[Show abstract][Hide abstract] ABSTRACT: We revisit the problem of searching for a target at an unknown location on a line when given upper and lower bounds on the distance D that separates the initial position of the searcher from the target. Prior to this work, only asymptotic bounds were known for the optimal competitive ratio achievable by any search strategy in the worst case. We present the first tight bounds on the exact optimal competitive ratio achievable, parameterized in terms of the given bounds on D, along with an optimal search strategy that achieves this competitive ratio. We prove that this optimal strategy is unique. We characterize the conditions under which an optimal strategy can be computed exactly and, when it cannot, we explain how numerical methods can be used efficiently. In addition, we answer several related open questions, including the maximal reach problem, and we discuss how to generalize these results to m rays, for any .
[Show abstract][Hide abstract] ABSTRACT: We consider several variations of the problems of covering a set of barriers (modeled as line segments) using sensors that can detect any intruder crossing any of the barriers. Sensors are initially located in the plane and they can relocate to the barriers. We assume that each sensor can detect any intruder in a circular area centered at the sensor. Given a set of barriers and a set of sensors located in the plane, we study three problems: the feasibility of barrier coverage, the problem of minimizing the largest relocation distance of a sensor (MinMax), and the problem of minimizing the sum of relocation distances of sensors (MinSum). When sensors are permitted to move to arbitrary positions on the barrier, the problems are shown to be NP-complete. We also study the case when sensors use perpendicular movement to one of the barriers. We show that when the barriers are parallel, both the MinMax and MinSum problems can be solved in polynomial time. In contrast, we show that even the feasibility problem is NP-complete if two perpendicular barriers are to be covered, even if the sensors are located at integer positions, and have only two possible sensing ranges. On the other hand, we give an O(n
3/2) algorithm for a natural special case of this last problem.
[Show abstract][Hide abstract] ABSTRACT: Let T be an edge weighted tree and let d min ,d max be two nonnegative real numbers. Then the pairwise compatibility graph (PCG) of T is a graph G such that each vertex of G corresponds to a distinct leaf of T and two vertices are adjacent in G if and only if the weighted distance between their corresponding leaves in T is in the interval [d min ,d max ]. Similarly, a given graph G is a PCG if there exist suitable T,d min ,d max , such that G is a PCG of T. Yanhaona, Bayzid and Rahman proved that there exists a graph with 15 vertices that is not a PCG. On the other hand, Calamoneri, Frascaria and Sinaimeri proved that every graph with at most seven vertices is a PCG. In this paper we construct a graph of eight vertices that is not a PCG, which strengthens the result of Yanhaona, Bayzid and Rahman, and implies optimality of the result of Calamoneri, Frascaria and Sinaimeri. We then construct a planar graph with sixteen vertices that is not a PCG. Finally, we prove a variant of the PCG recognition problem to be NP-complete.
[Show abstract][Hide abstract] ABSTRACT: Let $\set$ be a set of $n$ points in an $[n]^d$ grid, such that each point is assigned a color. Given a query range $\Q= [a_1, b_1] \times [a_2, b_2] \times \ldots \times [a_d, b_d]$,
the geometric range mode query problem asks to report the most frequent color (i.e., a mode) of the multiset of colors corresponding to points in $\set \cap \Q$. When $d=1$,
Chan et al.~(STACS 2012 \cite{chan2012linear}) gave a data structure that requires $O(n+(n/\Delta)^2/w)$ words of space and supports range mode queries in $O(\Delta)$ time for any $\Delta \geq 1$, where $w = \Omega(\log n)$ is the word size.
Chan et al.\ also proposed a data structures for higher dimensions
(i.e., $d \geq 2$) with $O(s_n+(n/\Delta)^{2d})$ space and $O(\Delta\cdot t_n)$
query time, where $s_n$ and $t_n$ denote the space and query time
of a data structure that supports
orthogonal range counting queries on the set $\set$.
In this paper we show that the space can be improved
without any increase to the query time, by presenting
an $O(s_n+(n/\Delta)^{2d}/w)$-space data structure
that supports orthogonal range mode queries on a set of $n$ points
in $d$ dimensions in $O(\Delta \cdot t_n)$ time, for any $\Delta \geq 1$.
When $d=1$, these space and query time costs match those
achieved by the current best known one-dimensional data structure.
