arXiv subjects

Stefan Dobrev

Publications and source records attributed to Stefan Dobrev.

5 recordsLinked to original sources

arXiv2026

Online Drone Coverage of Targets on a Line

We study a problem of online targets coverage by a drone or a sensor that is equipped with a camera or an antenna of fixed half-angle of view $α$. The targets to be monitored appear at arbitrary positions on a line barrier in an online manner. When a new target appears, the drone has to move to a location that covers the newly arrived target, as well as already existing targets. The objective is to design a coverage algorithm that optimizes the total length of the drone's trajectory. Our results are reported in terms of an algorithm's competitive ratio, i.e., the worst-case ratio (over all inputs) of its cost to that of an optimal offline algorithm. In terms of upper bounds, we present three online algorithms and prove bounds on their competitive ratios for every $α\in [0, π/2]$. The best of them, called \FA is significantly better than the other two for $π/6 < α< π/3$. In particular, for $α=π/4$, its worst case, \FA has competitive ratio $1.25$, while the other two have competitive ratio $\sqrt{2}$. Finally, we prove a lower bound on the competitive ratio of online algorithms for a drone with half-angle $α\in [0, π/4]$; this bound is a function of $α$ that achieves its maximum value at $α= π/4$ equal to $(1+\sqrt{2})/2 \approx 1.207$.

cs.DS↗

arXiv2019

Exploration of High-Dimensional Grids by Finite State Machines

We consider the problem of finding a treasure at an unknown point of an $n$-dimensional infinite grid, $n\geq 3$, by initially collocated finite state agents (scouts/robots). Recently, the problem has been well characterized for 2 dimensions for deterministic as well as randomized agents, both in synchronous and semi-synchronous models. It has been conjectured that $n+1$ randomized agents are necessary to solve this problem in the $n$-dimensional grid. In this paper we disprove the conjecture in a strong sense: we show that three randomized synchronous agents suffice to explore an $n$-dimensional grid for any $n$. Our algorithm is optimal in terms of the number of the agents. Our key insight is that a constant number of finite state machine agents can, by their positions and movements, implement a stack, which can store the path being explored. We also show how to implement our algorithm using: four randomized semi-synchronous agents; four deterministic synchronous agents; or five deterministic semi-synchronous agents. We give a different algorithm that uses $4$ deterministic semi-synchronous agents for the $3$-dimensional grid. This is provably optimal, and surprisingly, matches the result for $2$ dimensions. For $n\geq 4$, the time complexity of the solutions mentioned above is exponential in distance $D$ of the treasure from the starting point of the agents. We show that in the deterministic case, one additional agent brings the time down to a polynomial. Finally, we focus on algorithms that never venture much beyond the distance $D$. We describe an algorithm that uses $O(\sqrt{n})$ semi-synchronous deterministic agents that never go beyond $2D$, as well as show that any algorithm using $3$ synchronous deterministic agents in $3$ dimensions must travel beyond $Ω(D^{3/2})$ from the origin.

cs.MA↗

arXiv2018

Live Exploration of Dynamic Rings

In the graph exploration problem, a team of mobile computational entities, called agents, arbitrarily positioned at some nodes of a graph, must cooperate so that each node is eventually visited by at least one agent. In the literature, the main focus has been on graphs that are static; that is, the topology is either invariant in time or subject to localized changes. The few studies on exploration of dynamic graphs have been almost all limited to the centralized case (i.e., assuming complete a priori knowledge of the changes and the times of their occurrence). We investigate the decentralized exploration of dynamic graphs (i.e., when the agents are unaware of the location and timing of the changes) focusing, in this paper, on dynamic systems whose underlying graph is a ring. We first consider the fully-synchronous systems traditionally assumed in the literature; i.e., all agents are active at each time step. We then introduce the notion of semi-synchronous systems, where only a subset of agents might be active at each time step (the choice of the subset is made by an adversary); this model is common in the context of mobile agents in continuous spaces but has never been studied before for agents moving in graphs. Our main focus is on the impact that the level of synchrony as well as other factors such as anonymity, knowledge of the size of the ring, and chirality (i.e., common orientation) have on the solvability of the problem, focusing on the minimum number of agents necessary. We draw an extensive map of feasibility, and of complexity in terms of minimum number of agent movements. All our sufficiency proofs are constructive, and almost all our solution protocols are asymptotically optimal.

cs.DC↗

arXiv2017

Weak Coverage of a Rectangular Barrier

Assume n wireless mobile sensors are initially dispersed in an ad hoc manner in a rectangular region. They are required to move to final locations so that they can detect any intruder crossing the region in a direction parallel to the sides of the rectangle, and thus provide weak barrier coverage of the region. We study three optimization problems related to the movement of sensors to achieve weak barrier coverage: minimizing the number of sensors moved (MinNum), minimizing the average distance moved by the sensors (MinSum), and minimizing the maximum distance moved by the sensors (MinMax). We give an O(n^{3/2}) time algorithm for the MinNum problem for sensors of diameter 1 that are initially placed at integer positions; in contrast we show that the problem is NP-hard even for sensors of diameter 2 that are initially placed at integer positions. We show that the MinSum problem is solvable in O(n log n) time for homogeneous range sensors in arbitrary initial positions, while it is NP-hard for heterogeneous sensor ranges. Finally, we prove that even very restricted homogeneous versions of the MinMax problem are NP-hard.

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↗