Armando Castañeda

Renaming is weaker than set agreement but for perfect renaming: a map of sub-consensus tasks

In the wait-free shared memory model substantial attention has been devoted to understanding the relative power of sub-consensus tasks. Two important sub-consensus families of tasks have been identified: k-set agreement and M-renaming. When 2≤k≤n−1 and n≤M≤2n−2, these tasks are more powerful than read/write registers, but not strong enough to solve consensus for two processes. This paper studies the power of renaming with respect to set agreement. It shows that, in a system of n processes, n-renaming is strictly stronger than (n−1)-set agreement, but not stronger than (n−2)-set agreement. Furthermore, (n+1)-renaming cannot solve even (n−1)-set agreement. As a consequence, there are cases where set agreement and renaming are incomparable when looking at their power to implement each other.

Brief announcement: pareto optimal solutions to consensus and set consensus

A protocol P is Pareto-optimal if no protocol Q can decide as fast as P for all adversaries, while allowing at least one process to decide strictly earlier, in at least one instance. Pareto optimal protocols cannot be improved upon. We present the first Pareto-optimal solutions to consensus and k-set consensus for synchronous message-passing with crashes failures. Our k-set consensus protocol strictly dominates all known solutions, and our results expose errors in [1, 7, 8, 12]. Our proofs of Pareto optimality are completely constructive, and are devoid of any topological arguments or reductions.

Agreement via Symmetry Breaking: On the Structure of Weak Subconsensus Tasks

This paper is on the relative power and the relations linking two important synchronization problems in n-process wait-free shared memory models, namely, set agreement and renaming, which are two of the most studied subconsensus tasks. Since the 2006 seminal paper of Gafni, Rajsbaum and Herlihy, it is known that some renaming instances are strictly weaker than set agreement. Indeed, it was later on shown that not even (n + 1)-renaming (the strongest task in the renaming family, after perfect n-renaming) can implement (n - 1)-set agreement (the weakest non-trivial task in the set agreement family). These and other results seem to imply that renaming and, more generally, the tasks called generalized symmetry breaking tasks (GSB) are weaker than agreement tasks. This paper shows that this is not the case, namely, it shows that there is a large family of GSB tasks that are more powerful than (n - 1)-set agreement. Some of these tasks are equivalent to n-renaming, while others lie strictly between n-renaming and (n+1)-renaming. Moreover, none of these GSB tasks can solve (n - 2)-set agreement. Hence, these subconsensus tasks have a rich structure and are interesting in their own. The proofs of these results are based on algebraic topology techniques and new ideas about different notions of nondeterminism that can be associated with shared objects. Interestingly, this paper sheds a new light on the relations linking set agreement and renaming.

A non-topological proof for the impossibility of k-set agreement

In the k-set agreement task each process proposes a value, and it is required that each correct process has to decide a value which was proposed and at most k distinct values must be decided. Using topological arguments it has been proved that k-set agreement is unsolvable in the asynchronous wait-free read/write shared memory model, when k < n, the number of processes. This paper presents a simple, non-topological impossibility proof of k-set agreement. The proof depends on two simple properties of the immediate snapshot executions, a subset of all possible executions, and on the well known graph theory result stating that every graph has an even number of vertices with odd degree (the handshaking lemma).

Brief announcement: there are plenty of tasks weaker than perfect renaming and stronger than set agreement

In the asynchronous <i>wait-free</i> shared memory model, two families of tasks play a central role because of their implications in theory and in practice: <i>k-set agreement</i> and <i>M-renaming</i>. Let <i>n</i> denote the number of processes in the system. Previous research shows that (<i>n</i>-1)-set agreement can solve (2<i>n</i>-2)-renaming, for any value of <i>n</i>, while (2<i>n</i>-2)-renaming cannot solve (<i>n</i>-1)-set agreement, when <i>n</i> is odd. It is also known that, for every <i>n</i> ≥ 3, <i>n</i>-renaming, also called <i>perfect renaming</i>, is strictly stronger than (<i>n</i>-1)-set agreement. This paper shows that when <i>n</i> ≥ 4, there is a family of tasks that are strictly stronger than (<i>n</i>-1)-set agreement and strictly weaker than perfect renaming. This enlarges our view of both the nature and the structure of what are distributed computing tasks.

When and How Process Groups Can Be Used to Reduce the Renaming Space

Considering the M-renaming problem and process groups, this paper investigates the following question: Is there a relation between the number of groups and the size of the new name space M? This question can be rephrased as follows: Can the initial partitioning of the processes into m groups allows the size of the renaming space M to be reduced, and if yes, how much?

An Equivariance Theorem with Applications to Renaming

In the renaming problem, each process in a distributed system is issued a unique name from a large namespace, and the processes must coordinate with one another to choose unique names from a much smaller name space.We show that lower bounds on the solvability of renaming can be formulated as a purely topological question about the existence of an equivariant chain map from a “topological disk” to a “topological annulus”. Proving the non-existence of such a map implies the non-existence of a distributed renaming algorithm in several related models of computation.

On the Consensus Number of Non-adaptive Perfect Renaming

In the n,M-renaming problem, there are n processes, each with an initial name known only from itself, and these processes have to compute new names from the set {1,...,M}, despite asynchrony and any number of process crashes, such that no two processes have the same new name. If M=n, the renaming is said to be perfect. If M is only on function on the number n of the processes in the system, the renaming is said to be non-adaptive. In contrast, if M is on function on the number of processes that actually participate in a given execution, renaming is adaptive. The consensus number of a concurrent object is an integer which measures its synchronization power in presence of any number of process crashes. This paper investigates the consensus number of non-adaptive perfect renaming objects. It shows that, while a non-adaptive perfect renaming object for two processes ports has consensus number 2, its consensus number decreases to 1 when the number of processes which can access it increases beyond 2.