How slow, or fast, are standard random walks? - Analysis of hitting and cover times on trees

Yoshiaki Nonaka, Hirotaka Ono, Shuji Kijima, Masafumi Yamashita

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

Abstract

Random walk is a powerful tool, not only for modeling, but also for practical use such as the Internet crawlers. Standard random walks on graphs have been well studied; It is well-known that both hitting time and cover time of a standard random walk are bounded by O(n 3) for any graph with n vertices, besides the bound is tight for some graphs. Ikeda et al. (2003) provided "β-random walk," which realizes O(n 2) hitting time and O(n 2 log n) cover times for any graph, thus it archives, in a sense, "n-times improvement" compared to the standard random walk. This paper is concerned with optimizations of hitting and cover times, by drawing a comparison between the standard random walk and the fastest random walk. We show for any tree that the hitting time of the standard random walk is at most O(√n)-times longer than one of the fastest random walk. Similarly, the cover time of the standard random walk is at most O(√n log n)-times longer than the fastest one, for any tree. We also show that our bound for the hitting time is tight by giving examples, while we only give a lower bound Ω(√n= log n) for the cover time.

Original languageEnglish
Title of host publicationTheory of Computing 2011 - Proceedings of the 17th Computing
Subtitle of host publicationThe Australasian Theory Symposium, CATS 2011
Pages63-68
Number of pages6
Publication statusPublished - Dec 1 2011
EventTheory of Computing 2011 - 17th Computing: The Australasian Theory Symposium, CATS 2011 - Perth, WA, Australia
Duration: Jan 17 2011Jan 20 2011

Publication series

NameConferences in Research and Practice in Information Technology Series
Volume119
ISSN (Print)1445-1336

Other

OtherTheory of Computing 2011 - 17th Computing: The Australasian Theory Symposium, CATS 2011
CountryAustralia
CityPerth, WA
Period1/17/111/20/11

Fingerprint

Internet

All Science Journal Classification (ASJC) codes

  • Computer Networks and Communications
  • Computer Science Applications
  • Hardware and Architecture
  • Information Systems
  • Software

Cite this

Nonaka, Y., Ono, H., Kijima, S., & Yamashita, M. (2011). How slow, or fast, are standard random walks? - Analysis of hitting and cover times on trees. In Theory of Computing 2011 - Proceedings of the 17th Computing: The Australasian Theory Symposium, CATS 2011 (pp. 63-68). (Conferences in Research and Practice in Information Technology Series; Vol. 119).

How slow, or fast, are standard random walks? - Analysis of hitting and cover times on trees. / Nonaka, Yoshiaki; Ono, Hirotaka; Kijima, Shuji; Yamashita, Masafumi.

