arXiv subjects

Jurek Czyzowicz

Publications and source records attributed to Jurek Czyzowicz.

At least 19 recordsLinked to original sources

arXiv2021

Group Evacuation on a Line by Agents with Different Communication Abilities

We consider evacuation of a group of $n \geq 2$ autonomous mobile agents (or robots) from an unknown exit on an infinite line. The agents are initially placed at the origin of the line and can move with any speed up to the maximum speed $1$ in any direction they wish and they all can communicate when they are co-located. However, the agents have different wireless communication abilities: while some are fully wireless and can send and receive messages at any distance, a subset of the agents are senders, they can only transmit messages wirelessly, and the rest are receivers, they can only receive messages wirelessly. The agents start at the same time and their communication abilities are known to each other from the start. Starting at the origin of the line, the goal of the agents is to collectively find a target/exit at an unknown location on the line while minimizing the evacuation time, defined as the time when the last agent reaches the target. We investigate the impact of such a mixed communication model on evacuation time on an infinite line for a group of cooperating agents. In particular, we provide evacuation algorithms and analyze the resulting competitive ratio ($CR$) of the evacuation time for such a group of agents. If the group has two agents of two different types, we give an optimal evacuation algorithm with competitive ratio $CR=3+2 \sqrt{2}$. If there is a single sender or fully wireless agent, and multiple receivers we prove that $CR \in [2+\sqrt{5},5]$, and if there are multiple senders and a single receiver or fully wireless agent, we show that $CR \in [3,5.681319]$. Any group consisting of only senders or only receivers requires competitive ratio 9, and any other combination of agents has competitive ratio 3.

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arXiv2020

Evacuating Robots from a Disk Using Face-to-Face Communication

Assume that two robots are located at the centre of a unit disk. Their goal is to evacuate from the disk through an exit at an unknown location on the boundary of the disk. At any time the robots can move anywhere they choose on the disk, independently of each other, with maximum speed $1$. The robots can cooperate by exchanging information whenever they meet. We study algorithms for the two robots to minimize the evacuation time: the time when both robots reach the exit. In [CGGKMP14] the authors gave an algorithm defining trajectories for the two robots yielding evacuation time at most $5.740$ and also proved that any algorithm has evacuation time at least $3+ \fracπ{4} + \sqrt{2} \approx 5.199$. We improve both the upper and lower bound on the evacuation time of a unit disk. Namely, we present a new non-trivial algorithm whose evacuation time is at most $5.628$ and show that any algorithm has evacuation time at least $3+ \fracπ{6} + \sqrt{3} \approx 5.255$. To achieve the upper bound, we designed an algorithm which proposes a forced meeting between the two robots, even if the exit has not been found by either of them. We also show that such a strategy is provably optimal for a related problem of searching for an exit placed at the vertices of a regular hexagon.

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arXiv2020

The Bike Sharing Problem

Assume that $m \geq 1$ autonomous mobile agents and $0 \leq b \leq m$ single-agent transportation devices (called {\em bikes}) are initially placed at the left endpoint $0$ of the unit interval $[0,1]$. The agents are identical in capability and can move at speed one. The bikes cannot move on their own, but any agent riding bike $i$ can move at speed $v_i > 1$. An agent may ride at most one bike at a time. The agents can cooperate by sharing the bikes; an agent can ride a bike for a time, then drop it to be used by another agent, and possibly switch to a different bike. We study two problems. In the \BS problem, we require all agents and bikes starting at the left endpoint of the interval to reach the end of the interval as soon as possible. In the \RBS problem, we aim to minimize the arrival time of the agents; the bikes can be used to increase the average speed of the agents, but are not required to reach the end of the interval. Our main result is the construction of a polynomial time algorithm for the \BS problem that creates an arrival-time optimal schedule for travellers and bikes to travel across the interval. For the \RBS problem, we give an algorithm that gives an optimal solution for the case when at most one of the bikes can be abandoned.

