Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2013-02-15

AUTHORS

Giorgos Christodoulou, Kurt Mehlhorn, Evangelia Pyrga

ABSTRACT

We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou’s network. We improve upon the value 4/3 by means of Coordination Mechanisms.We increase the latency functions of the edges in the network, i.e., if ℓe(x) is the latency function of an edge e, we replace it by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat{\ell}_{e}(x)$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell_{e}(x) \le \hat{\ell}_{e}(x)$\end{document} for all x. Then an adversary fixes a demand rate as input. The engineered Price of Anarchy of the mechanism is defined as the worst-case ratio of the Nash social cost in the modified network over the optimal social cost in the original network. Formally, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat{C}_{N} (r)$\end{document} denotes the cost of the worst Nash flow in the modified network for rate r and Copt(r) denotes the cost of the optimal flow in the original network for the same rate then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathit{ePoA} = \max_{r \ge 0} \frac{\hat{C}_N(r)}{C_{\mathit{opt}}(r)}. $$\end{document}We first exhibit a simple coordination mechanism that achieves for any network of parallel links an engineered Price of Anarchy strictly less than 4/3. For the case of two parallel links our basic mechanism gives 5/4=1.25. Then, for the case of two parallel links, we describe an optimal mechanism; its engineered Price of Anarchy lies between 1.191 and 1.192. More... »

PAGES

619-640

References to SciGraph publications

  • 2009-09-11. On truthfulness and approximation for scheduling selfish tasks in JOURNAL OF SCHEDULING
  • 2007-01-01. Scheduling Selfish Tasks: About the Performance of Truthful Algorithms in COMPUTING AND COMBINATORICS
  • 2005. The Efficiency of Optimal Taxes in COMBINATORIAL AND ALGORITHMIC ASPECTS OF NETWORKING
  • 2008-01-01. Non-preemptive Coordination Mechanisms for Identical Machine Scheduling Games in STRUCTURAL INFORMATION AND COMMUNICATION COMPLEXITY
  • 2005. Coordination Mechanisms for Selfish Scheduling in INTERNET AND NETWORK ECONOMICS
  • 2011. Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms in ALGORITHMS – ESA 2011
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00453-013-9753-8

