Yossi Shiloach and Uzi Vishkin. Finding the Maximum, Merging and Sorting in a Parallel Computation Model. Journal of Algorithms, 2:88-- 102, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....communication between the different processing units, that makes it a more natural model to describe parallel algorithms. The two models, however, are not comparable in general. While some problems in the comparison model, e.g. finding the maximum [26] have similar algorithms in the PRAM model [25], for other prob 0020 0190 97 517.00 1997 Elsevier Science B.V. All rights reserved. PII S0020 0190(97)00032 X 104 D. Breslauer et al. lnfirmation Processing Letters 62 (1997) 03 110 lems, e.g. finding the median [1,5] only much slower PRAM algorithms [7] are known for any polynomial ....

Y. Shiloach and U. Vishkin, Finding the maximum, merging and sorting in a parallel computation model, J. Algorithms' 2 (1981) 88-102.

....CRCWs can take time O(t log p log log p) with O(p) processors [20] or time O(t) with O(p log p) processors [10] These results can be improved if we know the pattern of concurrent memory access. In particular, many uses of strong concurrent write reduce to finding a single maximum. Theorem 4 [68, 19, 63]. The maximum of n values can be found in the weak CRCW model (a) in time t = O(log log k n) using O(kn t) processors; b) in constant time using O(n 1 # ) processors; and (c) in constant time using O(n) processors, if the values can be represented by O(log n) bits each. 2. PREFIX COMPUTATION ....

Y. Shiloach and U. Vishkin, Finding the Maximum, Merging and Sorting in a Parallel Computation Model. J. Algorithms 2, 1981, 88--102.

....communication between the different processing units, what makes it a more natural model to describe parallel algorithms. The two models, however, are not comparable is general. While some problems in the comparison model, e.g. finding the maximum [24] have similar algorithms in the PRAM model [23], other problems, e.g. finding the median [1, 5] have only much slower PRAM algorithms [7] with any polynomial number of processors. On the other hand, there exist problems, e.g. element distinctness, that have slow comparison model algorithms [6] and constant time PRAM algorithms on integer ....

Y. Shiloach and U. Vishkin. Finding the maximum, merging and sorting in a parallel computation model. J. Algorithms, 2:88--102, 1981.

....by DFG Graduiertenkolleg Parallele Rechnernetzwerke in der Produktionstechnik , ME 872 4 1, and by DFG Grant DI 412 2. 1 The two models, however, are not comparable in general. While some problems in the comparison model, e.g. finding the maximum [26] have similar algorithms in the PRAM model [25], for other problems, e.g. finding the median [1, 5] only much slower PRAM algorithms [7] are known for any polynomial number of processors. On the other hand, there exist problems, e.g. element distinctness, that have slow comparison model algorithms [6] and constant time PRAM algorithms on ....

Y. Shiloach and U. Vishkin. Finding the maximum, merging and sorting in a parallel computation model. J. Algorithms, 2:88--102, 1981.

....bounded PRAMs (e.g [28] 20] 6] 19] It is reducible to sorting, and it contains the problem of finding the minimum, maximum and median. It also contains T n k as a special case. 2.4. 1 The MIN problem The MIN problem was investigated in bounded resource PRAMs by Shiloach and Vishkin [36]. They showed an O(log log n) algorithm for CRCW PRAM with O(n= log log n) many processors. In the PPRAM model it is trivial that one can get an O(n=p p=m log log m 2 p ) by using the Shiloach Vishkin algorithm for the final stage after reducing the problem to size m and remembering that ....

Y.Shiloach and U.Vishkin. Finding the maximum, merging and sorting in a parallel computation model. Journal of Algorithms, 2:88--102, 1981. 16

....intersection and differences) that leads to saving of a logarithmic factor in running time. 3 Some basic parallel routines Before we describe the parallel algorithm in details, we will briefly review some of the basic parallel routines that will be used frequently in our algorithm. Lemma 3. 1 ( SV81] Two sorted lists of sizes M and N can be sorted in O(log(M N) time using (M N) log(M N) processors. Definition: Suppose there are n tasks and i th task requests x i processors such that P n i=1 x i = O(n) Then, the processor allocation problem of size n is to allocate a given set ....

