arXiv subjects

Esther M. Arkin

Publications and source records attributed to Esther M. Arkin.

14 recordsLinked to original sources

arXiv2020

Cutting Polygons into Small Pieces with Chords: Laser-Based Localization

Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different objectives, either minimizing the maximum size of a piece for a given number of chords, or minimizing the number of chords that achieve a given size threshold for the pieces. We give hardness results for polygons with holes and approximation algorithms for multiple variants of the problem.

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

Data Races and the Discrete Resource-time Tradeoff Problem with Resource Reuse over Paths

A determinacy race occurs if two or more logically parallel instructions access the same memory location and at least one of them tries to modify its content. Races often lead to nondeterministic and incorrect program behavior. A data race is a special case of a determinacy race which can be eliminated by associating a mutual-exclusion lock or allowing atomic accesses to the memory location. However, such solutions can reduce parallelism by serializing all accesses to that location. For associative and commutative updates, reducers allow parallel race-free updates at the expense of using some extra space. We ask the following question. Given a fixed budget of extra space to mitigate the cost of races in a parallel program, which memory locations should be assigned reducers and how should the space be distributed among the reducers in order to minimize the overall running time? We argue that the races can be captured by a directed acyclic graph (DAG), with nodes representing memory cells and arcs representing read-write dependencies between cells. We then formulate our optimization problem on DAGs. We concentrate on a variation of this problem where space reuse among reducers is allowed by routing extra space along a source to sink path of the DAG and using it in the construction of reducers along the path. We consider two reducers and the corresponding duration functions (i.e., reduction time as a function of space budget). We generalize our race-avoiding space-time tradeoff problem to a discrete resource-time tradeoff problem with general non-increasing duration functions and resource reuse over paths. For general DAGs, the offline problem is strongly NP-hard under all three duration functions, and we give approximation algorithms. We also prove hardness of approximation for the general resource-time tradeoff problem and give a pseudo-polynomial time algorithm for series-parallel DAGs.

cs.DC↗

arXiv2017

Network Optimization on Partitioned Pairs of Points

Given $n$ pairs of points, $\mathcal{S} = \{\{p_1, q_1\}, \{p_2, q_2\}, \dots, \{p_n, q_n\}\}$, in some metric space, we study the problem of two-coloring the points within each pair, red and blue, to optimize the cost of a pair of node-disjoint networks, one over the red points and one over the blue points. In this paper we consider our network structures to be spanning trees, traveling salesman tours or matchings. We consider several different weight functions computed over the network structures induced, as well as several different objective functions. We show that some of these problems are NP-hard, and provide constant factor approximation algorithms in all cases.

cs.CG↗

arXiv2011

Probabilistic Bounds on the Length of a Longest Edge in Delaunay Graphs of Random Points in d-Dimensions

Motivated by low energy consumption in geographic routing in wireless networks, there has been recent interest in determining bounds on the length of edges in the Delaunay graph of randomly distributed points. Asymptotic results are known for random networks in planar domains. In this paper, we obtain upper and lower bounds that hold with parametric probability in any dimension, for points distributed uniformly at random in domains with and without boundary. The results obtained are asymptotically tight for all relevant values of such probability and constant number of dimensions, and show that the overhead produced by boundary nodes in the plane holds also for higher dimensions. To our knowledge, this is the first comprehensive study on the lengths of long edges in Delaunay graphs

cs.CG↗

arXiv2006

Minimum-Cost Coverage of Point Sets by Disks

We consider a class of geometric facility location problems in which the goal is to determine a set X of disks given by their centers (t_j) and radii (r_j) that cover a given set of demand points Y in the plane at the smallest possible cost. We consider cost functions of the form sum_j f(r_j), where f(r)=r^alpha is the cost of transmission to radius r. Special cases arise for alpha=1 (sum of radii) and alpha=2 (total area); power consumption models in wireless network design often use an exponent alpha>2. Different scenarios arise according to possible restrictions on the transmission centers t_j, which may be constrained to belong to a given discrete set or to lie on a line, etc. We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points t_j on a given line in order to cover demand points Y in the plane; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover Y; (c) a proof of NP-hardness for a discrete set of transmission points in the plane and any fixed alpha>1; and (d) a polynomial-time approximation scheme for the problem of computing a minimum cost covering tour (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.

cs.DS↗

arXiv2006

The Snowblower Problem

We introduce the snowblower problem (SBP), a new optimization problem that is closely related to milling problems and to some material-handling problems. The objective in the SBP is to compute a short tour for the snowblower to follow to remove all the snow from a domain (driveway, sidewalk, etc.). When a snowblower passes over each region along the tour, it displaces snow into a nearby region. The constraint is that if the snow is piled too high, then the snowblower cannot clear the pile. We give an algorithmic study of the SBP. We show that in general, the problem is NP-complete, and we present polynomial-time approximation algorithms for removing snow under various assumptions about the operation of the snowblower. Most commercially-available snowblowers allow the user to control the direction in which the snow is thrown. We differentiate between the cases in which the snow can be thrown in any direction, in any direction except backwards, and only to the right. For all cases, we give constant-factor approximation algorithms; the constants increase as the throw direction becomes more restricted. Our results are also applicable to robotic vacuuming (or lawnmowing) with bounded capacity dust bin and to some versions of material-handling problems, in which the goal is to rearrange cartons on the floor of a warehouse.