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arXiv2019

Time-Energy Tradeoffs for Evacuation by Two Robots in the Wireless Model

Two robots stand at the origin of the infinite line and are tasked with searching collaboratively for an exit at an unknown location on the line. They can travel at maximum speed $b$ and can change speed or direction at any time. The two robots can communicate with each other at any distance and at any time. The task is completed when the last robot arrives at the exit and evacuates. We study time-energy tradeoffs for the above evacuation problem. The evacuation time is the time it takes the last robot to reach the exit. The energy it takes for a robot to travel a distance $x$ at speed $s$ is measured as $xs^2$. The total and makespan evacuation energies are respectively the sum and maximum of the energy consumption of the two robots while executing the evacuation algorithm. Assuming that the maximum speed is $b$, and the evacuation time is at most $cd$, where $d$ is the distance of the exit from the origin, we study the problem of minimizing the total energy consumption of the robots. We prove that the problem is solvable only for $bc \geq 3$. For the case $bc=3$, we give an optimal algorithm, and give upper bounds on the energy for the case $bc>3$. We also consider the problem of minimizing the evacuation time when the available energy is bounded by $Δ$. Surprisingly, when $Δ$ is a constant, independent of the distance $d$ of the exit from the origin, we prove that evacuation is possible in time $O(d^{3/2}\log d)$, and this is optimal up to a logarithmic factor. When $Δ$ is linear in $d$, we give upper bounds on the evacuation time.

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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

Energy Consumption of Group Search on a Line

Consider two robots that start at the origin of the infinite line in search of an exit at an unknown location on the line. The robots can only communicate if they arrive at the same location at exactly the same time, i.e. they use the so-called face-to-face communication model. The group search time is defined as the worst-case time as a function of $d$, the distance of the exit from the origin, when both robots can reach the exit. It has long been known that for a single robot traveling at unit speed, the search time is at least $9d-o(d)$. It was shown recently that $k\geq2$ robots traveling at unit speed also require at least $9d$ group search time. We investigate energy-time trade-offs in group search by two robots, where the energy loss experienced by a robot traveling a distance $x$ at constant speed $s$ is given by $s^2 x$. Specifically, we consider the problem of minimizing the total energy used by the robots, under the constraints that the search time is at most a multiple $c$ of the distance $d$ and the speed of the robots is bounded by $b$. Motivation for this study is that for the case when robots must complete the search in $9d$ time with maximum speed one, a single robot requires at least $9d$ energy, while for two robots, all previously proposed algorithms consume at least $28d/3$ energy. When the robots have bounded memory, we generalize existing algorithms to obtain a family of optimal (and in some cases nearly optimal) algorithms parametrized by pairs of $b,c$ values that can solve the problem for the entire spectrum of these pairs for which the problem is solvable. We also propose a novel search algorithm, with unbounded memory, that simultaneously achieves search time $9d$ and consumes energy $8.42588d$. Our result shows that two robots can search on the line in optimal time $9d$ while consuming less total energy than a single robot within the same search time.

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arXiv2018

God Save the Queen

Queen Daniela of Sardinia is asleep at the center of a round room at the top of the tower in her castle. She is accompanied by her faithful servant, Eva. Suddenly, they are awakened by cries of "Fire". The room is pitch black and they are disoriented. There is exactly one exit from the room somewhere along its boundary. They must find it as quickly as possible in order to save the life of the queen. It is known that with two people searching while moving at maximum speed 1 anywhere in the room, the room can be evacuated (i.e., with both people exiting) in $1 + \frac{2π}{3} + \sqrt{3} \approx 4.8264$ time units and this is optimal~[Czyzowicz et al., DISC'14], assuming that the first person to find the exit can directly guide the other person to the exit using her voice. Somewhat surprisingly, in this paper we show that if the goal is to save the queen (possibly leaving Eva behind to die in the fire) there is a slightly better strategy. We prove that this "priority" version of evacuation can be solved in time at most $4.81854$. Furthermore, we show that any strategy for saving the queen requires time at least $3 + π/6 + \sqrt{3}/2 \approx 4.3896$ in the worst case. If one or both of the queen's other servants (Biddy and/or Lili) are with her, we show that the time bounds can be improved to $3.8327$ for two servants, and $3.3738$ for three servants. Finally we show lower bounds for these cases of $3.6307$ (two servants) and $3.2017$ (three servants). The case of $n\geq 4$ is the subject of an independent study by Queen Daniela's Royal Scientific Team.

