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Almost Optimal Algorithms for Token Collision in Anonymous Networks

Sirui Bai, Xinyu Fu, Xudong Wu, Penghui Yao, Chaodong Zheng

TL;DR

Token collision in anonymous networks is analyzed under the CONGEST model. The authors present near-optimal deterministic algorithms with time $\tilde{O}(D+kL/\log n)$ and a pipeline extension for large $L$, plus a near-optimal randomized algorithm, with matching lower bounds and impossibility results. The core technique builds BFS trees using identifier-based leadership and uses an identifier-induced forest to convergecast tokens and detect collisions, with a pipeline variant for large tokens. These results illuminate symmetry-breaking limits in anonymous networks and have implications for randomized unique identifiers, set-disjointness reductions, and practical distributed verification.

Abstract

In distributed systems, situations often arise where some nodes each holds a collection of tokens, and all nodes collectively need to determine whether all tokens are distinct. For example, if each token represents a logged-in user, the problem corresponds to checking whether there are duplicate logins. Similarly, if each token represents a data object or a timestamp, the problem corresponds to checking whether there are conflicting operations in distributed databases. In distributed computing theory, unique identifiers generation is also related to this problem: each node generates one token, which is its identifier, then a verification phase is needed to ensure all identifiers are unique. In this paper, we formalize and initiate the study of token collision. In this problem, a collection of $k$ tokens, each represented by some length-$L$ bit string, are distributed to $n$ nodes of an anonymous CONGEST network in an arbitrary manner. The nodes need to determine whether there are tokens with an identical value. We present near optimal deterministic algorithms for the token collision problem with $\tilde{O}(D+k\cdot L/\log{n})$ round complexity, where $D$ denotes the network diameter. Besides high efficiency, the prior knowledge required by our algorithms is also limited. For completeness, we further present a near optimal randomized algorithm for token collision.

Almost Optimal Algorithms for Token Collision in Anonymous Networks

TL;DR

Token collision in anonymous networks is analyzed under the CONGEST model. The authors present near-optimal deterministic algorithms with time and a pipeline extension for large , plus a near-optimal randomized algorithm, with matching lower bounds and impossibility results. The core technique builds BFS trees using identifier-based leadership and uses an identifier-induced forest to convergecast tokens and detect collisions, with a pipeline variant for large tokens. These results illuminate symmetry-breaking limits in anonymous networks and have implications for randomized unique identifiers, set-disjointness reductions, and practical distributed verification.

Abstract

In distributed systems, situations often arise where some nodes each holds a collection of tokens, and all nodes collectively need to determine whether all tokens are distinct. For example, if each token represents a logged-in user, the problem corresponds to checking whether there are duplicate logins. Similarly, if each token represents a data object or a timestamp, the problem corresponds to checking whether there are conflicting operations in distributed databases. In distributed computing theory, unique identifiers generation is also related to this problem: each node generates one token, which is its identifier, then a verification phase is needed to ensure all identifiers are unique. In this paper, we formalize and initiate the study of token collision. In this problem, a collection of tokens, each represented by some length- bit string, are distributed to nodes of an anonymous CONGEST network in an arbitrary manner. The nodes need to determine whether there are tokens with an identical value. We present near optimal deterministic algorithms for the token collision problem with round complexity, where denotes the network diameter. Besides high efficiency, the prior knowledge required by our algorithms is also limited. For completeness, we further present a near optimal randomized algorithm for token collision.
Paper Structure (29 sections, 26 theorems, 1 equation, 3 figures)

This paper contains 29 sections, 26 theorems, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. Results and contribution
  3. Deterministic scenario.
  4. Randomized scenario.
  5. Related work and discussion
  6. Preliminary
  7. The Deterministic Algorithm
  8. Build BFS-tree(s).
  9. Detect token collision.
  10. Analysis of the Deterministic Algorithm
  11. Correctness
  12. Complexity
  13. Proof of the main theorem
  14. Generalizing the Deterministic Algorithm when Tokens are Large
  15. Algorithm description
  16. ...and 14 more sections

Key Result

Theorem 2

In an $n$-node anonymous CONGEST network with diameter $D$, for any instance of the token collision problem in which $k$ tokens are encoded by length-$L$ bit strings, if every node knows the exact value of $n$ or $k$, then there exists an $O(D+k\cdot L/\log n)$-round deterministic algorithm when $L=

Figures (3)

  • Figure 2: Pseudocode of the deterministic token collision algorithm.
  • Figure 3: Pseudocode of the deterministic token collision algorithm for large tokens.
  • Figure 4: Procedure BFS-Tree-Building and Token-Collision-Detection for large tokens.

Theorems & Definitions (31)

  • Definition 1: Token Collision
  • Theorem 2: Deterministic Upper Bound, Part 1
  • Theorem 3: Deterministic Upper Bound, Part 2
  • Theorem 4: Deterministic Lower Bound
  • Theorem 5: Impossibility Result
  • Theorem 6: Randomized Upper and Lower Bound
  • Definition 8: Identifier-induced Graph
  • Lemma 9
  • Lemma 10
  • Lemma 11
  • ...and 21 more