Rectilinear Matching to the Integer Grid in Nearly-Linear Time

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 $p\ge1$.