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Register a new userA characterization of colorless anonymous $t$-resilient task computability
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Added by
Nayuta Yanagisawa
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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A task is a distributed problem for $n$ processes, in which each process starts with a private input value, communicates with other processes, and eventually decides an output value. A task is colorless if each process can adopt the input or output value of another process. Colorless tasks are well studied in the non-anonymous shared-memory model where each process has a distinct identifier that can be used to access a single-writer/multi-reader shared register. In the anonymous case, where processes have no identifiers and communicate through multi-writer/multi-reader registers, there is a recent topological characterization of the colorless tasks that are solvable when any number of asynchronous processes may crash. In this paper we study the case where at most $t$ processes may crash, where $1 le t < n$. We prove that a colorless task is $t$-resilient solvable non-anonymously if and only if it is $t$-resilient solvable anonymously. This implies a complete characterization of colorless anonymous t-resilient asynchronous task computability.
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An immediate snapshot object is a high level communication object, built on top of a read/write distributed system in which all except one processes may crash. It allows a process to write a value and obtain a set of values that represent a snapshot of the values written to the object, occurring immediately after the write step. Considering an $n$-process model in which up to $t$ processes may crash, this paper introduces first the $k$-resilient immediate snapshot object, which is a natural generalization of the basic immediate snapshot (which corresponds to the case $k=t=n-1$). In addition to the set containment properties of the basic immediate snapshot, a $k$-resilient immediate snapshot object requires that each set returned to a process contains at least $(n-k)$ pairs. The paper first shows that, for $k,t<n-1$, $k$-resilient immediate snapshot is impossible in asynchronous read/write systems. %Then the paper investigates the space of objects that %are impossible to solve in $n$-process $t$-crash read/write systems. Then the paper investigates a model of computation where the processes communicate with each other by accessing $k$-immediate snapshot objects, and shows that this model is stronger than the $t$-crash model. Considering the space of $x$-set agreement problems (which are impossible to solve in systems such that $xleq t$), the paper shows then that $x$-set agreement can be solved in read/write systems enriched with $k$-immediate snapshot objects for $x=max(1,t+k-(n-2))$. It also shows that, in these systems, $k$-resilient immediate snapshot and consensus are equivalent when $1leq t<n/2$ and $tleq kleq (n-1)-t$. Hence, %thanks to the problem map it provides, the paper establishes strong relations linking fundamental distributed computing objects (one related to communication, the other to agreement), which are impossible to solve in pure read/write systems.
We consider a swarm of $n$ autonomous mobile robots, distributed on a 2-dimensional grid. A basic task for such a swarm is the gathering process: All robots have to gather at one (not predefined) place. A common local model for extremely simple robots is the following: The robots do not have a common compass, only have a constant viewing radius, are autonomous and indistinguishable, can move at most a constant distance in each step, cannot communicate, are oblivious and do not have flags or states. The only gathering algorithm under this robot model, with known runtime bounds, needs $mathcal{O}(n^2)$ rounds and works in the Euclidean plane. The underlying time model for the algorithm is the fully synchronous $mathcal{FSYNC}$ model. On the other side, in the case of the 2-dimensional grid, the only known gathering algorithms for the same time and a similar local model additionally require a constant memory, states and flags to communicate these states to neighbors in viewing range. They gather in time $mathcal{O}(n)$. In this paper we contribute the (to the best of our knowledge) first gathering algorithm on the grid that works under the same simple local model as the above mentioned Euclidean plane strategy, i.e., without memory (oblivious), flags and states. We prove its correctness and an $mathcal{O}(n^2)$ time bound in the fully synchronous $mathcal{FSYNC}$ time model. This time bound matches the time bound of the best known algorithm for the Euclidean plane mentioned above. We say gathering is done if all robots are located within a $2times 2$ square, because in $mathcal{FSYNC}$ such configurations cannot be solved.
We show that asynchronous $t$ faults Byzantine system is equivalent to asynchronous $t$-resilient system, where unbeknownst to all, the private inputs of at most $t$ processors were altered and installed by a malicious oracle. The immediate ramification is that dealing with asynchronous Byzantine systems does not call for new topological methods, as was recently employed by various researchers: Asynchronous Byzantine is a standard asynchronous system with an input caveat. It also shows that two recent independent investigations of vector $epsilon$-agreement in the Byzantine model, and then in the fail-stop model, one was superfluous - in these problems the change of $t$ inputs allowed in the Byzantine has no effect compared to the fail-stop case. This result was motivated by the aim of casting any asynchronous system as a synchronous system where all processors are correct and it is the communication substrate in the form of message-adversary that misbehaves. Thus, in addition, we get such a characterization for the asynchronous Byzantine system.
We consider systems made of autonomous mobile robots evolving in highly dynamic discrete environment i.e., graphs where edges may appear and disappear unpredictably without any recurrence, stability, nor periodicity assumption. Robots are uniform (they execute the same algorithm), they are anonymous (they are devoid of any observable ID), they have no means allowing them to communicate together, they share no common sense of direction, and they have no global knowledge related to the size of the environment. However, each of them is endowed with persistent memory and is able to detect whether it stands alone at its current location. A highly dynamic environment is modeled by a graph such that its topology keeps continuously changing over time. In this paper, we consider only dynamic graphs in which nodes are anonymous, each of them is infinitely often reachable from any other one, and such that its underlying graph (i.e., the static graph made of the same set of nodes and that includes all edges that are present at least once over time) forms a ring of arbitrary size. In this context, we consider the fundamental problem of perpetual exploration: each node is required to be infinitely often visited by a robot. This paper analyzes the computability of this problem in (fully) synchronous settings, i.e., we study the deterministic solvability of the problem with respect to the number of robots. We provide three algorithms and two impossibility results that characterize, for any ring size, the necessary and sufficient number of robots to perform perpetual exploration of highly dynamic rings.
The TTE computability notion in effective metric spaces is usually defined by using Cauchy representations. Under some weak assumptions, we characterize this notion in a way which avoids using the representations.
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