 (1983), pp. 597--604.

....presented by T. Ottmann and D. Wood in the same way. 1 AVL Trees They are a basic data structure [1, 5] However it left open how to design massively parallel dictionaries on them. This problem was first solved for other classes of trees (2 3 trees) in 1983 by W. Paul, U. Vishkin and H. Wagener [8]. These ideas were applied on B trees by L. Higham and E. Schenk [3] in 1994 and on Skip lists by J. Gabarr o, C. Mart inez and X. Messeguer [2] in 1995. In AVL trees every node n stores a key key(n) and left(n) right(n) father(n) points to their sons and father. The height is height(nil) 0 ....

....new root of the rotated subtree we have: virtual ddepth(n) virtual ddepth(r 0 ) red(r 0 ) 2.3 Algorithms for insertions Massive parallel algorithm. Just, take the preceding set of rules and apply them in a synchronized way along the lines given in W. Paul, U. Vishkin and H. Wagener [8]. It holds: Theorem 1 Take a relaxed AVL having red nodes at the bottom. The nodes that belong to the front wave have the same virtual dynamic depth. Moreover, if we start from the bottom a virtual plane wave w moving up and (take for instance = 10) steps later we start another one w 0 moving ....

W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2--3 trees. In J. D'iaz, editor, Proc. 10th ICALP , LNCS 154, pages 597--609. SpringerVerlag, 1983.

....of keys, and has been applied to most search trees[EZG 82,BY95] Note that fringe analysis is the average case analysis of the fringe of the tree. We are interested on the fringe analysis of the synchronized parallel algorithm on 2 3 trees designed by Paul, Vishkin, and Wagener (PVW) PVW83] This kind of algorithms manage data types in a synchronized manner (PRAM algorithms [J aJ92] They can be envisaged as many sequential algorithms running simultaneously and executing the same operation at the same time. Therefore, it may happen that several processes read or write on the same ....

....1 : In a more compact form, the 1 OneStep can be rewritten as: T n;1 = 1 n 1 n Gamma 1 4 3 n Gamma 2 Later on we will give a direct proof of this compact expression. 3 Synchronized parallel insertion algorithms In this section we recall the algorithm of Paul, Vishkin, and Wagener [PVW83] and we introduce our MacroSplit algorithm. It is assumed that an array of k sorted keys a[1 : k] is inserted into a 2 3 tree having n leaves. The algorithms first hang the keys from the leaves and later rebalance the tree. The PVW algorithms differs from the MacroSplit algorithms on the ....

W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2--3 trees. In J. D'iaz, editor, Proc. 10th International Colloquium on Automata, Languages and Programming, LNCS 154, pages 597--609. Springer-Verlag, 1983. Also appeared as "Parallel computation on 2--3 trees" in RAIRO Informatique Th'eorique, pages 397-404, 1983.

....executed in parallel level by level. When paths from two (or three) recolored elements to the root meet at an internal node, the two (three) increments are combined, and only one proceeds to the next level. 2) For splitting m elements we can use a procedure similar to Paul, Vishkin and Wagener s [31] procedure for inserting into 2 3 trees. Since each leaf can only split into two leaves, the splitting may convert a node just above the leaf level into a 4, 5 or 6 node which then needs to be split. This in turn may split nodes at the next level, and the splitting can go up the tree level by ....

W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2--3 trees. In Lecture Notes in Computer Science 143: Proc. Colloquium on Automata, Languages and Programming, Barcelona, Spain, pages 597--609, Berlin/New York, July 1983. Springer-Verlag.

....to work in future versions of fork as well. The code has been tested using the current version of the fork compiler and a simulator for the SB PRAM. The code should therefore run immediately on the real hardware. 2. We present an implementation of the basic operations on 2 3 trees as described in [9]. The two crucial observations in the parallelization of the 2 3 tree operations are that operations (insertions or deletions) applied to different leaves can take place in parallel, and that a form of pipelining can be employed which makes it possible to perform any sequence of k operations (of ....

....also the operations of finding the item of minimum key, finding the item with the kth smallest key, splitting a dictionary according to a given key, and concatenating two dictionaries (the keys of the first dictionary must be smaller than the keys of the second) within the same time bound. In [9] it is shown how to implement the 2 3 tree operations on an EREW PRAM. This parallel implementation supports parallel searching for a set of k elements, and insertion and deletion of an ordered set of k elements in O(log n log k) parallel steps using k processors. The implementation given in ....

[Article contains additional citation context not shown here]

W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2-3 trees. In Proceedings of the 10th ICALP, volume 154 of Lecture Notes in Computer Science, pages 597--609, 1983.

