arXiv subjects

Andrzej Pelc

Publications and source records attributed to Andrzej Pelc.

At least 37 records · Page 2Linked to original sources

arXiv2020

Almost Universal Anonymous Rendezvous in the Plane

Two mobile agents represented by points freely moving in the plane and starting at two distinct positions, have to meet. The meeting, called rendezvous, occurs when agents are at distance at most $r$ of each other and never move after this time, where $r$ is a positive real unknown to them, called the visibility radius. Agents are anonymous and execute the same deterministic algorithm. Each agent has a set of private attributes, some or all of which can differ between agents. These attributes are: the initial position of the agent, its system of coordinates (orientation and chirality), the rate of its clock, its speed when it moves, and the time of its wake-up. If all attributes (except the initial positions) are identical and agents start at distance larger than $r$ then they can never meet. However, differences between attributes make it sometimes possible to break the symmetry and accomplish rendezvous. Such instances of the rendezvous problem (formalized as lists of attributes), are called feasible. Our contribution is three-fold. We first give an exact characterization of feasible instances. Thus it is natural to ask whether there exists a single algorithm that guarantees rendezvous for all these instances. We give a strong negative answer to this question: we show two sets $S_1$ and $S_2$ of feasible instances such that none of them admits a single rendezvous algorithm valid for all instances of the set. On the other hand, we construct a single algorithm that guarantees rendezvous for all feasible instances outside of sets $S_1$ and $S_2$. We observe that these exception sets $S_1$ and $S_2$ are geometrically very small, compared to the set of all feasible instances: they are included in low-dimension subspaces of the latter. Thus, our rendezvous algorithm handling all feasible instances other than these small sets of exceptions can be justly called almost universal.

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arXiv2020

Deterministic Leader Election in Anonymous Radio Networks

We consider leader election in anonymous radio networks modeled as simple undirected connected graphs. Nodes communicate in synchronous rounds. Nodes are anonymous and execute the same deterministic algorithm, so symmetry can be broken only in one way: by different wake-up times of the nodes. In which situations is it possible to break symmetry and elect a leader using time as symmetry breaker? To answer this question, we consider configurations. A configuration is the underlying graph with nodes tagged by non-negative integers with the following meaning. A node can either wake up spontaneously in the round shown on its tag, according to some global clock, or can be woken up hearing a message sent by one of its already awoken neighbours. The local clock of a node starts at its wakeup and nodes do not have access to the global clock determining their tags. A configuration is feasible if there exists a distributed algorithm that elects a leader for this configuration. Our main result is a complete algorithmic characterization of feasible configurations: we design a centralized decision algorithm, working in polynomial time, whose input is a configuration and which decides if the configuration is feasible. We also provide a dedicated deterministic distributed leader election algorithm for each feasible configuration that elects a leader for this configuration in time $O(n^2σ)$, where $n$ is the number of nodes and $σ$ is the difference between the largest and smallest tag of the configuration. We then prove that there cannot exist a universal deterministic distributed algorithm electing a leader for all feasible configurations. In fact, we show that such a universal algorithm cannot exist even for the class of 4-node feasible configurations. We also prove that a distributed version of our decision algorithm cannot exist.

