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Andrzej Pelc

Publications and source records attributed to Andrzej Pelc.

At least 19 recordsLinked to original sources
arXiv2026

Rendezvous of Mobile Deterministic Automata in Graphs

Two mobile agents, modeled as identical deterministic finite automata (DFA) navigating in synchronous rounds in a graph with unlabeled nodes, have to meet at some node. The well-researched task of meeting in a graph is known as rendezvous. Agents start at adversarially chosen distinct nodes in possibly different rounds. An instance of the rendezvous problem is the underlying graph, together with the initial nodes $u$ and $v$ of the agents. Such an instance is feasible, if there exists a DFA (possibly working only for this instance), such that its identical copies starting at nodes $u$ and $v$, with an arbitrary delay, accomplish rendezvous. A DFA is RV-universal for a class of instances, if it guarantees rendezvous of its copies starting at the designated nodes with arbitrary delay, for all feasible instances of this class. Our goal is to investigate the existence of RV-universal DFA. We start by observing that if agents cannot mark nodes in any way then there does not exist a RV-universal DFA even for the class of instances where the underlying graph is a line. Hence we allow the use of identical pebbles to mark the nodes by the agents. We consider stationary pebbles that can be dropped by agents at nodes but cannot be picked up, and movable pebbles that can be dropped by agents and later picked up. We observe that, even in the more powerful scenario of movable pebbles, if agents are equipped with any finite number of pebbles, there is no RV-universal DFA for the class of all instances. Hence we restrict attention to instances where the underlying graph is a tree. Our main contribution are two contrasting results showing that movability of pebbles is a crucial feature. We first prove that for any finite number of stationary pebbles there is no RV-universal DFA for trees, and then we design a RV-universal DFA for trees, where each agent is equipped with a single movable pebble.

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arXiv2025

Gathering Teams of Bounded Memory Agents on a Line

Several mobile agents, modelled as deterministic automata, navigate in an infinite line in synchronous rounds. All agents start in the same round. In each round, an agent can move to one of the two neighboring nodes, or stay idle. Agents have distinct labels which are integers from the set $\{1,\dots, L\}$. They start in teams, and all agents in a team have the same starting node. The adversary decides the compositions of teams, and their starting nodes. Whenever an agent enters a node, it sees the entry port number and the states of all collocated agents; this information forms the input of the agent on the basis of which it transits to the next state and decides the current action. The aim is for all agents to gather at the same node and stop. Gathering is feasible, if this task can be accomplished for any decisions of the adversary, and its time is the worst-case number of rounds from the start till gathering. We consider the feasibility and time complexity of gathering teams of agents, and give a complete solution of this problem. It turns out that both feasibility and complexity of gathering depend on the sizes of teams. We first concentrate on the case when all teams have the same size $x$. For the oriented line, gathering is impossible if $x=1$, and it can be accomplished in time $O(D)$, for $x>1$, where $D$ is the distance between the starting nodes of the most distant teams. This complexity is of course optimal. For the unoriented line, the situation is different. For $x=1$, gathering is also impossible, but for $x=2$, the optimal time of gathering is $Θ(D\log L)$, and for $x\geq 3$, the optimal time of gathering is $Θ(D)$. In the case when there are teams of different sizes, we show that gathering is always possible in time $O(D)$, even for the unoriented line. This complexity is of course optimal.

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arXiv2025

Impact of Knowledge on the Cost of Treasure Hunt in Trees

A mobile agent has to find an inert target in some environment that can be a graph or a terrain in the plane. This task is known as treasure hunt. We consider deterministic algorithms for treasure hunt in trees. Our goal is to establish the impact of different kinds of initial knowledge given to the agent on the cost of treasure hunt, defined as the total number of edge traversals until the agent reaches the treasure hidden in some node of the tree. The agent can be initially given either a complete map of the tree rooted at its starting node, with all port numbers marked, or a blind map of the tree rooted at its starting node but without port numbers. It may also be given, or not, the distance from the root to the treasure. This yields four different knowledge types that are partially ordered by their precision. (For example knowing the blind map and the distance is less precise than knowing the complete map and the distance). The penalty of a less precise knowledge type ${\cal T}_2$ over a more precise knowledge type ${\cal T}_1$ measures intuitively the worst-case ratio of the cost of an algorithm supplied with knowledge of type ${\cal T}_2$ over the cost of an algorithm supplied with knowledge of type ${\cal T}_1$. Our main results establish penalties for comparable knowledge types in this partial order. For knowledge types with known distance, the penalty for having a blind map over a complete map turns out to be very large. By contrast, for unknown distance, the penalty of having a blind map over having a complete map is small. When a map is provided (either complete or blind), the penalty of not knowing the distance over knowing it is medium.

