arXiv subjects

Paola Flocchini

Publications and source records attributed to Paola Flocchini.

At least 19 recordsLinked to original sources

arXiv2025

Universal Dancing by Luminous Robots under Sequential Schedulers

The Dancing problem requires a swarm of $n$ autonomous mobile robots to form a sequence of patterns, aka perform a choreography. Existing work has proven that some crucial restrictions on choreographies and initial configurations (e.g., on repetitions of patterns, periodicity, symmetries, contractions/expansions) must hold so that the Dancing problem can be solved under certain robot models. Here, we prove that these necessary constraints can be dropped by considering the LUMI model (i.e., where robots are endowed with a light whose color can be chosen from a constant-size palette) under the quite unexplored sequential scheduler. We formalize the class of Universal Dancing problems which require a swarm of $n$ robots starting from any initial configuration to perform a (periodic or finite) sequence of arbitrary patterns, only provided that each pattern consists of $n$ vertices (including multiplicities). However, we prove that, to be solvable under LUMI, the length of the feasible choreographies is bounded by the compositions of $n$ into the number of colors available to the robots. We provide an algorithm solving the Universal Dancing problem by exploiting the peculiar capability of sequential robots to implement a distributed counter mechanism. Even assuming non-rigid movements, our algorithm ensures spatial homogeneity of the performed choreography.

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arXiv2024

Universal Pattern Formation by Oblivious Robots Under Sequential Schedulers

We study the computational power that oblivious robots operating in the plane have under sequential schedulers. We show that this power is much stronger than the obvious capacity these schedulers offer of breaking symmetry, and thus to create a leader. In fact, we prove that under any sequential scheduler, robots are capable of solving problems that are unsolvable even with a leader under the fully synchronous scheduler FSYNC. More precisely, we consider the class of pattern formation problems, and focus on the most general problem in this class, Universal Pattern Formation (UPF), which requires the robots to form every pattern given in input, starting from any initial configuration (where some robots may occupy the same point, hence forming a multiplicity). We first show that UPF is unsolvable under FSYNC, even if the robots are endowed with additional strong capabilities (multiplicity detection, rigid movement, agreement on coordinate systems, presence of a unique leader). On the other hand, we prove that, except for point formation (Gathering), UPF is solvable under any sequential scheduler without any additional assumptions. We then turn our attention to the Gathering problem, and prove that weak multiplicity detection (the ability to detect a multiplicity but not the exact number of robots forming it) is necessary and sufficient for solvability under sequential schedulers. The results obtained show that the computational power of the robots under FSYNC (where Gathering is solvable without any multiplicity detection) and that under sequential schedulers are orthogonal.

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arXiv2024

Cops & Robber on Periodic Temporal Graphs

We consider the Cops and Robber pursuit-evasion game when the edge-set of the graph is allowed to change in time, possibly at every round. Specifically, the game is played on an infinite periodic sequence $\mathcal{G} = (G_0, \dots, G_{p-1})^*$ of graphs on the same set $V$ of $n$ vertices: in round $t$, the topology of $\mathcal{G}$ is $G_i=(V,E_i)$ where $i\equiv t \pmod{p}$. Concentrating on the case of a single cop, we provide a characterization of copwin periodic temporal graphs, establishing several basic properties on their nature, and extending to the temporal domain classical C&R concepts such as covers and corners. Based on these results, we design an efficient algorithm for determining if a periodic temporal graph is copwin. We also consider the case of $k>1$ cops. By shifting from a representation in terms of directed graphs to one in terms of directed multi-hypergraphs, we prove that all the fundamental properties established for $k=1$ continue to hold, providing a characterization of $k$-copwin periodic graphs, as well as a general strategy to determine if a periodic graph is $k$-copwin. Our results do not rely on any assumption on properties such as connectivity, symmetry, reflexivity held by the individual graphs in the sequence. They are established for a unified version of the game that includes the standard games studied in the literature, both for undirected and directed graphs, and both when the players are fully active and when they are not. They hold also for a variety of settings not considered in the literature.

