arXiv subjects

Anil Maheshwari

Publications and source records attributed to Anil Maheshwari.

At least 73 records · Page 4Linked to original sources

arXiv2015

Discrete Voronoi Games and $ε$-Nets, in Two and Three Dimensions

The one-round discrete Voronoi game, with respect to a $n$-point user set $U$, consists of two players Player 1 ($\mathcal{P}_1$) and Player 2 ($\mathcal{P}_2$). At first, $\mathcal{P}_1$ chooses a set of facilities $F_1$ following which $\mathcal{P}_2$ chooses another set of facilities $F_2$, disjoint from $F_1$. The payoff of $\mathcal{P}_2$ is defined as the cardinality of the set of points in $U$ which are closer to a facility in $F_2$ than to every facility in $F_1$, and the payoff of $\mathcal{P}_1$ is the difference between the number of users in $U$ and the payoff of $\mathcal{P}_2$. The objective of both the players in the game is to maximize their respective payoffs. In this paper we study the one-round discrete Voronoi game where $\mathcal{P}_1$ places $k$ facilities and $\mathcal{P}_2$ places one facility and we have denoted this game as $VG(k,1)$. Although the optimal solution of this game can be found in polynomial time, the polynomial has a very high degree. In this paper, we focus on achieving approximate solutions to $VG(k,1)$ with significantly better running times. We provide a constant-factor approximate solution to the optimal strategy of $\mathcal{P}_1$ in $VG(k,1)$ by establishing a connection between $VG(k,1)$ and weak $ε$-nets. To the best of our knowledge, this is the first time that Voronoi games are studied from the point of view of $ε$-nets.

cs.CG↗

arXiv2015

Packing Plane Perfect Matchings into a Point Set

Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? We prove that at least $\lceil\log_2{n}\rceil-2$ plane perfect matchings can be packed into any point set $P$. For some special configurations of point sets, we give the exact answer. We also consider some extensions of this problem.

cs.CG↗

arXiv2014

Optimal Data Structures for Farthest-Point Queries in Cactus Networks

Consider the continuum of points on the edges of a network, i.e., a connected, undirected graph with positive edge weights. We measure the distance between these points in terms of the weighted shortest path distance, called the network distance. Within this metric space, we study farthest points and farthest distances. We introduce optimal data structures supporting queries for the farthest distance and the farthest points on trees, cycles, uni-cyclic networks, and cactus networks.

cs.DS↗

arXiv2014

Matching in Gabriel Graphs

Given a set $P$ of $n$ points in the plane, the order-$k$ Gabriel graph on $P$, denoted by $k$-$GG$, has an edge between two points $p$ and $q$ if and only if the closed disk with diameter $pq$ contains at most $k$ points of $P$, excluding $p$ and $q$. We study matching problems in $k$-$GG$ graphs. We show that a Euclidean bottleneck perfect matching of $P$ is contained in $10$-$GG$, but $8$-$GG$ may not have any Euclidean bottleneck perfect matching. In addition we show that $0$-$GG$ has a matching of size at least $\frac{n-1}{4}$ and this bound is tight. We also prove that $1$-$GG$ has a matching of size at least $\frac{2(n-1)}{5}$ and $2$-$GG$ has a perfect matching. Finally we consider the problem of blocking the edges of $k$-$GG$.

cs.CG↗

arXiv2014

Higher-Order Triangular-Distance Delaunay Graphs: Graph-Theoretical Properties

We consider an extension of the triangular-distance Delaunay graphs (TD-Delaunay) on a set $P$ of points in the plane. In TD-Delaunay, the convex distance is defined by a fixed-oriented equilateral triangle $\triangledown$, and there is an edge between two points in $P$ if and only if there is an empty homothet of $\triangledown$ having the two points on its boundary. We consider higher-order triangular-distance Delaunay graphs, namely $k$-TD, which contains an edge between two points if the interior of the homothet of $\triangledown$ having the two points on its boundary contains at most $k$ points of $P$. We consider the connectivity, Hamiltonicity and perfect-matching admissibility of $k$-TD. Finally we consider the problem of blocking the edges of $k$-TD.

cs.CG↗

arXiv2013

Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph

A standard way to approximate the distance between any two vertices $p$ and $q$ on a mesh is to compute, in the associated graph, a shortest path from $p$ to $q$ that goes through one of $k$ sources, which are well-chosen vertices. Precomputing the distance between each of the $k$ sources to all vertices of the graph yields an efficient computation of approximate distances between any two vertices. One standard method for choosing $k$ sources, which has been used extensively and successfully for isometry-invariant surface processing, is the so-called Farthest Point Sampling (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources. In this paper, we analyze the stretch factor $\mathcal{F}_{FPS}$ of approximate geodesics computed using FPS, which is the maximum, over all pairs of distinct vertices, of their approximated distance over their geodesic distance in the graph. We show that $\mathcal{F}_{FPS}$ can be bounded in terms of the minimal value $\mathcal{F}^*$ of the stretch factor obtained using an optimal placement of $k$ sources as $\mathcal{F}_{FPS}\leq 2 r_e^2 \mathcal{F}^*+ 2 r_e^2 + 8 r_e + 1$, where $r_e$ is the ratio of the lengths of the longest and the shortest edges of the graph. This provides some evidence explaining why farthest point sampling has been used successfully for isometry-invariant shape processing. Furthermore, we show that it is NP-complete to find $k$ sources that minimize the stretch factor.

