Two mobile agents, starting from different nodes of a network modeled as a graph, and woken up at possibly different times, have to meet at the same node. This problem is known as rendezvous. Agents move in synchronous rounds. In each round, an agent can either stay idle or move to an adjacent node. We consider deterministic rendezvous in the infinite path, i.e., the infinite graph with all nodes of degree 2. Each node has a distinct label which is a positive integer. An agent currently located at a node can see its label and both ports 0 and 1 at the node. The time of rendezvous is the number of rounds until meeting, counted from the starting round of the earlier agent. We consider three scenarios. In the first scenario, each agent knows its position in the path, i.e., each of them knows its initial distance from the smallest-labeled node, on which side of this node it is located, and the direction towards it. For this scenario, we design a rendezvous algorithm working in time O(D), where D is the initial distance between the agents. This complexity is clearly optimal. In the second scenario, each agent initially knows only the label of its starting node and the initial distance D between the agents. In this scenario, we design a rendezvous algorithm working in time <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="446_2025_489_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$O(D\log ^\ell )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mi>D</mi> <msup> <mo>log</mo> <mo>∗</mo> </msup> <mi>ℓ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="446_2025_489_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation> is the larger label of the starting nodes. We also prove a matching lower bound <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="446_2025_489_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega (D\log ^\ell )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>D</mi> <msup> <mo>log</mo> <mo>∗</mo> </msup> <mi>ℓ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Finally, in the most general scenario, where each agent initially knows only the label of its starting node, we design a rendezvous algorithm working in time <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="446_2025_489_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">$O(D^2(\log ^*\ell )^3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mo>log</mo> <mo>∗</mo> </msup> <mi>ℓ</mi> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which is thus at most cubic in the lower bound. All our results remain valid (with small changes) for arbitrary finite paths and for cycles. Our algorithms are drastically better than approaches that use graph exploration, which have running times that depend on the size or diameter of the graph. Our main methodological tool, and the main novelty of the paper, is a two way reduction: from fast colouring of the infinite labeled path using a constant number of colours in the <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="446_2025_489_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {LOCAL}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">LOCAL</mi> </math></EquationSource> </InlineEquation> model to fast rendezvous in this path, and vice-versa. In one direction we use fast node colouring to quickly break symmetry between the identical agents. In the other direction, a lower bound on colouring time implies a lower bound on the time of breaking symmetry between the agents, and hence a lower bound on their meeting time.

Fast deterministic rendezvous in labeled lines

期刊论文
发表日期：

2025年6月24日

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Avery Miller,
Andrzej Pelc

摘要

Two mobile agents, starting from different nodes of a network modeled as a graph, and woken up at possibly different times, have to meet at the same node. This problem is known as rendezvous. Agents move in synchronous rounds. In each round, an agent can either stay idle or move to an adjacent node. We consider deterministic rendezvous in the infinite path, i.e., the infinite graph with all nodes of degree 2. Each node has a distinct label which is a positive integer. An agent currently located at a node can see its label and both ports 0 and 1 at the node. The time of rendezvous is the number of rounds until meeting, counted from the starting round of the earlier agent. We consider three scenarios. In the first scenario, each agent knows its position in the path, i.e., each of them knows its initial distance from the smallest-labeled node, on which side of this node it is located, and the direction towards it. For this scenario, we design a rendezvous algorithm working in time O(D), where D is the initial distance between the agents. This complexity is clearly optimal. In the second scenario, each agent initially knows only the label of its starting node and the initial distance D between the agents. In this scenario, we design a rendezvous algorithm working in time $O(D\log ^*\ell )$ $O (D {log}^{*} ℓ)$ , where $\ell $ $ℓ$ is the larger label of the starting nodes. We also prove a matching lower bound $\Omega (D\log ^*\ell )$ $Ω (D {log}^{*} ℓ)$ . Finally, in the most general scenario, where each agent initially knows only the label of its starting node, we design a rendezvous algorithm working in time $O(D^2(\log ^*\ell )^3)$ $O (D^{2} {({log}^{*} ℓ)}^{3})$ , which is thus at most cubic in the lower bound. All our results remain valid (with small changes) for arbitrary finite paths and for cycles. Our algorithms are drastically better than approaches that use graph exploration, which have running times that depend on the size or diameter of the graph. Our main methodological tool, and the main novelty of the paper, is a two way reduction: from fast colouring of the infinite labeled path using a constant number of colours in the $\mathcal {LOCAL}$ $LOCAL$ model to fast rendezvous in this path, and vice-versa. In one direction we use fast node colouring to quickly break symmetry between the identical agents. In the other direction, a lower bound on colouring time implies a lower bound on the time of breaking symmetry between the agents, and hence a lower bound on their meeting time.