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arXiv2024

Optimal-Length Labeling Schemes for Fast Deterministic Communication in Radio Networks

We consider two fundamental communication tasks in arbitrary radio networks: broadcasting (information from one source has to reach all nodes) and gossiping (every node has a message and all messages have to reach all nodes). Nodes are assigned labels that are (not necessarily different) binary strings. Each node knows its own label and can use it as a parameter in the same deterministic algorithm. The length of a labeling scheme is the largest length of a label. The goal is to find labeling schemes of asymptotically optimal length for the above tasks, and to design fast deterministic distributed algorithms for each of them, using labels of optimal length. Our main result concerns broadcasting. We show the existence of a labeling scheme of constant length that supports broadcasting in time $O(D+\log^2 n)$, where $D$ is the diameter of the network and $n$ is the number of nodes. This broadcasting time is an improvement over the best currently known $O(D\log n + \log^2 n)$ time of broadcasting with constant-length labels, due to Ellen and Gilbert (SPAA 2020). It also matches the optimal broadcasting time in radio networks of known topology. Hence, we show that appropriately chosen node labels of constant length permit to achieve, in a distributed way, the optimal centralized broadcasting time. This is, perhaps, the most surprising finding of this paper. We are able to obtain our result thanks to a novel methodological tool of propagating information in radio networks, that we call a 2-height respecting tree. Next, we apply our broadcasting algorithm to solve the gossiping problem. We get a gossiping algorithm working in time $O(D + Δ\log n + \log^2 n)$, using a labeling scheme of optimal length $O(\log Δ)$, where $Δ$ is the maximum degree. Our time is the same as the best known gossiping time in radio networks of known topology.

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arXiv2024

Sniffing Helps to Meet: Deterministic Rendezvous of Anonymous Agents in the Grid

Two identical anonymous mobile agents have to meet at a node of the infinite oriented grid whose nodes are unlabeled. This problem is known as rendezvous. The agents execute the same deterministic algorithm. Time is divided into rounds, and in each round each agent can either stay idle at the current node or move to an adjacent node. An adversary places the agents at two nodes of the grid at a distance at most $D$, and wakes them up in possibly different rounds. Each agent starts executing the algorithm in its wakeup round. If agents cannot leave any marks on visited nodes then they can never meet, even if they start simultaneously at adjacent nodes and know it. Hence, we assume that each agent marks any unmarked node it visits, and that an agent can distinguish if a node it visits has been previously marked or not. The time of a rendezvous algorithm is the number of rounds between the wakeup of the later agent and rendezvous. We ask the question whether the capability of marking nodes enables the agents to meet, and if so, what is the fastest rendezvous algorithm. We consider this problem under three scenarios. First, agents know $D$ but may start with arbitrary delay. Second, they start simultaneously but do not have any {\em a priori} knowledge. Third, most difficult scenario, we do not make any of the above facilitating assumptions. Agents start with arbitrary delay and they do not have any a priori knowledge. We prove that in the first two scenarios rendezvous can be accomplished in time $O(D)$. This is clearly optimal. For the third scenario, we prove that there does not exist any rendezvous algorithm working in time $o(D^{\sqrt{2}})$, and we show an algorithm working in time $O(D^2)$. The above negative result shows a separation between the optimal complexity in the two easier scenarios and the optimal complexity in the most difficult scenario.