cs.CG↗

arXiv2013

A Note on the Unsolvability of the Weighted Region Shortest Path Problem

Let S be a subdivision of the plane into polygonal regions, where each region has an associated positive weight. The weighted region shortest path problem is to determine a shortest path in S between two points s, t in R^2, where the distances are measured according to the weighted Euclidean metric-the length of a path is defined to be the weighted sum of (Euclidean) lengths of the sub-paths within each region. We show that this problem cannot be solved in the Algebraic Computation Model over the Rational Numbers (ACMQ). In the ACMQ, one can compute exactly any number that can be obtained from the rationals Q by applying a finite number of operations from +, -, \times, ÷, \sqrt[k]{}, for any integer k >= 2. Our proof uses Galois theory and is based on Bajaj's technique.

cs.CG↗

arXiv2013

An Optimal Algorithm for the Euclidean Bottleneck Full Steiner Tree Problem

Let $P$ and $S$ be two disjoint sets of $n$ and $m$ points in the plane, respectively. We consider the problem of computing a Steiner tree whose Steiner vertices belong to $S$, in which each point of $P$ is a leaf, and whose longest edge length is minimum. We present an algorithm that computes such a tree in $O((n+m)\log m)$ time, improving the previously best result by a logarithmic factor. We also prove a matching lower bound in the algebraic computation tree model.

cs.CG↗

arXiv2013

Similarity of Polygonal Curves in the Presence of Outliers

The Fréchet distance is a well studied and commonly used measure to capture the similarity of polygonal curves. Unfortunately, it exhibits a high sensitivity to the presence of outliers. Since the presence of outliers is a frequently occurring phenomenon in practice, a robust variant of Fréchet distance is required which absorbs outliers. We study such a variant here. In this modified variant, our objective is to minimize the length of subcurves of two polygonal curves that need to be ignored (MinEx problem), or alternately, maximize the length of subcurves that are preserved (MaxIn problem), to achieve a given Fréchet distance. An exact solution to one problem would imply an exact solution to the other problem. However, we show that these problems are not solvable by radicals over $\mathbb{Q}$ and that the degree of the polynomial equations involved is unbounded in general. This motivates the search for approximate solutions. We present an algorithm, which approximates, for a given input parameter $δ$, optimal solutions for the \MinEx\ and \MaxIn\ problems up to an additive approximation error $δ$ times the length of the input curves. The resulting running time is upper bounded by $\mathcal{O} \left(\frac{n^3}δ \log \left(\frac{n}δ \right)\right)$, where $n$ is the complexity of the input polygonal curves.

cs.CG↗

arXiv2013

Network Farthest-Point Diagrams

Consider the continuum of points along the edges of a network, i.e., an undirected graph with positive edge weights. We measure distance between these points in terms of the shortest path distance along the network, known as the network distance. Within this metric space, we study farthest points. We introduce network farthest-point diagrams, which capture how the farthest points---and the distance to them---change as we traverse the network. We preprocess a network G such that, when given a query point q on G, we can quickly determine the farthest point(s) from q in G as well as the farthest distance from q in G. Furthermore, we introduce a data structure supporting queries for the parts of the network that are farther away from q than some threshold R > 0, where R is part of the query. We also introduce the minimum eccentricity feed-link problem defined as follows. Given a network G with geometric edge weights and a point p that is not on G, connect p to a point q on G with a straight line segment pq, called a feed-link, such that the largest network distance from p to any point in the resulting network is minimized. We solve the minimum eccentricity feed-link problem using eccentricity diagrams. In addition, we provide a data structure for the query version, where the network G is fixed and a query consists of the point p.

cs.CG↗

arXiv2012

Visiting All Sites with Your Dog

Given a polygonal curve P, a pointset S, and an ε> 0, we study the problem of finding a polygonal curve Q whose vertices are from S and has a Frechet distance less or equal to εto curve P. In this problem, Q must visit every point in S and we are allowed to reuse points of pointset in building Q. First, we show that this problem in NP-Complete. Then, we present a polynomial time algorithm for a special cases of this problem, when P is a convex polygon.

cs.CG↗

arXiv2012

Fixed-Orientation Equilateral Triangle Matching of Point Sets

Given a point set $P$ and a class $\mathcal{C}$ of geometric objects, $G_\mathcal{C}(P)$ is a geometric graph with vertex set $P$ such that any two vertices $p$ and $q$ are adjacent if and only if there is some $C \in \mathcal{C}$ containing both $p$ and $q$ but no other points from $P$. We study $G_{\bigtriangledown}(P)$ graphs where $\bigtriangledown$ is the class of downward equilateral triangles (ie. equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half-$Θ_6$ graphs and TD-Delaunay graphs. The main result in our paper is that for point sets $P$ in general position, $G_{\bigtriangledown}(P)$ always contains a matching of size at least $\lceil\frac{n-2}{3}\rceil$ and this bound cannot be improved above $\lceil\frac{n-1}{3}\rceil$. We also give some structural properties of $G_{\davidsstar}(P)$ graphs, where $\davidsstar$ is the class which contains both upward and downward equilateral triangles. We show that for point sets in general position, the block cut point graph of $G_{\davidsstar}(P)$ is simply a path. Through the equivalence of $G_{\davidsstar}(P)$ graphs with $Θ_6$ graphs, we also derive that any $Θ_6$ graph can have at most $5n-11$ edges, for point sets in general position.