....to manage data bases, taking care, however,with the length of nodes. Finally, we have designed a concurrent algorithm on the fly approach. We leave for further research the design of massively parallel algorithms for Skip trees. The state of the art is that W. Paul, U. Vishkin and H. Wagener [19] designed a parallel algorithm for 2 3 trees, L. Higham and E. Schenks [9] designed one for B trees, and J. Gabarr o, C. Mart inez and X. Messeguer [7] designed a parallel algorithm for Skip lists. Thus, the parallel algorithm of Skip trees will arise from the right mixture of them. ....

W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2--3 trees. In J. D'iaz, editor, Proc. 10th International Colloquium on Automata, Programming and Languages, LNCS 154, pages 597--609. Springer-Verlag, 1983. Also appeared as "Parallel computation on 2--3 trees" in RAIRO Informatique Th'eorique, pages 397-404, 1983.

....for directed graphs similar to the tree data type is defined. Parallel operations include finding the connected components of a graph, and computing a (minimum) spanning tree 3. 5 Parallel dictionaries Currently PAD contains one non trivial parallel dictionary data structure based on 2 3 trees [15]. Dictionaries can be defined over base types ordered by an integer key. A parallel dictionary constructor makes it possible to build a dictionary from an ordered array of dictionary items. Dictionary operations include (parallel) searching and parallel (pipelined) insertion and deletion. ....

W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2-3 trees. In Proc. 10th ICALP, Springer LNCS 154, 1983.

....presented in [3] are slower in both computation and communication. In addition, our algorithms work for values of p much closer to n than those of [3] even when differences in the two models are taken into consideration. The first approach on parallel dictionaries is implied in the work of [36], where a parallel implementation of 2 3 trees is presented. Randomized algorithms for maintaining parallel dictionaries are derived in [38] These approaches fail, however, to deliver optimal performance when directly implemented on the BSP model of computation. The only previously known ....

....h level graphs can be extended (along the lines of [4] to handle dynamic operations. Unfortunately, the resulting algorithms are unduly complicated and of little practical significance. In this work, we improve upon the communication requirements of previous results on parallel dictionaries [12, 16, 36, 38], and extend the BSP algorithms of [12, 16] by presenting efficient and simple algorithms. We also show that optimality to within small multiplicative constant factors can be achieved for a wide range of the BSP parameters. The randomized algorithms we present are based on the ideas of [14, 15] ....

[Article contains additional citation context not shown here]

W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2-3 trees. In Proceeding of ICALP'83: Automate, Languages and Programming, pp. 597-609, LNCS 154, Springer Verlag, 1983.

....reading, but not writing, is permitted. See Karp and Ramachandran [1] for an overview of PRAM models and results. PRAM algorithms that manipulate dynamic data structures need to be able to secure and release memory from the global store in parallel. For example, Paul, Vishkin and Wagener [2] present algorithms for maintaining 2 3 trees that depend upon dynamic allocation and deallocation of memory. An extension to general B trees [3] similarly depends on parallel memory management. Section 2 describes a general scheme for an n log n processor EREW PRAM that dynamically allocates and ....

W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2-3 trees. In Lecture Notes in Computer Science 143: Proceedings of the 10th Colloquium on Automata, Languages and Programming, pages 597--609. Springer Verlag, 1983.

....runs in time O(log n) with high probability, for n = m = p. It is easily seen that large constant factors are involved and that it performs badly on the BSP model. Further it is not obvious how to generalize the algorithm work optimally for the case n m. The EREW PRAM algorithm of Paul et al. [11] is capable of executing n search queries on a search tree with m nodes in time O(logm log n) Unfortunately, their approach would require too much fine grained communication on the BSP model. Ranade [12] has developed a multisearch algorithm for the p processor butterfly network for n = p log ....

W. Paul, U. Vishkin and H. Wagener. Parallel dictionaries on 2-3 trees. Proc. of the 10th ICALP, Barcelona, Spain, 1983, 597--609.

....data type is defined. Parallel operations (not yet implemented) will include finding the connected components, and extracting a (minimum) spanning tree [3, 9, 10] Parallel dictionaries Currently PAD includes one nontrivial parallel data structure, namely a parallel dictionary based on 2 3 trees [14]. Dictionaries can be defined over base types ordered by an integer key. A parallel dictionary constructor makes it possible to build a dictionary from an ordered array of dictionary items. Dictionary operations include parallel searching and parallel (pipelined) insertion and deletion. ....