cs.DM↗

arXiv2024

Cop numbers of periodic graphs

A \emph{periodic graph} ${\cal G}=(G_0, G_1, G_2, \dots)$ with period $p$ is an infinite periodic sequence of graphs $G_i = G_{i + p} = (V,E_i)$, where $i \geq 0$. The graph $G=(V,\cup_i E_i)$ is called the footprint of ${\cal G}$. Recently, the arena where the Cops and Robber game is played has been extended from a graph to a periodic graph; in this case, the \emph{cop number} is also the minimum number of cops sufficient for capturing the robber. We study the connections and distinctions between the cop number $c({\cal G})$ of a periodic graph ${\cal G}$ and the cop number $c(G)$ of its footprint $G$ and establish several facts. For instance, we show that the smallest periodic graph with $c({\cal G}) = 3$ has at most $8$ nodes; in contrast, the smallest graph $G$ with $c(G) = 3$ has $10$ nodes. We push this investigation by generating multiple examples showing how the cop numbers of a periodic graph ${\cal G}$, the subgraphs $G_i$ and its footprint $G$ can be loosely tied. Based on these results, we derive upper bounds on the cop number of a periodic graph from properties of its footprint such as its treewidth.

math.CO↗

arXiv2024

Black Hole Search by Scattered Agents in Dynamic Rings

In this paper, we address the challenge of locating a black hole within a dynamic graph using a set of scattered agents, which start from arbitrary positions in the graph. A black hole is defined as a node that silently eliminates any agent that visits it, effectively modeling network failures such as a crashed host or a destructive virus. The black hole search problem is considered solved when at least one agent survives and possesses a complete map of the graph, including the precise location of the black hole. Our study focuses on the scenario where the underlying graph is a dynamic 1-interval connected ring: a ring graph where, in each round, one edge may be absent. Agents communicate with other agents using movable pebbles that can be placed on nodes. In this setting, we demonstrate that three agents are sufficient to identify the black hole in $O(n^2)$ moves. Furthermore, we prove that this number of agents is optimal. Additionally, we establish that the complexity bound is tight, requiring $\Omega(n^2)$ moves for any algorithm solving the problem with three agents, even when stronger communication mechanisms, such as unlimited-size whiteboards on nodes, are available.

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arXiv2023

On Asynchrony, Memory, and Communication: Separations and Landscapes

Research on distributed computing by a team of identical mobile computational entities, called robots, operating in a Euclidean space in $\mathit{Look}$-$\mathit{Compute}$-$\mathit{Move}$ ($\mathit{LCM}$) cycles, has recently focused on better understanding how the computational power of robots depends on the interplay between their internal capabilities (i.e., persistent memory, communication), captured by the four standard computational models (OBLOT, LUMI, FSTA, and FCOM) and the conditions imposed by the external environment, controlling the activation of the robots and their synchronization of their activities, perceived and modeled as an adversarial scheduler. We consider a set of adversarial asynchronous schedulers ranging from the classical semi-synchronous (SSYNCH) and fully asynchronous (ASYNCH) settings, including schedulers (emerging when studying the atomicity of the combination of operations in the $\mathit{LCM}$ cycles) whose adversarial power is in between those two. We ask the question: what is the computational relationship between a model $M_1$ under adversarial scheduler $K_1$ ($M_1(K_1)$) and a model $M_2$ under scheduler $K_2$ ($M_2(K_2)$)? For example, are the robots in $M_1(K_1)$ more powerful (i.e., they can solve more problems) than those in $M_2(K_2)$? We answer all these questions by providing, through cross-model analysis, a complete characterization of the computational relationship between the power of the four models of robots under the considered asynchronous schedulers. In this process, we also provide qualified answers to several open questions, including the outstanding one on the proper dominance of SSYNCH over ASYNCH in the case of unrestricted visibility.

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arXiv2022

On the Computational Power of Energy-Constrained Mobile Robots: Algorithms and Cross-Model Analysis

We consider distributed systems of identical autonomous computational entities, called robots, moving and operating in the plane in synchronous Look-Compute-Move (LCM) cycles. The algorithmic capabilities of these systems have been extensively investigated in the literature under four distinct models (OBLOT, FSTA, FCOM, LUMI), each identifying different levels of memory persistence and communication capabilities of the robots. Despite their differences, they all always assume that robots have unlimited amounts of energy. In this paper, we remove this assumption and start the study of the computational capabilities of robots whose energy is limited, albeit renewable. We first study the impact that memory persistence and communication capabilities have on the computational power of such energy-constrained systems of robots; we do so by analyzing the computational relationship between the four models under this energy constraint. We provide a complete characterization of this relationship. We then study the difference in computational power caused by the energy restriction and provide a complete characterization of the relationship between energy-constrained and unrestricted robots in each model. We prove that within LUMI there is no difference; an integral part of the proof is the design and analysis of an algorithm that in LUMI allows energy-constrained robots to execute correctly any protocol for robots with unlimited energy. We then show the (apparently counterintuitive) result that in all other models, the energy constraint actually provides the robots with a computational advantage.