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arXiv2020

Advice Complexity of Treasure Hunt in Geometric Terrains

Treasure hunt is the task of finding an inert target by a mobile agent in an unknown environment. We consider treasure hunt in geometric terrains with obstacles. Both the terrain and the obstacles are modeled as polygons and both the agent and the treasure are modeled as points. The agent navigates in the terrain, avoiding obstacles, and finds the treasure when there is a segment of length at most 1 between them, unobstructed by the boundary of the terrain or by the obstacles. The cost of finding the treasure is the length of the trajectory of the agent. We investigate the amount of information that the agent needs a priori in order to find the treasure at cost $O(L)$, where $L$ is the length of the shortest path in the terrain from the initial position of the agent to the treasure, avoiding obstacles. Following the paradigm of algorithms with advice, this information is given to the agent in advance as a binary string, by an oracle cooperating with the agent and knowing the whole environment: the terrain, the position of the treasure and the initial position of the agent. Advice complexity of treasure hunt is the minimum length of the advice string (up to multiplicative constants) that enables the agent to find the treasure at cost $O(L)$. We first consider treasure hunt in regular terrains which are defined as convex polygons with convex $c$-fat obstacles, for some constant $c>1$. A polygon is $c$-fat if the ratio of the radius of the smallest disc containing it to the radius of the largest disc contained in it is at most $c$. For the class of regular terrains, we establish the exact advice complexity of treasure hunt. We then show that advice complexity of treasure hunt for the class of arbitrary terrains (even for non-convex polygons without obstacles, and even for those with only horizontal or vertical sides) is exponentially larger than for regular terrains.

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arXiv2019

Want to Gather? No Need to Chatter!

A team of mobile agents, starting from different nodes of an unknown network, possibly at different times, have to meet at the same node and declare that they have all met. Agents have different labels and move in synchronous rounds along links of the network. The above task is known as gathering and was traditionally considered under the assumption that when some agents are at the same node then they can talk. In this paper we ask the question of whether this ability of talking is needed for gathering. The answer turns out to be no. Our main contribution are two deterministic algorithms that always accomplish gathering in a much weaker model. We only assume that at any time an agent knows how many agents are at the node that it currently occupies but agents do not see the labels of other co-located agents and cannot exchange any information with them. They also do not see other nodes than the current one. Our first algorithm works under the assumption that agents know a priori some upper bound N on the network size, and it works in time polynomial in N and in the length l of the smallest label. Our second algorithm does not assume any a priori knowledge about the network but its complexity is exponential in the network size and in the labels of agents. Its purpose is to show feasibility of gathering under this harsher scenario. As a by-product of our techniques we obtain, in the same weak model, the solution of the fundamental problem of leader election among agents. As an application of our result we also solve, in the same model, the well-known gossiping problem: if each agent has a message at the beginning, we show how to make all messages known to all agents, even without any a priori knowledge about the network. If agents know an upper bound N on the network size then our gossiping algorithm works in time polynomial in N, in l and in the length of the largest message.

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arXiv2019

Deterministic Rendezvous at a Node of Agents with Arbitrary Velocities

We consider the task of rendezvous in networks modeled as undirected graphs. Two mobile agents with different labels, starting at different nodes of an anonymous graph, have to meet. This task has been considered in the literature under two alternative scenarios: weak and strong. Under the weak scenario, agents may meet either at a node or inside an edge. Under the strong scenario, they have to meet at a node, and they do not even notice meetings inside an edge. Rendezvous algorithms under the strong scenario are known for synchronous agents. For asynchronous agents, rendezvous under the strong scenario is impossible even in the two-node graph, and hence only algorithms under the weak scenario were constructed. In this paper we show that rendezvous under the strong scenario is possible for agents with restricted asynchrony: agents have the same measure of time but the adversary can arbitrarily impose the speed of traversing each edge by each of the agents. We construct a deterministic rendezvous algorithm for such agents, working in time polynomial in the size of the graph, in the length of the smaller label, and in the largest edge traversal time.

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arXiv2019

Constant-Length Labeling Schemes for Deterministic Radio Broadcast

Broadcast is one of the fundamental network communication primitives. One node of a network, called the $\mathit{source}$, has a message that has to be learned by all other nodes. We consider the feasibility of deterministic broadcast in radio networks. If nodes of the network do not have any labels, deterministic broadcast is impossible even in the four-cycle. On the other hand, if all nodes have distinct labels, then broadcast can be carried out, e.g., in a round-robin fashion, and hence $O(\log n)$-bit labels are sufficient for this task in $n$-node networks. In fact, $O(\log Δ)$-bit labels, where $Δ$ is the maximum degree, are enough to broadcast successfully. Hence, it is natural to ask if very short labels are sufficient for broadcast. Our main result is a positive answer to this question. We show that every radio network can be labeled using 2 bits in such a way that broadcast can be accomplished by some universal deterministic algorithm that does not know the network topology nor any bound on its size. Moreover, at the expense of an extra bit in the labels, we get the additional strong property that there exists a common round in which all nodes know that broadcast has been completed. Finally, we show that 3-bit labels are also sufficient to solve both versions of broadcast in the case where the labeling scheme does not know which node is the source.

