arXiv subjects

Oscar Morales-Ponce

Publications and source records attributed to Oscar Morales-Ponce.

13 recordsLinked to original sources

arXiv2026

Search and Rescue on the Plane

We study a planar variant of the search and rescue problem whereby an agent starting at an arbitrary position $P_{θ,r} = (r\cosθ, r\sinθ)$ in the plane must locate an object at an unknown position on the positive $x$-axis and deliver it to the origin. Our main contribution is to characterize the optimal form of any competitive algorithm, derive closed-form expressions for the competitive ratio, and identify a critical angle $θ^* \approx 15.6^\circ$ which yields a phase transition to optimal competitive search and delivery in the following sense. For each angle $-π\leq θ\leq π$ we compute a checkpoint (landing position on the $x$-axis) where the agent must go first prior to initiating a search on the $x$-axis in order to optimize the competitive ratio of search and delivery. We show that if $|θ| \geq θ^*$ then the checkpoint is at the origin, while if $|θ| < θ^*$ then the agent should land at the checkpoint $(r \cdot k_{|θ|}, 0)$ on the $x$-axis, where $k_{|θ|}$ is a real number given by an explicit formula we present.

cs.DM↗

arXiv2025

Linear Search with Probabilistic Detection and Variable Speeds

We present results on new variants of the famous linear search (or cow-path) problem that involves an agent searching for a target with unknown position on the infinite line. We consider the variant where the agent can move either at speed $1$ or at a slower speed $v \in [0, 1)$. When traveling at the slower speed $v$, the agent is guaranteed to detect the target upon passing through its location. When traveling at speed $1$, however, the agent, upon passing through the target's location, detects it with probability $p \in [0, 1]$. We present algorithms and provide upper bounds for the competitive ratios for three cases separately: when $p=0$, $v=0$, and when $p,v \in (0,1)$. We also prove that the provided algorithm for the $p=0$ case is optimal.

cs.DM↗

arXiv2024

Optimal Delivery with a Faulty Drone

We introduce and study a new cooperative delivery problem inspired by drone-assisted package delivery. We consider a scenario where a drone, en route to deliver a package to a destination (a point on the plane), unexpectedly loses communication with its central command station. The command station cannot know whether the drone's system has wholly malfunctioned or merely experienced a communications failure. Consequently, a second, helper drone must be deployed to retrieve the package to ensure successful delivery. The central question of this study is to find the optimal trajectory for this second drone. We demonstrate that the optimal solution relies heavily on the relative spatial positioning of the command station, the destination point, and the last known location of the disconnected drone.

cs.DM↗

arXiv2024

Linear Search for an Escaping Target with Unknown Speed

We consider linear search for an escaping target whose speed and initial position are unknown to the searcher. A searcher (an autonomous mobile agent) is initially placed at the origin of the real line and can move with maximum speed $1$ in either direction along the line. An oblivious mobile target that is moving away from the origin with an unknown constant speed $v<1$ is initially placed by an adversary on the infinite line at distance $d$ from the origin in an unknown direction. We consider two cases, depending on whether $d$ is known or unknown. The main contribution of this paper is to prove a new lower bound and give algorithms leading to new upper bounds for search in these settings. This results in an optimal (up to lower order terms in the exponent) competitive ratio in the case where $d$ is known and improved upper and lower bounds for the case where $d$ is unknown. Our results solve an open problem proposed in [Coleman et al., Proc. OPODIS 2022].

cs.DM↗

arXiv2022

Line Search for an Oblivious Moving Target

Consider search on an infinite line involving an autonomous robot starting at the origin of the line and an oblivious moving target at initial distance $d \geq 1$ from it. The robot can change direction and move anywhere on the line with constant maximum speed $1$ while the target is also moving on the line with constant speed $v>0$ but is unable to change its speed or direction. The goal is for the robot to catch up to the target in as little time as possible. The classic case where $v=0$ and the target's initial distance $d$ is unknown to the robot is the well-studied ``cow-path problem''. Alpert and Gal gave an optimal algorithm for the case where a target with unknown initial distance $d$ is moving away from the robot with a known speed $v<1$. In this paper we design and analyze search algorithms for the remaining possible knowledge situations, namely, when $d$ and $v$ are known, when $v$ is known but $d$ is unknown, when $d$ is known but $v$ is unknown, and when both $v$ and $d$ are unknown. Furthermore, for each of these knowledge models we consider separately the case where the target is moving away from the origin and the case where it is moving toward the origin. We design algorithms and analyze competitive ratios for all eight cases above. The resulting competitive ratios are shown to be optimal when the target is moving towards the origin as well as when $v$ is known and the target is moving away from the origin.

