arXiv subjects
Publications and source records attributed to Prosenjit Bose.
We show an upper bound of $\frac{ \sin\left(\frac{3π}{10}\right) }{ \sin\left(\frac{2π}{5}\right)-\sin\left(\frac{3π}{10}\right) } <5.70$ on the spanning ratio of $Θ_5$-graphs, improving on the previous best known upper bound of $9.96$ [Bose, Morin, van Renssen, and Verdonschot. The Theta-5-graph is a spanner. Computational Geometry, 2015.]
In this paper we study local routing strategies on geometric graphs. Such strategies use geometric properties of the graph like the coordinates of the current and target nodes to route. Specifically, we study routing strategies in the presence of constraints which are obstacles that edges of the graph are not allowed to cross. Let $P$ be a set of $n$ points in the plane and let $S$ be a set of line segments whose endpoints are in $P$, with no two line segments intersecting properly. We present the first deterministic 1-local $O(1)$-memory routing algorithm that is guaranteed to find a path between two vertices in the visibility graph of $P$ with respect to a set of constraints $S$. The strategy never looks beyond the direct neighbors of the current node and does not store more than $O(1)$-information to reach the target. We then turn our attention to finding competitive routing strategies. We show that when routing on any triangulation $T$ of $P$ such that $S\subseteq T$, no $o(n)$-competitive routing algorithm exists when the routing strategy restricts its attention to the triangles intersected by the line segment from the source to the target (a technique commonly used in the unconstrained setting). Finally, we provide an $O(n)$-competitive deterministic 1-local $O(1)$-memory routing algorithm on any such $T$, which is optimal in the worst case, given the lower bound.
The fragile complexity of a comparison-based algorithm is $f(n)$ if each input element participates in $O(f(n))$ comparisons. In this paper, we explore the fragile complexity of algorithms adaptive to various restrictions on the input, i.e., algorithms with a fragile complexity parameterized by a quantity other than the input size n. We show that searching for the predecessor in a sorted array has fragile complexity $Θ(\log k)$, where $k$ is the rank of the query element, both in a randomized and a deterministic setting. For predecessor searches, we also show how to optimally reduce the amortized fragile complexity of the elements in the array. We also prove the following results: Selecting the $k$-th smallest element has expected fragile complexity $O(\log \log k)$ for the element selected. Deterministically finding the minimum element has fragile complexity $Θ(\log(Inv))$ and $Θ(\log(Runs))$, where $Inv$ is the number of inversions in a sequence and $Runs$ is the number of increasing runs in a sequence. Deterministically finding the median has fragile complexity $O(\log(Runs) + \log \log n)$ and $Θ(\log(Inv))$. Deterministic sorting has fragile complexity $Θ(\log(Inv))$ but it has fragile complexity $Θ(\log n)$ regardless of the number of runs.
We study affine invariant 2D triangulation methods. That is, methods that produce the same triangulation for a point set $S$ for any (unknown) affine transformation of $S$. Our work is based on a method by Nielson [A characterization of an affine invariant triangulation. Geom. Mod, 191-210. Springer, 1993] that uses the inverse of the covariance matrix of $S$ to define an affine invariant norm, denoted $A_{S}$, and an affine invariant triangulation, denoted ${DT}_{A_{S}}[S]$. We revisit the $A_{S}$-norm from a geometric perspective, and show that ${DT}_{A_{S}}[S]$ can be seen as a standard Delaunay triangulation of a transformed point set based on $S$. We prove that it retains all of its well-known properties such as being 1-tough, containing a perfect matching, and being a constant spanner of the complete geometric graph of $S$. We show that the $A_{S}$-norm extends to a hierarchy of related geometric structures such as the minimum spanning tree, nearest neighbor graph, Gabriel graph, relative neighborhood graph, and higher order versions of these graphs. In addition, we provide different affine invariant sorting methods of a point set $S$ and of the vertices of a polygon $P$ that can be combined with known algorithms to obtain other affine invariant triangulation methods of $S$ and of $P$.
A 1-bend boundary labelling problem consists of an axis-aligned rectangle $B$, $n$ points (called sites) in the interior, and $n$ points (called ports) on the labels along the boundary of $B$. The goal is to find a set of $n$ axis-aligned curves (called leaders), each having at most one bend and connecting one site to one port, such that the leaders are pairwise disjoint. A 1-bend boundary labelling problem is $k$-sided ($1\leq k\leq 4$) if the ports appear on $k$ different sides of $B$. Kindermann et al. ["Multi-Sided Boundary Labeling", Algorithmica, 76(1): 225-258, 2016] showed that the 1-bend three-sided and four-sided boundary labelling problems can be solved in $O(n^4)$ and $O(n^9)$ time, respectively. Bose et al. [SWAT, 12:1-12:14, 2018] improved the latter running time to $O(n^6)$ by reducing the problem to computing maximum independent set in an outerstring graph. In this paper, we improve both previous results by giving new algorithms with running times $O(n^3\log n)$ and $O(n^5)$ to solve the 1-bend three-sided and four-sided boundary labelling problems, respectively.