    DOI

    http://dx.doi.org/10.1007/s00453-013-9753-8

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1005888464


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/10", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Technology", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/1005", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Communications Technologies", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "University of Liverpool, Ashton Building, Ashton Street, L69 3BX, Liverpool, UK", 
              "id": "http://www.grid.ac/institutes/grid.10025.36", 
              "name": [
                "University of Liverpool, Ashton Building, Ashton Street, L69 3BX, Liverpool, UK"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Christodoulou", 
            "givenName": "Giorgos", 
            "id": "sg:person.012242176052.17", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012242176052.17"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Max-Planck-Institut f\u00fcr Informatik, Saarbr\u00fccken, Germany", 
              "id": "http://www.grid.ac/institutes/grid.419528.3", 
              "name": [
                "Max-Planck-Institut f\u00fcr Informatik, Saarbr\u00fccken, Germany"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Mehlhorn", 
            "givenName": "Kurt", 
            "id": "sg:person.011757371347.43", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011757371347.43"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Technische Universit\u00e4t M\u00fcnchen, Munich, Germany", 
              "id": "http://www.grid.ac/institutes/grid.6936.a", 
              "name": [
                "Technische Universit\u00e4t M\u00fcnchen, Munich, Germany"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Pyrga", 
            "givenName": "Evangelia", 
            "id": "sg:person.016052023445.47", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016052023445.47"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/978-3-540-73545-8_20", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1017293670", 
              "https://doi.org/10.1007/978-3-540-73545-8_20"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s10951-009-0118-8", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1044398600", 
              "https://doi.org/10.1007/s10951-009-0118-8"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-642-23719-5_11", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1035402197", 
              "https://doi.org/10.1007/978-3-642-23719-5_11"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/11527954_2", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1037617251", 
              "https://doi.org/10.1007/11527954_2"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/11600930_7", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1036428577", 
              "https://doi.org/10.1007/11600930_7"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-540-69355-0_17", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1022553650", 
              "https://doi.org/10.1007/978-3-540-69355-0_17"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2013-02-15", 
        "datePublishedReg": "2013-02-15", 
        "description": "We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou\u2019s network. We improve upon the value 4/3 by means of Coordination Mechanisms.We increase the latency functions of the edges in the network, i.e., if \u2113e(x) is the latency function of an edge e, we replace it by \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$\\hat{\\ell}_{e}(x)$\\end{document} with \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$\\ell_{e}(x) \\le \\hat{\\ell}_{e}(x)$\\end{document} for all x. Then an adversary fixes a demand rate as input. The engineered Price of Anarchy of the mechanism is defined as the worst-case ratio of the Nash social cost in the modified network over the optimal social cost in the original network. Formally, if \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$\\hat{C}_{N} (r)$\\end{document} denotes the cost of the worst Nash flow in the modified network for rate r and Copt(r) denotes the cost of the optimal flow in the original network for the same rate then \n\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathit{ePoA} = \\max_{r \\ge 0} \\frac{\\hat{C}_N(r)}{C_{\\mathit{opt}}(r)}. $$\\end{document}We first exhibit a simple coordination mechanism that achieves for any network of parallel links an engineered Price of Anarchy strictly less than\u00a04/3. For the case of two parallel links our basic mechanism gives 5/4=1.25. Then, for the case of two parallel links, we describe an optimal mechanism; its engineered Price of Anarchy lies between\u00a01.191 and\u00a01.192.", 
        "genre": "article", 
        "id": "sg:pub.10.1007/s00453-013-9753-8", 
        "isAccessibleForFree": true, 
        "isPartOf": [
          {
            "id": "sg:journal.1047644", 
            "issn": [
              "0178-4617", 
              "1432-0541"
            ], 
            "name": "Algorithmica", 
            "publisher": "Springer Nature", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "3", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "69"
          }
        ], 
        "keywords": [
          "parallel links", 
          "modified network", 
          "simple coordination mechanism", 
          "network", 
          "original network", 
          "affine latency functions", 
          "price of anarchy", 
          "routing", 
          "simple network", 
          "link", 
          "social costs", 
          "selfish routing", 
          "routing games", 
          "latency functions", 
          "cost", 
          "coordination mechanism", 
          "Nash flow", 
          "selfish routing game", 
          "optimal social cost", 
          "class of games", 
          "prices", 
          "demand rate", 
          "optimal mechanism", 
          "adversary", 
          "anarchy", 
          "exhibit", 
          "worst-case ratio", 
          "mechanism", 
          "first exhibit", 
          "edge", 
          "game", 
          "rate", 
          "ratio", 
          "basic mechanisms", 
          "rate R", 
          "optimal flow", 
          "function", 
          "maximum", 
          "means", 
          "same rate", 
          "flow", 
          "class", 
          "input", 
          "lies", 
          "cases", 
          "value 4/3", 
          "edge e"
        ], 
        "name": "Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms", 
        "pagination": "619-640", 
        "productId": [
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1005888464"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s00453-013-9753-8"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s00453-013-9753-8", 
          "https://app.dimensions.ai/details/publication/pub.1005888464"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2022-10-01T06:38", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-springernature-scigraph/baseset/20221001/entities/gbq_results/article/article_593.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://doi.org/10.1007/s00453-013-9753-8"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s00453-013-9753-8'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s00453-013-9753-8'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00453-013-9753-8'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00453-013-9753-8'


     

    This table displays all metadata directly associated to this object as RDF triples.

