52 research outputs found
Ants: Mobile Finite State Machines
Consider the Ants Nearby Treasure Search (ANTS) problem introduced by
Feinerman, Korman, Lotker, and Sereni (PODC 2012), where mobile agents,
initially placed at the origin of an infinite grid, collaboratively search for
an adversarially hidden treasure. In this paper, the model of Feinerman et al.
is adapted such that the agents are controlled by a (randomized) finite state
machine: they possess a constant-size memory and are able to communicate with
each other through constant-size messages. Despite the restriction to
constant-size memory, we show that their collaborative performance remains the
same by presenting a distributed algorithm that matches a lower bound
established by Feinerman et al. on the run-time of any ANTS algorithm
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 Z x 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
Brief Announcement: Energy Constrained Depth First Search
Depth first search is a natural algorithmic technique for constructing a closed route that visits all vertices of a graph. The length of such route equals, in an edge-weighted tree, twice the total weight of all edges of the tree and this is asymptotically optimal over all exploration strategies. This paper considers a variant of such search strategies where the length of each route is bounded by a positive integer B (e.g. due to limited energy resources of the searcher). The objective is to cover all the edges of a tree T using the minimum number of routes, each starting and ending at the root and each being of length at most B. To this end, we analyze the following natural greedy tree traversal process that is based on decomposing a depth first search traversal into a sequence of limited length routes. Given any arbitrary depth first search traversal R of the tree T, we cover R with routes R_1,...,R_l, each of length at most B such that: R_i starts at the root, reaches directly the farthest point of R visited by R_{i-1}, then R_i continues along the path R as far as possible, and finally R_i returns to the root. We call the above algorithm piecemeal-DFS and we prove that it achieves the asymptotically minimal number of routes l, regardless of the choice of R. Our analysis also shows that the total length of the traversal (and thus the traversal time) of piecemeal-DFS is asymptotically minimum over all energy-constrained exploration strategies. The fact that R can be chosen arbitrarily means that the exploration strategy can be constructed in an online fashion when the input tree T is not known in advance. Each route R_i can be constructed without any knowledge of the yet unvisited part of T. Surprisingly, our results show that depth first search is efficient for energy constrained exploration of trees, even though it is known that the same does not hold for energy constrained exploration of arbitrary graphs
Collaborative search on the plane without communication
We generalize the classical cow-path problem [7, 14, 38, 39] into a question
that is relevant for collective foraging in animal groups. Specifically, we
consider a setting in which k identical (probabilistic) agents, initially
placed at some central location, collectively search for a treasure in the
two-dimensional plane. The treasure is placed at a target location by an
adversary and the goal is to find it as fast as possible as a function of both
k and D, where D is the distance between the central location and the target.
This is biologically motivated by cooperative, central place foraging such as
performed by ants around their nest. In this type of search there is a strong
preference to locate nearby food sources before those that are further away.
Our focus is on trying to find what can be achieved if communication is limited
or altogether absent. Indeed, to avoid overlaps agents must be highly dispersed
making communication difficult. Furthermore, if agents do not commence the
search in synchrony then even initial communication is problematic. This holds,
in particular, with respect to the question of whether the agents can
communicate and conclude their total number, k. It turns out that the knowledge
of k by the individual agents is crucial for performance. Indeed, it is a
straightforward observation that the time required for finding the treasure is
(D + D 2 /k), and we show in this paper that this bound can be matched
if the agents have knowledge of k up to some constant approximation. We present
an almost tight bound for the competitive penalty that must be paid, in the
running time, if agents have no information about k. Specifically, on the
negative side, we show that in such a case, there is no algorithm whose
competitiveness is O(log k). On the other hand, we show that for every constant
\epsilon \textgreater{} 0, there exists a rather simple uniform search
algorithm which is -competitive. In addition, we give
a lower bound for the setting in which agents are given some estimation of k.
As a special case, this lower bound implies that for any constant \epsilon
\textgreater{} 0, if each agent is given a (one-sided)
-approximation to k, then the competitiveness is (log k).
Informally, our results imply that the agents can potentially perform well
without any knowledge of their total number k, however, to further improve,
they must be given a relatively good approximation of k. Finally, we propose a
uniform algorithm that is both efficient and extremely simple suggesting its
relevance for actual biological scenarios
Distributed Broadcasting and Mapping Protocols in Directed Anonymous Networks
We initiate the study of distributed protocols over directed anonymous networks that are not necessarily strongly connected. In such networks, nodes are aware only of their incoming and outgoing edges, have no unique identity, and have no knowledge of the network topology or even bounds on its parameters, like the number of nodes or the network diameter. Anonymous networks are of interest in various settings such as wireless ad-hoc networks and peer to peer networks. Our goal is to create distributed protocols that reduce the uncertainty by distributing the knowledge of the network topology to all the nodes.
We consider two basic protocols: broadcasting and unique label assignment. These two protocols enable a complete mapping of the network and can serve as key building blocks in more advanced protocols. We develop distributed asynchronous protocols as well as derive lower bounds on their communication complexity, total bandwidth complexity, and node label complexity. The resulting lower bounds are sometimes surprisingly high, exhibiting the complexity of topology extraction in directed anonymous networks
Deterministic Graph Exploration with Advice
We consider the task of graph exploration. An -node graph has unlabeled
nodes, and all ports at any node of degree are arbitrarily numbered
. A mobile agent has to visit all nodes and stop. The exploration
time is the number of edge traversals. We consider the problem of how much
knowledge the agent has to have a priori, in order to explore the graph in a
given time, using a deterministic algorithm. This a priori information (advice)
is provided to the agent by an oracle, in the form of a binary string, whose
length is called the size of advice. We consider two types of oracles. The
instance oracle knows the entire instance of the exploration problem, i.e., the
port-numbered map of the graph and the starting node of the agent in this map.
The map oracle knows the port-numbered map of the graph but does not know the
starting node of the agent.
We first consider exploration in polynomial time, and determine the exact
minimum size of advice to achieve it. This size is ,
for both types of oracles.
When advice is large, there are two natural time thresholds:
for a map oracle, and for an instance oracle, that can be achieved
with sufficiently large advice. We show that, with a map oracle, time
cannot be improved in general, regardless of the size of advice.
We also show that the smallest size of advice to achieve this time is larger
than , for any .
For an instance oracle, advice of size is enough to achieve time
. We show that, with any advice of size , the time of
exploration must be at least , for any , and with any
advice of size , the time must be .
We also investigate minimum advice sufficient for fast exploration of
hamiltonian graphs
Mobile agent rendezvous: A survey
Abstract. Recent results on the problem of mobile agent rendezvous on distributed networks are surveyed with an emphasis on outlining the various approaches taken by researchers in the theoretical computer science community.
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