Partial match queries constitute the most basic type of associative queries in multidimensional data structures such as K\documentclass[12pt]{minimal}
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\begin{document}$$K$$\end{document}-d trees or quadtrees. Given a query q=(q0,…,qK-1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {q}=(q_0,\ldots ,q_{K-1})$$\end{document} where s of the coordinates are specified and K-s\documentclass[12pt]{minimal}
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\begin{document}$$K-s$$\end{document} are left unspecified (qi=∗\documentclass[12pt]{minimal}
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\begin{document}$$q_i=*$$\end{document}), a partial match search returns the subset of data points x=(x0,…,xK-1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {x}=(x_0,\ldots ,x_{K-1})$$\end{document} in the data structure that match the given query, that is, the data points such that xi=qi\documentclass[12pt]{minimal}
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\begin{document}$$x_i=q_i$$\end{document} whenever qi≠∗\documentclass[12pt]{minimal}
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\begin{document}$$q_i\not =*$$\end{document}. There exists a wealth of results about the cost of partial match searches in many different multidimensional data structures, but most of these results deal with random queries. Only recently a few papers have begun to investigate the cost of partial match queries with a fixed query q\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {q}$$\end{document}. This paper represents a new contribution in this direction, giving a detailed asymptotic estimate of the expected cost Pn,q\documentclass[12pt]{minimal}
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\begin{document}$$P_{{n},\mathbf {q}}$$\end{document} for a given fixed query q\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {q}$$\end{document}. From previous results on the cost of partial matches with a fixed query and the ones presented here, a deeper understanding is emerging, uncovering the following functional shape for Pn,q\documentclass[12pt]{minimal}
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\begin{document}$$P_{{n},\mathbf {q}}$$\end{document}Pn,q=ν·∏i:qiis specifiedqi(1-qi)α/2·nα+l.o.t.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} P_{{n},\mathbf {q}} = \nu \cdot \left( \prod _{i:q_i\text { is specified}}\, q_i(1-q_i)\right) ^{\alpha /2}\cdot n^\alpha + \text {l.o.t.} \end{aligned}$$\end{document}(l.o.t. lower order terms, throughout this work) in many multidimensional data structures, which differ only in the exponent α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} and the constant ν\documentclass[12pt]{minimal}
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\begin{document}$$\nu $$\end{document}, both dependent on s and K, and, for some data structures, on the whole pattern of specified and unspecified coordinates in q\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {q}$$\end{document} as well. Although it is tempting to conjecture that this functional shape is “universal”, we have shown experimentally that it seems not to be true for a variant of K\documentclass[12pt]{minimal}
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\begin{document}$$K$$\end{document}-d trees called squarish K\documentclass[12pt]{minimal}
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\begin{document}$$K$$\end{document}-d trees.