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arXiv2019

Building a Nest by an Automaton

A robot modeled as a deterministic finite automaton has to build a structure from material available to it. The robot navigates in the infinite oriented grid $\mathbb{Z} \times \mathbb{Z}$. Some cells of the grid are full (contain a brick) and others are empty. The subgraph of the grid induced by full cells, called the field, is initially connected. The (Manhattan) distance between the farthest cells of the field is called its span. The robot starts at a full cell. It can carry at most one brick at a time. At each step it can pick a brick from a full cell, move to an adjacent cell and drop a brick at an empty cell. The aim of the robot is to construct the most compact possible structure composed of all bricks, i.e., a nest. That is, the robot has to move all bricks in such a way that the span of the resulting field be the smallest. Our main result is the design of a deterministic finite automaton that accomplishes this task and subsequently stops, for every initially connected field, in time $O(sz)$, where $s$ is the span of the initial field and $z$ is the number of bricks. We show that this complexity is optimal.

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arXiv2019

Cost vs. Information Tradeoffs for Treasure Hunt in the Plane

A mobile agent has to find an inert treasure hidden in the plane. Both the agent and the treasure are modeled as points. This is a variant of the task known as treasure hunt. The treasure is at a distance at most $D$ from the initial position of the agent, and the agent finds the treasure when it gets at distance $r$ from it, called the {\em vision radius}. However, the agent does not know the location of the treasure and does not know the parameters $D$ and $r$. The cost of finding the treasure is the length of the trajectory of the agent. We investigate the tradeoffs between the amount of information held {\em a priori} by the agent and the cost of treasure hunt. Following the well-established paradigm of {\em algorithms with advice}, this information is given to the agent in advance as a binary string, by an oracle cooperating with the agent and knowing the location of the treasure and the initial position of the agent. The size of advice given to the agent is the length of this binary string. For any size $z$ of advice and any $D$ and $r$, let $OPT(z,D,r)$ be the optimal cost of finding the treasure for parameters $z$, $D$ and $r$, if the agent has only an advice string of length $z$ as input. We design treasure hunt algorithms working with advice of size $z$ at cost $O(OPT(z,D,r))$ whenever $r\leq 1$ or $r\geq 0.9D$. For intermediate values of $r$, i.e., $1 0$, the treasure can be found at cost $O(OPT(z,D,r)^{1+α})$.

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arXiv2018

Latecomers Help to Meet: Deterministic Anonymous Gathering in the Plane

A team of anonymous mobile agents represented by points freely moving in the plane have to gather at a single point and stop. Agents start at different points of the plane and at possibly different times chosen by the adversary. They are equipped with compasses, a common unit of distance and clocks. They execute the same deterministic algorithm and travel at speed 1. When agents are at distance at most $ε$, for some positive constant $ε$ unknown to them, they can exchange all information. Due to the anonymity of the agents and the symmetry of the plane, gathering is impossible, e.g., if agents start simultaneously at distances larger than $ε$. However, if some agents start with a delay with respect to others, gathering may become possible. In which situations such latecomers can enable gathering? To answer this question we consider initial configurations formalized as sets of pairs $\{(p_1,t_1), (p_2,t_2),\dots , (p_n,t_n)\}$, for $n\geq 2$ where $p_i$ is the starting point of the $i$-th agent and $t_i$ is its starting time. An initial configuration is gatherable if agents starting at it can be gathered by some algorithm, even dedicated to this particular configuration. We characterize all gatherable initial configurations. Is there a universal deterministic algorithm that can gather all gatherable configurations of a given size. It turns out that the answer is no. We show that all gatherable configurations can be partitioned into two sets: bad and good configurations. We show that bad gatherable configurations (even of size 2) cannot be gathered by a common gathering algorithm, and we prove that there is a universal algorithm that gathers all good configurations of a given size.

