A walk on max-plus algebra

Sennosuke Watanabe; Akiko Fukuda; Etsuo Segawa; Iwao Sato

doi:10.1016/j.laa.2020.03.025

A walk on max-plus algebra

Sennosuke Watanabe, Akiko Fukuda, Etsuo Segawa, Iwao Sato

Research output: Contribution to journal › Article › peer-review

Abstract

Max-plus algebra is a kind of idempotent semiring over R_max:=R∪{−∞} with two operations ⊕:=max and ⊗:=+. In this paper, we introduce a new model of a walk on one dimensional lattice on Z, as an analogue of the quantum walk, over the max-plus algebra and we call it max-plus walk. In the conventional quantum walk, the summation of the ℓ²-norm of the states over all the positions is a conserved quantity. In contrast, the summation of eigenvalues of state decision matrices is a conserved quantity in the max-plus walk. Moreover, spectral analysis on the total time evolution operator is also given.

Original language	English
Pages (from-to)	29-48
Number of pages	20
Journal	Linear Algebra and Its Applications
Volume	598
DOIs	https://doi.org/10.1016/j.laa.2020.03.025
Publication status	Published - 2020 Aug 1

Keywords

Directed graph
Max-plus algebra
Quantum walk

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

Access to Document

10.1016/j.laa.2020.03.025

Cite this

@article{30790755ce054941978f7f0865eab24d,

title = "A walk on max-plus algebra",

abstract = "Max-plus algebra is a kind of idempotent semiring over Rmax:=R∪{−∞} with two operations ⊕:=max and ⊗:=+. In this paper, we introduce a new model of a walk on one dimensional lattice on Z, as an analogue of the quantum walk, over the max-plus algebra and we call it max-plus walk. In the conventional quantum walk, the summation of the ℓ2-norm of the states over all the positions is a conserved quantity. In contrast, the summation of eigenvalues of state decision matrices is a conserved quantity in the max-plus walk. Moreover, spectral analysis on the total time evolution operator is also given.",

keywords = "Directed graph, Max-plus algebra, Quantum walk",

author = "Sennosuke Watanabe and Akiko Fukuda and Etsuo Segawa and Iwao Sato",

note = "Funding Information: This work was partially supported by Grants-in-Aid for Scientific Research (C) No. 19K03616 and No. 19K03624 of the Japan Society for the Promotion of Science and Research Origin for Dressed Photon . Publisher Copyright: {\textcopyright} 2020 Elsevier Inc.",

year = "2020",

month = aug,

day = "1",

doi = "10.1016/j.laa.2020.03.025",

language = "English",

volume = "598",

pages = "29--48",

journal = "Linear Algebra and Its Applications",

issn = "0024-3795",

publisher = "Elsevier Inc.",

}

TY - JOUR

T1 - A walk on max-plus algebra

AU - Watanabe, Sennosuke

AU - Fukuda, Akiko

AU - Segawa, Etsuo

AU - Sato, Iwao

N1 - Funding Information: This work was partially supported by Grants-in-Aid for Scientific Research (C) No. 19K03616 and No. 19K03624 of the Japan Society for the Promotion of Science and Research Origin for Dressed Photon . Publisher Copyright: © 2020 Elsevier Inc.

PY - 2020/8/1

Y1 - 2020/8/1

N2 - Max-plus algebra is a kind of idempotent semiring over Rmax:=R∪{−∞} with two operations ⊕:=max and ⊗:=+. In this paper, we introduce a new model of a walk on one dimensional lattice on Z, as an analogue of the quantum walk, over the max-plus algebra and we call it max-plus walk. In the conventional quantum walk, the summation of the ℓ2-norm of the states over all the positions is a conserved quantity. In contrast, the summation of eigenvalues of state decision matrices is a conserved quantity in the max-plus walk. Moreover, spectral analysis on the total time evolution operator is also given.

AB - Max-plus algebra is a kind of idempotent semiring over Rmax:=R∪{−∞} with two operations ⊕:=max and ⊗:=+. In this paper, we introduce a new model of a walk on one dimensional lattice on Z, as an analogue of the quantum walk, over the max-plus algebra and we call it max-plus walk. In the conventional quantum walk, the summation of the ℓ2-norm of the states over all the positions is a conserved quantity. In contrast, the summation of eigenvalues of state decision matrices is a conserved quantity in the max-plus walk. Moreover, spectral analysis on the total time evolution operator is also given.

KW - Directed graph

KW - Max-plus algebra

KW - Quantum walk

UR - http://www.scopus.com/inward/record.url?scp=85082165633&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85082165633&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2020.03.025

DO - 10.1016/j.laa.2020.03.025

M3 - Article

AN - SCOPUS:85082165633

SN - 0024-3795

VL - 598

SP - 29

EP - 48

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

A walk on max-plus algebra

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this