arXiv subjects

Mirela Damian

Publications and source records attributed to Mirela Damian.

31 records · Page 2Linked to original sources

arXiv2010

Connectivity of Graphs Induced by Directional Antennas

This paper addresses the problem of finding an orientation and a minimum radius for directional antennas of a fixed angle placed at the points of a planar set S, that induce a strongly connected communication graph. We consider problem instances in which antenna angles are fixed at 90 and 180 degrees, and establish upper and lower bounds for the minimum radius necessary to guarantee strong connectivity. In the case of 90-degree angles, we establish a lower bound of 2 and an upper bound of 7. In the case of 180-degree angles, we establish a lower bound of sqrt(3) and an upper bound of 1+sqrt(3). Underlying our results is the assumption that the unit disk graph for S is connected.

cs.CG↗

arXiv2010

Pi/2-Angle Yao Graphs are Spanners

We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor 8(29+23sqrt(2)). Enroute to this, we also show that the Yao graph Y4 in the Linf metric is a planar spanner with stretch factor 8.

cs.CG↗

arXiv2008

Localized Spanners for Wireless Networks

We present a new efficient localized algorithm to construct, for any given quasi-unit disk graph G=(V,E) and any e > 0, a (1+e)-spanner for G of maximum degree O(1) and total weight O(w(MST)), where w(MST) denotes the weight of a minimum spanning tree for V. We further show that similar localized techniques can be used to construct, for a given unit disk graph G = (V, E), a planar Cdel(1+e)(1+pi/2)-spanner for G of maximum degree O(1) and total weight O(w(MST)). Here Cdel denotes the stretch factor of the unit Delaunay triangulation for V. Both constructions can be completed in O(1) communication rounds, and require each node to know its own coordinates.

cs.DC↗

arXiv2008

A Simple Yao-Yao-Based Spanner of Bounded Degree

It is a standing open question to decide whether the Yao-Yao structure for unit disk graphs (UDGs) is a length spanner of not. This question is highly relevant to the topology control problem for wireless ad hoc networks. In this paper we make progress towards resolving this question by showing that the Yao-Yao structure is a length spanner for UDGs of bounded aspect ratio. We also propose a new local algorithm, called Yao-Sparse-Sink, based on the Yao-Sink method introduced by Li, Wan, Wang and Frieder, that computes a (1+e)-spanner of bounded degree for a given UDG and for given e > 0. The Yao-Sparse-Sink method enables an efficient local computation of sparse sink trees. Finally, we show that all these structures for UDGs -- Yao, Yao-Yao, Yao-Sink and Yao-Sparse-Sink -- have arbitrarily large weight.

cs.CG↗

arXiv2008

Local Approximation Schemes for Topology Control

This paper presents a distributed algorithm on wireless ad-hoc networks that runs in polylogarithmic number of rounds in the size of the network and constructs a linear size, lightweight, (1+ε)-spanner for any given ε> 0. A wireless network is modeled by a d-dimensional α-quasi unit ball graph (α-UBG), which is a higher dimensional generalization of the standard unit disk graph (UDG) model. The d-dimensional α-UBG model goes beyond the unrealistic ``flat world'' assumption of UDGs and also takes into account transmission errors, fading signal strength, and physical obstructions. The main result in the paper is this: for any fixed ε> 0, 0 < α\le 1, and d \ge 2, there is a distributed algorithm running in O(\log n \log^* n) communication rounds on an n-node, d-dimensional α-UBG G that computes a (1+ε)-spanner G' of G with maximum degree Δ(G') = O(1) and total weight w(G') = O(w(MST(G)). This result is motivated by the topology control problem in wireless ad-hoc networks and improves on existing topology control algorithms along several dimensions. The technical contributions of the paper include a new, sequential, greedy algorithm with relaxed edge ordering and lazy updating, and clustering techniques for filtering out unnecessary edges.

cs.DS↗

arXiv2007

A New Lower Bound on Guard Placement for Wireless Localization

The problem of wireless localization asks to place and orient stations in the plane, each of which broadcasts a unique key within a fixed angular range, so that each point in the plane can determine whether it is inside or outside a given polygonal region. The primary goal is to minimize the number of stations. In this paper we establish a lower bound of 2n/3 - 1 stations for polygons in general position, for the case in which the placement of stations is restricted to polygon vertices, improving upon the existing n/2 lower bound.