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arXiv2020

Tight Bounds for Black Hole Search in Dynamic Rings

In this paper, we start the investigation of distributed computing by mobile agents in dangerous dynamic networks. The danger is posed by the presence in the network of a black hole BH, a harmful site that destroys all incoming agents without leaving any trace. The problem of determining the location of the black hole in a network, known as black hole search BHS, has been extensively studied in the literature, but always and only assuming that the network is static. At the same time, the existing results on mobile agents computing in dynamic networks never consider the presence of harmful sites. In this paper we start filling this research gap by studying black hole search in temporal rings, specifically focusing on 1-interval connectivity adversarial dynamics. The problem is solved if within finite time at least one agent survives and knows the location of BH. The main complexity parameters of BHS is the number of agents (called size) needed to solve the problem, and the number of moves (called cost) performed by the agents; in synchronous systems, such as temporal rings, an additional complexity measure is the amount of time until termination occurs. Feasibility and complexity depend on many parameters; in particular: whether the agents start from the same safe node or from possibly distinct safe locations, the size $n$ of the ring, whether or not $n$ is known, and the type of inter-agent communication (whiteboards, tokens, face-to-face, visual). In this paper, we provide a complete feasibility characterization for all instances of those parameters; all our algorithms are size optimal. Furthermore, we establish lower bounds on the cost (i.e., the number of moves) and time of size-optimal solutions for all instances of those parameters and show that our algorithms achieve those bound.

cs.DC↗

arXiv2020

A Characterization of Semi-Synchrony for Asynchronous Robots with Limited Visibility, and its Application to Luminous Synchronizer Design

A mobile robot system consists of anonymous mobile robots, each of which autonomously performs sensing, computation, and movement according to a common algorithm, so that the robots collectively achieve a given task. There are two main models of time and activation of the robots. In the semi-synchronous model (SSYNC), the robots share a common notion of time; at each time unit, a subset of the robots is activated, and each performs all three actions (sensing, computation, and movement) in that time unit. In the asynchronous model (ASYNC), there is no common notion of time, the robots are activated at arbitrary times, and the duration of each action is arbitrary but finite. In this paper, we investigate the problem of synchronizing ASNYC robots with limited sensing range, i.e., limited visibility. We first present a sufficient condition for an ASYNC execution of a common algorithm ${\cal A}$ to have a corresponding SSYNC execution of ${\cal A}$; our condition imposes timing constraints on the activation schedule of the robots and visibility constraints during movement. Then, we prove that this condition is necessary (with probability $1$) under a randomized ASYNC adversary. Finally, we present a synchronization algorithm for luminous ASYNC robots with limited visibility, each equipped with a light that can take a constant number of colors. Our algorithm enables luminous ASYNC robots to simulate any algorithm ${\cal A}$, designed for the (non-luminous) SSYNC robots and satisfying visibility constraints.

cs.DC↗

arXiv2020

Mobile RAM and Shape Formation by Programmable Particles

We investigate computational issues in the distributed model Amoebots of programmable matter. In this model, the computational entities, called particles, are anonymous finite-state machines that operate and move on an hexagonal tasselation of the plane. In this paper we show how a constant number of such weak particles can simulate a powerful Turing-complete entity that is able to move on the plane while computing. We then show an application of our tool to the classical Shape-Formation problem, providing a new and much more general distributed solution protocol. Indeed, the existing algorithms would allow to form only shapes made of arrangements of segments and triangles. Our algorithm allows the particles to form more abstract and general connected shapes, including circles and spirals, as well as fractal objects of non-integer dimension, such as the Sierpinski triangle or the Koch snowflake. In lieu of the existing limitation on the formability of a shape depending on the symmetry of the initial configuration of the particles, our result provides a complete characterization of the connected shapes that can be formed by an initially simply connected set of particles. Furthermore, in the case of non-connected shapes, we give almost-matching necessary and sufficient conditions for their formability.

