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 \(k\ge 3\) points within a set of n points in the plane, and for searching for an affine image of a set of \(k\ge d+2\) 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。这适用于真正的RAM模型用于搜索一个类似的一组的拷贝\（K \ GE 3 \）的一组内的点Ñ在平面上的点，和用于搜索一组的仿射图像\（K \ge d+2\)点在d空间中的一组n个点中。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。