Twenty-Sixth Canadian Conference on Computational Geometry (CCCG 2014).; 08/2014
[Show abstract][Hide abstract] ABSTRACT: Dujmovi, Eppstein, Suderman, and Wood showed that every 3-connected plane graph G with n vertices ad-mits a straight-line drawing with at most 2.5n − 3 seg-ments, which is also the best known upper bound when restricted to plane triangulations. On the other hand, they showed that there exist triangulations requiring 2n − 6 segments. In this paper we show that every plane triangulation admits a straight-line drawing with at most (7n − 2∆ 0 − 10)/3 ≤ 2.33n segments, where ∆ 0 is the number of cyclic faces in the minimum re-alizer of G. If the input triangulation is 4-connected, then our algorithm computes a drawing with at most (9n − 9)/4 ≤ 2.25n segments. For general plane graphs with n vertices and m edges, our algorithm requires at most (16n − 3m − 28)/3 ≤ 5.33n − m segments, which is smaller than 2.5n − 3 for all m ≥ 2.84n.
The 26th Canadian Conference on Computational Geometry (CCCG); 08/2014
[Show abstract][Hide abstract] ABSTRACT: A strict orthogonal drawing of a graph G=(V,E) in ℝ 2 is a drawing of G such that each vertex is mapped to a distinct point and each edge is mapped to a horizontal or vertical line segment. A graph G is HV-restricted if each of its edges is assigned a horizontal or vertical orientation. A strict orthogonal drawing of an HV-restricted graph G is good if it is planar and respects the edge orientations of G. In this paper we give a polynomial-time algorithm to check whether a given HV-restricted plane graph (i.e., a planar graph with a fixed combinatorial embedding) admits a good orthogonal drawing preserving the input embedding, which settles an open question posed by Maňuch, Patterson, Poon and Thachuk (GD 2010). We then examine HV-restricted planar graphs (i.e., when the embedding is not fixed). Here we completely characterize the 2-connected maximum-degree-three HV-restricted outerplanar graphs that admit good orthogonal drawings.
11th Latin American Symposium on Theoretical Informatics (LATIN 2014), Montevideo, Uruguay; 01/2014
[Show abstract][Hide abstract] ABSTRACT: We consider how to preprocess n colored points in the plane such that later, given a multiset of colors, we can quickly find an axis-aligned rectangle containing a subset of the points with exactly those colors, if one exists. We first give an index that uses o(n 4 ) space and o(n) query time when there are O(1) distinct colors. We then restrict our attention to the case in which there are only two distinct colors. We give an index that uses O(n) bits and O(1) query time to detect whether there exists a matching rectangle. Finally, we give a O(n)-space index that returns a matching rectangle, if one exists, in O(lg 2 n/lglgn) time.
25th Annual Symposium on Combinatorial Pattern Matching (CPM 2014), Moscow, Russia, June 16--18, 2014; 01/2014
[Show abstract][Hide abstract] ABSTRACT: The sequence of adjacent nodes (graph walk) visited by a routing algorithm on a graph G between given source and target nodes s and t is a c-competitive route if its length in G is at most c times the length of the shortest path from s to t in G. We present 21.766-, 17.982- and 15.479-competitive online routing algorithms on the Delaunay triangulation of an arbitrary given set of points in the plane. This improves the competitive ratio on Delaunay triangulations from the previous best of 45.749. We present a 7.621-competitive online routing algorithm for Delaunay triangulations of point sets in convex position.
[Show abstract][Hide abstract] ABSTRACT: In this paper we present a novel nonparametric method for simplifying piecewise linear curves and we apply this method as a statistical approximation of structure within sequential data in the plane. Specifically, given a sequence P of n points in the plane that determine a simple polygonal chain consisting of n-1 segments, we describe algorithms for selecting a subsequence Q subset of P (including the first and last points of P) that determines a second polygonal chain to approximate P, such that the number of crossings between the two polygonal chains is maximized, and the cardinality of Q is minimized among all such maximizing subsets of P. Our algorithms have respective running times O(n(2) log n) (respectively, O(n(2) root log n))when P is monotonic and O(n(2) log(2) n) (respectively, O(n(2) log(4/3) n)) when P is any simple polygonal chain in the Real RAM model (respectively, in the Word RAM model).