We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $Γ$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$, the ratio between the minimum length of any path from $u$ to $v$ and the Euclidean distance between $u$ and $v$ is small. The maximum such ratio, over all pairs of vertices of $G$, is the spanning ratio of $Γ$. First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio $1$, a proper straight-line drawing with spanning ratio $1$, and a planar straight-line drawing with spanning ratio $1$ are NP-complete, $\exists \mathbb R$-complete, and linear-time solvable problems, respectively, where a drawing is proper if no two vertices overlap and no edge overlaps a vertex. Second, we show that moving from spanning ratio $1$ to spanning ratio $1+ε$ allows us to draw every graph. Namely, we prove that, for every $ε>0$, every (planar) graph admits a proper (resp. planar) straight-line drawing with spanning ratio smaller than $1+ε$. Third, our drawings with spanning ratio smaller than $1+ε$ have large edge-length ratio, that is, the ratio between the length of the longest edge and the length of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that require exponential edge-length ratio in any straight-line drawing with constant spanning ratio.
A \emph{2-interval} is the union of two disjoint intervals on the real line. Two 2-intervals $D_1$ and $D_2$ are \emph{disjoint} if their intersection is empty (i.e., no interval of $D_1$ intersects any interval of $D_2$). There can be three different relations between two disjoint 2-intervals; namely, preceding ($<$), nested ($\sqsubset$) and crossing ($\between$). Two 2-intervals $D_1$ and $D_2$ are called \emph{$R$-comparable} for some $R\in\{<,\sqsubset,\between\}$, if either $D_1RD_2$ or $D_2RD_1$. A set $\mathcal{D}$ of disjoint 2-intervals is $\mathcal{R}$-comparable, for some $\mathcal{R}\subseteq\{<,\sqsubset,\between\}$ and $\mathcal{R}\neq\emptyset$, if every pair of 2-intervals in $\mathcal{R}$ are $R$-comparable for some $R\in\mathcal{R}$. Given a set of 2-intervals and some $\mathcal{R}\subseteq\{<,\sqsubset,\between\}$, the objective of the \emph{2-interval pattern problem} is to find a largest subset of 2-intervals that is $\mathcal{R}$-comparable. The 2-interval pattern problem is known to be $W[1]$-hard when $|\mathcal{R}|=3$ and $NP$-hard when $|\mathcal{R}|=2$ (except for $\mathcal{R}=\{<,\sqsubset\}$, which is solvable in quadratic time). In this paper, we fully settle the parameterized complexity of the problem by showing it to be $W[1]$-hard for both $\mathcal{R}=\{\sqsubset,\between\}$ and $\mathcal{R}=\{<,\between\}$ (when parameterized by the size of an optimal solution); this answers an open question posed by Vialette [Encyclopedia of Algorithms, 2008].
We study online routing algorithms on the $Θ$6-graph and the half-$Θ$6-graph (which is equivalent to a variant of the Delaunay triangulation). Given a source vertex s and a target vertex t in the $Θ$6-graph (resp. half-$Θ$6-graph), there exists a deterministic online routing algorithm that finds a path from s to t whose length is at most 2 st (resp. 2.89 st) which is optimal in the worst case [Bose et al., siam J. on Computing, 44(6)]. We propose alternative, slightly simpler routing algorithms that are optimal in the worst case and for which we provide an analysis of the average routing ratio for the $Θ$6-graph and half-$Θ$6-graph defined on a Poisson point process. For the $Θ$6-graph, our online routing algorithm has an expected routing ratio of 1.161 (when s and t random) and a maximum expected routing ratio of 1.22 (maximum for fixed s and t where all other points are random), much better than the worst-case routing ratio of 2. For the half-$Θ$6-graph, our memoryless online routing algorithm has an expected routing ratio of 1.43 and a maximum expected routing ratio of 1.58. Our online routing algorithm that uses a constant amount of additional memory has an expected routing ratio of 1.34 and a maximum expected routing ratio of 1.40. The additional memory is only used to remember the coordinates of the starting point of the route. Both of these algorithms have an expected routing ratio that is much better than their worst-case routing ratio of 2.89.
