arXiv subjects

Yehuda Afek

Publications and source records attributed to Yehuda Afek.

23 records · Page 2Linked to original sources

arXiv2012

Musical chairs

In the {\em Musical Chairs} game $MC(n,m)$ a team of $n$ players plays against an adversarial {\em scheduler}. The scheduler wins if the game proceeds indefinitely, while termination after a finite number of rounds is declared a win of the team. At each round of the game each player {\em occupies} one of the $m$ available {\em chairs}. Termination (and a win of the team) is declared as soon as each player occupies a unique chair. Two players that simultaneously occupy the same chair are said to be {\em in conflict}. In other words, termination (and a win for the team) is reached as soon as there are no conflicts. The only means of communication throughout the game is this: At every round of the game, the scheduler selects an arbitrary nonempty set of players who are currently in conflict, and notifies each of them separately that it must move. A player who is thus notified changes its chair according to its deterministic program. As we show, for $m\ge 2n-1$ chairs the team has a winning strategy. Moreover, using topological arguments we show that this bound is tight. For $m\leq 2n-2$ the scheduler has a strategy that is guaranteed to make the game continue indefinitely and thus win. We also have some results on additional interesting questions. For example, if $m \ge 2n-1$ (so that the team can win), how quickly can they achieve victory?

math.CO↗

arXiv2012

Beeping a Maximal Independent Set

We consider the problem of computing a maximal independent set (MIS) in an extremely harsh broadcast model that relies only on carrier sensing. The model consists of an anonymous broadcast network in which nodes have no knowledge about the topology of the network or even an upper bound on its size. Furthermore, it is assumed that an adversary chooses at which time slot each node wakes up. At each time slot a node can either beep, that is, emit a signal, or be silent. At a particular time slot, beeping nodes receive no feedback, while silent nodes can only differentiate between none of its neighbors beeping, or at least one of its neighbors beeping. We start by proving a lower bound that shows that in this model, it is not possible to locally converge to an MIS in sub-polynomial time. We then study four different relaxations of the model which allow us to circumvent the lower bound and find an MIS in polylogarithmic time. First, we show that if a polynomial upper bound on the network size is known, it is possible to find an MIS in O(log^3 n) time. Second, if we assume sleeping nodes are awoken by neighboring beeps, then we can also find an MIS in O(log^3 n) time. Third, if in addition to this wakeup assumption we allow sender-side collision detection, that is, beeping nodes can distinguish whether at least one neighboring node is beeping concurrently or not, we can find an MIS in O(log^2 n) time. Finally, if instead we endow nodes with synchronous clocks, it is also possible to find an MIS in O(log^2 n) time.

cs.DS↗

arXiv2012

Asynchrony from Synchrony

We consider synchronous dynamic networks which like radio networks may have asymmetric communication links, and are affected by communication rather than processor failures. In this paper we investigate the minimal message survivability in a per round basis that allows for the minimal global cooperation, i.e., allows to solve any task that is wait-free read-write solvable. The paper completely characterizes this survivability requirement. Message survivability is formalized by considering adversaries that have a limited power to remove messages in a round. Removal of a message on a link in one direction does not necessarily imply the removal of the message on that link in the other direction. Surprisingly there exist a single strongest adversary which solves any wait-free read/write task. Any different adversary that solves any wait-free read/write task is weaker, and any stronger adversary will not solve any wait-free read/write task. ABD \cite{ABD} who considered processor failure, arrived at an adversary that is $n/2$ resilient, consequently can solve tasks, such as $n/2$-set-consensus, which are not read/write wait-free solvable. With message adversaries, we arrive at an adversary which has exactly the read-write wait-free power. Furthermore, this adversary allows for a considerably simpler (simplest that we know of) proof that the protocol complex of any read/write wait-free task is a subdivided simplex, finally making this proof accessible for students with no algebraic-topology prerequisites, and alternatively dispensing with the assumption that the Immediate Snapshot complex is a subdivided simplex.

cs.DC↗

arXiv2011

Oblivious Collaboration

Communication is a crucial ingredient in every kind of collaborative work. But what is the least possible amount of communication required for a given task? We formalize this question by introducing a new framework for distributed computation, called {\em oblivious protocols}. We investigate the power of this model by considering two concrete examples, the {\em musical chairs} task $MC(n,m)$ and the well-known {\em Renaming} problem. The $MC(n,m)$ game is played by $n$ players (processors) with $m$ chairs. Players can {\em occupy} chairs, and the game terminates as soon as each player occupies a unique chair. Thus we say that player $P$ is {\em in conflict} if some other player $Q$ is occupying the same chair, i.e., termination means there are no conflicts. By known results from distributed computing, if $m \le 2n-2$, no strategy of the players can guarantee termination. However, there is a protocol with $m = 2n-1$ chairs that always terminates. Here we consider an oblivious protocol where in every time step the only communication is this: an adversarial {\em scheduler} chooses an arbitrary nonempty set of players, and for each of them provides only one bit of information, specifying whether the player is currently in conflict or not. A player notified not to be in conflict halts and never changes its chair, whereas a player notified to be in conflict changes its chair according to its deterministic program. Remarkably, even with this minimal communication termination can be guaranteed with only $m=2n-1$ chairs. Likewise, we obtain an oblivious protocol for the Renaming problem whose name-space is small as that of the optimal nonoblivious distributed protocol. Other aspects suggest themselves, such as the efficiency (program length) of our protocols. We make substantial progress here as well, though many interesting questions remain open.

cs.DC↗

arXiv2011

MIS on the fly

Humans are very good at optimizing solutions for specific problems. Biological processes, on the other hand, have evolved to handle multiple constrained distributed environments and so they are robust and adaptable. Inspired by observations made in a biological system we have recently presented a simple new randomized distributed MIS algorithm \cite{ZScience}. Here we extend these results by removing a number of strong assumptions that we made, making the algorithms more practical. Specifically we present an $O(\log^2 n)$ rounds synchronous randomized MIS algorithm which uses only 1 bit unary messages (a beeping signal with collision detection), allows for asynchronous wake up, does not assume any knowledge of the network topology, and assumes only a loose bound on the network size. We also present an extension with no collision detection in which the round complexity increases to $(\log^3 n)$. Finally, we show that our algorithm is optimal under some restriction, by presenting a tight lower bound of $Ω(\log^2 n)$ on the number of rounds required to construct a MIS for a restricted model.

cs.DC↗