arXiv subjects

Haitao Wang

Publications and source records attributed to Haitao Wang.

At least 163 records · Page 9Linked to original sources

arXiv2015

Algorithms on Minimizing the Maximum Sensor Movement for Barrier Coverage of a Linear Domain

In this paper, we study the problem of moving $n$ sensors on a line to form a barrier coverage of a specified segment of the line such that the maximum moving distance of the sensors is minimized. Previously, it was an open question whether this problem on sensors with arbitrary sensing ranges is solvable in polynomial time. We settle this open question positively by giving an $O(n^2 \log n)$ time algorithm. For the special case when all sensors have the same-size sensing range, the previously best solution takes $O(n^2)$ time. We present an $O(n \log n)$ time algorithm for this case; further, if all sensors are initially located on the coverage segment, our algorithm takes $O(n)$ time. Also, we extend our techniques to the cycle version of the problem where the barrier coverage is for a simple cycle and the sensors are allowed to move only along the cycle. For sensors with the same-size sensing range, we solve the cycle version in $O(n)$ time, improving the previously best $O(n^2)$ time solution.

cs.CG↗

arXiv2015

Linear Time Approximation Schemes for Geometric Maximum Coverage

We study approximation algorithms for the following geometric version of the maximum coverage problem: Let P be a set of n weighted points in the plane. We want to place m a * b rectangles such that the sum of the weights of the points in P covered by these rectangles is maximized.For any fixed > 0, we present efficient approximation schemes that can find (1-ε)-approximation to the optimal solution.In particular, for m = 1, our algorithm runs in linear time O(n log( 1/ε)), improving over the previous result. For m > 1, we present an algorithm that runs in O(n/εlog(1/ε)+m(1/ε)^(O(min(sqrt(m),1/ε))) time.

cs.CG↗

arXiv2015

An Optimal Algorithm for Minimum-Link Rectilinear Paths in Triangulated Rectilinear Domains

We consider the problem of finding minimum-link rectilinear paths in rectilinear polygonal domains in the plane. A path or a polygon is rectilinear if all its edges are axis-parallel. Given a set $\mathcal{P}$ of $h$ pairwise-disjoint rectilinear polygonal obstacles with a total of $n$ vertices in the plane, a minimum-link rectilinear path between two points is a rectilinear path that avoids all obstacles with the minimum number of edges. In this paper, we present a new algorithm for finding minimum-link rectilinear paths among $\mathcal{P}$. After the plane is triangulated, with respect to any source point $s$, our algorithm builds an $O(n)$-size data structure in $O(n+h\log h)$ time, such that given any query point $t$, the number of edges of a minimum-link rectilinear path from $s$ to $t$ can be computed in $O(\log n)$ time and the actual path can be output in additional time linear in the number of the edges of the path. The previously best algorithm computes such a data structure in $O(n\log n)$ time.

cs.CG↗

arXiv2015

Computing Shortest Paths among Curved Obstacles in the Plane

A fundamental problem in computational geometry is to compute an obstacle-avoiding Euclidean shortest path between two points in the plane. The case of this problem on polygonal obstacles is well studied. In this paper, we consider the problem version on curved obstacles, commonly modeled as splinegons. A splinegon can be viewed as replacing each edge of a polygon by a convex curved edge (polygons are special splinegons). Each curved edge is assumed to be of O(1) complexity. Given in the plane two points s and t and a set of $h$ pairwise disjoint splinegons with a total of $n$ vertices, we compute a shortest s-to-t path avoiding the splinegons, in $O(n+h\log^{1+ε}h+k)$ time, where k is a parameter sensitive to the structures of the input splinegons and is upper-bounded by $O(h^2)$. In particular, when all splinegons are convex, $k$ is proportional to the number of common tangents in the free space (called "free common tangents") among the splinegons. We develop techniques for solving the problem on the general (non-convex) splinegon domain, which also improve several previous results. In particular, our techniques produce an optimal output-sensitive algorithm for a basic visibility problem of computing all free common tangents among $h$ pairwise disjoint convex splinegons with a total of $n$ vertices. Our algorithm runs in $O(n+h\log h+k)$ time and $O(n)$ space, where $k$ is the number of all free common tangents. Even for the special case where all splinegons are convex polygons, the previously best algorithm for this visibility problem takes $O(n+h^2\log n)$ time.

cs.CG↗

arXiv2015

Computing the $L_1$ Geodesic Diameter and Center of a Simple Polygon in Linear Time

