Related papers: Rectangular Partitions of a Rectilinear Polygon

Let $\mathcal{T}$ be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles,…

We study approximation algorithms for the following geometric version of the maximum coverage problem: Let $\mathcal{P}$ be a set of $n$ weighted points in the plane. Let $D$ represent a planar object, such as a rectangle, or a disk. We…

We revisit a standard polygon containment problem: given a convex $k$-gon $P$ and a convex $n$-gon $Q$ in the plane, find a placement of $P$ inside $Q$ under translation and rotation (if it exists), or more generally, find the largest copy…

Let $G$ be a planar $3$-graph (i.e., a planar graph with vertex degree at most three) with $n$ vertices. We present the first $O(n^2)$-time algorithm that computes a planar orthogonal drawing of $G$ with the minimum number of bends in the…

The Mondrian problem consists of dissecting a square of side length $n\in \NN$ into non-congruent rectangles with natural length sides such that the difference $d(n)$ between the largest and the smallest areas of the rectangles partitioning…

For a set of n points in the plane, we consider the axis--aligned (p,k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain n-k points. In this paper, we consider the boxes to be either squares or…

Let G = (V, E) be a planar triangulated graph (PTG) having every face triangular. A rectilinear dual or an orthogonal floor plan (OFP) of G is obtained by partitioning a rectangle into \mid V \mid rectilinear regions (modules) where two…

We study a class of geometric covering and packing problems for bounded regions on the plane. We are given a set of axis-parallel line segments that induces a planar subdivision with a set of bounded (rectilinear) faces. We are interested…

Given a simple polygon P in the plane, we present new algorithms and data structures for computing the weak visibility polygon from any query line segment in P. We build a data structure in O(n) time and O(n) space that can compute the…

Given a set $P$ of $n$ points and a set $S$ of $m$ weighted disks in the plane, the disk coverage problem asks for a subset of disks of minimum total weight that cover all points of $P$. The problem is NP-hard. In this paper, we consider a…

Let $\mathcal{P}$ be the surface of a convex polyhedron with $n$ vertices. We consider the two-point shortest path query problem for $\mathcal{P}$: Constructing a data structure so that given any two query points $s$ and $t$ on…

Let $P$ be a planar set of $n$ points in general position. We consider the problem of computing an orientation of the plane for which the Rectilinear Convex Hull of $P$ has minimum area. Bae et al. (Computational Geometry: Theory and…

We consider the problem of finding all enclosing rectangles of minimum area that can contain a given set of rectangles without overlap. Our rectangle packer chooses the x-coordinates of all the rectangles before any of the y-coordinates. We…

We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let $R$ and $B$ be…

In the number partitioning problem (NPP) one aims to partition a given set of $N$ real numbers into two subsets with approximately equal sum. The NPP is a well-studied optimization problem and is famous for possessing a…

While well-known methods to list the intersections of either a list of segments or a complex polygon aim at achieving optimal time-complexity they often do so at the cost of memory comsumption and complex code. Real-life software…

We present an algorithm to find an {\it Euclidean Shortest Path} from a source vertex $s$ to a sink vertex $t$ in the presence of obstacles in $\Re^2$. Our algorithm takes $O(T+m(\lg{m})(\lg{n}))$ time and $O(n)$ space. Here, $O(T)$ is the…

We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of $n$ vertices and $h$ holes. We introduce a \emph{graph of oriented distances} to encode the distance between…

Consider a set $P$ of $n$ points on the boundary of an axis-aligned square $Q$. We study the boundary-anchored packing problem on $P$ in which the goal is to find a set of interior-disjoint axis-aligned rectangles in $Q$ such that each…

Given $n$ non-vertical lines in 3-space, their vertical depth (above/below) relation can contain cycles. We show that the lines can be cut into $O(n^{3/2}\mathop{\mathrm{polylog}} n)$ pieces, such that the depth relation among these pieces…