Y. Shiloach and U. Vishkin. Finding the maximum, merging and sorting in a parallel computation model. Journal of algorithms, 2(1):88 -- 102, 1981.

....We focus on the computation of the lower hull only, the computation of the upper hull being similar. For a sequence a 1 , a 2 , a n of items, the prefix maxima is the sequence a 1 , maxfa 1 ; a 2 g, maxfa 1 ; a 2 ; a n g. For later reference, we state the following result [20, 29]. Lemma 5.1 The task of computing the prefix maxima (prefix minima) of an n item sequence can be performed in O(log n) time using n log n EREW processors or in O(log log n) time using n log log n Common CRCW processors. Let P = fp 1 ; p 2 ; p n g be a set of n points in the plane ....

Y. Shiloach and U. Vishkin. Finding the maximum, merging and sorting in a parallel computation model. Journal of Algorithms, 2(1):88--102, 1981.

....algorithm is O(log n) not very much) The problem here is that the merge step remains sequential, and is the bottleneck. As mentioned earlier, the parallelism in a divide and conquer algorithm can often be enhanced by parallelizing the divide step and or the merge step. Using a parallel merge [67] two sorted sequences of n=2 keys can be merged with work O(n) and depth O(log n) Using this merge algorithm, the recurrence for the depth of mergesort becomes D(n) D(n=2) O(log n) which has solution D(n) O(log 2 n) Using a technique called pipelined divide and conquer the depth of ....

....planar convex hull problem. They are both based on divide and conquer, but one does most of the work before the divide step, and the other does most of the work after. 2 The depth of finding the minimum or maximum of a set of numbers can actually be improved to O(log log n) with concurrent reads [67]. Figure 12: An example of the QuickHull algorithm. 6.2.1 QuickHull The parallel QuickHull algorithm [20] is based on the sequential version [61] so named because of its similarity to the QuickSort algorithm. As with QuickSort, the strategy is to pick a pivot element, split the data based on ....

Yossi Shiloach and Uzi Vishkin. Finding the maximum, merging and sorting in a parallel computation model. Journal of Algorithms, 2(1):88--102, 1981.

....though the data must fan in on any real hardware and therefore take time that increases with the memory size. In spite of this inaccuracy in the model, the unit time assumption has served as an excellent basis for the analysis of algorithms. In the parallel random access machine (P RAM) models [16, 40, 42, 19, 20], memory references are again assumed to take unit time. In these parallel models, this unit time is large since there is no practical hardware known that does better than deterministic O(lg 2 n) or probabilistic O(lg n) bit times for an arbitrary memory reference from n processors. 1 ....

....n) When the p n= lg n, the algorithm is optimal. Although the split radix sort and the quicksort algorithms are variations of well known algorithms translated to a new model, the merging algorithm described here is original. The merging algorithm of Shiloach and Vishkin for the CRCW P RAM model [17, 42] has the same complexity but is quite different. Their algorithm is not recursive. The basic idea of the halving merge algorithm is to extract the odd indexed elements from each of the two vectors by packing them into smaller vectors, to recursively merge the half length vectors, and then to use ....

Y. Shiloach and U. Vishkin. Finding the maximum, merging and sorting in a parallel computation model. Journal of Algorithms, 2(1):88--102, 1981.

....input keys are divided in p n segments of p n keys, their maxima are found recursively, and nally, the maximum of all maxima is determined by direct comparisons. The algorithm can be executed sequentially in time O(n) If n (CRCW PRAM) processors are available, time O(log log n) is achieved [15], which is seen by the recurrence T (n) T ( p n) O(1) split max max parts part part (a) x1 xi xr xi all yes = xi is max (b) list in r in 1st in rth max of maxes test if test if maxr is max max max1 is output Figure 5: a) Maximum with p n = r divide and conquer. b) Subalgorithm for ....

Y. Shiloach and U. Vishkin. Finding the Maximum, Merging and Sorting in a Parallel Computation Model. Journal of Algorithms, 2(1):88102, 1981.