cs.DC↗

arXiv2022

Delivery to Safety with Two Cooperating Robots

Two cooperating, autonomous mobile robots with arbitrary nonzero max speeds are placed at arbitrary initial positions in the plane. A remotely detonated bomb is discovered at some source location and must be moved to a safe distance away from its initial location as quickly as possible. In the Bomb Squad problem, the robots cooperate by communicating face-to-face in order to pick up the bomb from the source and carry it away to the boundary of a disk centered at the source in the shortest possible time. The goal is to specify trajectories which define the robots' paths from start to finish and their meeting points which enable face-to-face collaboration by exchanging information and passing the bomb from robot to robot. We design algorithms reflecting the robots' knowledge about orientation and each other's speed and location. In the offline case, we design an optimal algorithm. For the limited knowledge cases, we provide online algorithms which consider robots' level of agreement on orientation as per OneAxis and NoAxis models, and knowledge of the boundary as per Visible, Discoverable, and Invisible. In all cases, we provide upper and lower bounds for the competitive ratios of the online problems.

cs.DC↗

arXiv2021

Message Delivery in the Plane by Robots with Different Speeds

We study a fundamental cooperative message-delivery problem on the plane. Assume $n$ robots which can move in any direction, are placed arbitrarily on the plane. Robots each have their own maximum speed and can communicate with each other face-to-face (i.e., when they are at the same location at the same time). There are also two designated points on the plane, $S$ (the source) and $D$ (the destination). The robots are required to transmit the message from the source to the destination as quickly as possible by face-to-face message passing. We consider both the offline setting where all information (the locations and maximum speeds of the robots) are known in advance and the online setting where each robot knows only its own position and speed along with the positions of $S$ and $D$. In the offline case, we discover an important connection between the problem for two-robot systems and the well-known Apollonius circle which we employ to design an optimal algorithm. We also propose a $\sqrt 2$ approximation algorithm for systems with any number of robots. In the online setting, we provide an algorithm with competitive ratio $\frac 17 \left( 5+ 4 \sqrt{2} \right)$ for two-robot systems and show that the same algorithm has a competitive ratio less than $2$ for systems with any number of robots. We also show these results are tight for the given algorithm. Finally, we give two lower bounds (employing different arguments) on the competitive ratio of any online algorithm, one of $1.0391$ and the other of $1.0405$.

cs.DC↗

arXiv2021

Computing the Optimal Longest Queue Length in Torus Networks

A collection of $k$ mobile agents is arbitrarily deployed in the edges of a directed torus network where agents perpetually move to the successor edge. Each node has a switch that allows one agent of the two incoming edges to pass to its successor edge in every round. The goal is to obtain a switch scheduling to reach and maintain a configuration where the longest queue length is minimum. We consider a synchronous system. We use the concept of conflict graphs to model the local conflicts that occur with incident links. We show that there does not exist an algorithm that can reduce the number of agents in any conflict cycle of the conflict graph providing that all the links have at least 2 agents at every round. Hence, the lower bound is at least the average queue length of the conflict cycle with the maximum average queue length. Next, we present a centralized algorithm that computes a strategy in $O(n\log n)$ time for each round that attains the optimal queue length in $O(σn)$ rounds where $n$ is the number of nodes in the network and $σ$ is the standard deviations of the queue lengths in the initial setting. Our technique is based on network flooding on conflict graphs. Next, we consider a distributed system where nodes have access to the length of their queues and use communication to self-coordinate with nearby nodes. We present a local algorithm using only the information of the queue lengths at distance two. We show that the algorithm attains the optimal queue length in $O(σC_{max}^2)$ rounds where $C_{max}$ is the length of the longest conflict cycle with the maximum average queue length.

cs.DS↗

arXiv2020

Minimizing The Maximum Distance Traveled To Form Patterns With Systems of Mobile Robots

In the pattern formation problem, robots in a system must self-coordinate to form a given pattern, regardless of translation, rotation, uniform-scaling, and/or reflection. In other words, a valid final configuration of the system is a formation that is \textit{similar} to the desired pattern. While there has been no shortage of research in the pattern formation problem under a variety of assumptions, models, and contexts, we consider the additional constraint that the maximum distance traveled among all robots in the system is minimum. Existing work in pattern formation and closely related problems are typically application-specific or not concerned with optimality (but rather feasibility). We show the necessary conditions any optimal solution must satisfy and present a solution for systems of three robots. Our work also led to an interesting result that has applications beyond pattern formation. Namely, a metric for comparing two triangles where a distance of $0$ indicates the triangles are similar, and $1$ indicates they are \emph{fully dissimilar}.

