A Massively Parallel Dynamic Programming for Approximate Rectangle Escape Problem

TL;DR

This paper introduces a parallel approximation algorithm for the rectangle escape problem, achieving improved time complexity and approximation ratios, and presents the first parallel algorithm for this NP-hard problem.

Contribution

It develops a massively parallel approximation algorithm for REP and SEP, with better time complexity and approximation ratios, including the first parallel solution for REP.

Findings

Achieves a 2-approximation for SEP with O(n^{3/2}k^2) time.

Provides an 8-approximation for REP with O(n log n + nk) time.

First parallel algorithm for REP in the MPC model.

Abstract

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.