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arXiv2019

Oblivious Permutations on the Plane

We consider a distributed system of n identical mobile robots operating in the two dimensional Euclidian plane. As in the previous studies, we consider the robots to be anonymous, oblivious, dis-oriented, and without any communication capabilities, operating based on the Look-Compute-Move model where the next location of a robot depends only on its view of the current configuration. Even in this seemingly weak model, most formation problems which require constructing specific configurations, can be solved quite easily when the robots are fully synchronized with each other. In this paper we introduce and study a new class of problems which, unlike the formation problems so far, cannot always be solved even in the fully synchronous model with atomic and rigid moves. This class of problems requires the robots to permute their locations in the plane. In particular, we are interested in implementing two special types of permutations -- permutations without any fixed points and permutations of order $n$. The former (called MOVE-ALL) requires each robot to visit at least two of the initial locations, while the latter (called VISIT-ALL) requires every robot to visit each of the initial locations in a periodic manner. We provide a characterization of the solvability of these problems, showing the main challenges in solving this class of problems for mobile robots. We also provide algorithms for the feasible cases, in particular distinguishing between one-step algorithms (where each configuration must be a permutation of the original configuration) and multi-step algorithms (which allow intermediate configurations). These results open a new research direction in mobile distributed robotics which has not been investigated before.

cs.DC↗

arXiv2019

Meeting in a Polygon by Anonymous Oblivious Robots

The Meeting problem for $k\geq 2$ searchers in a polygon $P$ (possibly with holes) consists in making the searchers move within $P$, according to a distributed algorithm, in such a way that at least two of them eventually come to see each other, regardless of their initial positions. The polygon is initially unknown to the searchers, and its edges obstruct both movement and vision. Depending on the shape of $P$, we minimize the number of searchers $k$ for which the Meeting problem is solvable. Specifically, if $P$ has a rotational symmetry of order $σ$ (where $σ=1$ corresponds to no rotational symmetry), we prove that $k=σ+1$ searchers are sufficient, and the bound is tight. Furthermore, we give an improved algorithm that optimally solves the Meeting problem with $k=2$ searchers in all polygons whose barycenter is not in a hole (which includes the polygons with no holes). Our algorithms can be implemented in a variety of standard models of mobile robots operating in Look-Compute-Move cycles. For instance, if the searchers have memory but are anonymous, asynchronous, and have no agreement on a coordinate system or a notion of clockwise direction, then our algorithms work even if the initial memory contents of the searchers are arbitrary and possibly misleading. Moreover, oblivious searchers can execute our algorithms as well, encoding information by carefully positioning themselves within the polygon. This code is computable with basic arithmetic operations, and each searcher can geometrically construct its own destination point at each cycle using only a compass. We stress that such memoryless searchers may be located anywhere in the polygon when the execution begins, and hence the information they initially encode is arbitrary. Our algorithms use a self-stabilizing map construction subroutine which is of independent interest.

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arXiv2018

TuringMobile: A Turing Machine of Oblivious Mobile Robots with Limited Visibility and its Applications

In this paper we investigate the computational power of a set of mobile robots with limited visibility. At each iteration, a robot takes a snapshot of its surroundings, uses the snapshot to compute a destination point, and it moves toward its destination. Each robot is punctiform and memoryless, it operates in $\mathbb{R}^m$, it has a local reference system independent of the other robots' ones, and is activated asynchronously by an adversarial scheduler. Moreover, robots are non-rigid, in that they may be stopped by the scheduler at each move before reaching their destination (but are guaranteed to travel at least a fixed unknown distance before being stopped). We show that despite these strong limitations, it is possible to arrange $3m+3k$ of these weak entities in $\mathbb{R}^m$ to simulate the behavior of a stronger robot that is rigid (i.e., it always reaches its destination) and is endowed with $k$ registers of persistent memory, each of which can store a real number. We call this arrangement a TuringMobile. In its simplest form, a TuringMobile consisting of only three robots can travel in the plane and store and update a single real number. We also prove that this task is impossible with fewer than three robots. Among the applications of the TuringMobile, we focused on Near-Gathering (all robots have to gather in a small-enough disk) and Pattern Formation (of which Gathering is a special case) with limited visibility. Interestingly, our investigation implies that both problems are solvable in Euclidean spaces of any dimension, even if the visibility graph of the robots is initially disconnected, provided that a small amount of these robots are arranged to form a TuringMobile. In the special case of the plane, a basic TuringMobile of only three robots is sufficient.

cs.DC↗

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.