International Journal of Computational Geometry & Applications 01/2014; 23(06). DOI:10.1142/S021819591360011X · 0.08 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We conjecture that the balanced complete bipartite graph $K_{\lfloor n/2
\rfloor,\lceil n/2 \rceil}$ contains more cycles than any other $n$-vertex
triangle-free graph, and we make some progress toward proving this. We give
equivalent conditions for cycle-maximal triangle-free graphs; show bounds on
the numbers of cycles in graphs depending on numbers of vertices and edges,
girth, and homomorphisms to small fixed graphs; and use the bounds to show that
among regular graphs, the conjecture holds. We also consider graphs that are
close to being regular, with the minimum and maximum degrees differing by at
most a positive integer $k$. For $k=1$, we show that any such counterexamples
have $n\le 91$ and are not homomorphic to $C_5$; and for any fixed $k$ there
exists a finite upper bound on the number of vertices in a counterexample.
Finally, we describe an algorithm for efficiently computing the matrix
permanent (a $#P$-complete problem in general) in a special case used by our
bounds.
[Show abstract][Hide abstract] ABSTRACT: We revisit the problem of searching for a target at an unknown location on a
line when given upper and lower bounds on the distance D that separates the
initial position of the searcher from the target. Prior to this work, only
asymptotic bounds were known for the optimal competitive ratio achievable by
any search strategy in the worst case. We present the first tight bounds on the
exact optimal competitive ratio achievable, parameterized in terms of the given
bounds on D, along with an optimal search strategy that achieves this
competitive ratio. We prove that this optimal strategy is unique. We
characterize the conditions under which an optimal strategy can be computed
exactly and, when it cannot, we explain how numerical methods can be used
efficiently. In addition, we answer several related open questions, including
the maximal reach problem, and we discuss how to generalize these results to m
rays, for any m >= 2.
[Show abstract][Hide abstract] ABSTRACT: Consider a sliding camera that travels back and forth along an orthogonal
line segment $s$ inside an orthogonal polygon $P$ with $n$ vertices. The camera
can see a point $p$ inside $P$ if and only if there exists a line segment
containing $p$ that crosses $s$ at a right angle and is completely contained in
$P$. In the minimum sliding cameras (MSC) problem, the objective is to guard
$P$ with the minimum number of sliding cameras. In this paper, we give an
$O(n^{5/2})$-time $(7/2)$-approximation algorithm to the MSC problem on any
simple orthogonal polygon with $n$ vertices, answering a question posed by Katz
and Morgenstern (2011). To the best of our knowledge, this is the first
constant-factor approximation algorithm for this problem.
[Show abstract][Hide abstract] ABSTRACT: We give an O(nlog3n)-time linear-space algorithm that, given a plane 3-tree G with n vertices and a set S of n points in the plane, determines whether G has a point-set embedding on S (i.e., a planar straight-line drawing of G where each vertex is mapped to a distinct point of S), improving the O(n4/3+ε)-time O(n4/3)-space algorithm of Moosa and Rahman. Given an arbitrary plane graph G and a point set S, Di Giacomo and Liotta gave an algorithm to compute 2-bend point-set embeddings of G on S using O(W3) area, where W is the length of the longest edge of the bounding box of S. Their algorithm uses O(W3) area even when the input graphs are restricted to plane 3-trees. We introduce new techniques for computing 2-bend point-set embeddings of plane 3-trees that takes only O(W2) area. We also give approximation algorithms for point-set embeddings of plane 3-trees. Our results on 2-bend point-set embeddings and approximate point-set embeddings hold for partial plane 3-trees (e.g., series-parallel graphs and Halin graphs).
Proceedings of the 13th international conference on Algorithms and Data Structures; 08/2013
[Show abstract][Hide abstract] ABSTRACT: Let P be an orthogonal polygon. Consider a sliding camera that travels back and forth along an orthogonal line segment s⊆P as its trajectory. The camera can see a point p∈P if there exists a point q∈s such that pq is a line segment normal to s that is completely contained in P. In the minimum-cardinality sliding cameras problem, the objective is to find a set S of sliding cameras of minimum cardinality to guard P (i.e., every point in P can be seen by some sliding camera in S) while in the minimum-length sliding cameras problem the goal is to find such a set S so as to minimize the total length of trajectories along which the cameras in S travel. In this paper, we first settle the complexity of the minimum-length sliding cameras problem by showing that it is polynomial tractable even for orthogonal polygons with holes, answering a question posed by Katz and Morgenstern [9]. Next we show that the minimum-cardinality sliding cameras problem is NP-hard when P is allowed to have holes, which partially answers another question posed by Katz and Morgenstern [9].