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arXiv2018

Using Time to Break Symmetry: Universal Deterministic Anonymous Rendezvous

Two anonymous mobile agents navigate synchronously in an anonymous graph and have to meet at a node, using a deterministic algorithm. This is a symmetry breaking task called rendezvous, equivalent to the fundamental task of leader election between the agents. When is this feasible in a completely anonymous environment? It is known that agents can always meet if their initial positions are nonsymmetric, and that if they are symmetric and agents start simultaneously then rendezvous is impossible. What happens for symmetric initial positions with non-simultaneous start? Can symmetry between the agents be broken by the delay between their starting times? In order to answer these questions, we consider {\em space-time initial configurations} (abbreviated by STIC). A STIC is formalized as $[(u,v),δ]$, where $u$ and $v$ are initial nodes of the agents in some graph and $δ$ is a non-negative integer that represents the difference between their starting times. A STIC is {\em feasible} if there exists a deterministic algorithm, even dedicated to this particular STIC, which accomplishes rendezvous for it. Our main result is a characterization of all feasible STICs and the design of a universal deterministic algorithm that accomplishes rendezvous for all of them without {\em any } a priori knowledge of the agents. Thus, as far as feasibility is concerned, we completely solve the problem of symmetry breaking between two anonymous agents in anonymous graphs. Moreover, we show that such a universal algorithm cannot work for all feasible STICs in time polynomial in the initial distance between the agents.

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arXiv2018

Deterministic Computations on a PRAM with Static Processor and Memory Faults

We consider Parallel Random Access Machine (PRAM) which has some processors and memory cells faulty. The faults considered are static, i.e., once the machine starts to operate, the operational/faulty status of PRAM components does not change. We develop a deterministic simulation of a fully operational PRAM on a similar faulty machine which has constant fractions of faults among processors and memory cells. The simulating PRAM has $n$ processors and $m$ memory cells, and simulates a PRAM with $n$ processors and a constant fraction of $m$ memory cells. The simulation is in two phases: it starts with preprocessing, which is followed by the simulation proper performed in a step-by-step fashion. Preprocessing is performed in time $O((\frac{m}{n}+ \log n)\log n)$. The slowdown of a step-by-step part of the simulation is $O(\log m)$.

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arXiv2017

Reaching a Target in the Plane with no Information

A mobile agent has to reach a target in the Euclidean plane. Both the agent and the target are modeled as points. In the beginning, the agent is at distance at most $D>0$ from the target. Reaching the target means that the agent gets at a {\em sensing distance} at most $r>0$ from it. The agent has a measure of length and a compass. We consider two scenarios: in the {\em static} scenario the target is inert, and in the {\em dynamic} scenario it may move arbitrarily at any (possibly varying) speed bounded by $v$. The agent has no information about the parameters of the problem, in particular it does not know $D$, $r$ or $v$. The goal is to reach the target at lowest possible cost, measured by the total length of the trajectory of the agent. Our main result is establishing the minimum cost (up to multiplicative constants) of reaching the target under both scenarios, and providing the optimal algorithm for the agent. For the static scenario the minimum cost is $Θ((\log D + \log \frac{1}{r}) D^2/r)$, and for the dynamic scenario it is $Θ((\log M + \log \frac{1}{r}) M^2/r)$, where $M=\max(D,v)$. Under the latter scenario, the speed of the agent in our algorithm grows exponentially with time, and we prove that for an agent whose speed grows only polynomially with time, this cost is impossible to achieve.