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arXiv2024

Deterministic Collision-Free Exploration of Unknown Anonymous Graphs

We consider the fundamental task of network exploration. A network is modeled as a simple connected undirected n-node graph with unlabeled nodes, and all ports at any node of degree d are arbitrarily numbered 0,.....,d-1. Each of two identical mobile agents, initially situated at distinct nodes, has to visit all nodes and stop. Agents execute the same deterministic algorithm and move in synchronous rounds: in each round, an agent can either remain at the same node or move to an adjacent node. Exploration must be collision-free: in every round at most one agent can be at any node. We assume that agents have vision of radius 2: an awake agent situated at a node v can see the subgraph induced by all nodes at a distance at most 2 from v, sees all port numbers in this subgraph, and the agents located at these nodes. Agents do not know the entire graph but they know an upper bound n on its size. The time of an exploration is the number of rounds since the wakeup of the later agent to the termination by both agents. We show a collision-free exploration algorithm working in time polynomial in n, for arbitrary graphs of size larger than 2. Moreover, we show that if agents have only vision of radius 1, then collision-free exploration is impossible, e.g., in any tree of diameter 2.

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arXiv2023

Fast Deterministic Rendezvous in Labeled Lines

Two mobile agents, starting from different nodes of a network modeled as a graph, and woken up at possibly different times, have to meet at the same node. This problem is known as rendezvous. We consider deterministic distributed rendezvous in the infinite path. Each node has a distinct label which is a positive integer. The time of rendezvous is the number of rounds until meeting, counted from the starting round of the earlier agent. We consider three scenarios. In the first scenario, each agent knows its position in the line, i.e., each of them knows its initial distance from the smallest-labeled node, on which side of this node it is located, and the direction towards it. For this scenario, we give a rendezvous algorithm working in time $O(D)$, where $D$ is the initial distance between the agents. This complexity is clearly optimal. In the second scenario, each agent initially knows only the label of its starting node and the initial distance $D$ between the agents. In this scenario, we give a rendezvous algorithm working in time $O(D\log^*\ell)$, where $\ell$ is the larger label of the starting nodes. We prove a matching lower bound $Ω(D\log^*\ell)$. Finally, in the most general scenario, where each agent initially knows only the label of its starting node, we give a rendezvous algorithm working in time $O(D^2(\log^*\ell)^3)$, which is at most cubic in the lower bound. All our results remain valid (with small changes) for arbitrary finite paths and for cycles. Our algorithms are drastically better than approaches that use graph exploration, whose running times depend on the graph's size or diameter. Our main methodological tool, and the main novelty of the paper, is a two way reduction: from fast colouring of the infinite labeled path using a constant number of colours in the LOCAL model to fast rendezvous in this path, and vice-versa.

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arXiv2023

Deterministic Treasure Hunt and Rendezvous in Arbitrary Connected Graphs

Treasure hunt and rendezvous are fundamental tasks performed by mobile agents in graphs. In treasure hunt, an agent has to find an inert target (called treasure) situated at an unknown node of the graph. In rendezvous, two agents, initially located at distinct nodes of the graph, traverse its edges in synchronous rounds and have to meet at some node. We assume that the graph is connected (otherwise none of these tasks is feasible) and consider deterministic treasure hunt and rendezvous algorithms. The time of a treasure hunt algorithm is the worst-case number of edge traversals performed by the agent until the treasure is found. The time of a rendezvous algorithm is the worst-case number of rounds since the wakeup of the earlier agent until the meeting. To the best of our knowledge, all known treasure hunt and rendezvous algorithms rely on the assumption that degrees of all nodes are finite, even when the graph itself may be infinite. In the present paper we remove this assumption for the first time, and consider both above tasks in arbitrary connected graphs whose nodes can have either finite or countably infinite degrees. Our main result is the first universal treasure hunt algorithm working for arbitrary connected graphs. We prove that the time of this algorithm has optimal order of magnitude among all possible treasure hunt algorithms working for arbitrary connected graphs. As a consequence of this result we obtain the first universal rendezvous algorithm working for arbitrary connected graphs. The time of this algorithm is polynomial in a lower bound holding in many graphs, in particular in the tree all of whose degrees are infinite.

