The k-set agreement problem is a generalization of the consensus problem. Namely, assuming that each process proposes a value, every non-faulty process should decide one of the proposed values, and no more than k different values should be decided. This is a hard problem in the sense that we cannot solve it in an asynchronous system, as soon as k or more processes may crash. One way to sidestep this impossibility result consists in weakening the termination property, requiring that a process must decide a value only if it executes alone during a long enough period of time. This is the well-known obstruction-freedom progress condition. Consider a system of n anonymous asynchronous processes that communicate through atomic read/write registers, and such that any number of them may crash. In this paper, we address and solve the challenging open problem of designing an obstruction-free k-set agreement algorithm using only (n-k+1) atomic registers. From a shared memory cost point of view, our algorithm is the best algorithm known so far, thereby establishing a new upper bound on the number of registers needed to solve the problem, and in comparison to the previous upper bound, its gain is (n-k) registers. We then extend this algorithm into a space-optimal solution for the repeated version of k-set agreement, and an x-obstruction-free solution that employs 0(n-k+x) atomic registers (with 1 <= x <= k < n).

Solving the consensus problem requires in one way or another that the underlying system satisfies some synchrony assumption. Considering an asynchronous message-passing system of n processes where (a) up to t< n/3 may commit Byzantine failures, and (b) each pair of processes is connected by two uni-directional channels (with possibly different timing properties), this paper investigates the synchrony assumption required to solve consensus, and presents a signature-free consensus algorithm whose synchrony requirement is the existence of a process that is an eventual {t+1}bisource. Such a process p is a correct process that eventually has (a) timely input channels from t correct processes and (b) timely output channels to t correct processes (these input and output channels can connect p to different subsets of processes). As this synchrony condition was shown to be necessary and sufficient in the stronger asynchronous system model (a) enriched with message authentication, and (b) where the channels are bidirectional and have the same timing properties in both directions, it follows that it is also necessary and sufficient in the weaker system model considered in the paper. In addition to the fact that it closes a long-lasting problem related to Byzantine agreement, a noteworthy feature of the proposed algorithm lies in its design simplicity, which is a first-class property.

We study distributed coordination among autonomous mobile robots, focussing on the problem of gathering the robots at a single location. The gathering problem has been solved previously using deterministic algorithms even for robots that are anonymous, oblivious, disoriented, and operate in the semi-synchronous ATOM model. However these solutions require all robots to be fault-free. The recent results of Agmon and Peleg [1] show how to gather all correct robots when one of the robots may crash permanently. We study gathering in n-robot systems with f crashes for any f <; n. In such a scenario, no robot can wait for another robot, i.e., the algorithm must be wait-free. We provide such a wait-free algorithm to gather all correct robots assuming the capabilities of strong multiplicity detection and chirality. Unlike previous solutions, our algorithm does not impose the requirement of initially distinct locations, and works for any arbitrary initial configuration of robots (except the bivalent configuration where deterministic gathering is not possible).

In the traditional consensus task, processes are required to agree on a common value chosen among the initial values of the participating processes. It is well known that consensus cannot be solved in crash-prone, asynchronous distributed systems. Two generalizations of the consensus tasks have been introduced: k-set agreement and k-parallel consensus. The k-set agreement task has the same requirements as consensus except that processes are allowed to decide up to k distinct values. In the k-parallel consensus task, each process participates simultaneously in k instances of consensus and is required to decide in at least one of them; any two processes deciding in the same instance must decide the same value. It is known that both tasks are equivalent in the wait-free shared memory model. Perhaps surprisingly, this paper shows that this is no longer the case in the n-process asynchronous message passing model with at most t process crashes. Specifically, the paper establishes that for parameters t, n, k such that t > n+k-2/2 , k-parallel consensus is strictly harder than k-set agreement. The proof compares the information on failures necessary to solve each task in the failure detector framework and relies on a result in topological combinatorics, namely, the chromatic number of Kneser graphs. The paper also introduces the new failure detector class VΣk , which is a generalization of the quorum failures detector class Σ suited to k-parallel consensus.