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arXiv2017

Gathering in the Plane of Location-Aware Robots in the Presence of Spies

A set of mobile robots (represented as points) is distributed in the Cartesian plane. The collection contains an unknown subset of byzantine robots which are indistinguishable from the reliable ones. The reliable robots need to gather, i.e., arrive to a configuration in which at the same time, all of them occupy the same point on the plane. The robots are equipped with GPS devices and at the beginning of the gathering process they communicate the Cartesian coordinates of their respective positions to the central authority. On the basis of this information, without the knowledge of which robots are faulty, the central authority designs a trajectory for every robot. The central authority aims to provide the trajectories which result in the shortest possible gathering time of the healthy robots. The efficiency of a gathering strategy is measured by its competitive ratio, i.e., the maximal ratio between the time required for gathering achieved by the given trajectories and the optimal time required for gathering in the offline case, i.e., when the faulty robots are known to the central authority in advance. The role of the byzantine robots, controlled by the adversary, is to act so that the gathering is delayed and the resulting competitive ratio is maximized. The objective of our paper is to propose efficient algorithms when the central authority is aware of an upper bound on the number of byzantine robots. We give optimal algorithms for collections of robots known to contain at most one faulty robot. When the proportion of byzantine robots is known to be less than one half or one third, we provide algorithms with small constant competitive ratios. We also propose algorithms with bounded competitive ratio in the case where the proportion of faulty robots is arbitrary.

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arXiv2017

Patrolling a Path Connecting a Set of Points with Unbalanced Frequencies of Visits

Patrolling consists of scheduling perpetual movements of a collection of mobile robots, so that each point of the environment is regularly revisited by any robot in the collection. In previous research, it was assumed that all points of the environment needed to be revisited with the same minimal frequency. In this paper we study efficient patrolling protocols for points located on a path, where each point may have a different constraint on frequency of visits. The problem of visiting such divergent points was recently posed by Gasieniec et al. in [13], where the authors study protocols using a single robot patrolling a set of $n$ points located in nodes of a complete graph and in Euclidean spaces. The focus in this paper is on patrolling with two robots. We adopt a scenario in which all points to be patrolled are located on a line. We provide several approximation algorithms concluding with the best currently known $\sqrt 3$-approximation.

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arXiv2017

Exploring Graphs with Time Constraints by Unreliable Collections of Mobile Robots

A graph environment must be explored by a collection of mobile robots. Some of the robots, a priori unknown, may turn out to be unreliable. The graph is weighted and each node is assigned a deadline. The exploration is successful if each node of the graph is visited before its deadline by a reliable robot. The edge weight corresponds to the time needed by a robot to traverse the edge. Given the number of robots which may crash, is it possible to design an algorithm, which will always guarantee the exploration, independently of the choice of the subset of unreliable robots by the adversary? We find the optimal time, during which the graph may be explored. Our approach permits to find the maximal number of robots, which may turn out to be unreliable, and the graph is still guaranteed to be explored. We concentrate on line graphs and rings, for which we give positive results. We start with the case of the collections involving only reliable robots. We give algorithms finding optimal times needed for exploration when the robots are assigned to fixed initial positions as well as when such starting positions may be determined by the algorithm. We extend our consideration to the case when some number of robots may be unreliable. Our most surprising result is that solving the line exploration problem with robots at given positions, which may involve crash-faulty ones, is NP-hard. The same problem has polynomial solutions for a ring and for the case when the initial robots' positions on the line are arbitrary. The exploration problem is shown to be NP-hard for star graphs, even when the team consists of only two reliable robots.

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arXiv2017

Rendezvous on a Line by Location-Aware Robots Despite the Presence of Byzantine Faults

A set of mobile robots is placed at points of an infinite line. The robots are equipped with GPS devices and they may communicate their positions on the line to a central authority. The collection contains an unknown subset of "spies", i.e., byzantine robots, which are indistinguishable from the non-faulty ones. The set of the non-faulty robots need to rendezvous in the shortest possible time in order to perform some task, while the byzantine robots may try to delay their rendezvous for as long as possible. The problem facing a central authority is to determine trajectories for all robots so as to minimize the time until the non-faulty robots have rendezvoused. The trajectories must be determined without knowledge of which robots are faulty. Our goal is to minimize the competitive ratio between the time required to achieve the first rendezvous of the non-faulty robots and the time required for such a rendezvous to occur under the assumption that the faulty robots are known at the start. We provide a bounded competitive ratio algorithm, where the central authority is informed only of the set of initial robot positions, without knowing which ones or how many of them are faulty. When an upper bound on the number of byzantine robots is known to the central authority, we provide algorithms with better competitive ratios. In some instances we are able to show these algorithms are optimal.

