We prove that some exact geometric pattern matching problems reduce in linear time to $k$-SUM when the pattern has a fixed size $k$. This holds in the real RAM model for searching for a similar copy of a set of $k\geq 3$ points within a set of $n$ points in the plane, and for searching for an affine image of a set of $k\geq d+2$ points within a set of $n$ points in $d$-space. As corollaries, we obtain improved real RAM algorithms and decision trees for the two problems. In particular, they can be solved by algebraic decision trees of near-linear height.

中文翻译：

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几何模式匹配减少到 k-SUM

我们证明，当模式具有固定大小 $k$ 时，一些精确的几何模式匹配问题在线性时间内减少到 $k$-SUM。这适用于真实的 RAM 模型，用于在平面中的一组 $n$ 点中搜索一组 $k\geq 3$ 点的相似副本，以及搜索一组 $k\ 的仿射图像在 $d$-space 中一组 $n$ 点中的 geq d+2$ 点。作为推论，我们获得了针对这两个问题的改进的真实 RAM 算法和决策树。特别是，它们可以通过接近线性高度的代数决策树来解决。