In this paper, we show that the $L_1$ geodesic diameter and center of a simple polygon can be computed in linear time. For the purpose, we focus on revealing basic geometric properties of the $L_1$ geodesic balls, that is, the metric balls with respect to the $L_1$ geodesic distance. More specifically, in this paper we show that any family of $L_1$ geodesic balls in any simple polygon has Helly number two, and the $L_1$ geodesic center consists of midpoints of shortest paths between diametral pairs. These properties are crucial for our linear-time algorithms, and do not hold for the Euclidean case.

cs.CG↗

arXiv2015

Range Queries on Uncertain Data

Given a set $P$ of $n$ uncertain points on the real line, each represented by its one-dimensional probability density function, we consider the problem of building data structures on $P$ to answer range queries of the following three types for any query interval $I$: (1) top-$1$ query: find the point in $P$ that lies in $I$ with the highest probability, (2) top-$k$ query: given any integer $k\leq n$ as part of the query, return the $k$ points in $P$ that lie in $I$ with the highest probabilities, and (3) threshold query: given any threshold $τ$ as part of the query, return all points of $P$ that lie in $I$ with probabilities at least $τ$. We present data structures for these range queries with linear or nearly linear space and efficient query time.

cs.CG↗

arXiv2014

Minimizing the Aggregate Movements for Interval Coverage

We consider an interval coverage problem. Given $n$ intervals of the same length on a line $L$ and a line segment $B$ on $L$, we want to move the intervals along $L$ such that every point of $B$ is covered by at least one interval and the sum of the moving distances of all intervals is minimized. As a basic geometry problem, it has applications in mobile sensor barrier coverage in wireless sensor networks. The previous work solved the problem in $O(n^2)$ time. In this paper, by discovering many interesting observations and developing new algorithmic techniques, we present an $O(n\log n)$ time algorithm. We also show an $Ω(n\log n)$ time lower bound for this problem, which implies the optimality of our algorithm.

cs.CG↗

arXiv2014

On Top-$k$ Weighted SUM Aggregate Nearest and Farthest Neighbors in the $L_1$ Plane

In this paper, we study top-$k$ aggregate (or group) nearest neighbor queries using the weighted SUM operator under the $L_1$ metric in the plane. Given a set $P$ of $n$ points, for any query consisting of a set $Q$ of $m$ weighted points and an integer $k$, $ 1 \le k \le n$, the top-$k$ aggregate nearest neighbor query asks for the $k$ points of $P$ whose aggregate distances to $Q$ are the smallest, where the aggregate distance of each point $p$ of $P$ to $Q$ is the sum of the weighted distances from $p$ to all points of $Q$. We build an $O(n\log n\log\log n)$-size data structure in $O(n\log n \log\log n)$ time, such that each top-$k$ query can be answered in $O(m\log m+(k+m)\log^2 n)$ time. We also obtain other results with trade-off between preprocessing and query. Even for the special case where $k=1$, our results are better than the previously best method (in PODS 2012), which requires $O(n\log^2 n)$ preprocessing time, $O(n\log^2 n)$ space, and $O(m^2\log^3 n)$ query time. In addition, for the one-dimensional version of this problem, our approach can build an $O(n)$-size data structure in $O(n\log n)$ time that can support $O(\min\{k,\log m\}\cdot m+k+\log n)$ time queries. Further, we extend our techniques to the top-$k$ aggregate farthest neighbor queries, with the same bounds.

cs.CG↗

arXiv2014

Two-Point $L_1$ Shortest Path Queries in the Plane

Let $\mathcal{P}$ be a set of $h$ pairwise-disjoint polygonal obstacles with a total of $n$ vertices in the plane. We consider the problem of building a data structure that can quickly compute an $L_1$ shortest obstacle-avoiding path between any two query points $s$ and $t$. Previously, a data structure of size $O(n^2\log n)$ was constructed in $O(n^2\log^2 n)$ time that answers each two-point query in $O(\log^2 n+k)$ time, i.e., the shortest path length is reported in $O(\log^2 n)$ time and an actual path is reported in additional $O(k)$ time, where $k$ is the number of edges of the output path. In this paper, we build a new data structure of size $O(n+h^2\cdot \log h \cdot 4^{\sqrt{\log h}})$ in $O(n+h^2\cdot \log^{2} h \cdot 4^{\sqrt{\log h}})$ time that answers each query in $O(\log n+k)$ time. Note that $n+h^2\cdot \log^{2} h \cdot 4^{\sqrt{\log h}}=O(n+h^{2+ε})$ for any constant $ε>0$. Further, we extend our techniques to the weighted rectilinear version in which the "obstacles" of $\mathcal{P}$ are rectilinear regions with "weights" and allow $L_1$ paths to travel through them with weighted costs. Our algorithm answers each query in $O(\log n+k)$ time with a data structure of size $O(n^2\cdot \log n\cdot 4^{\sqrt{\log n}})$ that is built in $O(n^2\cdot \log^{2} n\cdot 4^{\sqrt{\log n}})$ time (note that $n^2\cdot \log^{2} n\cdot 4^{\sqrt{\log n}}= O(n^{2+ε})$ for any constant $ε>0$).

