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<p><b>Abstract</b>—For 2 ≤<it>k</it>≤<it>n</it>, the <it>k</it>-merge problem is to merge a collection of <it>k</it> sorted sequences of total length <it>n</it> into a new sorted sequence. The <it>k</it>-merge problem is fundamental as it provides a common generalization of both merging and sorting. The main contribution of this work is to give simple and intuitive work-time optimal algorithms for the <it>k</it>-merge problem on three PRAM models, thus settling the status of the <it>k</it>-merge problem. We first prove that Ω(<it>n</it> log <it>k</it>) work is required to solve the <it>k</it>-merge problem on the PRAM models. We then show that the EREW-PRAM and both the CREW-PRAM and the CRCW require Ω(log <it>n</it>) time and Ω(log log <it>n</it> + log <it>k</it>) time, respectively, provided that the amount of work is bounded by <it>O</it>(<it>n</it> log <it>k</it>). Our first <it>k</it>-merge algorithm runs in Θ(log <it>n</it>) time and performs Θ(<it>n</it> log <it>k</it>) work on the EREW-PRAM. Finally, we design a work-time optimal CREW-PRAM <it>k</it>-merge algorithm that runs in Θ(log log <it>n</it> + log <it>k</it>) time and performs Θ(<it>n</it> log <it>k</it>) work. This latter algorithm is also work-time optimal on the CRCW-PRAM model. Our algorithms completely settle the status of the <it>k</it>-merge problem on the three main PRAM models.</p>
Merging, sorting, parallel algorithms, work-time optimal algorithms, information retrieval, databases, query processing.

S. Olariu, T. Hayashi and K. Nakano, "Work-Time Optimal k-Merge Algorithms on the PRAM," in IEEE Transactions on Parallel & Distributed Systems, vol. 9, no. , pp. 275-282, 1998.
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