Revisionist Simulations: A New Approach to Proving Space Lower Bounds
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Determining the space complexity of x-obstruction-free k-set agreement for x≤ k is an open problem. In x-obstruction-free protocols, processes are required to return in executions where at most x processes take steps. The best known upper bound on the number of registers needed to solve this problem among n>k processes is n-k+x registers. No general lower bound better than 2 was known. We prove that any x-obstruction-free protocol solving k-set agreement among n>k processes uses at least (n-x)/(k+1-x)+1 registers. Our main tool is a simulation that serves as a reduction from the impossibility of deterministic wait-free k-set agreement: if a protocol uses fewer registers, then it is possible for k+1 processes to simulate the protocol and deterministically solve k-set agreement in a wait-free manner, which is impossible. A critical component of the simulation is the ability of simulating processes to revise the past of simulated processes. We introduce a new augmented snapshot object, which facilitates this. We also prove that any space lower bound on the number of registers used by obstruction-free protocols applies to protocols that satisfy nondeterministic solo termination. Hence, our lower bound of (n-1)/k+1 for the obstruction-free (x=1) case also holds for randomized wait-free free protocols. In particular, this gives a tight lower bound of exactly n registers for solving obstruction-free and randomized wait-free consensus. Finally, our new techniques can be applied to get a space lower of n/2+1 for ϵ-approximate agreement, for sufficiently small ϵ. It requires participating processes to return values within ϵ of each other. The best known upper bounds are (1/ϵ) and n, while no general lower bounds were known.
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