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<p><b>Abstract</b>—Query processing is a crucial component of various application domains including information retrieval, database design and management, pattern recognition, robotics, and VLSI. Many of these applications involve data stored in a matrix satisfying a number of properties. One property that occurs time and again specifies that the rows and the columns of the matrix are independently sorted. It is customary to refer to such a matrix as <it>sorted</it>. An instance of the Batched Searching and Ranking problem, (BSR, for short) involves a sorted matrix <it>A</it> of <it>items</it> from a totally ordered universe, along with a collection <it>Q</it> of <it>queries</it>. <it>Q</it> is an arbitrary mix of the following query types: For a <it>search</it> query <it>q</it> <sub><it>j</it></sub>, one is interested in an item of <it>A</it> that is closest to <it>q</it> <sub><it>j</it></sub>; for a <it>rank</it> query <it>q</it> <sub><it>j</it></sub> one is interested in the number of items of <it>A</it> that are strictly smaller than <it>q</it> <sub><it>j</it></sub>. The BSR problem asks for solving all queries in <it>Q</it>. In this work, we consider the BSR problem in the following context: The matrix <it>A</it> is pretiled, one item per processor, onto an enhanced mesh of size <tmath>$\sqrt n\times \sqrt n$</tmath>; the <it>m</it> queries are stored, one per processor, in the first <tmath>${{m \over {\sqrt n}}}$</tmath> columns of the platform. Our main contribution is twofold. First, we show that any algorithm that solves the BSR problem must take at least <tmath>$\Omega ({\rm max\{log}n,\sqrt m\})$</tmath> time in the worst case. Second, we show that this time lower bound is tight on meshes of size <tmath>$\sqrt n\times \sqrt n$</tmath> enhanced with multiple broadcasting, by exhibiting an algorithm solving the BSR problem in <tmath>$\Theta ({\rm max\{log}\!\!n,\sqrt m\})$</tmath> time on such a platform.</p>
Searching, ranking, parallel algorithms, time-optimal algorithms, enhanced meshes, VLSI, database design, pattern recognition, robotics.

S. Olariu, H. Gurla, J. L. Schwing, L. Wilson and V. Bokka, "Time-Optimal Domain-Specific Querying on Enhanced Meshes," in IEEE Transactions on Parallel & Distributed Systems, vol. 8, no. , pp. 13-24, 1997.
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