Symmetry breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open. In this paper we work in the LOCAL model (where the input graph and underlying distributed network are identical) and study the randomized complexity of four fundamental symmetry breaking problems on graphs: computing MISs (maximal independent sets), maximal matchings, vertex colorings, and ruling sets. A small sample of our results includes• An MIS algorithm running in O(log 2 ∆ + 2 O( √ log log n) ) time, where ∆ is the maximum degree. This is the first MIS algorithm to improve on the 1986 algorithms of Luby and Alon, Babai, and Itai, when log n ∆ 2 √ log n , and comes close to the Ω(log ∆) lower bound of Kuhn, Moscibroda, and Wattenhofer.• A maximal matching algorithm running in O(log ∆ + log 4 log n) time. This is the first significant improvement to the 1986 algorithm of Israeli and Itai. Moreover, its dependence on ∆ is provably optimal. • A (∆ + 1)-coloring algorithm requiring O(log ∆ + 2 O( IntroductionBreaking symmetry is one of the central themes in the theory of distributed computing. At initialization the nodes of a distributed system are assumed to be in the same state, possibly with distinct * A preliminary version of this paper appeared in the

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.

Abstract. An (α, β)-spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)-spanner of size O(n 1+1/k ) and an (additive) (1, 2)-spanner of size O(n 3/2 ). However no other additive spanners are known to exist.In this article we develop a couple of new techniques for constructing (α, β)-spanners. Our first result is an additive (1, 6)-spanner of size O(n 4/3 ). The construction algorithm can be understood as an economical agent that assigns costs and values to paths in the graph, purchasing affordable paths and ignoring expensive ones, which are intuitively well approximated by paths already purchased. We show that this path buying algorithm can be parameterized in different ways to yield other sparseness-distortion tradeoffs. Our second result addresses the problem of which (α, β)-spanners can be computed efficiently, ideally in linear time. We show that, for any k, a (k, k − 1)-spanner with size O(kn 1+1/k ) can be found in linear time, and, further, that in a distributed network the algorithm terminates in a constant number of rounds. Previous spanner constructions with similar performance had roughly twice the multiplicative distortion.

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