A Massively Parallel Dynamic Programming for Approximate Rectangle Escape Problem

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.