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arXiv2017

Temporal flows in Temporal networks

We introduce temporal flows on temporal networks, i.e., networks the links of which exist only at certain moments of time. Such networks are ephemeral in the sense that no link exists after some time. Our flow model is new and differs from the "flows over time" model, also called "dynamic flows" in the literature. We show that the problem of finding the maximum amount of flow that can pass from a source vertex s to a sink vertex t up to a given time is solvable in Polynomial time, even when node buffers are bounded. We then examine mainly the case of unbounded node buffers. We provide a simplified static Time-Extended network (STEG), which is of polynomial size to the input and whose static flow rates are equivalent to the respective temporal flow of the temporal network, using STEG, we prove that the maximum temporal flow is equal to the minimum temporal s-t cut. We further show that temporal flows can always be decomposed into flows, each of which moves only through a journey, i.e., a directed path whose successive edges have strictly increasing moments of existence. We partially characterise networks with random edge availabilities that tend to eliminate the s-t temporal flow. We then consider mixed temporal networks, which have some edges with specified availabilities and some edges with random availabilities, we show that it is #P-hard to compute the tails and expectations of the maximum temporal flow (which is now a random variable) in a mixed temporal network.

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arXiv2017

Linear Search with Terrain-Dependent Speeds

We revisit the linear search problem where a robot, initially placed at the origin on an infinite line, tries to locate a stationary target placed at an unknown position on the line. Unlike previous studies, in which the robot travels along the line at a constant speed, we consider settings where the robot's speed can depend on the direction of travel along the line, or on the profile of the terrain, e.g. when the line is inclined, and the robot can accelerate. Our objective is to design search algorithms that achieve good competitive ratios for the time spent by the robot to complete its search versus the time spent by an omniscient robot that knows the location of the target. We consider several new robot mobility models in which the speed of the robot depends on the terrain.These include 1) different constant speeds for different directions, 2) speed with constant acceleration and/or variability depending on whether a certain segment has already been searched, 3) speed dependent on the incline of the terrain. We provide both upper and lower bounds on the competitive ratios of search algorithms for these models, and in many cases, we derive optimal algorithms for the search time.

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arXiv2016

Search on a Line by Byzantine Robots

We consider the problem of fault-tolerant parallel search on an infinite line by $n$ robots. Starting from the origin, the robots are required to find a target at an unknown location. The robots can move with maximum speed $1$ and can communicate in wireless mode among themselves. However, among the $n$ robots, there are $f$ robots that exhibit {\em byzantine faults}. A faulty robot can fail to report the target even after reaching it, or it can make malicious claims about having found the target when in fact it has not. Given the presence of such faulty robots, the search for the target can only be concluded when the non-faulty robots have sufficient verification that the target has been found. We aim to design algorithms that minimize the value of $S_d(n,f)$, the time to find a target at a distance $d$ from the origin by $n$ robots among which $f$ are faulty. We give several different algorithms whose running time depends on the ratio $f/n$, the density of faulty robots, and also prove lower bounds. Our algorithms are optimal for some densities of faulty robots.

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arXiv2016

Convergecast and Broadcast by Power-Aware Mobile Agents

A set of identical, mobile agents is deployed in a weighted network. Each agent has a battery -- a power source allowing it to move along network edges. An agent uses its battery proportionally to the distance traveled. We consider two tasks : convergecast, in which at the beginning, each agent has some initial piece of information, and information of all agents has to be collected by some agent; and broadcast in which information of one specified agent has to be made available to all other agents. In both tasks, the agents exchange the currently possessed information when they meet. The objective of this paper is to investigate what is the minimal value of power, initially available to all agents, so that convergecast or broadcast can be achieved. We study this question in the centralized and the distributed settings. In the centralized setting, there is a central monitor that schedules the moves of all agents. In the distributed setting every agent has to perform an algorithm being unaware of the network. In the centralized setting, we give a linear-time algorithm to compute the optimal battery power and the strategy using it, both for convergecast and for broadcast, when agents are on the line. We also show that finding the optimal battery power for convergecast or for broadcast is NP-hard for the class of trees. On the other hand, we give a polynomial algorithm that finds a 2-approximation for convergecast and a 4-approximation for broadcast, for arbitrary graphs. In the distributed setting, we give a 2-competitive algorithm for convergecast in trees and a 4-competitive algorithm for broadcast in trees. The competitive ratio of 2 is proved to be the best for the problem of convergecast, even if we only consider line networks. Indeed, we show that there is no (2 -- $ε$)-competitive algorithm for convergecast or for broadcast in the class of lines, for any $ε$ \textgreater{} 0.

