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Tight Bounds on Window Size and Time for Single-Agent Graph Exploration under T-Interval Connectivity

Yuichi Sudo, Naoki Kitamura, Masahiro Shibata, Junya Nakamura, Sébastien Tixeuil, Toshimitsu Masuzawa, Koichi Wada

Abstract

We study deterministic exploration by a single agent in $T$-interval-connected graphs, a standard model of dynamic networks in which, for every time window of length $T$, the intersection of the graphs within the window is connected. The agent does not know the window size $T$, nor the number of nodes $n$ or edges $m$, and must visit all nodes of the graph. We consider two visibility models, $KT_0$ and $KT_1$, depending on whether the agent can observe the identifiers of neighboring nodes. We investigate two fundamental questions: the minimum window size that guarantees exploration, and the optimal exploration time under sufficiently large window size. For both models, we show that a window size $T = Ω(m)$ is necessary. We also present deterministic algorithms whose required window size is $O(ε(n,m)\cdot m + n \log^2 n)$, where $ε(n,m) = \frac{\ln n}{1 + \ln m - \ln n}$. These bounds are tight for a wide range of $m$, in particular when $m = n^{1+Θ(1)}$. The same algorithms also yield optimal or near-optimal exploration time: we prove lower bounds of $Ω((m - n + 1)n)$ in the $KT_0$ model and $Ω(m)$ in the $KT_1$ model, and show that our algorithms match these bounds up to a polylogarithmic factor, while being fully time-optimal when $m = n^{1+Θ(1)}$. This yields tight bounds when parameterized solely by $n$: $Θ(n^3)$ for $KT_0$ and $Θ(n^2)$ for $KT_1$.

Tight Bounds on Window Size and Time for Single-Agent Graph Exploration under T-Interval Connectivity

Abstract

We study deterministic exploration by a single agent in -interval-connected graphs, a standard model of dynamic networks in which, for every time window of length , the intersection of the graphs within the window is connected. The agent does not know the window size , nor the number of nodes or edges , and must visit all nodes of the graph. We consider two visibility models, and , depending on whether the agent can observe the identifiers of neighboring nodes. We investigate two fundamental questions: the minimum window size that guarantees exploration, and the optimal exploration time under sufficiently large window size. For both models, we show that a window size is necessary. We also present deterministic algorithms whose required window size is , where . These bounds are tight for a wide range of , in particular when . The same algorithms also yield optimal or near-optimal exploration time: we prove lower bounds of in the model and in the model, and show that our algorithms match these bounds up to a polylogarithmic factor, while being fully time-optimal when . This yields tight bounds when parameterized solely by : for and for .

Paper Structure

This paper contains 10 sections, 20 theorems, 19 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Models and Notations
  4. Our Contribution
  5. Upper Bounds
  6. With 1-hop view
  7. With 0-hop view
  8. Lower Bounds
  9. Conclusion
  10. Supplemental Lemmas

Key Result

Theorem 1

In the $KT_0$ model, there exist a function $\tau(n,m) = O(\epsilon(n,m)\cdot m + n \log^2 n)$ and a deterministic algorithm $\mathcal{A}$ such that, for any $n \ge 3$ and $m\in[n..\binom{n}{2}]$, a single agent running $\mathcal{A}$ explores any $\tau(n,m)$-interval-connected graph with $n$ nodes a

Figures (3)

  • Figure 1: $K_5(u,v)$ (left) and $K_6(u,v)$ (right)
  • Figure 2: The underlying graph for the proof of Theorem \ref{['thm:psi_nm_lower']}, excluding null edges
  • Figure 3: The underlying graph for the proof of Theorem \ref{['thm:kt_zero_lower']}, excluding null edges

Theorems & Definitions (37)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Corollary 2
  • Theorem 4
  • Theorem 5
  • Remark 1
  • Lemma 1
  • ...and 27 more