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≥2 with time
complexity O(n3/2k2). This improves the time complexity of existing
algorithms which are at least quadratic. Also, the approximation ratio of our
algorithm for k≥3 is 3/2 which is tight. We also give a
8-approximation algorithm for REP with time complexity O(nlogn+nk) and
give a MPC version of this algorithm for k=O(1) which is the first parallel
algorithm for this problem