arXiv subjects

Hugo A. Akitaya

Publications and source records attributed to Hugo A. Akitaya.

At least 37 records · Page 2Linked to original sources

arXiv2021

Compacting Squares: Input-Sensitive In-Place Reconfiguration of Sliding Squares

A well-established theoretical model for modular robots in two dimensions are edge-connected configurations of square modules, which can reconfigure through so-called sliding moves. Dumitrescu and Pach [Graphs and Combinatorics, 2006] proved that it is always possible to reconfigure one edge-connected configuration of $n$ squares into any other using at most $O(n^2)$ sliding moves, while keeping the configuration connected at all times. For certain pairs of configurations, reconfiguration may require $Ω(n^2)$ sliding moves. However, significantly fewer moves may be sufficient. We prove that it is NP-hard to minimize the number of sliding moves for a given pair of edge-connected configurations. On the positive side we present Gather&Compact, an input-sensitive in-place algorithm that requires only $O(\bar{P} n)$ sliding moves to transform one configuration into the other, where $\bar{P}$ is the maximum perimeter of the two bounding boxes. The squares move within the bounding boxes only, with the exception of at most one square at a time which may move through the positions adjacent to the bounding boxes. The $O(\bar{P} n)$ bound never exceeds $O(n^2)$, and is optimal (up to constant factors) among all bounds parameterized by just $n$ and $\bar{P}$. Our algorithm is built on the basic principle that well-connected components of modular robots can be transformed efficiently. Hence we iteratively increase the connectivity within a configuration, to finally arrive at a single solid $xy$-monotone component. We implemented Gather&Compact and compared it experimentally to the in-place modification by Moreno and Sacristán [EuroCG 2020] of the Dumitrescu and Pach algorithm (MSDP). Our experiments show that Gather&Compact consistently outperforms MSDP by a significant margin, on all types of square configurations.

cs.CG↗

arXiv2021

Circumscribing Polygons and Polygonizations for Disjoint Line Segments

Given a planar straight-line graph $G=(V,E)$ in $\mathbb{R}^2$, a \emph{circumscribing polygon} of $G$ is a simple polygon $P$ whose vertex set is $V$, and every edge in $E$ is either an edge or an internal diagonal of $P$. A circumscribing polygon is a \emph{polygonization} for $G$ if every edge in $E$ is an edge of $P$. We prove that every arrangement of $n$ disjoint line segments in the plane has a subset of size $Ω(\sqrt{n})$ that admits a circumscribing polygon, which is the first improvement on this bound in 20 years. We explore relations between circumscribing polygons and other problems in combinatorial geometry, and generalizations to $\mathbb{R}^3$. We show that it is NP-complete to decide whether a given graph $G$ admits a circumscribing polygon, even if $G$ is 2-regular. Settling a 30-year old conjecture by Rappaport, we also show that it is NP-complete to determine whether a geometric matching admits a polygonization.

cs.CG↗

arXiv2020

Characterizing Universal Reconfigurability of Modular Pivoting Robots

We give both efficient algorithms and hardness results for reconfiguring between two connected configurations of modules in the hexagonal grid. The reconfiguration moves that we consider are "pivots", where a hexagonal module rotates around a vertex shared with another module. Following prior work on modular robots, we define two natural sets of hexagon pivoting moves of increasing power: restricted and monkey moves. When we allow both moves, we present the first universal reconfiguration algorithm, which transforms between any two connected configurations using $O(n^3)$ monkey moves. This result strongly contrasts the analogous problem for squares, where there are rigid examples that do not have a single pivoting move preserving connectivity. On the other hand, if we only allow restricted moves, we prove that the reconfiguration problem becomes PSPACE-complete. Moreover, we show that, in contrast to hexagons, the reconfiguration problem for pivoting squares is PSPACE-complete regardless of the set of pivoting moves allowed. In the process, we strengthen the reduction framework of Demaine et al. [FUN'18] that we consider of independent interest.

