Consider a distributed system with @math processors out of which @math can be Byzantine faulty. In the approximate agreement task, each processor @math receives an input value @math and has to decide on an output value @math such that - the output values are in the convex hull of the non-faulty processors' input values, - the output values are within distance @math of each other. Classically, the values are assumed to be from an @math -dimensional Euclidean space, where @math . In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph @math and the goal is to output vertices that are within distance @math of each other in @math , but still remain in the graph-induced convex hull of the input values. For @math , the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any @math , we show that the task is solvable in asynchronous systems when @math is chordal and @math , where @math is the clique number of @math . In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures.
Another class of structured agreement problems in the wait-free asynchronous setting are tasks @cite_32 , which generalise @math -set agreement and approximate agreement (e.g., @math -set agreement and one-dimensional approximate agreement). In loop agreement, the set of inputs consists of three distinct vertices on a loop in a 2-dimensional simplicial complex and the outputs are vertices of the complex with certain constraints, whereas are a generalisation of loop agreement to higher dimensions @cite_39 . These tasks are part of large body of work exploring the deep connection of asynchronous computability and combinatorial topology, which has successfully been used to characterise the of various distributed tasks @cite_11 . Gafni and Kuznetsov's @math -reconciliation task @cite_17 achieves geodesic approximate agreement on a graph of system configurations.
@cite_17 @cite_11 @cite_32 @cite_39
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In the classical consensus problem, each of n processors receives a private input value and produces a decision value which is one of the original input values, with the requirement that all processors decide the same value. A central result in distributed computing is that, in several standard models including the asynchronous shared-memory model, this problem has no deterministic solution. The k-set agreement problem is a generalization of the classical consensus proposed by Chaudhuri [ Inform. and Comput., 105 (1993), pp. 132--158], where the agreement condition is weakened so that the decision values produced may be different, as long as the number of distinct values is at most k. For @math it was not known whether this problem is solvable deterministically in the asynchronous shared memory model. In this paper, we resolve this question by showing that for any k < n, there is no deterministic wait-free protocol for n processors that solves the k-set agreement problem. The proof technique is new: it is based on the development of a topological structure on the set of possible processor schedules of a protocol. This topological structure has a natural interpretation in terms of the knowledge of the processors of the state of the system. This structure reveals a close analogy between the impossibility of wait-free k-set agreement and the Brouwer fixed point theorem for the k-dimensional ball. In the classical consensus problem,each of n processors receives a private input value and produces a decision value which is one of the original input values,with the requirement that all processors decide the same value. A central result in distributed computing is that,in several standard models including the asynchronous shared-memory model,this problem has no determinis- tic solution. The k-set agreement problem is a generalization of the classical consensus proposed by Chaudhuri (Inform. and Comput.,105 (1993),pp. 132-158),where the agreement condition is weak- ened so that the decision values produced may be different,as long as the number of distinct values is at most k .F or n>k ≥ 2 it was not known whether this problem is solvable deterministically in the asynchronous shared memory model. In this paper,we resolve this question by showing that for any k<n ,there is no deterministic wait-free protocol for n processors that solves the k-set agreement problem. The proof technique is new: it is based on the development of a topological structure on the set of possible processor schedules of a protocol. This topological structure has a natural interpretation in terms of the knowledge of the processors of the state of the system. This structure reveals a close analogy between the impossibility of wait-free k-set agreement and the Brouwer fixed point theorem for the k-dimensional ball. This paper studies the lattice agreement problem and the generalized lattice agreement problem in distributed message passing systems. In the lattice agreement problem, given input values from a lattice, processes have to non-trivially decide output values that lie on a chain. We consider the lattice agreement problem in both synchronous and asynchronous systems. For synchronous lattice agreement, we present two algorithms which run in @math and @math rounds, respectively, where @math denotes the height of the input sublattice @math , @math is the number of crash failures the system can tolerate, and @math is the number of processes in the system. These algorithms have significant better round complexity than previously known algorithms. The algorithm by attiya1995atomic takes @math synchronous rounds, and the algorithm by Mavronicolasa mavronicolasabound takes @math rounds. For asynchronous lattice agreement, we propose an algorithm which has time complexity of @math message delays which improves on the previously known time complexity of @math message delays. The generalized lattice agreement problem defined by in faleiro2012generalized is a generalization of the lattice agreement problem where it is applied for the replicated state machine. We propose an algorithm which guarantees liveness when a majority of the processes are correct in asynchronous systems. Our algorithm requires @math units of time in the worst case which is better than @math units of time required by the algorithm of faleiro2012generalized . Loop agreement is a family of wait-free tasks that includes instances of set agreement and approximate agreement tasks. A task G implements task F if one can construct a solution to F from a solution to G, possibly followed by access to a read write memory. Loop agreement tasks form a lattice under this notion of implementation.This paper presents a classification of loop agreement tasks. Each loop agreement task can be assigned an algebraic signature consisting of a finitely presented group G and a distinguished element g in G. This signature characterizes the task's power to implement other tasks. If F and G are loop agreement tasks with respective signatures 〈F,f〉 and 〈G,g〉, then F implements G if and only if there exists a group homomorphism h : F → G carrying f to g.