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<p><b>Abstract</b>—In this paper, we efficiently map a priority queue on the hypercube architecture in a load balanced manner, with no additional communication overhead, and present optimal parallel algorithms for performing insert and deletemin operations. Two implementations for such operations are proposed on the single-port hypercube model. In a <it>b</it>-bandwidth, <it>n</it>-item priority queue in which every node contains <it>b</it> items in sorted order, the first implementation achieves optimal speed-up of <tmath>$O({\rm min}\{{\rm log}\,\,n,{\textstyle{{b\,\,{\rm log}\,\,n} \over {{\rm log}\,\,b\,\,+\,\,{\rm log}\,{\rm log}\,\,n}}}\})$</tmath> for inserting <it>b</it> presorted items or deleting <it>b</it> smallest items, where <tmath>$b = O(n^{{1 \mathord{\left/ {\vphantom {1 c}} \right. \kern-\nulldelimiterspace} c}})$</tmath> with <it>c</it> > 1. In particular, single insertion and deletion operations are cost-optimal and require <tmath>$O({\textstyle{{{\rm log}\,n} \over p}} + {\rm log} \,\, p)$</tmath> time using <tmath>$O({\textstyle{{{\rm log}^{}\,\,n} \over {{\rm log}\,{\rm log}\,\,n}}})$</tmath> processors.</p><p>The second implementation is more scalable since it uses a larger number of processors, and attains a "nearly" optimal speed-up on the single hypercube. Namely, the insertion of log <it>n</it> presorted items or the deletion of the log <it>n</it> smallest items is accomplished in <it>O</it>(log log <it>n</it><super>2</super>)time using <tmath>$O({\textstyle{{{\rm log}^2\,\,n} \over {{\rm log}\,{\rm log}\,\,n}}})$</tmath> processors. Finally, on the slightly more powerful pipelined hypercube model, the second implementation performs log <it>n</it> operations in <it>O</it>(log log <it>n</it>) time using <tmath>$O({\textstyle{{{\rm log}^2\,\,n} \over {{\rm log}\,{\rm log}\,\,n}}})$</tmath> processors, thus achieving an optimal speed-up. To the best of our knowledge, our algorithms are the first implementations of <it>b</it>-bandwidth distributed priority queues, which are load balanced and yet guarantee optimal speed-ups.</p>
Hamiltonian path, heap, hypercube, load balance, priority queue, slope-tree, b-bandwidth slope-heap, speed-up.

S. K. Das, M. C. Pinotti and F. Sarkar, "Optimal and Load Balanced Mapping of Parallel Priority Queues in Hypercubes," in IEEE Transactions on Parallel & Distributed Systems, vol. 7, no. , pp. 555-564, 1996.
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