## Abstract

An anonymous shared memory REG can be seen as an array of atomic registers such that there is no a priori agreement among the processes on the names of the registers. As an example a very same physical register can be known as REG[x] by a process p and as REG[y] (where y ≠ x) by another process q. Moreover, the register known as REG[a] by a process p and the register known as REG[b] by a process q can be the same physical register. It is assumed that each process has a unique identifier that can only be compared for equality. This article is on solving the d-election problem, in which it is required to elect at least one and at most d leaders, in such an anonymous shared memory system. We notice that the 1-election problem is the familiar leader election problem. Let n be the number of processes and m the size of the anonymous memory (number of atomic registers). The article shows that the condition gcd(m, n) ≤ d is necessary and sufficient for solving the d-election problem, where communication is through read/write or read+modify+write registers. The algorithm used to prove the sufficient condition relies on Bezout's Identity - a Diophantine equation relating numbers according to their Greatest Common Divisor. Furthermore, in the process of proving the sufficient condition, it is shown that 1-leader election can be solved using only a single read/write register (which refutes a 1989 conjecture stating that three non-anonymous registers are necessary), and that the exact d-election problem, where exactly d leaders must be elected, can be solved if and only if gcd(m, n) divides d.

Original language | English |
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Title of host publication | PODC 2020 - Proceedings of the 39th Symposium on Principles of Distributed Computing |

Publisher | Association for Computing Machinery |

Pages | 41-50 |

Number of pages | 10 |

ISBN (Electronic) | 9781450375825 |

DOIs | |

State | Published - 31 Jul 2020 |

Externally published | Yes |

Event | 39th Symposium on Principles of Distributed Computing, PODC 2020 - Virtual, Online, Italy Duration: 3 Aug 2020 → 7 Aug 2020 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Principles of Distributed Computing |
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### Conference

Conference | 39th Symposium on Principles of Distributed Computing, PODC 2020 |
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Country/Territory | Italy |

City | Virtual, Online |

Period | 3/08/20 → 7/08/20 |

### Bibliographical note

Publisher Copyright:© 2020 ACM.

## Keywords

- RW register
- anonymous register
- asynchrony
- atomic register
- bezout identity
- bounded register
- concurrent algorithm
- equality-based comparison
- leader election
- process identity
- symmetric algorithm
- symmetry breaking