cs.CG↗

arXiv2014

Efficient Algorithms for One-Dimensional k-Center Problems

We consider the problem of finding k centers for n weighted points on a real line. This (weighted) k-center problem was solved in O(n log n) time previously by using Cole's parametric search and other complicated approaches. In this paper, we present an easier O(n log n) time algorithm that avoids the parametric search, and in certain special cases our algorithm solves the problem in O(n) time. In addition, our techniques involve developing interesting data structures for processing queries that find a lowest point in the common intersection of a certain subset of half-planes. This subproblem is interesting in its own right and our solution for it may find other applications as well.

cs.CG↗

arXiv2013

Aggregate-Max Nearest Neighbor Searching in the Plane

We study the aggregate/group nearest neighbor searching for the MAX operator in the plane. For a set $P$ of $n$ points and a query set $Q$ of $m$ points, the query asks for a point of $P$ whose maximum distance to the points in $Q$ is minimized. We present data structures for answering such queries for both $L_1$ and $L_2$ distance measures. Previously, only heuristic and approximation algorithms were given for both versions. For the $L_1$ version, we build a data structure of O(n) size in $O(n\log n)$ time, such that each query can be answered in $O(m+\log n)$ time. For the $L_2$ version, we build a data structure in $O(n\log n)$ time and $O(n\log \log n)$ space, such that each query can be answered in $O(m\sqrt{n}\log^{O(1)} n)$ time, and alternatively, we build a data structure in $O(n^{2+ε})$ time and space for any $ε>0$, such that each query can be answered in $O(m\log n)$ time. Further, we extend our result for the $L_1$ version to the top-$k$ queries where each query asks for the $k$ points of $P$ whose maximum distances to $Q$ are the smallest for any $k$ with $1\leq k\leq n$: We build a data structure of O(n) size in $O(n\log n)$ time, such that each top-$k$ query can be answered in $O(m+k\log n)$ time.

cs.CG↗

arXiv2013

Matroid and Knapsack Center Problems

In the classic $k$-center problem, we are given a metric graph, and the objective is to open $k$ nodes as centers such that the maximum distance from any vertex to its closest center is minimized. In this paper, we consider two important generalizations of $k$-center, the matroid center problem and the knapsack center problem. Both problems are motivated by recent content distribution network applications. Our contributions can be summarized as follows: 1. We consider the matroid center problem in which the centers are required to form an independent set of a given matroid. We show this problem is NP-hard even on a line. We present a 3-approximation algorithm for the problem on general metrics. We also consider the outlier version of the problem where a given number of vertices can be excluded as the outliers from the solution. We present a 7-approximation for the outlier version. 2. We consider the (multi-)knapsack center problem in which the centers are required to satisfy one (or more) knapsack constraint(s). It is known that the knapsack center problem with a single knapsack constraint admits a 3-approximation. However, when there are at least two knapsack constraints, we show this problem is not approximable at all. To complement the hardness result, we present a polynomial time algorithm that gives a 3-approximate solution such that one knapsack constraint is satisfied and the others may be violated by at most a factor of $1+ε$. We also obtain a 3-approximation for the outlier version that may violate the knapsack constraint by $1+ε$.

cs.DS↗

arXiv2012

Weak Visibility Queries of Line Segments in Simple Polygons

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 visibility polygon for any query line segment s in O(k log n) time, where k is the size of the visibility polygon of s and n is the number of vertices of P. Alternatively, we build a data structure in O(n^3) time and O(n^3) space that can compute the visibility polygon for any query line segment in O(k + log n) time.

