On rectangular covering problems

Many applications like picture processing, data compression or pattern recognition require a covering of a set of points most often located in the (discrete) plane by rectangles due to specific cost constraints. In this paper we provide exact dynamic programming algorithms for covering point sets by regular rectangles, that have to obey certain (parameterized) boundary conditions. The concrete representative out of a class of objective functions that is studied is to minimize sum of area, circumference and number of patches used. This objective function may be motivated by requirements of numerically solving PDE's by discretization over (adaptive multi-)grids. More precisely, we propose exact deterministic algorithms for such problems based on a (set theoretic) dynamic programming approach yielding a time bound of O(n^23^n) . In a second step this bound is (asymptotically) decreased to O(n^62^n) by exploiting the underlying rectangular and lattice structures. Finally, a generalization of the problem and its solution methods is discussed for the case of arbitrary (finite) space dimension

On rectangular covering problems

Many applications like picture processing, data compression or pattern recognition require a covering of a set of points most often located in the (discrete) plane by rectangles due to specific cost constraints. In this paper we provide exact dynamic programming algorithms for covering point sets by regular rectangles, that have to obey certain (parameterized) boundary conditions. The concrete representative out of a class of objective functions that is studied is to minimize sum of area, circumference and number of patches used. This objective function may be motivated by requirements of numerically solving PDE's by discretization over (adaptive multi-)grids. More precisely, we propose exact deterministic algorithms for such problems based on a (set theoretic) dynamic programming approach yielding a time bound of O(n^23^n) . In a second step this bound is (asymptotically) decreased to O(n^62^n) by exploiting the underlying rectangular and lattice structures. Finally, a generalization of the problem and its solution methods is discussed for the case of arbitrary (finite) space dimension

Средства математического моделирования задач покрытия

The article considers theorems which are useful for the mathematical and computer modeling of the covering problems of a compact canonical multiconnected polygonal set with a finite family of convex polygons. Based on peculiarities of a rectangular cover, theorems concerning a partition of the translation vector space and an upper bound of the number of local extrema of the Г-function are formulated

Orientation-Constrained Rectangular Layouts

We construct partitions of rectangles into smaller rectangles from an input consisting of a planar dual graph of the layout together with restrictions on the orientations of edges and junctions of the layout. Such an orientation-constrained layout, if it exists, may be constructed in polynomial time, and all orientation-constrained layouts may be listed in polynomial time per layout.Comment: To appear at Algorithms and Data Structures Symposium, Banff, Canada, August 2009. 12 pages, 5 figure

Approximation Algorithm for Line Segment Coverage for Wireless Sensor Network

The coverage problem in wireless sensor networks deals with the problem of covering a region or parts of it with sensors. In this paper, we address the problem of covering a set of line segments in sensor networks. A line segment ` is said to be covered if it intersects the sensing regions of at least one sensor distributed in that region. We show that the problem of finding the minimum number of sensors needed to cover each member in a given set of line segments in a rectangular area is NP-hard. Next, we propose a constant factor approximation algorithm for the problem of covering a set of axis-parallel line segments. We also show that a PTAS exists for this problem.Comment: 16 pages, 5 figures

On the Duality of Semiantichains and Unichain Coverings

We study a min-max relation conjectured by Saks and West: For any two posets $P$ and $Q$ the size of a maximum semiantichain and the size of a minimum unichain covering in the product $P\times Q$ are equal. For positive we state conditions on $P$ and $Q$ that imply the min-max relation. Based on these conditions we identify some new families of posets where the conjecture holds and get easy proofs for several instances where the conjecture had been verified before. However, we also have examples showing that in general the min-max relation is false, i.e., we disprove the Saks-West conjecture.Comment: 10 pages, 3 figure