Sublinear time complexity is required by the massively parallel computation (MPC) model. Breaking dynamic programs into a set of sparse dynamic programs that can be divided, solved, and merged in sublinear time. The rectangle escape problem (REP) is defined as follows: For $n$ axis-aligned rectangles inside an axis-aligned bounding box $B$, extend each rectangle in only one of the four directions: up, down, left, or right until it reaches $B$ and the density $k$ is minimized, where $k$ is the maximum number of extensions of rectangles to the boundary that pass through a point inside bounding box $B$. REP is NP-hard for $k>1$. If the rectangles are points of a grid (or unit squares of a grid), the problem is called the square escape problem (SEP) and it is still NP-hard. We give a $2$-approximation algorithm for SEP with $k\geq2$ with time complexity $O(n^{3/2}k^2)$. This improves the time complexity of existing algorithms which are at least quadratic. Also, the approximation ratio of our algorithm for $k\geq 3$ is $3/2$ which is tight. We also give a $8$-approximation algorithm for REP with time complexity $O(n\log n+nk)$ and give a MPC version of this algorithm for $k=O(1)$ which is the first parallel algorithm for this problem.

翻译：大规模并行计算（MPC）模型要求亚线性时间复杂度。本文方法将动态规划分解为一组可划分、求解及合并的稀疏动态规划，使其在亚线性时间内完成。矩形逃逸问题（REP）定义为：给定轴对齐边界框$B$内的$n$个轴对齐矩形，每个矩形沿上、下、左、右四个方向之一扩展至边界$B$，最小化密度$k$，其中$k$表示穿过边界框$B$内任意点的矩形扩展至边界的最大数量。REP在$k>1$时为NP难问题。若矩形为网格点（或网格单位正方形），则问题称为正方形逃逸问题（SEP），该问题同样为NP难。针对$k\geq2$的SEP，本文提出时间复杂度为$O(n^{3/2}k^2)$的2-近似算法，改进了现有算法至少二次的时间复杂度。此外，对$k\geq3$的情况，算法近似比为$3/2$且已达紧界。针对REP，本文提出时间复杂度为$O(n\log n+nk)$的8-近似算法，并给出该算法在$k=O(1)$条件下的MPC版本，这是该问题的首个并行算法。