cs.CG↗

arXiv2012

Computing L1 Shortest Paths among Polygonal Obstacles in the Plane

Given a point $s$ and a set of $h$ pairwise disjoint polygonal obstacles of totally $n$ vertices in the plane, we present a new algorithm for building an $L_1$ shortest path map of size O(n) in $O(T)$ time and O(n) space such that for any query point $t$, the length of the $L_1$ shortest obstacle-avoiding path from $s$ to $t$ can be reported in $O(\log n)$ time and the actual shortest path can be found in additional time proportional to the number of edges of the path, where $T$ is the time for triangulating the free space. It is currently known that $T=O(n+h\log^{1+ε}h)$ for an arbitrarily small constant $ε>0$. If the triangulation can be done optimally (i.e., $T=O(n+h\log h)$), then our algorithm is optimal. Previously, the best algorithm computes such an $L_1$ shortest path map in $O(n\log n)$ time and O(n) space. Our techniques can be extended to obtain improved results for other related problems, e.g., computing the $L_1$ geodesic Voronoi diagram for a set of point sites in a polygonal domain, finding shortest paths with fixed orientations, finding approximate Euclidean shortest paths, etc.

cs.CG↗

arXiv2011

Optimal Point Movement for Covering Circular Regions

Given $n$ points in a circular region $C$ in the plane, we study the problems of moving the $n$ points to its boundary to form a regular $n$-gon such that the maximum (min-max) or the sum (min-sum) of the Euclidean distances traveled by the points is minimized. The problems have applications, e.g., in mobile sensor barrier coverage of wireless sensor networks. The min-max problem further has two versions: the decision version and optimization version. For the min-max problem, we present an $O(n\log^2 n)$ time algorithm for the decision version and an $O(n\log^3 n)$ time algorithm for the optimization version. The previously best algorithms for the two problem versions take $O(n^{3.5})$ time and $O(n^{3.5}\log n)$ time, respectively. For the min-sum problem, we show that a special case with all points initially lying on the boundary of the circular region can be solved in $O(n^2)$ time, improving a previous $O(n^4)$ time solution. For the general min-sum problem, we present a 3-approximation $O(n^2)$ time algorithm, improving the previous $(1+π)$-approximation $O(n^2)$ time algorithm. A by-product of our techniques is an algorithm for dynamically maintaining the maximum matching of a circular convex bipartite graph; our algorithm can handle each vertex insertion or deletion on the graph in $O(\log^2 n)$ time. This result is interesting in its own right.

cs.CG↗

arXiv2010

An Improved Algorithm for Reconstructing a Simple Polygon from the Visibility Angles

In this paper, we study the following problem of reconstructing a simple polygon: Given a cyclically ordered vertex sequence of an unknown simple polygon P of n vertices and, for each vertex v of P, the sequence of angles defined by all the visible vertices of v in P, reconstruct the polygon P (up to similarity). An O(n^3 log n) time algorithm has been proposed for this problem. We present an improved algorithm with running time O(n^2), based on new observations on the geometric structures of the problem. Since the input size (i.e., the total number of input visibility angles) is O(n^2) in the worst case, our algorithm is worst-case optimal.

cs.CG↗

arXiv2009

Perfect Reflection of Chiral Fermions in Gated Graphene Nanoribbons

We describe the results of a theoretical study of transport through gated metallic graphene nanoribbons using a non-equilibrium Green function method. Although analogies with quantum field theory predict perfect transmission of chiral fermions through gated regions in one dimension, we find \emph{perfect reflection} of chiral fermions in armchair ribbons for specific configurations of the gate. This effect should be measurable in narrow graphene constrictions gated by a charged carbon nanotube.

cond-mat.mes-hall↗

arXiv2008

Astrodynamical Space Test of Relativity using Optical Devices I (ASTROD I) - A class-M fundamental physics mission proposal for Cosmic Vision 2015-2025

ASTROD I is a planned interplanetary space mission with multiple goals. The primary aims are: to test General Relativity with an improvement in sensitivity of over 3 orders of magnitude, improving our understanding of gravity and aiding the development of a new quantum gravity theory; to measure key solar system parameters with increased accuracy, advancing solar physics and our knowledge of the solar system and to measure the time rate of change of the gravitational constant with an order of magnitude improvement and the anomalous Pioneer acceleration, thereby probing dark matter and dark energy gravitationally. It is an international project, with major contributions from Europe and China and is envisaged as the first in a series of ASTROD missions. ASTROD I will consist of one spacecraft carrying a telescope, four lasers, two event timers and a clock. Two-way, two-wavelength laser pulse ranging will be used between the spacecraft in a solar orbit and deep space laser stations on Earth, to achieve the ASTROD I goals. A second mission, ASTROD II is envisaged as a three-spacecraft mission which would test General Relativity to one part per billion, enable detection of solar g-modes, measure the solar Lense-Thirring effect to 10 parts per million, and probe gravitational waves at frequencies below the LISA bandwidth. In the third phase (ASTROD III or Super-ASTROD), larger orbits could be implemented to map the outer solar system and to probe primordial gravitational-waves at frequencies below the ASTROD II bandwidth.

astro-ph↗