[Paper Review] Shared Randomness Helps with Local Distributed Problems

Alkida Balliu, Mohsen Ghaffari+6 more|arXiv (Cornell University)|Jul 7, 2024

Optimization and Search Problems1 citations

TL;DR

This paper demonstrates that shared randomness can exponentially reduce the round complexity of a locally checkable labeling (LCL) problem Π in the distributed LOCAL model: while private randomness requires Ω(√n) rounds, shared randomness enables a solution in O(log n) rounds. The result resolves long-standing open questions by showing shared randomness is not just helpful but necessary for certain LCL problems, and it establishes separations between quantum, classical, and distributional models of computation.

ABSTRACT

By prior work, we have many results related to distributed graph algorithms for problems that can be defined with local constraints; the formal framework used in prior work is locally checkable labeling problems (LCLs), introduced by Naor and Stockmeyer in the 1990s. It is known, for example, that if we have a deterministic algorithm that solves an LCL in $o(\log n)$ rounds, we can speed it up to $O(\log^*n)$ rounds, and if we have a randomized $O(\log^*n)$ rounds algorithm, we can derandomize it for free. It is also known that randomness helps with some LCL problems: there are LCL problems with randomized complexity $Θ(\log\log n)$ and deterministic complexity $Θ(\log n)$. However, so far there have not been any LCL problems in which the use of shared randomness has been necessary; in all prior algorithms it has been enough that the nodes have access to their own private sources of randomness. Could it be the case that shared randomness never helps with LCLs? Could we have a general technique that takes any distributed graph algorithm for any LCL that uses shared randomness, and turns it into an equally fast algorithm where private randomness is enough? In this work we show that the answer is no. We present an LCL problem $Π$ such that the round complexity of $Π$ is $Ω(\sqrt n)$ in the usual randomized \local model with private randomness, but if the nodes have access to a source of shared randomness, then the complexity drops to $O(\log n)$. As corollaries, we also resolve several other open questions related to the landscape of distributed computing in the context of LCL problems. In particular, problem $Π$ demonstrates that distributed quantum algorithms for LCL problems strictly benefit from a shared quantum state. Problem $Π$ also gives a separation between finitely dependent distributions and non-signaling distributions.

Motivation & Objective

• To investigate whether shared randomness provides a computational advantage for locally checkable labeling (LCL) problems in distributed systems.
• To determine if there exists an LCL problem where shared randomness is provably necessary, beyond what private randomness can achieve.
• To resolve open questions about the power of shared randomness, finitely dependent distributions, and non-signaling distributions in the context of distributed LCL problems.
• To establish a separation between classical randomized algorithms with private vs. shared randomness in the LOCAL model.

Proposed method

• Construct an LCL problem Π defined on grid-structured graphs with specific local constraints on node outputs.
• Prove that in the private-randomness LOCAL model, any algorithm solving Π requires Ω(√n) rounds using a probabilistic argument based on independence of left- and right-most column nodes.
• Show that with access to shared randomness, a randomized algorithm can solve Π in O(log n) rounds by coordinating outputs across distant nodes using a common random string.
• Use the bounded-dependence and online-LOCAL models to establish lower bounds, demonstrating that the complexity gap persists even under stronger assumptions.
• Leverage the problem Π to derive implications for quantum algorithms and distributional models, showing that shared randomness enables advantages not achievable with private randomness.
• Apply combinatorial and probabilistic arguments, including union bounds and concentration inequalities, to upper- and lower-bound success probabilities in adversarial settings.

Experimental results

Research questions

• RQ1 Is there an LCL problem for which shared randomness provides a superpolynomial speedup over private randomness in the LOCAL model?
• RQ2 Can shared randomness be necessary for solving some LCL problems, or is private randomness always sufficient?
• RQ3 Does the existence of such a problem imply a separation between finitely dependent and non-signaling distributions in distributed computing?
• RQ4 Can distributed quantum algorithms for LCL problems benefit from shared quantum states in a way that classical algorithms cannot?
• RQ5 Is there a general derandomization technique that removes the need for shared randomness in LCL algorithms?

Key findings

• The LCL problem Π requires Ω(√n) rounds in the private-randomness LOCAL model, even for randomized algorithms.
• With access to shared randomness, the same problem Π can be solved in O(log n) rounds, demonstrating an exponential speedup.
• The problem Π establishes a separation between finitely dependent distributions and non-signaling distributions in distributed computing.
• The problem Π shows that distributed quantum algorithms for LCL problems strictly benefit from shared quantum states, as classical algorithms cannot achieve the same efficiency without shared randomness.
• The result implies that shared randomness is not just helpful but necessary for certain LCL problems, contradicting prior intuition that private randomness suffices.
• The lower bounds hold even in stronger models such as the deterministic online-LOCAL and bounded-dependence models, confirming the robustness of the complexity gap.