13 research outputs found
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 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 at speed is measured as . 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 , and the evacuation time is at most
, where 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 . For the case , we give an
optimal algorithm, and give upper bounds on the energy for the case .
We also consider the problem of minimizing the evacuation time when the
available energy is bounded by . Surprisingly, when is a
constant, independent of the distance of the exit from the origin, we prove
that evacuation is possible in time , and this is optimal up
to a logarithmic factor. When is linear in , we give upper bounds
on the evacuation time.Comment: This is the full version of the paper with the same title which will
appear in the proceedings of the 26th International Colloquium on Structural
Information and Communication Complexity (SIROCCO'19) L'Aquila, Italy during
July 1-4, 201
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
Parallel Search with no Coordination
We consider a parallel version of a classical Bayesian search problem.
agents are looking for a treasure that is placed in one of the boxes indexed by
according to a known distribution . The aim is to minimize
the expected time until the first agent finds it. Searchers run in parallel
where at each time step each searcher can "peek" into a box. A basic family of
algorithms which are inherently robust is \emph{non-coordinating} algorithms.
Such algorithms act independently at each searcher, differing only by their
probabilistic choices. We are interested in the price incurred by employing
such algorithms when compared with the case of full coordination. We first show
that there exists a non-coordination algorithm, that knowing only the relative
likelihood of boxes according to , has expected running time of at most
, where is the expected running time of the best
fully coordinated algorithm. This result is obtained by applying a refined
version of the main algorithm suggested by Fraigniaud, Korman and Rodeh in
STOC'16, which was designed for the context of linear parallel search.We then
describe an optimal non-coordinating algorithm for the case where the
distribution is known. The running time of this algorithm is difficult to
analyse in general, but we calculate it for several examples. In the case where
is uniform over a finite set of boxes, then the algorithm just checks boxes
uniformly at random among all non-checked boxes and is essentially times
worse than the coordinating algorithm.We also show simple algorithms for Pareto
distributions over boxes. That is, in the case where for
, we suggest the following algorithm: at step choose uniformly
from the boxes unchecked in ,
where . It turns out this algorithm is asymptotically
optimal, and runs about times worse than the case of full coordination
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
Overcoming Probabilistic Faults in Disoriented Linear Search
We consider search by mobile agents for a hidden, idle target, placed on the
infinite line. Feasible solutions are agent trajectories in which all agents
reach the target sooner or later. A special feature of our problem is that the
agents are -faulty, meaning that every attempt to change direction is an
independent Bernoulli trial with known probability , where is the
probability that a turn fails. We are looking for agent trajectories that
minimize the worst-case expected termination time, relative to competitive
analysis.
First, we study linear search with one deterministic -faulty agent, i.e.,
with no access to random oracles, . For this problem, we provide
trajectories that leverage the probabilistic faults into an algorithmic
advantage. Our strongest result pertains to a search algorithm (deterministic,
aside from the adversarial probabilistic faults) which, as , has
optimal performance , up to the additive term that
can be arbitrarily small. Additionally, it has performance less than for
. When , our algorithm has performance
, which we also show is optimal up to a constant factor.
Second, we consider linear search with two -faulty agents, ,
for which we provide three algorithms of different advantages, all with a
bounded competitive ratio even as . Indeed, for this problem,
we show how the agents can simulate the trajectory of any -faulty agent
(deterministic or randomized), independently of the underlying communication
model. As a result, searching with two agents allows for a solution with a
competitive ratio of , or a competitive ratio of
. Our final contribution is a novel algorithm for searching
with two -faulty agents that achieves a competitive ratio
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 tar-get placed at an unknown position on the line. Unlike previous studies, in which the robot travels along the line at a constant speed, we con-sider 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 con-stant 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 pro-vide both upper and lower bounds on the competitive ratios of search algorithms for these models, and in many cases, we derive optimal algo-rithms for the search time
Evacuation of Equilateral Triangles by Mobile Agents of Limited Communication Range
We consider the problem of evacuating mobile agents from a unit-sided equilateral triangle through an exit located at an unknown location on the perimeter of the triangle. The agents are initially located at the centroid of the triangle. An agent can move at speed at most one, and finds the exit only when it reaches the point where the exit is located. The agents can collaborate in the search for the exit. The goal of the {\em evacuation problem} is to minimize the evacuation time, defined as the worst-case time for {\em all} the agents to reach the exit.
Two models of communication between agents have been studied before; {\em non-wireless} or {\em face-to-face communication} model and {\em wireless communication} model. In the former model, agents can exchange information about the location of the exit only if they are at the same point at the same time, whereas in the latter model, the agents can send and receive information about the exit at any time regardless of their positions in the domain. In this thesis, we propose a new and more realistic communication model: agents can communicate with other agents at distance at most with .
We propose and analyze several algorithms for the problem of evacuation by agents in this model; our results indicate that the best strategy to be used varies depending on the values of and . For two agents, we give five strategies, the last of which achieves the best performance among all the five strategies for
all sub-ranges of in the range .
We also show a lower bound on the evacuation time of two agents for any agents, we study three strategies for evacuation: in the first strategy, called {\sf X3C}, agents explore all three sides of the triangle before connecting to exchange information; in the second strategy, called {\sf X1C}, agents explore a single side of the triangle before connecting; in the third strategy, called {\sf CXP}, the agents travel to the perimeter to locations in which they are connected, and explore it while always staying connected. For 3 or 4 agents, we show that X3C works better than X1C for small values of , while X1C works better for larger values of . Finally, we show that for any , evacuation of agents can be done using the CXP strategy in time , which is optimal in terms of time, and asymptotically optimal in terms of the number of agents