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The Topology of Local Computing in Networks

Pierre Fraigniaud, Ami Paz

TL;DR

This work tackles the challenge of applying algebraic topology to network computing in the LOCAL model, where node identifiers can cause combinatorial blowups. It introduces local protocol complexes whose size is independent of the network, and proves a topological characterization of LCL tasks on bounded-degree graphs: a $t$-round solution exists if and only if there is a name-preserving, name-independent simplicial map from a local protocol complex to the output complex. The main theorem formalizes a tight link between distributed solvability and topological maps, with a derived corollary reproof of Linial's $\Omega(\log^* n)$ lower bound for 3-coloring on rings. The framework extends to colorings, MIS, and related tasks, and highlights the utility of finite local models for understanding locality in networked distributed computation.

Abstract

Modeling distributed computing in a way enabling the use of formal methods is a challenge that has been approached from different angles, among which two techniques emerged at the turn of the century: protocol complexes, and directed algebraic topology. In both cases, the considered computational model generally assumes communication via shared objects, typically a shared memory consisting of a collection of read-write registers. Our paper is concerned with network computing, where the processes are located at the nodes of a network, and communicate by exchanging messages along the edges of that network. Applying the topological approach for verification in network computing is a considerable challenge, mainly because the presence of identifiers assigned to the nodes yields protocol complexes whose size grows exponentially with the size of the underlying network. However, many of the problems studied in this context are of local nature, and their definitions do not depend on the identifiers or on the size of the network. We leverage this independence in order to meet the above challenge, and present $\textit{local}$ protocol complexes, whose sizes do not depend on the size of the network. As an application of the design of "compact" protocol complexes, we reformulate the celebrated lower bound of $Ω(\log^*n)$ rounds for 3-coloring the $n$-node ring, in the algebraic topology framework.

The Topology of Local Computing in Networks

TL;DR

This work tackles the challenge of applying algebraic topology to network computing in the LOCAL model, where node identifiers can cause combinatorial blowups. It introduces local protocol complexes whose size is independent of the network, and proves a topological characterization of LCL tasks on bounded-degree graphs: a -round solution exists if and only if there is a name-preserving, name-independent simplicial map from a local protocol complex to the output complex. The main theorem formalizes a tight link between distributed solvability and topological maps, with a derived corollary reproof of Linial's lower bound for 3-coloring on rings. The framework extends to colorings, MIS, and related tasks, and highlights the utility of finite local models for understanding locality in networked distributed computation.

Abstract

Modeling distributed computing in a way enabling the use of formal methods is a challenge that has been approached from different angles, among which two techniques emerged at the turn of the century: protocol complexes, and directed algebraic topology. In both cases, the considered computational model generally assumes communication via shared objects, typically a shared memory consisting of a collection of read-write registers. Our paper is concerned with network computing, where the processes are located at the nodes of a network, and communicate by exchanging messages along the edges of that network. Applying the topological approach for verification in network computing is a considerable challenge, mainly because the presence of identifiers assigned to the nodes yields protocol complexes whose size grows exponentially with the size of the underlying network. However, many of the problems studied in this context are of local nature, and their definitions do not depend on the identifiers or on the size of the network. We leverage this independence in order to meet the above challenge, and present protocol complexes, whose sizes do not depend on the size of the network. As an application of the design of "compact" protocol complexes, we reformulate the celebrated lower bound of rounds for 3-coloring the -node ring, in the algebraic topology framework.

Paper Structure

This paper contains 20 sections, 4 theorems, 18 equations, 12 figures.

Table of Contents

  1. Context and Objective
  2. Objective.
  3. Our Results
  4. Models and Definitions
  5. The LOCAL model
  6. Locally Checkable Labelings (LCL)
  7. Warm Up: Coloring and MIS in the Ring
  8. Reduction from 3-coloring to MIS
  9. Remark.
  10. Task specification.
  11. Name-Independent Algorithms
  12. Impossibility in Zero Rounds
  13. Impossibility in One Round
  14. The 2-Round Algorithm
  15. General Case with IDs
  16. ...and 5 more sections

Key Result

Lemma 1

A task $(\mathcal{I},\mathcal{O},\Delta)$ is solvable in time $t$ if and only if there exists a simplicial map $\delta:\mathcal{P}^{(t)}\to \mathcal{O}$ such that, for every $\sigma\in\mathcal{I}$, $\delta(\Xi_t(\sigma))\subseteq \Delta(\sigma)$.

Figures (12)

  • Figure 1: (a) A chromatic subdivision of a 3-process simplex; (b) Subdivision for 1-resiliency; a triangle labeled, e.g., $\{i\}\{jk\}$ corresponds to the case in which $p_i$ writes and reads the memory without seeing $p_j$ and $p_k$, while $p_j$ and $p_k$ saw $p_i$ when they read after they wrote, and they also saw each other; all possible interleavings for one write-read instruction are displayed.
  • Figure 2: (a) The input complex of binary consensus for three processes; (b) The scissor cuts for the consistently directed 3-process cycle $C_3$ after one round; (c) The scissor cuts for the directed 3-process star $S_3$, where edges are directed from the center to the leaves, after one round.
  • Figure 3: The commutative diagrams of Lemma \ref{['lem:generic']} (left), and Theorem \ref{['theo:main-simplified']} (right).
  • Figure 4: Three consecutive nodes in the $n$-node ring.
  • Figure 5: The local complex $\mathcal{M}_2$ of MIS in the ring. (a) the vertices are labeled with the index of the processes and the values; (b) the indexes of the processes are replaced by colors.
  • ...and 7 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 2: A simplified version of Theorem \ref{['theo:main']}
  • Theorem 3: Ramsey's Theorem GRSS2015
  • Definition 4
  • Definition 5
  • Theorem 6