New classes of distributed time complexity

Abstract

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(log∗ n), Θ(log n), Θ(n1/k), and Θ(n). It is also known that there are two gaps: one between ω(1) and o(log log∗ n), and another between ω(log∗ n) and o(log n). It has been conjectured that many more gapsexist, and that the overall time hierarchy is 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(logα n) for any α ≥ 1, 2Θ(log^α n) for any α ≤ 1, and Θ(nα) for any α < 1/2 in the high end of the complexity spectrum, and Θ(logα log∗ n) for any α ≥ 1, 2Θ(log^α log^∗ n) for any α ≤ 1, and Θ((log∗ n)α) for any α ≤ 1 in the low end of the complexity spectrum; here α is a positive rational number.