W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2-3 trees. In Proceedings of the 10th ICALP, volume 154 of Lecture Notes in Computer Science, pages 597--609, 1983.

....dictionary data structure. An adaptive allocation and deallocation of space, so as to maintain at each step space linear in the number of representatives in the S data structure, or in the number of program variables, can be implemented with p processors and logarithmic time on the erew pram [PVW83] and in O(lg p) time and linear work with high probability, on a crcw pram [GMV91] In a recent work [BGM95] we have developed data structures that are weaker than the general dictionary data structure, yet sufficient for our purposes, that obtain a simpler execution on the crcw pram; ....

W.J. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2-3 trees. In Proc. 10th Int. Colloquium on Automata Languages and Programming, Springer LNCS 154, pages 597--609, 1983.

....All asymptotic bounds refer to the problem size as n 1. 1.2 Known Results There exists a vast variety of dynamic data structures (e.g. 2 3 trees, B trees, treaps) for the RAM model, see e.g. Mehlhorn [5] They all need time O(log m) per operation, if the search tree has size m. Paul et al. [7] give an EREW PRAM implementation of 2 3 trees. With n processors, they can process n search(x) insert(x) or delete(x) operations in time O(log m log n) By simulating their PRAM algorithm on the BSP n operations can be executed in computation time O( n log m p ) and communication time O(g n ....

....where the ranges become larger with growing input sizes m and n. The amortized approach allows much larger ranges. Our results improve the results of Ranade [6] and Baumker et al. 2] by supporting a fully dynamic search tree rather than a static one, and the result of Paul et al. [7] by significantly reducing the communication time. Further, our algorithms improve upon the algorithms of Ranade [6] and Paul et al. 7] by communicating in a blockwise fashion. We proceed in the following way. In Section 4, we introduce the BB (a) tree, which is a mixture of a B tree and a ....

[Article contains additional citation context not shown here]

W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2-3 trees. In J. Diaz, editor, ICALP 83: Automata Languages and Programming. Volume 154 of Lecture Notes in Computer Science, pp. 597--609. Springer Verlag, New York, NY, 1983.

....into a 2 3 tree having n leaves. The MacroSplit insertions algorithm has two main successive phases. Percolation Phase. In a top down strategy, the set of keys to be inserted is split into several packets and these packets are routed down. Finally, these packets are attached to the leaves [PVW83,GMM96,GM97] Reconstruction Phase. In a bottom up phase the packets attached to the leaves are really inserted and the tree is reconstructed. This reconstruction is based in just one unique wave moving bottom up. First, the packets are incorporated at the bottom internal nodes of the tree. In ....

....a maximum number of internal nodes holding two keys. Another possibility consists on generate a maximum number of nodes with one key. The MacroSplit algorithm can be seen as a height level description of the well known parallel insertion algorithm given by W. Paul, U. Vishkin and H. Wagener in [PVW83] whose reconstruction phase has been refined (in order to avoid concurrent readings) This refinement take place splitting a MacroSplit step into several more basic steps chained together in a pipeline. 3 Qualitative behavior of insertion algorithms In the further sections we will develop the ....

W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2--3 trees. In J. D'iaz, editor, Proc. 10th International Colloquium on Automata, Programming and Languages, LNCS 154, pages 597--609. Springer-Verlag, 1983. Also appeared as "Parallel computation on 2--3 trees" in RAIRO Informatique Th'eorique, pages 397-404, 1983.

....using a dynamic dictionary data structure. Hence, an adaptive allocation and deallocation of space, so as to maintain at each step space which is linear in the number of live processes, or in the number of program variables, can be implemented in logarithmic time on a p processor erew pram [PVW83] and in O(lg p) time and linear work with high probability, on a crcw pram [GMV91] Acknowledgements. The authors thank the anonymous referees for comments that helped improve the presentation of this paper. ....

W.J. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2-3 trees. In Proc. 10th Int. Colloquium on Automata Languages and Programming, Springer LNCS 154, pages 597--609, 1983.

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W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2-3 trees. In Proc. of 10th ICALP, Springer LNCS 154, pages 597--609, 1983.

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U. Vishkin, W. J. Paul, and H. Wagener. Parallel dictionaries on 2-3 trees. In ICALP 1983, pages 597--609.

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Paul, W., U. Vishkin and H. Wagenet, Parallel dictionaries on 2-3 trees, Pro<:. 10th ICALP

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W. Paul, U. Vishkin, and H. Wagener. Parallel dictionaries on 2--3 trees. In J. D'iaz, editor, Proc. 10th ICALP, LNCS 154, pages 597--609. SpringerVerlag,

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