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arXiv2023

Computing Functions by Teams of Deterministic Finite Automata

We consider the task of computing functions $f: \mathbb{N}^k\to \mathbb{N}$, where $ \mathbb{N}$ is the set of natural numbers, by finite teams of agents modelled as deterministic finite automata. The computation is carried out in a distributed way, using the {\em discrete half-line}, which is the infinite graph with one node of degree 1 (called the root) and infinitely many nodes of degree 2. The node at distance $j$ from the root represents the integer $j$. We say that a team $\mathcal{A}^f$ of automata computes a function $f$, if in the beginning of the computation all automata from $\mathcal{A}^f$ are located at the arguments $x_1,\dots,x_k$ of the function $f$, in groups $\mathcal{A}^f _j$ at $x_j$, and at the end, all automata of the team gather at $f(x_1,\dots,x_k)$ and transit to a special state $STOP$. At each step of the computation, an automaton $a$ can ``see'' states of all automata colocated at the same node: the set of these states forms an input of $a$. Our main result shows that, for every primitive recursive function, there exists a finite team of automata that computes this function. We prove this by showing that basic primitive recursive functions can be computed by teams of automata, and that functions resulting from the operations of composition and of primitive recursion can be computed by teams of automata, provided that the ingredient functions of these operations can be computed by teams of automata. We also observe that cooperation between automata is necessary: even some very simple functions $f: \mathbb{N}\to \mathbb{N}$ cannot be computed by a single automaton.

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arXiv2023

Deterministic Rendezvous Algorithms

The task of rendezvous (also called {\em gathering}) calls for a meeting of two or more mobile entities, starting from different positions in some environment. Those entities are called mobile agents or robots, and the environment can be a network modeled as a graph or a terrain in the plane, possibly with obstacles. The rendezvous problem has been studied in many different scenarios. Two among many adopted assumptions particularly influence the methodology to be used to accomplish rendezvous. One of the assumptions specifies whether the agents in their navigation can see something apart from parts of the environment itself, for example other agents or marks left by them. The other assumption concerns the way in which the entities move: it can be either deterministic or randomized. In this paper we survey results on deterministic rendezvous of agents that cannot see the other agents prior to meeting them, and cannot leave any marks.

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arXiv2023

Explorable families of graphs

Graph exploration is one of the fundamental tasks performed by a mobile agent in a graph. An $n$-node graph has unlabeled nodes, and all ports at any node of degree $d$ are arbitrarily numbered $0,\dots, d-1$. A mobile agent, initially situated at some starting node $v$, has to visit all nodes of the graph and stop. In the absence of any initial knowledge of the graph the task of deterministic exploration is often impossible. On the other hand, for some families of graphs it is possible to design deterministic exploration algorithms working for any graph of the family. We call such families of graphs {\em explorable}. Examples of explorable families are all finite families of graphs, as well as the family of all trees. In this paper we study the problem of which families of graphs are explorable. We characterize all such families, and then ask the question whether there exists a universal deterministic algorithm that, given an explorable family of graphs, explores any graph of this family, without knowing which graph of the family is being explored. The answer to this question turns out to depend on how the explorable family is given to the hypothetical universal algorithm. If the algorithm can get the answer to any yes/no question about the family, then such a universal algorithm can be constructed. If, on the other hand, the algorithm can be only given an algorithmic description of the input explorable family, then such a universal deterministic algorithm does not exist.

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arXiv2023

Exploring Wedges of an Oriented Grid by an Automaton with Pebbles

A mobile agent, modeled as a deterministic finite automaton, navigates in the infinite anonymous oriented grid $\mathbb{Z} \times \mathbb{Z}$. It has to explore a given infinite subgraph of the grid by visiting all of its nodes. We focus on the simplest subgraphs, called {\em wedges}, spanned by all nodes of the grid located between two half-lines in the plane, with a common origin. Many wedges turn out to be impossible to explore by an automaton that cannot mark nodes of the grid. Hence, we study the following question: Given a wedge $W$, what is the smallest number $p$ of (movable) pebbles for which there exists an automaton that can explore $W$ using $p$ pebbles? Our main contribution is a complete solution of this problem. For each wedge $W$ we determine this minimum number $p$, show an automaton that explores it using $p$ pebbles and show that fewer pebbles are not enough. We show that this smallest number of pebbles can vary from 0 to 3, depending on the angle between half-lines limiting the wedge and depending on whether the automaton can cross these half-lines or not.