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arXiv2017

Deterministic rendezvous with detection using beeps

Two mobile agents, starting at arbitrary, possibly different times from arbitrary nodes of an unknown network, have to meet at some node. Agents move in synchronous rounds: in each round an agent can either stay at the current node or move to one of its neighbors. Agents have different labels which are positive integers. Each agent knows its own label, but not the label of the other agent. In traditional formulations of the rendezvous problem, meeting is accomplished when the agents get to the same node in the same round. We want to achieve a more demanding goal, called rendezvous with detection: agents must become aware that the meeting is accomplished, simultaneously declare this and stop. This awareness depends on how an agent can communicate to the other agent its presence at a node. We use two variations of the arguably weakest model of communication, called the beeping model, introduced in [8]. In each round an agent can either listen or beep. In the local beeping model, an agent hears a beep in a round if it listens in this round and if the other agent is at the same node and beeps. In the global beeping model, an agent hears a loud beep in a round if it listens in this round and if the other agent is at the same node and beeps, and it hears a soft beep in a round if it listens in this round and if the other agent is at some other node and beeps. We first present a deterministic algorithm of rendezvous with detection working, even for the local beeping model, in an arbitrary unknown network in time polynomial in the size of the network and in the length of the smaller label (i.e., in the logarithm of this label). However, in this algorithm, agents spend a lot of energy: the number of moves that an agent must make, is proportional to the time of rendezvous. It is thus natural to ask if bounded-energy agents can always achieve rendezvous with detection as well...

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arXiv2017

Deterministic meeting of sniffing agents in the plane

Two mobile agents, starting at arbitrary, possibly different times from arbitrary locations in the plane, have to meet. Agents are modeled as discs of diameter 1, and meeting occurs when these discs touch. Agents have different labels which are integers from the set of 0 to L-1. Each agent knows L and knows its own label, but not the label of the other agent. Agents are equipped with compasses and have synchronized clocks. They make a series of moves. Each move specifies the direction and the duration of moving. This includes a null move which consists in staying inert for some time, or forever. In a non-null move agents travel at the same constant speed, normalized to 1. We assume that agents have sensors enabling them to estimate the distance from the other agent (defined as the distance between centers of discs), but not the direction towards it. We consider two models of estimation. In both models an agent reads its sensor at the moment of its appearance in the plane and then at the end of each move. This reading (together with the previous ones) determines the decision concerning the next move. In both models the reading of the sensor tells the agent if the other agent is already present. Moreover, in the monotone model, each agent can find out, for any two readings in moments t1 and t2, whether the distance from the other agent at time t1 was smaller, equal or larger than at time t2. In the weaker binary model, each agent can find out, at any reading, whether it is at distance less than \r{ho} or at distance at least \r{ho} from the other agent, for some real \r{ho} > 1 unknown to them. Such distance estimation mechanism can be implemented, e.g., using chemical sensors. Each agent emits some chemical substance (scent), and the sensor of the other agent detects it, i.e., sniffs. The intensity of the scent decreases with the distance.

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arXiv2017

Use of Information, Memory and Randomization in Asynchronous Gathering

We investigate initial information, unbounded memory and randomization in gathering mobile agents on a grid. We construct a state machine, such that it is possible to gather, with probability 1, all configurations of its copies. This machine has initial input, unbounded memory, and is randomized. We show that no machine having any two of these capabilities but not the third, can be used to gather, with high probability, all configurations. We construct deterministic Turing Machines that are used to gather all connected configurations, and we construct deterministic finite automata that are used to gather all contractible connected configurations.

