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The Local Information Cost of Distributed Graph Spanners

Peter Robinson

TL;DR

The paper introduces local information cost (LIC) as a fundamental, problem-agnostic measure of how much information nodes in a distributed network must learn to solve a graph problem, and proves LIC lower bounds translate into concrete communication and time lower bounds in KT1 CONGEST, asynchronous KT1, and related models. It develops a general technique to bound LIC by embedding a two-part graph into a lower-bound construction for graph spanners, using entropy-based arguments and traversal sequences to show many random edges are essential while most static edges are not, yielding a nontrivial lower bound for constructing a (2t−1)-spanner with O(n^{1+1/t+ε}) edges. The main result shows LIC_{1/n}((2t−1)-spanner) = Ω((1/t^2) n^{1+1/(2t)} log n) for suitable t, implying that any polynomial-time spanner algorithm in KT1 CONGEST must incur a super-constant, n-dependent communication burden and cannot be both time- and communication-optimal. Together with reductions to node-capacitated clique and gossip models, the work demonstrates a sharp time–communication trade-off and highlights fundamental limits of spanner construction under KT1, guiding future exploration of optimality gaps and information-theoretic barriers in distributed graph algorithms.

Abstract

We introduce the \emph{local information cost} (LIC), which quantifies the amount of information that nodes in a network need to learn when solving a graph problem. We show that the local information cost presents a natural lower bound on the communication complexity of distributed algorithms. For the synchronous CONGEST KT1 model, where each node has initial knowledge of its neighbors' IDs, we prove that $Ω(\frac{\text{LIC}_γ(P)}{\logτ\log n})$ bits are required for solving a graph problem $P$ with a $τ$-round algorithm that errs with probability at most $γ$. Our result is the first lower bound that yields a general trade-off between communication and time for graph problems in the CONGEST KT1 model. We demonstrate how to apply the local information cost by deriving a lower bound on the communication complexity of computing a spanner with multiplicative stretch $2t-1$ that consists of at most $O(n^{1+\frac{1}{t} + ε})$ edges, where $ε= O( {1}/{t^2} )$. More concretely, we show that any $O(\text{poly}(n))$-time spanner algorithm must send at least $\tildeΩ(\tfrac{1}{t^2} n^{1+{1}/{2t}})$ bits. Previously, only a trivial lower bound of $\tilde Ω(n)$ bits was known for this problem. (See PDF for the full abstract.)

The Local Information Cost of Distributed Graph Spanners

TL;DR

The paper introduces local information cost (LIC) as a fundamental, problem-agnostic measure of how much information nodes in a distributed network must learn to solve a graph problem, and proves LIC lower bounds translate into concrete communication and time lower bounds in KT1 CONGEST, asynchronous KT1, and related models. It develops a general technique to bound LIC by embedding a two-part graph into a lower-bound construction for graph spanners, using entropy-based arguments and traversal sequences to show many random edges are essential while most static edges are not, yielding a nontrivial lower bound for constructing a (2t−1)-spanner with O(n^{1+1/t+ε}) edges. The main result shows LIC_{1/n}((2t−1)-spanner) = Ω((1/t^2) n^{1+1/(2t)} log n) for suitable t, implying that any polynomial-time spanner algorithm in KT1 CONGEST must incur a super-constant, n-dependent communication burden and cannot be both time- and communication-optimal. Together with reductions to node-capacitated clique and gossip models, the work demonstrates a sharp time–communication trade-off and highlights fundamental limits of spanner construction under KT1, guiding future exploration of optimality gaps and information-theoretic barriers in distributed graph algorithms.

Abstract

We introduce the \emph{local information cost} (LIC), which quantifies the amount of information that nodes in a network need to learn when solving a graph problem. We show that the local information cost presents a natural lower bound on the communication complexity of distributed algorithms. For the synchronous CONGEST KT1 model, where each node has initial knowledge of its neighbors' IDs, we prove that bits are required for solving a graph problem with a -round algorithm that errs with probability at most . Our result is the first lower bound that yields a general trade-off between communication and time for graph problems in the CONGEST KT1 model. We demonstrate how to apply the local information cost by deriving a lower bound on the communication complexity of computing a spanner with multiplicative stretch that consists of at most edges, where . More concretely, we show that any -time spanner algorithm must send at least bits. Previously, only a trivial lower bound of bits was known for this problem. (See PDF for the full abstract.)

Paper Structure

This paper contains 17 sections, 18 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Results
  3. Technical Approach of the Lower Bound for Graph Spanners
  4. Can We Use 2-Party Communication Complexity?
  5. Additional Related Work
  6. Roadmap
  7. Distributed Computing Models
  8. Local Information Cost
  9. Lower Bounds for Distributed Algorithms via Local Information Cost
  10. Limitations of the $\mathop{\mathrm{\mathsf{LIC}}}\nolimits$ Approach
  11. A Lower Bound Construction for Spanners
  12. Reachability and Critical Edges
  13. Critical Edges
  14. Proof of Lemma \ref{['lem:reachability']}
  15. Traversal Sequences and Reachable Sets
  16. ...and 2 more sections

Key Result

Lemma 1

Consider any $\gamma$-error algorithm $\mathcal{A}$ as stated in Definition def:lic. Then,

Figures (1)

  • Figure 1: A simplified instance of the lower bound graph sampled according to $\mathcal{G}_k$ with randomly assigned node IDs. The region of $u_1$ consists of nodes $u_1,\dots,u_4,v_1,\dots,v_4$. Assuming that $k=5$, the edge $(u_1,u_{i+1})$ is critical as any other path from $u_1$ to $u_{i+1}$ has length greater than $5$. On the other hand, the edges $(u_1,u_5)$ and $(u_1,u_{10})$ are not critical as they are both reachable by traversal sequences of length (at most) $k$: after removing $(u_1,u_5$), node $u_5$ is reachable via the traversal sequence $RBBR$. Similarly, if we discard $(u_1,u_{10})$, then $u_{10}$ is still in the reachable set $\mathcal{R}(RRBB)$.

Theorems & Definitions (30)

  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 20 more