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arXiv2022

Deterministic Rendezvous in Infinite Trees

The rendezvous task calls for two mobile agents, starting from different nodes of a network modeled as a graph to meet at the same node. Agents have different labels which are integers from a set $\{1,\dots,L\}$. They wake up at possibly different times and move in synchronous rounds. In each round, an agent can either stay idle or move to an adjacent node. We consider deterministic rendezvous algorithms. The time of such an algorithm is the number of rounds since the wakeup of the earlier agent till the meeting. In this paper we consider rendezvous in infinite trees. Our main goal is to study the impact of orientation of a tree on the time of rendezvous. We first design a rendezvous algorithm working for unoriented regular trees, whose time is in $O(z(D) \log L)$, where $z(D)$ is the size of the ball of radius $D$, i.e, the number of nodes at distance at most $D$ from a given node. The algorithm works for arbitrary delay between waking times of agents and does not require any initial information about parameters $L$ or $D$. Its disadvantage is its complexity: $z(D)$ is exponential in $D$ for any degree $d>2$ of the tree. We prove that this high complexity is inevitable: $Ω(z(D))$ turns out to be a lower bound on rendezvous time in unoriented regular trees, even for simultaneous start and even when agents know $L$ and $D$. Then we turn attention to oriented trees. While for arbitrary delay between waking times of agents the lower bound $Ω(z(D))$ still holds, for simultaneous start the time of rendezvous can be dramatically shortened. We show that if agents know either a polynomial upper bound on $L$ or a linear upper bound on $D$, then rendezvous can be accomplished in oriented trees in time $O(D\log L)$, which is optimal. When no such extra knowledge is available, we design an algorithm working in time $O(D^2+\log ^2L)$.

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arXiv2021

Deterministic Size Discovery and Topology Recognition in Radio Networks with Short Labels

We consider the fundamental problems of size discovery and topology recognition in radio networks modeled by simple undirected connected graphs. Size discovery calls for all nodes to output the number of nodes in the graph, called its size, and in the task of topology recognition each node has to learn the topology of the graph and its position in it. In radio networks, nodes communicate in synchronous rounds and start in the same round. In each round a node can either transmit the same message to all its neighbors, or stay silent and listen. At the receiving end, a node $v$ hears a message from a neighbor $w$ in a given round, if $v$ listens in this round, and if $w$ is its only neighbor that transmits in this round. If more than one neighbor of a node $v$ transmits in a given round, there is a collision at $v$. We do not assume collision detection: in case of a collision, node $v$ does not hear anything. The time of a deterministic algorithm for each of the above problems is the worst-case number of rounds it takes to solve it. Our goal is to construct short labeling schemes for size discovery and topology recognition in arbitrary radio networks, and to design efficient deterministic algorithms using these schemes. For size discovery, we construct a labeling scheme of length $O(\log\logΔ)$ and we design an algorithm for this problem using this scheme and working in time $O(\log^2 n)$, where $n$ is the size of the graph. We also show that time complexity $O(\log^2 n)$ is optimal for the problem of size discovery, whenever the labeling scheme is of optimal length. For topology recognition, we construct a labeling scheme of length $O(\logΔ)$, and we design an algorithm for this problem using this scheme working in time $O\left(DΔ+\min(Δ^2,n)\right)$. We also show that the length of our labeling scheme is asymptotically optimal.

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arXiv2021

Almost-Optimal Deterministic Treasure Hunt in Arbitrary Graphs

A mobile agent navigating along edges of a simple connected graph, either finite or countably infinite, has to find an inert target (treasure) hidden in one of the nodes. This task is known as treasure hunt. The agent has no a priori knowledge of the graph, of the location of the treasure or of the initial distance to it. The cost of a treasure hunt algorithm is the worst-case number of edge traversals performed by the agent until finding the treasure. Awerbuch, Betke, Rivest and Singh [3] considered graph exploration and treasure hunt for finite graphs in a restricted model where the agent has a fuel tank that can be replenished only at the starting node $s$. The size of the tank is $B=2(1+α)r$, for some positive real constant $α$, where $r$, called the radius of the graph, is the maximum distance from $s$ to any other node. The tank of size $B$ allows the agent to make at most $\lfloor B\rfloor$ edge traversals between two consecutive visits at node $s$. Let $e(d)$ be the number of edges whose at least one extremity is at distance less than $d$ from $s$. Awerbuch, Betke, Rivest and Singh [3] conjectured that it is impossible to find a treasure hidden in a node at distance at most $d$ at cost nearly linear in $e(d)$. We first design a deterministic treasure hunt algorithm working in the model without any restrictions on the moves of the agent at cost $\mathcal{O}(e(d) \log d)$, and then show how to modify this algorithm to work in the model from [3] with the same complexity. Thus we refute the above twenty-year-old conjecture. We observe that no treasure hunt algorithm can beat cost $Θ(e(d))$ for all graphs and thus our algorithms are also almost optimal.