cs.CG↗

arXiv2007

Connecting Polygonizations via Stretches and Twangs

We show that the space of polygonizations of a fixed planar point set S of n points is connected by O(n^2) ``moves'' between simple polygons. Each move is composed of a sequence of atomic moves called ``stretches'' and ``twangs''. These atomic moves walk between weakly simple ``polygonal wraps'' of S. These moves show promise to serve as a basis for generating random polygons.

cs.CG↗

arXiv2007

Unfolding Manhattan Towers

We provide an algorithm for unfolding the surface of any orthogonal polyhedron that falls into a particular shape class we call Manhattan Towers, to a nonoverlapping planar orthogonal polygon. The algorithm cuts along edges of a 4x5x1 refinement of the vertex grid.

cs.CG↗

arXiv2006

Grid Vertex-Unfolding Orthogonal Polyhedra

An edge-unfolding of a polyhedron is produced by cutting along edges and flattening the faces to a *net*, a connected planar piece with no overlaps. A *grid unfolding* allows additional cuts along grid edges induced by coordinate planes passing through every vertex. A vertex-unfolding permits faces in the net to be connected at single vertices, not necessarily along edges. We show that any orthogonal polyhedron of genus zero has a grid vertex-unfolding. (There are orthogonal polyhedra that cannot be vertex-unfolded, so some type of "gridding" of the faces is necessary.) For any orthogonal polyhedron P with n vertices, we describe an algorithm that vertex-unfolds P in O(n^2) time. Enroute to explaining this algorithm, we present a simpler vertex-unfolding algorithm that requires a 3 x 1 refinement of the vertex grid.

cs.CG↗

arXiv2006

Epsilon-Unfolding Orthogonal Polyhedra

An unfolding of a polyhedron is produced by cutting the surface and flattening to a single, connected, planar piece without overlap (except possibly at boundary points). It is a long unsolved problem to determine whether every polyhedron may be unfolded. Here we prove, via an algorithm, that every orthogonal polyhedron (one whose faces meet at right angles) of genus zero may be unfolded. Our cuts are not necessarily along edges of the polyhedron, but they are always parallel to polyhedron edges. For a polyhedron of n vertices, portions of the unfolding will be rectangular strips which, in the worst case, may need to be as thin as epsilon = 1/2^{Omega(n)}.

cs.CG↗

arXiv2005

An O(n log n)-Time Algorithm for the Restricted Scaffold Assignment

The assignment problem takes as input two finite point sets S and T and establishes a correspondence between points in S and points in T, such that each point in S maps to exactly one point in T, and each point in T maps to at least one point in S. In this paper we show that this problem has an O(n log n)-time solution, provided that the points in S and T are restricted to lie on a line (linear time, if S and T are presorted).

cs.CG↗

arXiv2004

Partitioning Regular Polygons into Circular Pieces II:Nonconvex Partitions

We explore optimal circular nonconvex partitions of regular k-gons. The circularity of a polygon is measured by its aspect ratio: the ratio of the radii of the smallest circumscribing circle to the largest inscribed disk. An optimal circular partition minimizes the maximum ratio over all pieces in the partition. We show that the equilateral triangle has an optimal 4-piece nonconvex partition, the square an optimal 13-piece nonconvex partition, and the pentagon has an optimal nonconvex partition with more than 20 thousand pieces. For hexagons and beyond, we provide a general algorithm that approaches optimality, but does not achieve it.

cs.CG↗

arXiv2003

A Note on Objects Built From Bricks without Corners

We report a small advance on a question raised by Robertson, Schweitzer, and Wagon in [RSW02]. They constructed a genus-13 polyhedron built from bricks without corners, and asked whether every genus-0 such polyhedron must have a corner. A brick is a parallelopiped, and a corner is a brick of degree three or less in the brick graph. We describe a genus-3 polyhedron built from bricks with no corner, narrowing the genus gap.

cs.CG↗