cs.CG↗

arXiv2012

Localized Geometric Query Problems

A new class of geometric query problems are studied in this paper. We are required to preprocess a set of geometric objects $P$ in the plane, so that for any arbitrary query point $q$, the largest circle that contains $q$ but does not contain any member of $P$, can be reported efficiently. The geometric sets that we consider are point sets and boundaries of simple polygons.

cs.CG↗

arXiv2012

Space-efficient Algorithms for Visibility Problems in Simple Polygon

Given a simple polygon $P$ consisting of $n$ vertices, we study the problem of designing space-efficient algorithms for computing (i) the visibility polygon of a point inside $P$, (ii) the weak visibility polygon of a line segment inside $P$ and (iii) the minimum link path between a pair of points inside $P$. For problem (i) two algorithms are proposed. The first one is an in-place algorithm where the input array may be lost. It uses only O(1) extra space apart from the input array. The second one assumes that the input is given in a read-only array, and it needs $O(\sqrt{n})$ extra space. The time complexity of both the algorithms are O(n). For problem (ii), we have assumed that the input polygon is given in a read-only array. Our proposed algorithm runs in $O(n^2)$ time using O(1) extra space. For problem (iii) the time and space complexities of our proposed algorithm are $O(kn)$ and O(1) respectively; $k$ is the length (number of links) in a minimum link path between the given pair of points.

cs.CG↗

arXiv2011

An Approximation Algorithm for Computing Shortest Paths in Weighted 3-d Domains

We present the first polynomial time approximation algorithm for computing shortest paths in weighted three-dimensional domains. Given a polyhedral domain $\D$, consisting of $n$ tetrahedra with positive weights, and a real number $\eps\in(0,1)$, our algorithm constructs paths in $\D$ from a fixed source vertex to all vertices of $\D$, whose costs are at most $1+\eps$ times the costs of (weighted) shortest paths, in $O(\C(\D)\frac{n}{\eps^{2.5}}\log\frac{n}{\eps}\log^3\frac{1}{\eps})$ time, where $\C(\D)$ is a geometric parameter related to the aspect ratios of tetrahedra. The efficiency of the proposed algorithm is based on an in-depth study of the local behavior of geodesic paths and additive Voronoi diagrams in weighted three-dimensional domains, which are of independent interest. The paper extends the results of Aleksandrov, Maheshwari and Sack [JACM 2005] to three dimensions.

cs.CG↗

arXiv2010

Querying for the Largest Empty Geometric Object in a Desired Location

We study new types of geometric query problems defined as follows: given a geometric set $P$, preprocess it such that given a query point $q$, the location of the largest circle that does not contain any member of $P$, but contains $q$ can be reported efficiently. The geometric sets we consider for $P$ are boundaries of convex and simple polygons, and point sets. While we primarily focus on circles as the desired shape, we also briefly discuss empty rectangles in the context of point sets.

cs.CG↗

arXiv2010

Customer Appeasement Scheduling

Almost all of the current process scheduling algorithms which are used in modern operating systems (OS) have their roots in the classical scheduling paradigms which were developed during the 1970's. But modern computers have different types of software loads and user demands. We think it is important to run what the user wants at the current moment. A user can be a human, sitting in front of a desktop machine, or it can be another machine sending a request to a server through a network connection. We think that OS should become intelligent to distinguish between different processes and allocate resources, including CPU, to those processes which need them most. In this work, as a first step to make the OS aware of the current state of the system, we consider process dependencies and interprocess communications. We are developing a model, which considers the need to satisfy interactive users and other possible remote users or customers, by making scheduling decisions based on process dependencies and interprocess communications. Our simple proof of concept implementation and experiments show the effectiveness of this approach in the real world applications. Our implementation does not require any change in the software applications nor any special kind of configuration in the system, Moreover, it does not require any additional information about CPU needs of applications nor other resource requirements. Our experiments show significant performance improvement for real world applications. For example, almost constant average response time for Mysql data base server and constant frame rate for mplayer under different simulated load values.

cs.OS↗

arXiv2010

Improved Methods For Generating Quasi-Gray Codes

Consider a sequence of bit strings of length d, such that each string differs from the next in a constant number of bits. We call this sequence a quasi-Gray code. We examine the problem of efficiently generating such codes, by considering the number of bits read and written at each generating step, the average number of bits read while generating the entire code, and the number of strings generated in the code. Our results give a trade-off between these constraints, and present algorithms that do less work on average than previous results, and that increase the number of bit strings generated.

cs.DM↗