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Publication# The Inherent Price of Indulgence

Abstract

This paper presents a tight lower bound on the time complexity of indulgent consensus algorithms, i.e., consensus algorithms that use unreliable failure detectors. We state and prove our tight lower bound in the unifying framework of round-by-round fault detectors. We show that any P -based t-resilient consensus algorithm requires at least t+2 rounds for a global decision even in runs that are synchronous. We then prove the bound to be tight by exhibiting a new P-based t-resilient consensus algorithm that reaches a global decision at round t + 2 in every synchronous run. Our new algorithm is in this sense significantly faster than the most efficient indulgent algorithm we knew of (which requires 2t + 2 rounds). We contrast our lower bound with the well-known t + 1 round tight lower bound on consensus for the synchronous model, pointing out the price of indulgence.

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Related concepts (25)

Time complexity

In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor.

Complexity class

In computational complexity theory, a complexity class is a set of computational problems "of related resource-based complexity". The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time or memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements.

P (complexity)

In computational complexity theory, P, also known as PTIME or DTIME(nO(1)), is a fundamental complexity class. It contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. Cobham's thesis holds that P is the class of computational problems that are "efficiently solvable" or "tractable". This is inexact: in practice, some problems not known to be in P have practical solutions, and some that are in P do not, but this is a useful rule of thumb.

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