AI 中文总结
研究直角匹配到整数网格问题，该问题目标集无限，需确定有限相关网格点集。本文证明几何压缩定理，\(O(n\log^2 n)\)时间构造\(O(n)\)大小集合\(\mathcal{C}\)，结合\(\ell_1\)距离网络表示，给出近线性时间随机精确算法及近似算法，改进标准方法。
AI 中文摘要
直角匹配到整数网格问题要求将\(\mathbb{R}^2\)中的\(n\)个点分配到\(\mathbb{Z}^2\)中不同的点，使总\(\ell_1\)移动最小。主要困难在于目标集是无限的，需先确定有限的相关网格点集且不损失最优性。本文证明了此无限目标问题的几何压缩定理，在\(O(n\log^2 n)\)时间内构造渐近最优大小为\(O(n)\)的集合\(\mathcal{C}\)。结合\(\ell_1\)距离的线性大小稀疏网络表示，在特定字-RAM模型下，通过近线性时间最小成本流算法得到期望运行时间为\(\widetilde O(n)\)的随机精确算法，改进了标准的\(\widetilde O(n^2)\)方法，还给出了固定整数\(p\geq1\)时的\(\widetilde O(n\sqrt n\log(1/\varepsilon))\)时间\((1 + \varepsilon)\)近似算法。
英文摘要
Rectilinear matching to the integer grid asks to assign each of $n$ points in $\mathbb R^2$ to a distinct point of $\mathbb Z^2$, minimizing total $\ell_1$ movement. The main difficulty is that the target set is infinite: one must first identify a finite set of relevant grid points without losing optimality. We prove a geometric compression theorem for this infinite-target problem. In $O(n\log^2 n)$ time, we construct a set $\mathcal{C}$ of asymptotically optimal size $O(n)$ such that, simultaneously for every $p\in[1,\infty]$, some optimal $\ell_p$ assignment uses only points of $\mathcal{C}$. The construction is independent of the subsequent optimization algorithm and of the coordinate spread. For the rectilinear case, we combine this candidate set with a linear-size sparse network representation of $\ell_1$ distances. In the word-RAM model with $O(1)$-word dyadic coordinates and $O(\log n)$ fractional bits, a nearly-linear time minimum-cost flow algorithm then gives a randomized exact algorithm with expected running time $\widetilde O(n)$. This improves the standard $\widetilde O(n^2)$ approach. Combined with existing finite geometric matching algorithms, the same candidate set also gives an $\widetilde O(n\sqrt n\log(1/\varepsilon))$-time $(1+\varepsilon)$ approximation for every fixed integer $p\ge1$.