cs.CG↗

arXiv2020

Reconfiguration of Connected Graph Partitions via Recombination

Motivated by applications in gerrymandering detection, we study a reconfiguration problem on connected partitions of a connected graph $G$. A partition of $V(G)$ is \emph{connected} if every part induces a connected subgraph. In many applications, it is desirable to obtain parts of roughly the same size, possibly with some slack $s$. A \emph{Balanced Connected $k$-Partition with slack $s$}, denoted \emph{$(k,s)$-BCP}, is a partition of $V(G)$ into $k$ nonempty subsets, of sizes $n_1,\ldots , n_k$ with $|n_i-n/k|\leq s$, each of which induces a connected subgraph (when $s=0$, the $k$ parts are perfectly balanced, and we call it \emph{$k$-BCP} for short). A \emph{recombination} is an operation that takes a $(k,s)$-BCP of a graph $G$ and produces another by merging two adjacent subgraphs and repartitioning them. Given two $k$-BCPs, $A$ and $B$, of $G$ and a slack $s\geq 0$, we wish to determine whether there exists a sequence of recombinations that transform $A$ into $B$ via $(k,s)$-BCPs. We obtain four results related to this problem: (1) When $s$ is unbounded, the transformation is always possible using at most $6(k-1)$ recombinations. (2) If $G$ is Hamiltonian, the transformation is possible using $O(kn)$ recombinations for any $s \ge n/k$, and (3) we provide negative instances for $s \leq n/(3k)$. (4) We show that the problem is PSPACE-complete when $k \in O(n^{\varepsilon})$ and $s \in O(n^{1-\varepsilon})$, for any constant $0 < \varepsilon \le 1$, even for restricted settings such as when $G$ is an edge-maximal planar graph or when $k=3$ and $G$ is planar.

cs.DM↗

arXiv2020

Folding Polyominoes with Holes into a Cube

When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special \emph{simple} holes guarantee foldability.

cs.CG↗

arXiv2020

Negative Instance for the Edge Patrolling Beacon Problem

Can an infinite-strength magnetic beacon always ``catch'' an iron ball, when the beacon is a point required to be remain nonstrictly outside a polygon, and the ball is a point always moving instantaneously and maximally toward the beacon subject to staying nonstrictly within the same polygon? Kouhestani and Rappaport [JCDCG 2017] gave an algorithm for determining whether a ball-capturing beacon strategy exists, while conjecturing that such a strategy always exists. We disprove this conjecture by constructing orthogonal and general-position polygons in which the ball and the beacon can never be united.

cs.CG↗

arXiv2019

Face flips in origami tessellations

Given a flat-foldable origami crease pattern $G=(V,E)$ (a straight-line drawing of a planar graph on a region of the plane) with a mountain-valley (MV) assignment $μ:E\to\{-1,1\}$ indicating which creases in $E$ bend convexly (mountain) or concavely (valley), we may \emph{flip} a face $F$ of $G$ to create a new MV assignment $μ_F$ which equals $μ$ except for all creases $e$ bordering $F$, where we have $μ_F(e)=-μ(e)$. In this paper we explore the configuration space of face flips for a variety of crease patterns $G$ that are tilings of the plane, proving examples where $μ_F$ results in a MV assignment that is either never, sometimes, or always flat-foldable for various choices of $F$. We also consider the problem of finding, given two foldable MV assignments $μ_1$ and $μ_2$ of a given crease pattern $G$, a minimal sequence of face flips to turn $μ_1$ into $μ_2$. We find polynomial-time algorithms for this in the cases where $G$ is either a square grid or the Miura-ori, and show that this problem is NP-hard in the case where $G$ is the triangle lattice.

math.CO↗

arXiv2019

Distance Measures for Embedded Graphs

We introduce new distance measures for comparing straight-line embedded graphs based on the Fréchet distance and the weak Fréchet distance. These graph distances are defined using continuous mappings and thus take the combinatorial structure as well as the geometric embeddings of the graphs into account. We present a general algorithmic approach for computing these graph distances. Although we show that deciding the distances is NP-hard for general embedded graphs, we prove that our approach yields polynomial time algorithms if the graphs are trees, and for the distance based on the weak Fréchet distance if the graphs are planar embedded. Moreover, we prove that deciding the distances based on the Fréchet distance remains NP-hard for planar embedded graphs and show how our general algorithmic approach yields an exponential time algorithm and a polynomial time approximation algorithm for this case.

