[Paper Review] An Exponential Separation Between Randomized and Deterministic Complexity in the LOCAL Model

Yi‐Jun Chang, Tsvi Kopelowitz+1 more|arXiv (Cornell University)|Feb 26, 2016

Complexity and Algorithms in Graphs25 references18 citations

TL;DR

This paper establishes an exponential separation between randomized and deterministic algorithms in the LOCAL model by proving that Δ-coloring trees requires Θ(logΔ log n) rounds with randomization but Θ(logΔ n) rounds deterministically for Δ ≥ 55. The result demonstrates that randomness provides a superpolynomial speedup for symmetry-breaking problems and shows that graph shattering is essential for achieving sub-polynomial randomized complexity.

ABSTRACT

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 connections between the deterministic and randomized complexities in the LOCAL model. Each result has a very compelling take-away message: 1. Fast $Δ$-coloring of trees requires random bits: Building on the recent lower bounds of Brandt et al., we prove that the randomized complexity of $Δ$-coloring a tree with maximum degree $Δ\ge 55$ is $Θ(\log_Δ\log n)$, whereas its deterministic complexity is $Θ(\log_Δn)$ for any $Δ\ge 3$. This also establishes a large separation between the deterministic complexity of $Δ$-coloring and $(Δ+1)$-coloring trees. 2. Randomized lower bounds imply deterministic lower bounds: We prove that any deterministic algorithm for a natural class of problems that runs in $O(1)+o(\log_Δn)$ rounds can be transformed to run in $O(\log^*n-\log^*Δ+1)$ rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires $Ω(\log_Δn)$ time deterministically. 3. 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 $\sqrt{\log n}$. This shows that a deterministic $Ω(\log_Δn)$ lower bound for any problem implies a randomized $Ω(\log_Δ\log n)$ lower bound. It also illustrates that the graph shattering technique is absolutely essential to the LOCAL model.

Motivation & Objective

• To resolve the long-standing question of whether randomness provides a superpolynomial speedup in the LOCAL model for symmetry-breaking problems.
• To formally establish that the exponential gap between randomized and deterministic complexity is necessary, not just observed in practice.
• To prove that randomized lower bounds imply deterministic lower bounds and vice versa, revealing deep structural connections between the two complexity classes.
• To demonstrate that the graph shattering technique is essential for achieving fast randomized algorithms in the LOCAL model.
• To show that for Δ ≥ 55, Δ-coloring of trees is exponentially faster in the randomized setting than in the deterministic setting.

Proposed method

• The authors analyze the randomized complexity of Δ-coloring trees using a three-phase algorithm: (1) iteratively coloring vertices via local minima and maximal independent sets (MIS), (2) 3-coloring residual components of bounded size using Theorem 9, and (3) completing the coloring via another MIS application.
• They use probabilistic arguments and Chernoff bounds to show that uncolored vertices with exactly three uncolored neighbors form small connected components with high probability.
• They prove that the probability a distant-3 set of size t is fully contained in the residual set S is at most (1−ε)/(4Δ³)^t, enabling a union bound over all such sets.
• They apply Lemma 3 to bound the number of distant-3 sets and use a high-probability concentration argument to show that S forms components of size O(log n) with high probability.
• They derive a randomized lower bound of Ω(logΔ log n) from a deterministic lower bound of Ω(logΔ n) on smaller instances, using a reduction from √log n-sized graphs.
• They introduce a transformation that converts any deterministic O(1)+o(logΔ n)-round algorithm into one running in O(log*n − log*Δ + 1) rounds, enabling deterministic lower bound derivation from randomized lower bounds.

Experimental results

Research questions

• RQ1 Is the exponential gap between randomized and deterministic complexity in the LOCAL model necessary, or could deterministic algorithms eventually match randomized performance?
• RQ2 Can randomized lower bounds be used to infer deterministic lower bounds, and vice versa, in the LOCAL model?
• RQ3 Why is the graph shattering technique essential for achieving fast randomized algorithms in the LOCAL model?
• RQ4 What is the precise randomized complexity of Δ-coloring trees with maximum degree Δ ≥ 55?
• RQ5 Can deterministic complexity lower bounds be used to derive nontrivial randomized lower bounds?

Key findings

• For Δ ≥ 55, Δ-coloring of trees has a randomized complexity of Θ(logΔ log n), which is exponentially smaller than the deterministic complexity of Θ(logΔ n).
• The randomized complexity of any problem is at least its deterministic complexity on instances of size √log n, implying that a deterministic Ω(logΔ n) lower bound implies a randomized Ω(logΔ log n) lower bound.
• The graph shattering technique is provably essential: improving the 2^{O(√log n)}-round complexity of the Panconesi-Srinivasan algorithm would require improving the 2^{O(√log log n)} term in the best-known randomized algorithms.
• Any deterministic algorithm for a natural class of problems that runs in O(1)+o(logΔ n) rounds can be transformed to run in O(log*n − log*Δ + 1) rounds, enabling deterministic lower bounds to be derived from randomized lower bounds.
• The set of vertices with exactly three uncolored neighbors after Phase 1 forms connected components of size O(log n) with high probability, enabling efficient 3-coloring in Phase 2.
• The probability that a distant-3 set of size t is fully contained in the residual set S is at most ((1−ε)/(4Δ³))^t, which decays exponentially with t, ensuring small component sizes with high probability.