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Faster Symmetric Rendezvous on Four or More Locations

Javier Cembrano, Felix Fischer, Max Klimm

Abstract

In the symmetric rendezvous problem two players follow the same (randomized) strategy to visit one of $n$ locations in each time step $t=0,1,2,\dots$. Their goal is to minimize the expected time until they visit the same location and thus meet. Anderson and Weber [J. Appl. Prob., 1990] proposed a strategy that operates in rounds of $n-1$ steps: a player either remains in one location for $n-1$ steps or visits the other $n-1$ locations in random order; the choice between these two options is made with a probability that depends only on $n$. The strategy is known to be optimal for $n=2$ and $n=3$, and there is convincing evidence that it is not optimal for $n=4$. We show that it is not optimal for any $n\geq 4$, by constructing a strategy with a smaller expected meeting time.

Faster Symmetric Rendezvous on Four or More Locations

Abstract

In the symmetric rendezvous problem two players follow the same (randomized) strategy to visit one of locations in each time step . Their goal is to minimize the expected time until they visit the same location and thus meet. Anderson and Weber [J. Appl. Prob., 1990] proposed a strategy that operates in rounds of steps: a player either remains in one location for steps or visits the other locations in random order; the choice between these two options is made with a probability that depends only on . The strategy is known to be optimal for and , and there is convincing evidence that it is not optimal for . We show that it is not optimal for any , by constructing a strategy with a smaller expected meeting time.

Paper Structure

This paper contains 28 sections, 26 theorems, 138 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Structure of the Paper
  4. Open Questions
  5. Further Related Work
  6. Preliminaries
  7. Home Location and Round-based Strategies
  8. Shifted Derangements
  9. Warm-Up: Correlating Consecutive Permutations
  10. Rendezvous with Visibility
  11. Beating the Uniform Strategy
  12. Beating the Anderson--Weber Strategy
  13. Meeting Probabilities
  14. Expected Meeting Time
  15. Proofs Deferred from \ref{['sec:prelims']}
  16. ...and 13 more sections

Key Result

Lemma 1

For all $n \in \mathbb{N}_0$ and $k \in \{0,\dots,n\}$, the entry $d_n^k$ is equal to the number of permutations $\pi \in \mathcal{P}(\{1+k,\dots,n+k\})$ with $\pi(i) \neq i$ for all $i\in [n]$.

Figures (1)

  • Figure 1: Examples of original and contracted derangement graphs. The original graphs contain edges between pairs of permutations that do not meet; the contracted graphs have edges between vertices that represent a set of permutations, and the label corresponds to the non-meeting probability when choosing one permutation from each endpoint uniformly at random.

Theorems & Definitions (40)

  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Corollary 1
  • Lemma 3
  • Theorem 1
  • Corollary 2
  • Lemma 4
  • proof : Proof of \ref{['thm:meeting-prob']}
  • Lemma 5
  • ...and 30 more