cs.CG↗

arXiv2019

Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The $O(1)$ Musketeers

We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra "helper" modules ("musketeers") suffice to reconfigure the remaining $n$ modules between any two given configurations. Our algorithm uses $O(n^2)$ pivot moves, which is worst-case optimal. Previous reconfiguration algorithms either require less restrictive "sliding" moves, do not preserve facet-connectivity, or for the setting we consider, could only handle a small subset of configurations defined by a local forbidden pattern. Configurations with the forbidden pattern do have disconnected reconfiguration graphs (discrete configuration spaces), and indeed we show that they can have an exponential number of connected components. But forbidding the local pattern throughout the configuration is far from necessary, as we show that just a constant number of added modules (placed to be freely reconfigurable) suffice for universal reconfigurability. We also classify three different models of natural pivot moves that preserve facet-connectivity, and show separations between these models.

cs.CG↗

arXiv2019

Recognizing Weak Embeddings of Graphs

We present an efficient algorithm for a problem in the interface between clustering and graph embeddings. An embedding $φ:G\rightarrow M$ of a graph $G$ into a 2-manifold $M$ maps the vertices in $V(G)$ to distinct points and the edges in $E(G)$ to interior-disjoint Jordan arcs between the corresponding vertices. In applications in clustering, cartography, and visualization, nearby vertices and edges are often bundled to the same point or overlapping arcs, due to data compression or low resolution. This raises the computational problem of deciding whether a given map $φ:G\rightarrow M$ comes from an embedding. A map $φ:G\rightarrow M$ is a \textbf{weak embedding} if it can be perturbed into an embedding $ψ_\varepsilon:G\rightarrow M$ with $\|φ-ψ_\varepsilon\|<\varepsilon$ for every $\varepsilon>0$, where $\|.\|$ is the unform norm. A polynomial-time algorithm for recognizing weak embeddings has recently been found by Fulek and Kynčl. It reduces the problem to solving a system of linear equations over $\mathbb{Z}_2$. It runs in $O(n^{2ω})\leq O(n^{4.75})$ time, where $ω\in [2,2.373)$ is the matrix multiplication exponent and $n$ is the number of vertices and edges of $G$. We improve the running time to $O(n\log n)$. Our algorithm is also conceptually simpler: We perform a sequence of \emph{local operations} that gradually "untangles" the image $φ(G)$ into an embedding $ψ(G)$, or reports that $φ$ is not a weak embedding. It combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.

cs.CG↗

arXiv2019

Rock Climber Distance: Frogs versus Dogs

The classical measure of similarity between two polygonal chains in Euclidean space is the Fréchet distance, which corresponds to the coordinated motion of two mobile agents along the chains while minimizing their maximum distance. As computing the Fréchet distance takes near-quadratic time under the Strong Exponential Time Hypothesis (SETH), we explore two new distance measures, called rock climber distance and $k$-station distance, in which the agents move alternately in their coordinated motion that traverses the polygonal chains. We show that the new variants are equivalent to the Fréchet or the Hausdorff distance if the number of moves is unlimited. When the number of moves is limited to a given parameter $k$, we show that it is NP-hard to determine the distance between two curves. We also describe a 2-approximation algorithm to find the minimum $k$ for which the distance drops below a given threshold.

cs.CG↗

arXiv2019

Infinite All-Layers Simple Foldability

We study the problem of deciding whether a crease pattern can be folded by simple folds (folding along one line at a time) under the infinite all-layers model introduced by [Akitaya et al., 2017], in which each simple fold is defined by an infinite line and must fold all layers of paper that intersect this line. This model is motivated by folding in manufacturing such as sheet-metal bending. We improve on [Arkin et al., 2004] by giving a deterministic $O(n)$-time algorithm to decide simple foldability of 1D crease patterns in the all-layers model. Then we extend this 1D result to 2D, showing that simple foldability in this model can be decided in linear time for unassigned axis-aligned orthogonal crease patterns on axis-aligned 2D orthogonal paper. On the other hand, we show that simple foldability is strongly NP-complete if a subset of the creases have a mountain-valley assignment, even for an axis-aligned rectangle of paper.

