1,048,273 research outputs found
On the Space Complexity of Set Agreement
The -set agreement problem is a generalization of the classical consensus
problem in which processes are permitted to output up to different input
values. In a system of processes, an -obstruction-free solution to the
problem requires termination only in executions where the number of processes
taking steps is eventually bounded by . This family of progress conditions
generalizes wait-freedom () and obstruction-freedom (). In this
paper, we prove upper and lower bounds on the number of registers required to
solve -obstruction-free -set agreement, considering both one-shot and
repeated formulations. In particular, we show that repeated set agreement
can be solved using registers and establish a nearly matching lower
bound of
On the Round Complexity of Randomized Byzantine Agreement
We prove lower bounds on the round complexity of randomized Byzantine agreement (BA) protocols, bounding the halting probability of such protocols after one and two rounds. In particular, we prove that:
1) BA protocols resilient against n/3 [resp., n/4] corruptions terminate (under attack) at the end of the first round with probability at most o(1) [resp., 1/2+ o(1)].
2) BA protocols resilient against n/4 corruptions terminate at the end of the second round with probability at most 1-Theta(1).
3) For a large class of protocols (including all BA protocols used in practice) and under a plausible combinatorial conjecture, BA protocols resilient against n/3 [resp., n/4] corruptions terminate at the end of the second round with probability at most o(1) [resp., 1/2 + o(1)].
The above bounds hold even when the parties use a trusted setup phase, e.g., a public-key infrastructure (PKI).
The third bound essentially matches the recent protocol of Micali (ITCS\u2717) that tolerates up to n/3 corruptions and terminates at the end of the third round with constant probability
Communication Complexity of the Secret Key Agreement in Algorithmic Information Theory
It is known that the mutual information, in the sense of Kolmogorov
complexity, of any pair of strings x and y is equal to the length of the
longest shared secret key that two parties can establish via a probabilistic
protocol with interaction on a public channel, assuming that the parties hold
as their inputs x and y respectively. We determine the worst-case communication
complexity of this problem for the setting where the parties can use private
sources of random bits. We show that for some x, y the communication complexity
of the secret key agreement does not decrease even if the parties have to agree
on a secret key whose size is much smaller than the mutual information between
x and y. On the other hand, we discuss examples of x, y such that the
communication complexity of the protocol declines gradually with the size of
the derived secret key. The proof of the main result uses spectral properties
of appropriate graphs and the expander mixing lemma, as well as information
theoretic techniques.Comment: 33 pages, 6 figures. v3: the full version of the MFCS 2020 pape
Complexity of Multi-Value Byzantine Agreement
In this paper, we consider the problem of maximizing the throughput of
Byzantine agreement, given that the sum capacity of all links in between nodes
in the system is finite. We have proposed a highly efficient Byzantine
agreement algorithm on values of length l>1 bits. This algorithm uses error
detecting network codes to ensure that fault-free nodes will never disagree,
and routing scheme that is adaptive to the result of error detection. Our
algorithm has a bit complexity of n(n-1)l/(n-t), which leads to a linear cost
(O(n)) per bit agreed upon, and overcomes the quadratic lower bound
(Omega(n^2)) in the literature. Such linear per bit complexity has only been
achieved in the literature by allowing a positive probability of error. Our
algorithm achieves the linear per bit complexity while guaranteeing agreement
is achieved correctly even in the worst case. We also conjecture that our
algorithm can be used to achieve agreement throughput arbitrarily close to the
agreement capacity of a network, when the sum capacity is given
The Step Complexity of Multidimensional Approximate Agreement
Approximate agreement allows a set of n processes to obtain outputs that are within a specified distance ? > 0 of one another and within the convex hull of the inputs.
When the inputs are real numbers, there is a wait-free shared-memory approximate agreement algorithm [Moran, 1995] whose step complexity is in O(n log(S/?)), where S, the spread of the inputs, is the maximal distance between inputs. There is another wait-free algorithm [Schenk, 1995] that avoids the dependence on n and achieves O(log(M/?)) step complexity where M, the magnitude of the inputs, is the absolute value of the maximal input.
This paper considers whether it is possible to obtain an approximate agreement algorithm whose step complexity depends on neither n nor the magnitude of the inputs, which can be much larger than their spread. On the negative side, we prove that ?(min{(log M)/(log log M), (?log n)/(log log n)}) is a lower bound on the step complexity of approximate agreement, even when the inputs are real numbers. On the positive side, we prove that a polylogarithmic dependence on n and S/? can be achieved, by presenting an approximate agreement algorithm with O(log n (log n + log(S/?))) step complexity. Our algorithm works for multidimensional domains. The step complexity can be further restricted to be in O(min{log n (log n + log (S/?)), log(M/?)}) when the inputs are real numbers
N-Person Bargaining and Strategic Complexity
We investigate the effect of introducing costs of complexity in the n -person unanimity bargaining game. In particular, the paper provides a justification for stationary equilibrium strategies in the class of games where complexity costs matter. As is well-known, in this game every individually rational allocation is sustainable as a Nash equilibrium (also as a subgame perfect equilibrium if players are sufficiently patient and if n > 2). Moreover, delays in agreement are also possible in such equilibria. By limiting ourselves to strategies that can be implemented by a machine (automaton) and by suitably modifying the definition of complexity for the purpose of analysing a single extensive form, we find that complexity costs do not reduce the range of possible allocations but they do limit the amount of delay that can occur in any agreement. In particular, we show that in any n-player game, for any allocation z; an agreement on z at any period t can be sustained as a Nash equilibrium of the machine game with complexity costs if and only if t · n: We use the limit on delay result to establish that, in equilibrium, the machines implement stationary strategies. Finally, we also show that noisy Nash equilibrium” with complexity costs sustain only the unique stationary subgame perfect equilibrium allocation.
- …