Let $P$ be a simple polygon with $n$ vertices, and let $q \in P$ be a point in $P$. Let $k \in \{0, \dots, n - 1\}$. A point $p \in P$ is $k$-visible from $q$ if and only if the line segment $pq$ crosses the boundary of $P$ at most $k$ times. The $k$-visibility region of $q$ in $P$ is the set of all points that are $k$-visible from $q$. We study the problem of computing the $k$-visibility region in the limited workspace model, where the input resides in a random-access read-only memory of $O(n)$ words, each with $Ω(\log{n})$ bits. The algorithm can read and write $O(s)$ additional words of workspace, where $s \in \mathbb{N}$ is a parameter of the model. The output is written to a write-only stream. Given a simple polygon $P$ with $n$ vertices and a point $q \in P$, we present an algorithm that reports the $k$-visibility region of $q$ in $P$ in $O(cn/s+c\log{s} + \min\{\lceil k/s \rceil n,n \log{\log_s{n}}\})$ expected time using $O(s)$ words of workspace. Here, $c \in \{1, \dots, n\}$ is the number of critical vertices of $P$ for $q$ where the $k$-visibility region of $q$ may change. We generalize this result for polygons with holes and for sets of non-crossing line segments.
We consider the design of adaptive data structures for searching elements of a tree-structured space. We use a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree. This model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. The model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online $O(\log \log n)$-competitive search tree data structure in this model, matching the best known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one which we call Steiner-closed search trees, which may be of independent interest. Moreover our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees, and secondly from these trees into paths.
We present improved upper bounds on the spanning ratio of constrained $θ$-graphs with at least 6 cones and constrained Yao-graphs with 5 or at least 7 cones. Given a set of points in the plane, a Yao-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $θ= 2 π/m$, and adds an edge to the closest vertex in each cone. Constrained Yao-graphs have the additional property that no edge properly intersects any of the given line segment constraints. Constrained $θ$-graphs are similar to constrained Yao-graphs, but use a different method to determine the closest vertex. We present tight bounds on the spanning ratio of a large family of constrained $θ$-graphs. We show that constrained $θ$-graphs with $4k + 2$ ($k \geq 1$ and integer) cones have a tight spanning ratio of $1 + 2 \sin(θ/2)$, where $θ$ is $2 π/ (4k + 2)$. We also present improved upper bounds on the spanning ratio of the other families of constrained $θ$-graphs. These bounds match the current upper bounds in the unconstrained setting. We also show that constrained Yao-graphs with an even number of cones ($m \geq 8$) have spanning ratio at most $1 / ( 1 - 2 \sin (θ/2) )$ and constrained Yao-graphs with an odd number of cones ($m \geq 5$) have spanning ratio at most $1 / ( 1 - 2 \sin (3θ/8) )$. As is the case with constrained $θ$-graphs, these bounds match the current upper bounds in the unconstrained setting, which implies that like in the unconstrained setting using more cones can make the spanning ratio worse.
An $ω$-wedge is the closed set of points contained between two rays that are emanating from a single point (the apex), and are separated by an angle $ω< π$. Given a convex polygon $P$, we place the $ω$-wedge such that $P$ is inside the wedge and both rays are tangent to $P$. The set of apex positions of all such placements of the $ω$-wedge is called the $ω$-cloud of $P$. We investigate reconstructing a polygon $P$ from its $ω$-cloud. Previous work on reconstructing $P$ from probes with the $ω$-wedge required knowledge of the points of tangency between $P$ and the two rays of the $ω$-wedge in addition to the location of the apex. Here we consider the setting where the maximal $ω$-cloud alone is given. We give two conditions under which it uniquely defines $P$: (i) when $ω< π$ is fixed/given, or (ii) when what is known is that $ω< π/2$. We show that if neither of these two conditions hold, then $P$ may not be unique. We show that, when the uniqueness conditions hold, the polygon $P$ can be reconstructed in $O(n)$ time with $O(1)$ working space in addition to the input, where $n$ is the number of arcs in the input $ω$-cloud.