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arXiv2015

On Convergence and Threshold Properties of Discrete Lotka-Volterra Population Protocols

In this work we focus on a natural class of population protocols whose dynamics are modelled by the discrete version of Lotka-Volterra equations. In such protocols, when an agent $a$ of type (species) $i$ interacts with an agent $b$ of type (species) $j$ with $a$ as the initiator, then $b$'s type becomes $i$ with probability $P\_{ij}$. In such an interaction, we think of $a$ as the predator, $b$ as the prey, and the type of the prey is either converted to that of the predator or stays as is. Such protocols capture the dynamics of some opinion spreading models and generalize the well-known Rock-Paper-Scissors discrete dynamics. We consider the pairwise interactions among agents that are scheduled uniformly at random. We start by considering the convergence time and show that any Lotka-Volterra-type protocol on an $n$-agent population converges to some absorbing state in time polynomial in $n$, w.h.p., when any pair of agents is allowed to interact. By contrast, when the interaction graph is a star, even the Rock-Paper-Scissors protocol requires exponential time to converge. We then study threshold effects exhibited by Lotka-Volterra-type protocols with 3 and more species under interactions between any pair of agents. We start by presenting a simple 4-type protocol in which the probability difference of reaching the two possible absorbing states is strongly amplified by the ratio of the initial populations of the two other types, which are transient, but "control" convergence. We then prove that the Rock-Paper-Scissors protocol reaches each of its three possible absorbing states with almost equal probability, starting from any configuration satisfying some sub-linear lower bound on the initial size of each species. That is, Rock-Paper-Scissors is a realization of a "coin-flip consensus" in a distributed system. Some of our techniques may be of independent value.

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arXiv2013

The Beachcombers' Problem: Walking and Searching with Mobile Robots

We introduce and study a new problem concerning the exploration of a geometric domain by mobile robots. Consider a line segment $[0,I]$ and a set of $n$ mobile robots $r_1,r_2,..., r_n$ placed at one of its endpoints. Each robot has a {\em searching speed} $s_i$ and a {\em walking speed} $w_i$, where $s_i <w_i$. We assume that each robot is aware of the number of robots of the collection and their corresponding speeds. At each time moment a robot $r_i$ either walks along a portion of the segment not exceeding its walking speed $w_i$ or searches a portion of the segment with the speed not exceeding $s_i$. A search of segment $[0,I]$ is completed at the time when each of its points have been searched by at least one of the $n$ robots. We want to develop {\em mobility schedules} (algorithms) for the robots which complete the search of the segment as fast as possible. More exactly we want to maximize the {\em speed} of the mobility schedule (equal to the ratio of the segment length versus the time of the completion of the schedule). We analyze first the offline scenario when the robots know the length of the segment that is to be searched. We give an algorithm producing a mobility schedule for arbitrary walking and searching speeds and prove its optimality. Then we propose an online algorithm, when the robots do not know in advance the actual length of the segment to be searched. The speed $S$ of such algorithm is defined as $S = \inf_{I_L} S(I_L)$ where $S(I_L)$ denotes the speed of searching of segment $I_L=[0,L]$. We prove that the proposed online algorithm is 2-competitive. The competitive ratio is shown to be better in the case when the robots' walking speeds are all the same.

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arXiv2010

How to meet asynchronously (almost) everywhere

Two mobile agents (robots) with distinct labels have to meet in an arbitrary, possibly infinite, unknown connected graph or in an unknown connected terrain in the plane. Agents are modeled as points, and the route of each of them only depends on its label and on the unknown environment. The actual walk of each agent also depends on an asynchronous adversary that may arbitrarily vary the speed of the agent, stop it, or even move it back and forth, as long as the walk of the agent in each segment of its route is continuous, does not leave it and covers all of it. Meeting in a graph means that both agents must be at the same time in some node or in some point inside an edge of the graph, while meeting in a terrain means that both agents must be at the same time in some point of the terrain. Does there exist a deterministic algorithm that allows any two agents to meet in any unknown environment in spite of this very powerfull adversary? We give deterministic rendezvous algorithms for agents starting at arbitrary nodes of any anonymous connected graph (finite or infinite) and for agents starting at any interior points with rational coordinates in any closed region of the plane with path-connected interior. While our algorithms work in a very general setting ? agents can, indeed, meet almost everywhere ? we show that none of the above few limitations imposed on the environment can be removed. On the other hand, our algorithm also guarantees the following approximate rendezvous for agents starting at arbitrary interior points of a terrain as above: agents will eventually get at an arbitrarily small positive distance from each other.

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