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arXiv2020

Finding the Size and the Diameter of a Radio Network Using Short Labels

The number of nodes of a network, called its size, and the largest distance between nodes of a network, called its diameter, are among the most important network parameters. Knowing the size and/or diameter is a prerequisite of many distributed network algorithms. A radio network is a collection of nodes, with wireless transmission and receiving capabilities. It is modeled as a simple undirected graph whose nodes communicate in synchronous rounds. In each round, a node can either transmit a message to all its neighbors, or stay silent and listen. At the receiving end, a node $v$ hears a message from a neighbor $w$ in a round $i$, if $v$ listens in round $i$, and if $w$ is its only neighbor that transmits in round $i$. If $v$ listens in a round, and multiple neighbors of $v$ transmit in this round, a collision occurs at $v$. If $v$ transmits in a round, it does not hear anything. If listening nodes can distinguish collision from silence, we say that the network has collision detection capability, otherwise there is no collision detection. We consider the tasks of size discovery and diameter discovery: finding the size (resp. the diameter) of an unknown radio network with collision detection. All nodes have to output the size (resp. the diameter) of the network, using a deterministic algorithm. Nodes have labels which are binary strings. The length of a labeling scheme is the largest length of a label. We concentrate on the following problems: 1. What is the shortest labeling scheme that permits size discovery in all radio networks of maximum degree $Δ$? 2. What is the shortest labeling scheme that permits diameter discovery in all radio networks? We show that the minimum length of a labeling scheme that permits size discovery is $Θ(\log\log Δ)$. By contrast, we show that diameter discovery can be done using a labeling scheme of constant length.

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arXiv2020

Four Shades of Deterministic Leader Election in Anonymous Networks

Leader election is one of the fundamental problems in distributed computing: a single node, called the leader, must be specified. This task can be formulated either in a weak way, where one node outputs 'leader' and all other nodes output 'non-leader', or in a strong way, where all nodes must also learn which node is the leader. If the nodes of the network have distinct identifiers, then such an agreement means that all nodes have to output the identifier of the elected leader. For anonymous networks, the strong version of leader election requires that all nodes must be able to find a path to the leader, as this is the only way to identify it. For any network in which leader election (weak or strong) is possible knowing the map of the network, there is a minimum time in which this can be done. We consider four formulations of leader election discussed in the literature in the context of anonymous networks : one is the weak formulation, and the three others specify three different ways of finding the path to the leader in the strong formulation. Our aim is to compare the amount of initial information needed to accomplish each of these "four shades" of leader election in minimum time. We show that the amount of information required to accomplish leader election in the weak formulation in minimum time is exponentially smaller than that needed for any of the strong formulations. Thus, if the required amount of advice is used as a measure of the difficulty of the task, the weakest version of leader election in minimum time is drastically easier than any version of the strong formulation in minimum time.

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arXiv2020

Deterministic Treasure Hunt in the Plane with Angular Hints

A mobile agent equipped with a compass and a measure of length has to find an inert treasure in the Euclidean plane. Both the agent and the treasure are modeled as points. In the beginning, the agent is at a distance at most $D>0$ from the treasure, but knows neither the distance nor any bound on it. Finding the treasure means getting at distance at most 1 from it. The agent makes a series of moves. Each of them consists in moving straight in a chosen direction at a chosen distance. In the beginning and after each move the agent gets a hint consisting of a positive angle smaller than $2π$ whose vertex is at the current position of the agent and within which the treasure is contained. We investigate the problem of how these hints permit the agent to lower the cost of finding the treasure, using a deterministic algorithm, where the cost is the worst-case total length of the agent's trajectory. It is well known that without any hint the optimal (worst case) cost is $Θ(D^2)$. We show that if all angles given as hints are at most $π$, then the cost can be lowered to $O(D)$, which is optimal. If all angles are at most $β$, where $β<2π$ is a constant unknown to the agent, then the cost is at most $O(D^{2-ε})$, for some $ε>0$. For both these positive results we present deterministic algorithms achieving the above costs. Finally, if angles given as hints can be arbitrary, smaller than $2π$, then we show that cost $Θ(D^2)$ cannot be beaten.

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