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Optimal and Load Balanced Mapping of Parallel Priority Queues in Hypercubes
June 1996 (vol. 7 no. 6)
pp. 555-564

Abstract—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 b-bandwidth, n-item priority queue in which every node contains b items in sorted order, the first implementation achieves optimal speed-up of $O({\rm min}\{{\rm log}\,\,n,{\textstyle{{b\,\,{\rm log}\,\,n} \over {{\rm log}\,\,b\,\,+\,\,{\rm log}\,{\rm log}\,\,n}}}\})$ for inserting b presorted items or deleting b smallest items, where $b = O(n^{{1 \mathord{\left/ {\vphantom {1 c}} \right. \kern-\nulldelimiterspace} c}})$ with c > 1. In particular, single insertion and deletion operations are cost-optimal and require $O({\textstyle{{{\rm log}\,n} \over p}} + {\rm log} \,\, p)$ time using $O({\textstyle{{{\rm log}^{}\,\,n} \over {{\rm log}\,{\rm log}\,\,n}}})$ processors.

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 n presorted items or the deletion of the log n smallest items is accomplished in O(log log n2)time using $O({\textstyle{{{\rm log}^2\,\,n} \over {{\rm log}\,{\rm log}\,\,n}}})$ processors. Finally, on the slightly more powerful pipelined hypercube model, the second implementation performs log n operations in O(log log n) time using $O({\textstyle{{{\rm log}^2\,\,n} \over {{\rm log}\,{\rm log}\,\,n}}})$ processors, thus achieving an optimal speed-up. To the best of our knowledge, our algorithms are the first implementations of b-bandwidth distributed priority queues, which are load balanced and yet guarantee optimal speed-ups.

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Index Terms:
Hamiltonian path, heap, hypercube, load balance, priority queue, slope-tree, b-bandwidth slope-heap, speed-up.
Sajal K. Das, M. Cristina Pinotti, Falguni Sarkar, "Optimal and Load Balanced Mapping of Parallel Priority Queues in Hypercubes," IEEE Transactions on Parallel and Distributed Systems, vol. 7, no. 6, pp. 555-564, June 1996, doi:10.1109/71.506694
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