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arXiv2017

Deterministic Distributed Construction of $T$-Dominating Sets in Time $T$

A $k$-dominating set is a set $D$ of nodes of a graph such that, for each node $v$, there exists a node $w \in D$ at distance at most $k$ from $v$. Our aim is the deterministic distributed construction of small $T$-dominating sets in time $T$ in networks modeled as undirected $n$-node graphs and under the $\cal{LOCAL}$ communication model. For any positive integer $T$, if $b$ is the size of a pairwise disjoint collection of balls of radii at least $T$ in a graph, then $b$ is an obvious lower bound on the size of a $T$-dominating set. Our first result shows that, even on rings, it is impossible to construct a $T$-dominating set of size $s$ asymptotically $b$ (i.e., such that $s/b \rightarrow 1$) in time $T$. In the range of time $T \in Θ(\log^* n)$, the size of a $T$-dominating set turns out to be very sensitive to multiplicative constants in running time. Indeed, it follows from \cite{KP}, that for time $T=γ\log^* n$ with large constant $γ$, it is possible to construct a $T$-dominating set whose size is a small fraction of $n$. By contrast, we show that, for time $T=α\log^* n $ for small constant $α$, the size of a $T$-dominating set must be a large fraction of $n$. Finally, when $T \in o (\log^* n)$, the above lower bound implies that, for any constant $x<1$, it is impossible to construct a $T$-dominating set of size smaller than $xn$, even on rings. On the positive side, we provide an algorithm that constructs a $T$-dominating set of size $n- Θ(T)$ on all graphs.

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arXiv2017

Deterministic Gathering with Crash Faults

A team consisting of an unknown number of mobile agents, starting from different nodes of an unknown network, have to meet at the same node and terminate. This problem is known as {\em gathering}. We study deterministic gathering algorithms under the assumption that agents are subject to {\em crash faults} which can occur at any time. Two fault scenarios are considered. A {\em motion fault} immobilizes the agent at a node or inside an edge but leaves intact its memory at the time when the fault occurred. A more severe {\em total fault} immobilizes the agent as well, but also erases its entire memory. Of course, we cannot require faulty agents to gather. Thus the gathering problem for fault prone agents calls for all fault-free agents to gather at a single node, and terminate. When agents move completely asynchronously, gathering with crash faults of any type is impossible. Hence we consider a restricted version of asynchrony, where each agent is assigned by the adversary a fixed speed, possibly different for each agent. Agents have clocks ticking at the same rate. Each agent can wait for a time of its choice at any node, or decide to traverse an edge but then it moves at constant speed assigned to it. Moreover, agents have different labels. Each agent knows its label and speed but not those of other agents. We construct a gathering algorithm working for any team of at least two agents in the scenario of motion faults, and a gathering algorithm working in the presence of total faults, provided that at least two agents are fault free all the time. If only one agent is fault free, the task of gathering with total faults is sometimes impossible. Both our algorithms work in time polynomial in the size of the graph, in the logarithm of the largest label, in the inverse of the smallest speed, and in the ratio between the largest and the smallest speed.

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arXiv2017

Short Labeling Schemes for Topology Recognition in Wireless Tree Networks

We consider the problem of topology recognition in wireless (radio) networks modeled as undirected graphs. Topology recognition is a fundamental task in which every node of the network has to output a map of the underlying graph i.e., an isomorphic copy of it, and situate itself in this map. In wireless networks, 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 given round, if $v$ listens in this round, and if $w$ is its only neighbor that transmits in this round. Nodes have labels which are (not necessarily different) binary strings. The length of a labeling scheme is the largest length of a label. We concentrate on wireless networks modeled by trees, and we investigate two problems. \begin{itemize} \item What is the shortest labeling scheme that permits topology recognition in all wireless tree networks of diameter $D$ and maximum degree $Δ$? \item What is the fastest topology recognition algorithm working for all wireless tree networks of diameter $D$ and maximum degree $Δ$, using such a short labeling scheme? \end{itemize} We are interested in deterministic topology recognition algorithms. For the first problem, we show that the minimum length of a labeling scheme allowing topology recognition in all trees of maximum degree $Δ\geq 3$ is $Θ(\log\log Δ)$. For such short schemes, used by an algorithm working for the class of trees of diameter $D\geq 4$ and maximum degree $Δ\geq 3$, we show almost matching bounds on the time of topology recognition: an upper bound $O(DΔ)$, and a lower bound $Ω(DΔ^ε)$, for any constant $ε<1$.

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