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In the rendezvous problem, two computing entities (called *agents*) located at different vertices in a graph have to meet at the same vertex. In this paper, we consider the synchronous *neighborhood rendezvous problem*, where the agents are initially located at two adjacent vertices. While this problem can be trivially solved in *O*(Δ) rounds (Δ is the maximum degree of the graph), it is highly challenging to reveal whether that problem can be solved in *o*(Δ) rounds, even assuming the rich computational capability of agents. The only known result is that the time complexity of *O*($O(sqrt{n})$) rounds is achievable if the graph is complete and agents are probabilistic, asymmetric, and can use whiteboards placed at vertices. Our main contribution is to clarify the situation (with respect to computational models and graph classes) admitting such a sublinear-time rendezvous algorithm. More precisely, we present two algorithms achieving fast rendezvous additionally assuming bounded minimum degree, unique vertex identifier, accessibility to neighborhood IDs, and randomization. The first algorithm runs within $ ilde{O}(sqrt{nDelta/delta} + n/delta)$ rounds for graphs of the minimum degree larger than $sqrt{n}$, where *n* is the number of vertices in the graph, and δ is the minimum degree of the graph. The second algorithm assumes that the largest vertex ID is *O*(*n*), and achieves $ ilde{O}left( rac{n}{sqrt{delta}}
ight)$-round time complexity without using whiteboards. These algorithms attain *o*(Δ)-round complexity in the case of $delta = {omega}(sqrt{n} log n)$ and δ=ω(*n*^{2/3}log^{4/3}*n*) respectively. We also prove that four unconventional assumptions of our algorithm, bounded minimum degree, accessibility to neighborhood IDs, initial distance one, and randomization are all inherently necessary for attaining fast rendezvous. That is, one can obtain the Ω(*n*)-round lower bound if either one of them is removed.

- Publication
- IEICE TRANSACTIONS on Information Vol.E105-D No.3 pp.597-610

- Publication Date
- 2022/03/01

- Publicized
- 2021/12/17

- Online ISSN
- 1745-1361

- DOI
- 10.1587/transinf.2021EDP7104

- Type of Manuscript
- PAPER

- Category
- Fundamentals of Information Systems

Ryota EGUCHI

Nagoya Institute of Technology

Naoki KITAMURA

Nagoya Institute of Technology

Taisuke IZUMI

Osaka University

The copyright of the original papers published on this site belongs to IEICE. Unauthorized use of the original or translated papers is prohibited. See IEICE Provisions on Copyright for details.

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Ryota EGUCHI, Naoki KITAMURA, Taisuke IZUMI, "Fast Neighborhood Rendezvous" in IEICE TRANSACTIONS on Information,
vol. E105-D, no. 3, pp. 597-610, March 2022, doi: 10.1587/transinf.2021EDP7104.

Abstract: In the rendezvous problem, two computing entities (called *agents*) located at different vertices in a graph have to meet at the same vertex. In this paper, we consider the synchronous *neighborhood rendezvous problem*, where the agents are initially located at two adjacent vertices. While this problem can be trivially solved in *O*(Δ) rounds (Δ is the maximum degree of the graph), it is highly challenging to reveal whether that problem can be solved in *o*(Δ) rounds, even assuming the rich computational capability of agents. The only known result is that the time complexity of *O*($O(sqrt{n})$) rounds is achievable if the graph is complete and agents are probabilistic, asymmetric, and can use whiteboards placed at vertices. Our main contribution is to clarify the situation (with respect to computational models and graph classes) admitting such a sublinear-time rendezvous algorithm. More precisely, we present two algorithms achieving fast rendezvous additionally assuming bounded minimum degree, unique vertex identifier, accessibility to neighborhood IDs, and randomization. The first algorithm runs within $ ilde{O}(sqrt{nDelta/delta} + n/delta)$ rounds for graphs of the minimum degree larger than $sqrt{n}$, where *n* is the number of vertices in the graph, and δ is the minimum degree of the graph. The second algorithm assumes that the largest vertex ID is *O*(*n*), and achieves $ ilde{O}left( rac{n}{sqrt{delta}}
ight)$-round time complexity without using whiteboards. These algorithms attain *o*(Δ)-round complexity in the case of $delta = {omega}(sqrt{n} log n)$ and δ=ω(*n*^{2/3}log^{4/3}*n*) respectively. We also prove that four unconventional assumptions of our algorithm, bounded minimum degree, accessibility to neighborhood IDs, initial distance one, and randomization are all inherently necessary for attaining fast rendezvous. That is, one can obtain the Ω(*n*)-round lower bound if either one of them is removed.