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arXiv2017

Shape Formation by Programmable Particles

Shape formation is a basic distributed problem for systems of computational mobile entities. Intensively studied for systems of autonomous mobile robots, it has recently been investigated in the realm of programmable matter. Namely, it has been studied in the geometric Amoebot model, where the anonymous entities, called particles, operate on a hexagonal tessellation of the plane and have limited computational power (they have constant memory), strictly local interaction and communication capabilities (only with particles in neighboring nodes of the grid), and limited motorial capabilities (from a grid node to an empty neighboring node); their activation is controlled by an adversarial scheduler. Recent investigations have shown how, starting from a well-structured configuration in which the particles form a (not necessarily complete) triangle, the particles can form a large class of shapes. This result has been established under several assumptions: agreement on the clockwise direction (i.e., chirality), a sequential activation schedule, and randomization (i.e., particles can flip coins). In this paper we provide a characterization of which shapes can be formed deterministically starting from any simply connected initial configuration of $n$ particles. As a byproduct, if randomization is allowed, then any input shape can be formed from any initial (simply connected) shape by our algorithm, provided that $n$ is large enough. Our algorithm works without chirality, proving that chirality is computationally irrelevant for shape formation. Furthermore, it works under a strong adversarial scheduler, not necessarily sequential. We also consider the complexity of shape formation in terms of both the number of rounds and of moves performed by the particles. We prove that our solution has a complexity of $O(n^2)$ rounds and moves: this number of moves is also asymptotically optimal.

cs.DC↗

arXiv2017

Line-Recovery by Programmable Particles

Shape formation has been recently studied in distributed systems of programmable particles. In this paper we consider the shape recovery problem of restoring the shape when $f$ of the $n$ particles have crashed. We focus on the basic line shape, used as a tool for the construction of more complex configurations. We present a solution to the line recovery problem by the non-faulty anonymous particles; the solution works regardless of the initial distribution and number $f<n-4$ of faults, of the local orientations of the non-faulty entities, and of the number of non-faulty entities activated in each round (i.e., semi-synchronous adversarial scheduler).

cs.DC↗

arXiv2017

Fault-Induced Dynamics of Oblivious Robots on a Line

The study of computing in presence of faulty robots in the Look-Compute-Move model has been the object of extensive investigation, typically with the goal of designing algorithms tolerant to as many faults as possible. In this paper, we initiate a new line of investigation on the presence of faults, focusing on a rather different issue. We are interested in understanding the dynamics of a group of robots when they execute an algorithm designed for a fault-free environment, in presence of some undetectable crashed robots. We start this investigation focusing on the classic point-convergence algorithm by Ando et al. for robots with limited visibility, in a simple setting (which already presents serious challenges): the robots operate fully synchronously on a line, and at most two of them are faulty. Interestingly, and perhaps surprisingly, the presence of faults induces the robots to perform some form of scattering, rather than point-convergence. In fact, we discover that they arrange themselves inside the segment delimited by the two faults in interleaved sequences of equidistant robots. The structure that they form has a hierarchical nature: robots organize themselves in groups where a group of some level converges to an equidistant distribution only after all groups of lower levels have converged. This is the first study on the unintended dynamics of oblivious robots induced by the presence of faults.

cs.DC↗

arXiv2017

Gathering in Dynamic Rings

The gathering problem requires a set of mobile agents, arbitrarily positioned at different nodes of a network to group within finite time at the same location, not fixed in advanced. The extensive existing literature on this problem shares the same fundamental assumption: the topological structure does not change during the rendezvous or the gathering; this is true also for those investigations that consider faulty nodes. In other words, they only consider static graphs. In this paper we start the investigation of gathering in dynamic graphs, that is networks where the topology changes continuously and at unpredictable locations. We study the feasibility of gathering mobile agents, identical and without explicit communication capabilities, in a dynamic ring of anonymous nodes; the class of dynamics we consider is the classic 1-interval-connectivity. We focus on the impact that factors such as chirality (i.e., a common sense of orientation) and cross detection (i.e., the ability to detect, when traversing an edge, whether some agent is traversing it in the other direction), have on the solvability of the problem. We provide a complete characterization of the classes of initial configurations from which the gathering problem is solvable in presence and in absence of cross detection and of chirality. The feasibility results of the characterization are all constructive: we provide distributed algorithms that allow the agents to gather. In particular, the protocols for gathering with cross detection are time optimal. We also show that cross detection is a powerful computational element. We prove that, without chirality, knowledge of the ring size is strictly more powerful than knowledge of the number of agents; on the other hand, with chirality, knowledge of n can be substituted by knowledge of k, yielding the same classes of feasible initial configurations.

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