....model on our model. 3. We prove upper bounds in the model for merging and sorting. In particular we give a parallel algorithm that merges two sorted sequences of size n stored as balanced trees with O(n) work and O(log n) steps. The algorithm borrows ideas from algorithms designed for the PRAM [35] but has some substantial changes to make up for the lack of random access. Based on this algorithm we can sort a sequence stored as a balanced tree with O(n log n) work and O(log 2 n) steps. For sequences stored as a list any algorithm would require Omega Gamma n) steps just to traverse the ....

....n= lg n splitters that partition the result exactly and uses these splitters to extract the appropriate subsequences of the two inputs, appending the results. Note that algorithms based on partitioning each input sequence into equal sized blocks, such as the PRAM algorithm of Shiloach and Vishkin [35] cannot be directly implemented efficiently on the A PAL model. This is because it is hard without side effects to do the patching between the two sequences. Also note that given a solution of the ranking problem (each element in a has its rank in b and vice versa) it remains nontrivial to ....

Yossi Shiloach and Uzi Vishkin. Finding the maximum, merging and sorting in a parallel computation model. Journal of Algorithms, 2(1):88--102, 1981.

....become the new value. Nothing is known about the selection. 10 Other memory access models for PRAMs has also been proposed. See e.g. 37, 59] Tolerant ( 25] The contents of a memory location does not change, if two or more processors try to write into that location in a given step. Common ([63]) If two or more processors attempt to write into a given memory location in a given step, then they must be attempting to write the same value, which then gets stored. Weak ( 41] Same as Common except that the common value must be 0. Collision ( 20] If two or more processors attempt to write ....

Y. Shiloach and U. Vishkin. Finding the Maximum, Merging and Sorting in a Parallel Computation Model. Journal of Algorithms, 2:88--102, 1981.

....geometry and computer graphics. The scan model allows a particularly simple definition of the algorithms. The Model Researchers have suggested several synchronous parallel models of computation. The most popular of these models are the parallel random access machine (P RAM) and related models [18, 34, 21]. A P RAM consists of a set of processors (random access machines) attached to a single shared memory. Processors communicate through the shared memory: one processor can write a value into the memory and another can read this value. Several variations of the P RAM models have been suggested that ....

Y. Shiloach and U. Vishkin, Finding the Maximum, Merging and Sorting in a Parallel Computation Model, Journal of Algorithms, 2 (1981), pp. 88--102.

....algorithm would require Omega Gamma n) depth just to traverse the list. This accentuates the importance of storing data as trees rather than lists to take advantage of parallel implementations of functional languages. The merging in mergesort borrows ideas from algorithms designed for the PRAM [41], but has some substantial changes to make up for the lack of random access. Both sorting algorithms require O(n log n) work and O(log 2 n) depth, and our work bounds are optimal for both merging and sorting, and our depth bounds are optimal for merging. Applicative order evaluation is used ....

Yossi Shiloach and Uzi Vishkin. Finding the maximum, merging and sorting in a parallel computation model. Journal of Algorithms, 2(1):88--102, 1981.

....we report actually exists, and the inter thread parallelism that the Spawn MT (or EMT) instruction code provides is already in an imperative form. This last contribution resembles the contribution of the PRAM model in providing imperative algorithms; see for instance the contributions of [SV81] and [BH85] in turning Valiant s [Va75] decision tree algorithms imperative, and the contribution of [SV82] in proposing to use Brent s non imperative theorem in the work time methodology for presenting (imperative) PRAM algorithms where all overheads are taken into account . The paper [Vi97] ....

Y. Shiloach and U. Vishkin. Finding the maximum, merging and sorting in a parallel computation model. J. of Algorithms, 2 (1981), 88--102.

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Yossi Shiloach and Uzi Vishkin. Finding the Maximum, Merging and Sorting in a Parallel Computation Model. Journal of Algorithms, 2:88-- 102, 1981.

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Y. Shiloach and U. Vishkin. Finding the maximum, merging and sorting in a parallel computation model. Journal of algorithms, 2(1):88 -- 102, 1981.

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