Merge algorithm

Abstract: We give a parallel implementation of merge sort on a CREW PRAM that uses n processors and $O(\log n)$ time; the constant in the running time is small. We also give a more complex version of the algorithm for the EREW PRAM; it also uses n processors and $O(\log n)$ time. The constant in the running time is still moderate, though not as small.

Abstract: : The paper asserts that the hidden-surface problem is mainly one of sorting. The various surfaces of an object to be shown in hidden-surface or hidden-line form must be sorted to find out which ones are visible at various places on the screen. Surfaces may be sorted by lateral position in the picture (XY), by depth (Z), or by other criteria. The paper shows that the order of sorting and the types of sorting used form differences among the existing hidden-surface algorithms. (Modified author abstract)

Abstract: Sorting is arguably the most studied problem in computer science, both because it is used as a substep in many applications and because it is a simple, combinatorial problem with many interesting and diverse solutions. Sorting is also an important benchmark for parallel supercomputers. It requires significant communication bandwidth among processors, unlike many other supercomputer benchmarks, and the most efficient sorting algorithms communicate data in irregular patterns. Parallel algorithms for sorting have been studied since at least the 1960’s. An early advance in parallel sorting came in 1968 when Batcher discovered the elegant U(lg2 n)-depth bitonic sorting network [3]. For certain families of fixed interconnection networks, such as the hypercube and shuffle-exchange, Batcher’s bitonic sorting technique provides a parallel algorithm for sorting n numbers in U(lg2 n) time with n processors. The question of existence of a o(lg2 n)-depth sorting network remained open until 1983, when Ajtai, Komlos, and Szemeredi [1] provided an optimal U(lg n)-depth sorting network, but unfortunately, their construction leads to larger networks than those given by bitonic sort for all “practical” values of n. Leighton [15] has shown that any U(lg n)-depth family of sorting networks can be used to sort n numbers in U(lg n) time in the bounded-degree fixed interconnection network domain. Not surprisingly, the optimal U(lg n)-time fixed interconnection sorting networks implied by the AKS construction are also impractical. In 1983, Reif and Valiant proposed a more practical O(lg n)-time randomized algorithm for sorting [19], called flashsort. Many other parallel sorting algorithms have been proposed in the literature, including parallel versions of radix sort and quicksort [5], a variant of quicksort called hyperquicksort [23], smoothsort [18], column sort [15], Nassimi and Sahni’s sort [17], and parallel merge sort [6]. This paper reports the findings of a project undertaken at Thinking Machines Corporation to develop a fast sorting algorithm for the Connection Machine Supercomputer model CM-2. The primary goals of this project were: