Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish numerous connections between the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message:Fast ∆-coloring of trees requires random bits. Building on the recent randomized lower bounds of Brandt et al.[11], we prove that the randomized complexity of ∆-coloring a tree with maximum degree ∆ is Θ(log ∆ log n), for any ∆ ≥ 55, whereas its deterministic complexity is Θ(log ∆ n) for any ∆ ≥ 3. 1 This also establishes a large separation between the deterministic complexity of ∆-coloring and (∆ + 1)-coloring trees.Randomized lower bounds imply deterministic lower bounds. We prove that any deterministic algorithm for a natural class of problems that runs in O(1) + o(log ∆ n) rounds can be transformed to run in O(log * n − log * ∆ + 1) rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires Ω(log ∆ n) time deterministically. (This gives an alternate proof that deterministically ∆-coloring a tree with small ∆ takes Ω(log ∆ n) rounds.)Deterministic lower bounds imply randomized lower bounds. We prove that the randomized complexity of any natural problem on instances of size n is at least its deterministic complexity on instances of size √ log n. This shows that a deterministic Ω(log ∆ n) lower bound for any problem (∆-coloring a tree, for example) implies a randomized Ω(log ∆ log n) lower bound. It also illustrates that the graph shattering technique employed in recent randomized symmetry breaking algorithms is absolutely essential to the LOCAL model. For example, it is provably impossible to improve the 2 O( √ log log n) terms in the complexities of the best MIS and (∆ + 1)-coloring algorithms without also improving the 2 O( √ log n) -round Panconesi-Srinivasan algorithms.

Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish numerous connections between the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message:Fast ∆-coloring of trees requires random bits. Building on the recent randomized lower bounds of Brandt et al. [11], we prove that the randomized complexity of ∆-coloring a tree with maximum degree ∆ is Θ(log ∆ log n), for any ∆ ≥ 55, whereas its deterministic complexity is Θ(log ∆ n) for any ∆ ≥ 3. 1 This also establishes a large separation between the deterministic complexity of ∆-coloring and (∆ + 1)-coloring trees.Randomized lower bounds imply deterministic lower bounds. We prove that any deterministic algorithm for a natural class of problems that runs in O(1) + o(log ∆ n) rounds can be transformed to run in O(log * n − log * ∆ + 1) rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires Ω(log ∆ n) time deterministically. (This gives an alternate proof that deterministically ∆-coloring a tree with small ∆ takes Ω(log ∆ n) rounds.)Deterministic lower bounds imply randomized lower bounds. We prove that the randomized complexity of any natural problem on instances of size n is at least its deterministic complexity on instances of size √ log n. This shows that a deterministic Ω(log ∆ n) lower bound for any problem (∆-coloring a tree, for example) implies a randomized Ω(log ∆ log n) lower bound. It also illustrates that the graph shattering technique employed in recent randomized symmetry breaking algorithms is absolutely essential to the LOCAL model. For example, it is provably impossible to improve the 2 O( √ log log n) terms in the complexities of the best MIS and (∆ + 1)-coloring algorithms without also improving the 2 O( √ log n) -round Panconesi-Srinivasan algorithms.

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The celebrated Time Hierarchy Theorem for Turing machines states, informally, that more problems can be solved given more time. The extent to which a time hierarchy-type theorem holds in the classic distributed LOCAL model has been open for many years. In particular, it is consistent with previous results that all natural problems in the LOCAL model can be classified according to a small constant number of complexities, such as Op1q, Oplog˚nq, Oplog nq, 2 Op ? log nq , etc. In this paper we establish the first time hierarchy theorem for the LOCAL model and prove that several gaps exist in the LOCAL time hierarchy. Our main results are as follows:• We define an infinite set of simple coloring problems called Hierarchical 2 1 2 -Coloring. A correctly colored graph can be confirmed by simply checking the neighborhood of each vertex, so this problem fits into the class of locally checkable labeling (LCL) problems. However, the complexity of the klevel Hierarchical 2 1 2 -Coloring problem is Θpn 1{k q, for k P Z`. The upper and lower bounds hold for both general graphs and trees, and for both randomized and deterministic algorithms.• Consider any LCL problem on bounded degree trees. We prove an automatic-speedup theorem that states that any randomized n op1q -time algorithm solving the LCL can be transformed into a deterministic Oplog nq-time algorithm. Together with a previous result [6], this establishes that on trees, there are no natural deterministic complexities in the ranges ωplog˚nq-oplog nq or ωplog nqn op1q .• We expose a gap in the randomized time hierarchy on general graphs. Roughly speaking, any randomized algorithm that solves an LCL problem in sublogarithmic time can be sped up to run in OpTLLLq time, which is the complexity of the distributed Lovász local lemma problem, currently known to be Ωplog log nq and Oplog nq.Finally, we revisit Naor and Stockmeyer's characterization of Op1q-time LOCAL algorithms for LCL problems (as order-invariant w.r.t. vertex IDs) and calculate the complexity gaps that are directly implied by their proof. For n-rings we see a ωp1q-oplog˚nq complexity gap, for p ? nˆ?nq-tori an ωp1qop a log˚nq gap, and for bounded degree trees and general graphs, an ωp1q-oplogplog˚nqq complexity gap.The goal of this paper is to understand the spectrum of natural problem complexities that can exist in the LOCAL model [31,37] of distributed computation, and to quantify the value of randomness in this model. Whereas the time hierarchy of Turing machines is known 1 to be very "smooth", recent work [6,5] has exhibited strange gaps in the LOCAL complexity hierarchy. Indeed, prior to this work it was not even known if the LOCAL model could support more than a small constant number of problem complexities. Before surveying prior work in this area, let us formally define the deterministic and randomized variants of the LOCAL model, and the class of locally checkable labeling (LCL) problems, which are intuitively those graph problems that can be computed locally in nondeterministic constant time.In bo...

Vertex coloring is one of the classic symmetry breaking problems studied in distributed computing. In this paper we present a new algorithm for (∆ + 1)-list coloring in the randomized LOCAL model running in O(Detd(poly log n)) time, where Detd(n ) is the deterministic complexity of (deg +1)-list coloring on n -vertex graphs. (In this problem, each v has a palette of size deg(v) + 1.) This improves upon a previous randomized algorithm of Harris, Schneider, and Su [STOC 2016, J. ACM 2018] with complexity O( √ log ∆ + log log n + Detd(poly log n)), and, for some range of ∆, is much faster than the best known deterministic algorithm of Fraigniaud, Heinrich, and Kosowski [FOCS 2016] and Barenboim, Elkin, and Goldenberg [PODC 2018], with complexity O( √ ∆ log ∆ log * ∆ + log * n). Our algorithm appears to be optimal, in view of the Ω(Det(poly log n)) randomized lower bound due to Chang, Kopelowitz, and Pettie [FOCS 2016, SIAM J. Comput. 2019], where Det is the deterministic complexity of (∆ + 1)-list coloring. At present, the best upper bounds on Detd(n ) and Det(n ) are both 2 O √ log n and use a black box application of network decompositions due to Panconesi and Srinivasan [J.Algorithms 1996]. It is quite possible that the true deterministic complexities of both problems are the same, asymptotically, which would imply the randomized optimality of our (∆ + 1)-list coloring algorithm.1 In the case of MIS, the subproblems actually have size poly(∆) log n, but satisfy the additional property that they contain distance-5 dominating sets of size O(log n), which is often just as good as having poly log(n) size. See [9, §3] or [21, §4] for more discussion of this.2 See [33,14,12] for the formal definition of the class of locally checkable labeling (LCL) problems.