Compressing rectilinear pictures and minimizing access control lists

We consider a geometric model for the problem of minimizing access control lists (ACLs) in network routers, a model that also has applications to rectilinear picture compression and figure drawing in common graphics software packages. Here the goal is to create a colored rectilinear pattern within an initially white rectangular canvas, and the basic operation is to choose a subrectangle and paint it a single color, overwriting all previous colors in the rectangle. Rectangle Rule List (RRL) minimization is the problem of finding the shortest list of rules needed to create a given pattern. ACL minimization is a restricted version of this problem where the set of allowed rectangles must correspond to pairs of IP address prefixes. Motivated by the ACL application, we study the special cases of RRL and ACL minimization in which all rectangles must be strips that extend either the full width or the full height of the canvas (strip-rules). We provide several equivalent characterizations of the patterns achievable using strip-rules and present polynomial-time algorithms for optimally constructing such patterns when, as in the ACL application, the only colors are black and white (permit or deny). We also show that RRL minimization is NP-hard in general and provide O (min (n^ (1/3), OPT^ (1/2) ) ) -approximation algorithms for general RRL and ACL minimization by exploiting our results about strip-rule patterns.