cs.CG↗

arXiv2020

Synchronous Robotic Framework

We present a synchronous robotic testbed called SyROF that allows fast implementation of robotic swarms. Our main goal is to lower the entry barriers to cooperative-robot systems for undergraduate and graduate students. The testbed provides a high-level programming environment that allows the implementation of Timed Input/Output Automata (TIOA). SyROF offers the following unique characteristics: 1) a transparent mechanism to synchronize robot maneuvers, 2) a membership service with a failure detector, and 3) a transparent service to provide common knowledge in every round. These characteristics are fundamental to simplifying the implementation of robotic swarms. The software is organized in five layers: The lower layer consists of a real-time publish-subscribe system that allows efficient communication between tasks. The next layer is an implementation of a Kalman filter to estimate the position, orientation, and speed of the robot. The third layer consists of a synchronizer that synchronously executes the robot maneuvers, provides common knowledge to all the active participants, and handles failures. The fifth layer consists of the programming environment.

cs.RO↗

arXiv2019

Optimal Patrolling of High Priority Segments While Visiting the Unit Interval with a Set of Mobile Robots

Consider a region that requires to be protected from unauthorized penetrations. The border of the region, modeled as a unit line segment, consists of high priority segments that require the highest level of protection separated by low priority segments that require to be visited infinitely often. We study the problem of patrolling the border with a set of $k$ robots. The goal is to obtain a strategy that minimizes the maximum idle time (the time that a point is left unattended) of the high priority points while visiting the low priority points infinitely often. We use the concept of single lid cover (segments of fixed length) where each high priority point is covered with at least one lid, and then we extend it to strong double-lid cover where each high priority point is covered with at least two lids, and the unit line segment is fully covered. Let $λ_{k-1}$ be the minimum lid length that accepts a single $λ_{k-1}$-lid cover with $k-1$ lids and $Λ_{2k}$ be the minimum lid length that accepts a strong double $Λ_{2k}$-lid cover with $2k$ lids. We show that $2\min(Λ_{2k}, λ_{k-1})$ is the lower bound of the idle time when the max speed of the robots is one. To compute $Λ_{2k}$ and $λ_{k-1}$, we present an algorithm with time complexity $O(\max(k, n)\log{n})$ where $n$ is the number of high priority sections. For the upper bound, first we present a strategy with idle time $λ_{k-1}$ where one robot covers the unit line, and the remaining robots cover the lids of a single $λ_{k-1}$-lid cover with $k-1$ lids. Then, we present a simple strategy with idle time $3Λ_{2k}$ that splits the unit line into not-disjoint $k$ segments of equal length that robots synchronously cover. Then, we present a complex strategy that split the unit line into $k$ non-disjoint segments that robots asynchronously cover.

cs.DC↗

arXiv2015

Cooperation with Disagreement Correction in the Presence of Communication Failures

Vehicle-to-vehicle communication is a fundamental requirement in cooperative vehicular systems to achieve high performance while keeping high safety standards. Vehicles periodically exchange critical information with nearby vehicles to determine their maneuvers according to the information quality and established strategies. However, wireless communication is prone to failures. Thus, participants can be unaware that other participants have not received the information on time resulting in conflicting trajectories that may not be safe. We present a deterministic solution that allows all participants to use a default strategy when other participants have not received on time the complete information. We base our solution on a timed distributed protocol that adapts its output according to the effect of message omission failures so that the disagreement period occurs for no longer than a constant time (of the order of milliseconds) that only depends on the message delay. We formally show the correctness and perform experiments to corroborate its efficiency. We explain how the proposed solution can be used on vehicular platooning to attain high performance and still guarantee high safety standards despite communication failures. We believe that this work can facilitate the implementation of cooperative driving systems that have to deal with inherent (communication) uncertainties.

cs.DC↗

arXiv2011

Approximating the Edge Length of 2-Edge Connected Planar Geometric Graphs on a Set of Points

Given a set $P$ of $n$ points in the plane, we solve the problems of constructing a geometric planar graph spanning $P$ 1) of minimum degree 2, and 2) which is 2-edge connected, respectively, and has max edge length bounded by a factor of 2 times the optimal; we also show that the factor 2 is best possible given appropriate connectivity conditions on the set $P$, respectively. First, we construct in $O(n\log{n})$ time a geometric planar graph of minimum degree 2 and max edge length bounded by 2 times the optimal. This is then used to construct in $O(n\log n)$ time a 2-edge connected geometric planar graph spanning $P$ with max edge length bounded by $\sqrt{5}$ times the optimal, assuming that the set $P$ forms a connected Unit Disk Graph. Second, we prove that 2 times the optimal is always sufficient if the set of points forms a 2 edge connected Unit Disk Graph and give an algorithm that runs in $O(n^2)$ time. We also show that for $k \in O(\sqrt{n})$, there exists a set $P$ of $n$ points in the plane such that even though the Unit Disk Graph spanning $P$ is $k$-vertex connected, there is no 2-edge connected geometric planar graph spanning $P$ even if the length of its edges is allowed to be up to 17/16.

cs.DM↗