cs.CG↗

arXiv2018

How to Fit a Tree in a Box

We study compact straight-line embeddings of trees. We show that perfect binary trees can be embedded optimally: a tree with $n$ nodes can be drawn on a $\sqrt n$ by $\sqrt n$ grid. We also show that testing whether a given binary tree has an upward embedding with a given combinatorial embedding in a given grid is NP-hard.

cs.CG↗

arXiv2018

Maximum Area Axis-Aligned Square Packings

Given a point set $S=\{s_1,\ldots , s_n\}$ in the unit square $U=[0,1]^2$, an anchored square packing is a set of $n$ interior-disjoint empty squares in $U$ such that $s_i$ is a corner of the $i$th square. The reach $R(S)$ of $S$ is the set of points that may be covered by such a packing, that is, the union of all empty squares anchored at points in $S$. It is shown that area$(R(S))\geq \frac12$ for every finite set $S\subset U$, and this bound is the best possible. The region $R(S)$ can be computed in $O(n\log n)$ time. Finally, we prove that finding a maximum area anchored square packing is NP-complete. This is the first hardness proof for a geometric packing problem where the size of geometric objects in the packing is unrestricted.

cs.CG↗

arXiv2017

Upward Partitioned Book Embeddings

We analyze a directed variation of the book embedding problem when the page partition is prespecified and the nodes on the spine must be in topological order (upward book embedding). Given a directed acyclic graph and a partition of its edges into $k$ pages, can we linearly order the vertices such that the drawing is upward (a topological sort) and each page avoids crossings? We prove that the problem is NP-complete for $k\ge 3$, and for $k\ge 4$ even in the special case when each page is a matching. By contrast, the problem can be solved in linear time for $k=2$ pages when pages are restricted to matchings. The problem comes from Jack Edmonds (1997), motivated as a generalization of the map folding problem from computational origami.

cs.CG↗

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↗

arXiv2016

Reconstruction of weakly simple polygons from their edges

Given $n$ line segments in the plane, do they form the edge set of a \emph{weakly simple polygon}; that is, can the segment endpoints be perturbed by at most $\varepsilon$, for any $\varepsilon>0$, to obtain a simple polygon? While the analogous question for \emph{simple polygons} can easily be answered in $O(n\log n)$ time, we show that it is NP-complete for weakly simple polygons. We give $O(n)$-time algorithms in two special cases: when all segments are collinear, or the segment endpoints are in general position. These results extend to the variant in which the segments are \emph{directed}, and the counterclockwise traversal of a polygon should follow the orientation. We study related problems for the case that the union of the $n$ input segments is connected. (i) If each segment can be subdivided into several segments, find the minimum number of subdivision points to form a weakly simple polygon. (ii) If new line segments can be added, find the minimum total length of new segments that creates a weakly simple polygon. We give worst-case upper and lower bounds for both problems.

cs.CG↗

arXiv2016

Multi-Colored Spanning Graphs

We study a problem proposed by Hurtado et al. (2016) motivated by sparse set visualization. Given $n$ points in the plane, each labeled with one or more primary colors, a \emph{colored spanning graph} (CSG) is a graph such that for each primary color, the vertices of that color induce a connected subgraph. The \textsc{Min-CSG} problem asks for the minimum sum of edge lengths in a colored spanning graph. We show that the problem is NP-hard for $k$ primary colors when $k\ge 3$ and provide a $(2-\frac{1}{3+2\varrho})$-approximation algorithm for $k=3$ that runs in polynomial time, where $\varrho$ is the Steiner ratio. Further, we give a $O(n)$ time algorithm in the special case that the input points are collinear and $k$ is constant.

cs.CG↗