Inproved Bounds for Rectangular and Guillotine Partitions

We study the problem of partitioning a rectangle S with a set of interior points Q into rectangles by introducing a set of line segments of least total length. The set of partitioning line segments must include every point in Q. Since this prob/em is computationally intractable (NP-hard), several approximation algorithms for its solution have been developed. In this paper we show that the length of an optimal guillotine partition is not greater than 1.75 times the length of an optimal rectangular partition. Since an optimal guillotine partition can be obtained on O(n n) time, we have a polynomial time approximation algorithm for finding near-optimal rectangular partitions. Given a rectangular boundary S and a set Q of points inside S, we study the problem of partitioning S into rectangles in such a way that every point in Q lies on at least one of the partitioning line segments and the total length of the partitioning line segments is least possible. Such a partition is called an optimal rectangular partition. The proofs given by Lingas et al. (1982) can be trivially extended to show that finding an optimal rectangular partition is a computationally intractable problem (NP-hard). Since then, several approximation algorithms have been proposed, i.e. algorithms that guarantee for every problem instance I that/2(E~px(I)) < cE(Eopt(I)), where E,p • is the set of partitioning line segments given by the approximation algorithm, Eo_at(I) is the set of partitioning line segments in an optimal solution, c is some constant, and L(E(I)) is the sum of the length of the partitioning line segments in E(I). Gonzalez & Zheng (1985a) present a divide-and-conquer approximation algorithm that generates solutions with E(E~px(I))_<(3+x//3)/S(Eopt(I)). The time complexity for their algorithm is O(n2), where n is the number of points in set Q. Levcopoulos (1986) showed that it is possible to implement this approximation algorithm in O(n log n) time. Gonzalez & Zheng (1985b) give an O(n 4) approximation algorithm that guarantees solutions with /2(Eap,(I)) < 3/_S(Eopt(I)). The approximation bound is smaller than the one for the algorithm given in Gonzalez & Zheng (1985a); however, there is a substantial difference between the time complexities of these two algorithms. The second algorithm (Gonzalez & Zheng,