New Classes of Distributed Time Complexity
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A number of recent papers -- e.g. Brandt et al. (STOC 2016), Chang et al. (FOCS 2016), Ghaffari & Su (SODA 2017), Brandt et al. (PODC 2017), and Chang & Pettie (FOCS 2017) -- have advanced our understanding of one of the most fundamental questions in theory of distributed computing: what are the possible time complexity classes of LCL problems in the LOCAL model? In essence, we have a graph problem Π in which a solution can be verified by checking all radius-O(1) neighbourhoods, and the question is what is the smallest T such that a solution can be computed so that each node chooses its own output based on its radius-T neighbourhood. Here T is the distributed time complexity of Π. The time complexity classes for deterministic algorithms in bounded-degree graphs that are known to exist by prior work are Θ(1), Θ(^* n), Θ( n), Θ(n^1/k), and Θ(n). It is also known that there are two gaps: one between ω(1) and o(^* n), and another between ω(^* n) and o( n). It has been conjectured that many more gaps would exist, and that the overall time hierarchy would be relatively simple -- indeed, this is known to be the case in restricted graph families such as cycles and grids. We show that the picture is much more diverse than previously expected. We present a general technique for engineering LCL problems with numerous different deterministic time complexities, including Θ( ^p/q n ), 2^Θ( ^q/p n ), and Θ(n^pq/(p+q)^2) in the high end and Θ( ^p/q^* n ), 2^Θ( ^q/p^* n ), and Θ((^* n)^q/p) in the low end of the complexity spectrum (p > q).
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