URL: https://global.ieice.org/en_transactions/information/10.1587/transinf.2021EDP7104/_p

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@ARTICLE{e105-d_3_597,

author={Ryota EGUCHI, Naoki KITAMURA, Taisuke IZUMI, },

journal={IEICE TRANSACTIONS on Information},

title={Fast Neighborhood Rendezvous},

year={2022},

volume={E105-D},

number={3},

pages={597-610},

abstract={In the rendezvous problem, two computing entities (called *agents*) located at different vertices in a graph have to meet at the same vertex. In this paper, we consider the synchronous *neighborhood rendezvous problem*, where the agents are initially located at two adjacent vertices. While this problem can be trivially solved in *O*(Δ) rounds (Δ is the maximum degree of the graph), it is highly challenging to reveal whether that problem can be solved in *o*(Δ) rounds, even assuming the rich computational capability of agents. The only known result is that the time complexity of *O*($O(sqrt{n})$) rounds is achievable if the graph is complete and agents are probabilistic, asymmetric, and can use whiteboards placed at vertices. Our main contribution is to clarify the situation (with respect to computational models and graph classes) admitting such a sublinear-time rendezvous algorithm. More precisely, we present two algorithms achieving fast rendezvous additionally assuming bounded minimum degree, unique vertex identifier, accessibility to neighborhood IDs, and randomization. The first algorithm runs within $ ilde{O}(sqrt{nDelta/delta} + n/delta)$ rounds for graphs of the minimum degree larger than $sqrt{n}$, where *n* is the number of vertices in the graph, and δ is the minimum degree of the graph. The second algorithm assumes that the largest vertex ID is *O*(*n*), and achieves $ ilde{O}left( rac{n}{sqrt{delta}}
ight)$-round time complexity without using whiteboards. These algorithms attain *o*(Δ)-round complexity in the case of $delta = {omega}(sqrt{n} log n)$ and δ=ω(*n*^{2/3}log^{4/3}*n*) respectively. We also prove that four unconventional assumptions of our algorithm, bounded minimum degree, accessibility to neighborhood IDs, initial distance one, and randomization are all inherently necessary for attaining fast rendezvous. That is, one can obtain the Ω(*n*)-round lower bound if either one of them is removed.},

keywords={},

doi={10.1587/transinf.2021EDP7104},

ISSN={1745-1361},

month={March},}

Copy

TY - JOUR

TI - Fast Neighborhood Rendezvous

T2 - IEICE TRANSACTIONS on Information

SP - 597

EP - 610

AU - Ryota EGUCHI

AU - Naoki KITAMURA

AU - Taisuke IZUMI

PY - 2022

DO - 10.1587/transinf.2021EDP7104

JO - IEICE TRANSACTIONS on Information

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AB - In the rendezvous problem, two computing entities (called *agents*) located at different vertices in a graph have to meet at the same vertex. In this paper, we consider the synchronous *neighborhood rendezvous problem*, where the agents are initially located at two adjacent vertices. While this problem can be trivially solved in *O*(Δ) rounds (Δ is the maximum degree of the graph), it is highly challenging to reveal whether that problem can be solved in *o*(Δ) rounds, even assuming the rich computational capability of agents. The only known result is that the time complexity of *O*($O(sqrt{n})$) rounds is achievable if the graph is complete and agents are probabilistic, asymmetric, and can use whiteboards placed at vertices. Our main contribution is to clarify the situation (with respect to computational models and graph classes) admitting such a sublinear-time rendezvous algorithm. More precisely, we present two algorithms achieving fast rendezvous additionally assuming bounded minimum degree, unique vertex identifier, accessibility to neighborhood IDs, and randomization. The first algorithm runs within $ ilde{O}(sqrt{nDelta/delta} + n/delta)$ rounds for graphs of the minimum degree larger than $sqrt{n}$, where *n* is the number of vertices in the graph, and δ is the minimum degree of the graph. The second algorithm assumes that the largest vertex ID is *O*(*n*), and achieves $ ilde{O}left( rac{n}{sqrt{delta}}
ight)$-round time complexity without using whiteboards. These algorithms attain *o*(Δ)-round complexity in the case of $delta = {omega}(sqrt{n} log n)$ and δ=ω(*n*^{2/3}log^{4/3}*n*) respectively. We also prove that four unconventional assumptions of our algorithm, bounded minimum degree, accessibility to neighborhood IDs, initial distance one, and randomization are all inherently necessary for attaining fast rendezvous. That is, one can obtain the Ω(*n*)-round lower bound if either one of them is removed.

ER -