arXiv subjects

Rajasekhar Inkulu

Publications and source records attributed to Rajasekhar Inkulu.

5 recordsLinked to original sources

arXiv2016

Minimum Weight Connectivity Augmentation for Planar Straight-Line Graphs

We consider edge insertion and deletion operations that increase the connectivity of a given planar straight-line graph (PSLG), while minimizing the total edge length of the output. We show that every connected PSLG $G=(V,E)$ in general position can be augmented to a 2-connected PSLG $(V,E\cup E^+)$ by adding new edges of total Euclidean length $\|E^+\|\leq 2\|E\|$, and this bound is the best possible. An optimal edge set $E^+$ can be computed in $O(|V|^4)$ time; however the problem becomes NP-hard when $G$ is disconnected. Further, there is a sequence of edge insertions and deletions that transforms a connected PSLG $G=(V,E)$ into a planar straight-line cycle $G'=(V,E')$ such that $\|E'\|\leq 2\|{\rm MST}(V)\|$, and the graph remains connected with edge length below $\|E\|+\|{\rm MST}(V)\|$ at all stages. These bounds are the best possible.

cs.CG↗

arXiv2014

ANN queries: covering Voronoi diagram with hyperboxes

Given a set $S$ of $n$ points in $d$-dimensional Euclidean metric space $X$ and a small positive real number $ε$, we present an algorithm to preprocess $S$ and answer queries that require finding a set $S' \subseteq S$ of $ε$-approximate nearest neighbors (ANNs) to a given query point $q \in X$. The following are the characteristics of points belonging to set $S'$: - $\forall s \in S'$, $\exists$ a point $p \in X$ such that $|pq| \le ε$ and the nearest neighbor of $p$ is $s$, and - $\exists$ a $s' \in S'$ such that $s'$ is a nearest neighbor of $q$. During the preprocessing phase, from the Voronoi diagram of $S$ we construct a set of box trees of size $O(4^d\frac{V}δ(\fracπε)^{d-1})$ which facilitate in querying ANNs of any input query point in $O(\frac{1}{d}lg \frac{V}δ + (\fracπε)^{d-1})$ time. Here $δ$ equals to $(\fracε{2\sqrt{d}})^d$, and $V$ is the volume of a large bounding box that contains all the points of set $S$. The average case cardinality of $S'$ is shown to rely on $S$ and $ε$.

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↗

arXiv2010

A near optimal algorithm for finding Euclidean shortest path in polygonal domain

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 time to triangulate the polygonal region, $m$ is the number of obstacles, and $n$ is the number of vertices. This bound is close to the known lower bound of $O(n+m\lg{m})$ time and $O(n)$ space. Our approach involve progressing shortest path wavefront as in continuous Dijkstra-type method, and confining its expansion to regions of interest.

cs.CG↗

arXiv2010

Approximate Shortest Path through a Weighted Planar Subdivision

This paper presents an approximation algorithm for finding a shortest path between two points $s$ and $t$ in a weighted planar subdivision $\PS$. Each face $f$ of $\PS$ is associated with a weight $w_f$, and the cost of travel along a line segment on $f$ is $w_f$ multiplied by the Euclidean norm of that line segment. The cost of a path which traverses across several faces of the subdivision is the sum of the costs of travel along each face. Our algorithm progreeses the discretized shortest path wavefront from source $s$, and takes polynomial time in finding an $ε$-approximate shortest path.

cs.CG↗