cs.DS↗

arXiv2005

The Freeze-Tag Problem: How to Wake Up a Swarm of Robots

An optimization problem that naturally arises in the study of swarm robotics is the Freeze-Tag Problem (FTP) of how to awaken a set of ``asleep'' robots, by having an awakened robot move to their locations. Once a robot is awake, it can assist in awakening other slumbering robots.The objective is to have all robots awake as early as possible. While the FTP bears some resemblance to problems from areas in combinatorial optimization such as routing, broadcasting, scheduling, and covering, its algorithmic characteristics are surprisingly different. We consider both scenarios on graphs and in geometric environments.In graphs, robots sleep at vertices and there is a length function on the edges. Awake robots travel along edges, with time depending on edge length. For most scenarios, we consider the offline version of the problem, in which each awake robot knows the position of all other robots. We prove that the problem is NP-hard, even for the special case of star graphs. We also establish hardness of approximation, showing that it is NP-hard to obtain an approximation factor better than 5/3, even for graphs of bounded degree.These lower bounds are complemented with several positive algorithmic results, including: (1) We show that the natural greedy strategy on star graphs has a tight worst-case performance of 7/3 and give a polynomial-time approximation scheme (PTAS) for star graphs. (2) We give a simple O(log D)-competitive online algorithm for graphs with maximum degree D and locally bounded edge weights. (3) We give a PTAS, running in nearly linear time, for geometrically embedded instances.

cs.DS↗

arXiv2005

Optimal Covering Tours with Turn Costs

We give the first algorithmic study of a class of ``covering tour'' problems related to the geometric Traveling Salesman Problem: Find a polygonal tour for a cutter so that it sweeps out a specified region (``pocket''), in order to minimize a cost that depends mainly on the number of em turns. These problems arise naturally in manufacturing applications of computational geometry to automatic tool path generation and automatic inspection systems, as well as arc routing (``postman'') problems with turn penalties. We prove the NP-completeness of minimum-turn milling and give efficient approximation algorithms for several natural versions of the problem, including a polynomial-time approximation scheme based on a novel adaptation of the m-guillotine method.

cs.DS↗

arXiv2003

When Can You Fold a Map?

We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds; the simplest one-layer simple fold rotates a portion of paper about a crease in the paper by +-180 degrees. We first consider the analogous questions in one dimension lower -- bending a segment into a flat object -- which lead to interesting problems on strings. We develop efficient algorithms for the recognition of simply foldable 1D crease patterns, and reconstruction of a sequence of simple folds. Indeed, we prove that a 1D crease pattern is flat-foldable by any means precisely if it is by a sequence of one-layer simple folds. Next we explore simple foldability in two dimensions, and find a surprising contrast: ``map'' folding and variants are polynomial, but slight generalizations are NP-complete. Specifically, we develop a linear-time algorithm for deciding foldability of an orthogonal crease pattern on a rectangular piece of paper, and prove that it is (weakly) NP-complete to decide foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper, (2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a square piece of paper, and (3) crease patterns without a mountain/valley assignment.

cs.CG↗

arXiv2002

Algorithms for Rapidly Dispersing Robot Swarms in Unknown Environments

We develop and analyze algorithms for dispersing a swarm of primitive robots in an unknown environment, R. The primary objective is to minimize the makespan, that is, the time to fill the entire region. An environment is composed of pixels that form a connected subset of the integer grid. There is at most one robot per pixel and robots move horizontally or vertically at unit speed. Robots enter R by means of k>=1 door pixels Robots are primitive finite automata, only having local communication, local sensors, and a constant-sized memory. We first give algorithms for the single-door case (i.e., k=1), analyzing the algorithms both theoretically and experimentally. We prove that our algorithms have optimal makespan 2A-1, where A is the area of R. We next give an algorithm for the multi-door case (k>1), based on a wall-following version of the leader-follower strategy. We prove that our strategy is O(log(k+1))-competitive, and that this bound is tight for our strategy and other related strategies.

cs.RO↗

arXiv2002

The Lazy Bureaucrat Scheduling Problem

We introduce a new class of scheduling problems in which the optimization is performed by the worker (single ``machine'') who performs the tasks. A typical worker's objective is to minimize the amount of work he does (he is ``lazy''), or more generally, to schedule as inefficiently (in some sense) as possible. The worker is subject to the constraint that he must be busy when there is work that he can do; we make this notion precise both in the preemptive and nonpreemptive settings. The resulting class of ``perverse'' scheduling problems, which we denote ``Lazy Bureaucrat Problems,'' gives rise to a rich set of new questions that explore the distinction between maximization and minimization in computing optimal schedules.

cs.DS↗

arXiv2002

On the Reflexivity of Point Sets

We introduce a new measure for planar point sets S that captures a combinatorial distance that S is from being a convex set: The reflexivity rho(S) of S is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general). Our study considers also some closely related quantities, such as the convex cover number kappa_c(S) of a planar point set, which is the smallest number of convex chains that cover S, and the convex partition number kappa_p(S), which is given by the smallest number of convex chains with pairwise-disjoint convex hulls that cover S. We have proved that it is NP-complete to determine the convex cover or the convex partition number and have given logarithmic-approximation algorithms for determining each.

cs.CG↗

arXiv2002

An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures

We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple non-self-intersecting wire structure (a carpenter's ruler) can be straightened, a problem that was open for several years and has only recently been solved in the affirmative. If we impose some of the constraints that are imposed by the manufacturing process, we obtain quite different results. In particular, we study the variant of the carpenter's ruler problem in which there is a restriction that only one joint can be modified at a time. For a linkage that does not self-intersect or self-touch, the recent results of Connelly et al. and Streinu imply that it can always be straightened, modifying one joint at a time. However, we show that for a linkage with even a single vertex degeneracy, it becomes NP-hard to decide if it can be straightened while altering only one joint at a time. If we add the restriction that each joint can be altered at most once, we show that the problem is NP-complete even without vertex degeneracies. In the special case, arising in wire forming manufacturing, that each joint can be altered at most once, and must be done sequentially from one or both ends of the linkage, we give an efficient algorithm to determine if a linkage can be straightened.

cs.CG↗