Theory of Computing 2011 - Proceedings of the 17th Computing: The Australasian Theory Symposium, CATS 2011. 2011. p. 63-68 (Conferences in Research and Practice in Information Technology Series; Vol. 119).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Nonaka, Y, Ono, H, Kijima, S & Yamashita, M 2011, How slow, or fast, are standard random walks? - Analysis of hitting and cover times on trees. in Theory of Computing 2011 - Proceedings of the 17th Computing: The Australasian Theory Symposium, CATS 2011. Conferences in Research and Practice in Information Technology Series, vol. 119, pp. 63-68, Theory of Computing 2011 - 17th Computing: The Australasian Theory Symposium, CATS 2011, Perth, WA, Australia, 1/17/11.
Nonaka Y, Ono H, Kijima S, Yamashita M. How slow, or fast, are standard random walks? - Analysis of hitting and cover times on trees. In Theory of Computing 2011 - Proceedings of the 17th Computing: The Australasian Theory Symposium, CATS 2011. 2011. p. 63-68. (Conferences in Research and Practice in Information Technology Series).
Nonaka, Yoshiaki ; Ono, Hirotaka ; Kijima, Shuji ; Yamashita, Masafumi. / How slow, or fast, are standard random walks? - Analysis of hitting and cover times on trees. Theory of Computing 2011 - Proceedings of the 17th Computing: The Australasian Theory Symposium, CATS 2011. 2011. pp. 63-68 (Conferences in Research and Practice in Information Technology Series).
@inproceedings{fd7bb40ab5654914be4763e0964db755,
title = "How slow, or fast, are standard random walks? - Analysis of hitting and cover times on trees",
abstract = "Random walk is a powerful tool, not only for modeling, but also for practical use such as the Internet crawlers. Standard random walks on graphs have been well studied; It is well-known that both hitting time and cover time of a standard random walk are bounded by O(n 3) for any graph with n vertices, besides the bound is tight for some graphs. Ikeda et al. (2003) provided {"}β-random walk,{"} which realizes O(n 2) hitting time and O(n 2 log n) cover times for any graph, thus it archives, in a sense, {"}n-times improvement{"} compared to the standard random walk. This paper is concerned with optimizations of hitting and cover times, by drawing a comparison between the standard random walk and the fastest random walk. We show for any tree that the hitting time of the standard random walk is at most O(√n)-times longer than one of the fastest random walk. Similarly, the cover time of the standard random walk is at most O(√n log n)-times longer than the fastest one, for any tree. We also show that our bound for the hitting time is tight by giving examples, while we only give a lower bound Ω(√n= log n) for the cover time.",
author = "Yoshiaki Nonaka and Hirotaka Ono and Shuji Kijima and Masafumi Yamashita",
year = "2011",
month = "12",
day = "1",
language = "English",
isbn = "9781920682989",
series = "Conferences in Research and Practice in Information Technology Series",
pages = "63--68",
booktitle = "Theory of Computing 2011 - Proceedings of the 17th Computing",

}

TY - GEN

T1 - How slow, or fast, are standard random walks? - Analysis of hitting and cover times on trees

AU - Nonaka, Yoshiaki

AU - Ono, Hirotaka

AU - Kijima, Shuji

AU - Yamashita, Masafumi

PY - 2011/12/1

Y1 - 2011/12/1

N2 - Random walk is a powerful tool, not only for modeling, but also for practical use such as the Internet crawlers. Standard random walks on graphs have been well studied; It is well-known that both hitting time and cover time of a standard random walk are bounded by O(n 3) for any graph with n vertices, besides the bound is tight for some graphs. Ikeda et al. (2003) provided "β-random walk," which realizes O(n 2) hitting time and O(n 2 log n) cover times for any graph, thus it archives, in a sense, "n-times improvement" compared to the standard random walk. This paper is concerned with optimizations of hitting and cover times, by drawing a comparison between the standard random walk and the fastest random walk. We show for any tree that the hitting time of the standard random walk is at most O(√n)-times longer than one of the fastest random walk. Similarly, the cover time of the standard random walk is at most O(√n log n)-times longer than the fastest one, for any tree. We also show that our bound for the hitting time is tight by giving examples, while we only give a lower bound Ω(√n= log n) for the cover time.

AB - Random walk is a powerful tool, not only for modeling, but also for practical use such as the Internet crawlers. Standard random walks on graphs have been well studied; It is well-known that both hitting time and cover time of a standard random walk are bounded by O(n 3) for any graph with n vertices, besides the bound is tight for some graphs. Ikeda et al. (2003) provided "β-random walk," which realizes O(n 2) hitting time and O(n 2 log n) cover times for any graph, thus it archives, in a sense, "n-times improvement" compared to the standard random walk. This paper is concerned with optimizations of hitting and cover times, by drawing a comparison between the standard random walk and the fastest random walk. We show for any tree that the hitting time of the standard random walk is at most O(√n)-times longer than one of the fastest random walk. Similarly, the cover time of the standard random walk is at most O(√n log n)-times longer than the fastest one, for any tree. We also show that our bound for the hitting time is tight by giving examples, while we only give a lower bound Ω(√n= log n) for the cover time.

UR - http://www.scopus.com/inward/record.url?scp=84864631238&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864631238&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84864631238

SN - 9781920682989

T3 - Conferences in Research and Practice in Information Technology Series

SP - 63

EP - 68

BT - Theory of Computing 2011 - Proceedings of the 17th Computing

ER -