For a set of points in the plane and a fixed integer $k > 0$, the Yao graph $Y_k$ partitions the space around each point into $k$ equiangular cones of angle $θ=2π/k$, and connects each point to a nearest neighbor in each cone. It is known for all Yao graphs, with the sole exception of $Y_5$, whether or not they are geometric spanners. In this paper we close this gap by showing that for odd $k \geq 5$, the spanning ratio of $Y_k$ is at most $1/(1-2\sin(3θ/8))$, which gives the first constant upper bound for $Y_5$, and is an improvement over the previous bound of $1/(1-2\sin(θ/2))$ for odd $k \geq 7$. We further reduce the upper bound on the spanning ratio for $Y_5$ from $10.9$ to $2+\sqrt{3} \approx 3.74$, which falls slightly below the lower bound of $3.79$ established for the spanning ratio of $Θ_5$ ($Θ$-graphs differ from Yao graphs only in the way they select the closest neighbor in each cone). This is the first such separation between a Yao and $Θ$-graph with the same number of cones. We also give a lower bound of $2.87$ on the spanning ratio of $Y_5$. Finally, we revisit the $Y_6$ graph, which plays a particularly important role as the transition between the graphs ($k > 6$) for which simple inductive proofs are known, and the graphs ($k \le 6$) whose best spanning ratios have been established by complex arguments. Here we reduce the known spanning ratio of $Y_6$ from $17.6$ to $5.8$, getting closer to the spanning ratio of 2 established for $Θ_6$.
For any constants $d\ge 1$, $ε>0$, $t>1$, and any $n$-point set $P\subset\mathbb{R}^d$, we show that there is a geometric graph $G=(P,E)$ having $O(n\log^2 n\log\log n)$ edges with the following property: For any $F\subseteq P$, there exists $F^+\supseteq F$, $|F^+| \le (1+ε)|F|$ such that, for any pair $p,q\in P\setminus F^+$, the graph $G-F$ contains a path from $p$ to $q$ whose (Euclidean) length is at most $t$ times the Euclidean distance between $p$ and $q$. In the terminology of robust spanners (Bose \et al, SICOMP, 42(4):1720--1736, 2013) the graph $G$ is a $(1+ε)k$-robust $t$-spanner of $P$. This construction is sparser than the recent constructions of Buchin, Olàh, and Har-Peled (arXiv:1811.06898) who prove the existence of $(1+ε)k$-robust $t$-spanners with $n\log^{O(d)} n$ edges.
We consider the problem of routing on a network in the presence of line segment constraints (i.e., obstacles that edges in our network are not allowed to cross). Let $P$ be a set of $n$ points in the plane and let $S$ be a set of non-crossing line segments whose endpoints are in $P$. We present two deterministic 1-local $O(1)$-memory routing algorithms that are guaranteed to find a path of at most linear size between any pair of vertices of the \emph{visibility graph} of $P$ with respect to a set of constraints $S$ (i.e., the algorithms never look beyond the direct neighbours of the current location and store only a constant amount of additional information). Contrary to {\em all} existing deterministic local routing algorithms, our routing algorithms do not route on a plane subgraph of the visibility graph. Additionally, we provide lower bounds on the routing ratio of any deterministic local routing algorithm on the visibility graph.
We study the problem of placing a set $T$ of broadcast towers in a simple polygon $P$ in order for any point to locate itself in the interior of $P$. Let $V(p)$ denote the visibility polygon of a point $p$, as the set of all points $q \in P$ that are visible to $p$. For any point $p \in P$: for each tower $t \in T \cap V(p)$ the point $p$ receives the coordinates of $t$ and the Euclidean distance between $t$ and $p$. From this information $p$ can determine its coordinates. We show a tower-positioning algorithm that computes such a set $T$ of size at most $\lfloor 2n/3\rfloor$, where $n$ is the size of $P$. This improves the previous upper bound of $\lfloor 8n/9\rfloor$ towers. We also show that $\lfloor 2n/3\rfloor$ towers are sometimes necessary.
Art Gallery Localization (AGL) is the problem of placing a set $T$ of broadcast towers in a simple polygon $P$ in order for a point to locate itself in the interior. For any point $p \in P$: for each tower $t \in T \cap V(p)$ (where $V(p)$ denotes the visibility polygon of $p$) the point $p$ receives the coordinates of $t$ and the Euclidean distance between $t$ and $p$. From this information $p$ can determine its coordinates. We study the computational complexity of AGL problem. We show that the problem of determining the minimum number of broadcast towers that can localize a point anywhere in a simple polygon $P$ is NP-hard. We show a reduction from Boolean Three Satisfiability problem to our problem and give a proof that the reduction takes polynomial time.
We consider a repulsion actuator located in an $n$-sided convex environment full of point particles. When the actuator is activated, all the particles move away from the actuator. We study the problem of gathering all the particles to a point. We give an $O(n^2)$ time algorithm to compute all the actuator locations that gather the particles to one point with one activation, and an $O(n)$ time algorithm to find a single such actuator location if one exists. We then provide an $O(n)$ time algorithm to place the optimal number of actuators whose sequential activation results in the gathering of the particles when such a placement exists.