38th International Symposium on Mathematical Foundations of Computer Science (MFCS 2013), Klosterneuburg, Austria; 08/2013
[Show abstract][Hide abstract] ABSTRACT: Let $P$ be an orthogonal polygon. Consider a sliding camera that travels back
and forth along an orthogonal line segment $s\in P$ as its \emph{trajectory}.
The camera can see a point $p\in P$ if there exists a point $q\in s$ such that
$pq$ is a line segment normal to $s$ that is completely inside $P$. In the
\emph{minimum-cardinality sliding cameras problem}, the objective is to find a
set $S$ of sliding cameras of minimum cardinality to guard $P$ (i.e., every
point in $P$ can be seen by some sliding camera) while in the
\emph{minimum-length sliding cameras problem} the goal is to find such a set
$S$ so as to minimize the total length of trajectories along which the cameras
in $S$ travel.
In this paper, we first settle the complexity of the minimum-length sliding
cameras problem by showing that it is polynomial tractable even for orthogonal
polygons with holes, answering a question asked by Katz and Morgenstern (2011).
We next show that the minimum-cardinality sliding cameras problem is
\textsc{NP}-hard when $P$ is allowed to have holes, which partially answers
another question asked by Katz and Morgenstern (2011).
[Show abstract][Hide abstract] ABSTRACT: The Twenty-Second Annual Canadian Conference on Computational Geometry (CCCG 2010) was held at the University of Manitoba in Winnipeg, Canada, from August 9-11, 2010, and co-chaired by Stephane Durocher and Jason Morrison. This was the first time this conference was hosted in the province of Manitoba. As in previous years, the focus of the conference was on current topics in discrete and computational geometry, including both theoretical and applied results. Submitted manuscripts were refereed by a program committee, which in 2010 consisted of twenty-six international researchers. The conference included three invited plenary lectures, presented respectively by David Avis (who gave the annual Erdos Memorial Lecture), David Eppstein, and David Kirkpatrick. The conference had 99 attendees and received 90 paper submissions, of which three were withdrawn and 68 were accepted. Accepted papers were published in the conference proceedings (available electronically at cccg.ca). Authors of exceptional papers were invited to submit a revised version to this special issue of Computational Geometry: Theory and Applications. The conference organizers would like to thank the program committee for their meticulous reviews and discussion of submitted papers, and for promoting the conference. As well, the guest editors of this issue would like to thank the authors for carefully preparing and revising submissions and the referees for their efforts in providing thorough and prompt reviews of the submitted manuscripts.
[Show abstract][Hide abstract] ABSTRACT: We introduce the exact coloured κ-enclosing object problem: given a set P of n points in R2, each of which has an associated colour in {1, . . . , t}, and a vector c = (c1, . . . , ct), where ci ε Z+ for each 1 ≤ i ≤ t, find a region that contains exactly ci points of P of colour i for each i. We examine the problems of finding exact coloured κ-enclosing axis-aligned rectangles, squares, discs, and two-sided dominating regions in a t-coloured point set.
25th Canadian Conference on Computational Geometry (CCCG 2013), Waterloo, Ontario, August 8--10, 2013; 01/2013
[Show abstract][Hide abstract] ABSTRACT: We present $O(n)$-space data structures to support various range frequency queries on a given array $A[0:n-1]$ or tree $T$ with $n$ nodes. Given a query consisting of an arbitrary pair of pre-order rank indices $(i,j)$, our data structures return a least frequent element, mode, or $\alpha$-minority of the multiset of elements in the unique path with endpoints at indices $i$ and $j$ in $A$ or $T$. We describe a data structure that supports range least frequent element queries on arrays in $O(\sqrt{n / w})$ time, improving the $\Theta(\sqrt{n})$ worst-case time required by the data structure of Chan et al.\ (SWAT 2012), where $w \in \Omega(\log n)$ is the word size in bits. We describe a data structure that supports range mode queries on trees in $O(\log\log n \sqrt{n / w})$ time, improving the $\Theta(\sqrt{n} \log n)$ worst-case time required by the data structure of Krizanc et al.\ (ISAAC 2003). Finally, we describe a data structure that supports range $\alpha$-minority queries on trees in $O(\alpha^{-1} \log\log n)$ time, where $\alpha \in [0,1]$ is specified at query time.
38th International Symposium on Mathematical Foundations of Computer Science (MFCS 2013), Klosterneuberg, Austria, August 26--30, 2013; 01/2013