    148 TRIPLES      21 PREDICATES      77 URIs      63 LITERALS      6 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s00453-013-9753-8 schema:about anzsrc-for:10
    2 anzsrc-for:1005
    3 schema:author Nb3af1d162893426f8261aa3a0ad558ea
    4 schema:citation sg:pub.10.1007/11527954_2
    5 sg:pub.10.1007/11600930_7
    6 sg:pub.10.1007/978-3-540-69355-0_17
    7 sg:pub.10.1007/978-3-540-73545-8_20
    8 sg:pub.10.1007/978-3-642-23719-5_11
    9 sg:pub.10.1007/s10951-009-0118-8
    10 schema:datePublished 2013-02-15
    11 schema:datePublishedReg 2013-02-15
    12 schema:description We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou’s network. We improve upon the value 4/3 by means of Coordination Mechanisms.We increase the latency functions of the edges in the network, i.e., if ℓe(x) is the latency function of an edge e, we replace it by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat{\ell}_{e}(x)$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell_{e}(x) \le \hat{\ell}_{e}(x)$\end{document} for all x. Then an adversary fixes a demand rate as input. The engineered Price of Anarchy of the mechanism is defined as the worst-case ratio of the Nash social cost in the modified network over the optimal social cost in the original network. Formally, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat{C}_{N} (r)$\end{document} denotes the cost of the worst Nash flow in the modified network for rate r and Copt(r) denotes the cost of the optimal flow in the original network for the same rate then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathit{ePoA} = \max_{r \ge 0} \frac{\hat{C}_N(r)}{C_{\mathit{opt}}(r)}. $$\end{document}We first exhibit a simple coordination mechanism that achieves for any network of parallel links an engineered Price of Anarchy strictly less than 4/3. For the case of two parallel links our basic mechanism gives 5/4=1.25. Then, for the case of two parallel links, we describe an optimal mechanism; its engineered Price of Anarchy lies between 1.191 and 1.192.
    13 schema:genre article
    14 schema:isAccessibleForFree true
    15 schema:isPartOf Nae6f8751272d4c2aad7401135a698097
    16 Nafa891706d6e4341935bfbdaa6d9bded
    17 sg:journal.1047644
    18 schema:keywords Nash flow
    19 adversary
    20 affine latency functions
    21 anarchy
    22 basic mechanisms
    23 cases
    24 class
    25 class of games
    26 coordination mechanism
    27 cost
    28 demand rate
    29 edge
    30 edge e
    31 exhibit
    32 first exhibit
    33 flow
    34 function
    35 game
    36 input
    37 latency functions
    38 lies
    39 link
    40 maximum
    41 means
    42 mechanism
    43 modified network
    44 network
    45 optimal flow
    46 optimal mechanism
    47 optimal social cost
    48 original network
    49 parallel links
    50 price of anarchy
    51 prices
    52 rate
    53 rate R
    54 ratio
    55 routing
    56 routing games
    57 same rate
    58 selfish routing
    59 selfish routing game
    60 simple coordination mechanism
    61 simple network
    62 social costs
    63 value 4/3
    64 worst-case ratio
    65 schema:name Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms
    66 schema:pagination 619-640
    67 schema:productId N122f12c705874088a41e7181113d30e8
    68 N4af3e49d331c420ca3dceb9496976312
    69 schema:sameAs https://app.dimensions.ai/details/publication/pub.1005888464
    70 https://doi.org/10.1007/s00453-013-9753-8
    71 schema:sdDatePublished 2022-10-01T06:38
    72 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    73 schema:sdPublisher N35002600d8a2495e8cf40e9cd2fe787a
    74 schema:url https://doi.org/10.1007/s00453-013-9753-8
    75 sgo:license sg:explorer/license/
    76 sgo:sdDataset articles
    77 rdf:type schema:ScholarlyArticle
    78 N0666a6ee7b0243baaa7367a6399fcb51 rdf:first sg:person.016052023445.47
    79 rdf:rest rdf:nil
    80 N122f12c705874088a41e7181113d30e8 schema:name dimensions_id
    81 schema:value pub.1005888464
    82 rdf:type schema:PropertyValue
    83 N35002600d8a2495e8cf40e9cd2fe787a schema:name Springer Nature - SN SciGraph project
    84 rdf:type schema:Organization
    85 N4af3e49d331c420ca3dceb9496976312 schema:name doi
    86 schema:value 10.1007/s00453-013-9753-8
    87 rdf:type schema:PropertyValue
    88 N87e1d8dd797441d9a2d3f1cf7c0d5c85 rdf:first sg:person.011757371347.43
    89 rdf:rest N0666a6ee7b0243baaa7367a6399fcb51
    90 Nae6f8751272d4c2aad7401135a698097 schema:volumeNumber 69
    91 rdf:type schema:PublicationVolume
    92 Nafa891706d6e4341935bfbdaa6d9bded schema:issueNumber 3
    93 rdf:type schema:PublicationIssue
    94 Nb3af1d162893426f8261aa3a0ad558ea rdf:first sg:person.012242176052.17
    95 rdf:rest N87e1d8dd797441d9a2d3f1cf7c0d5c85
    96 anzsrc-for:10 schema:inDefinedTermSet anzsrc-for:
    97 schema:name Technology
    98 rdf:type schema:DefinedTerm
    99 anzsrc-for:1005 schema:inDefinedTermSet anzsrc-for:
    100 schema:name Communications Technologies
    101 rdf:type schema:DefinedTerm
    102 sg:journal.1047644 schema:issn 0178-4617
    103 1432-0541
    104 schema:name Algorithmica
    105 schema:publisher Springer Nature
    106 rdf:type schema:Periodical
    107 sg:person.011757371347.43 schema:affiliation grid-institutes:grid.419528.3
    108 schema:familyName Mehlhorn
    109 schema:givenName Kurt
    110 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011757371347.43
    111 rdf:type schema:Person
    112 sg:person.012242176052.17 schema:affiliation grid-institutes:grid.10025.36
    113 schema:familyName Christodoulou
    114 schema:givenName Giorgos
    115 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012242176052.17
    116 rdf:type schema:Person
    117 sg:person.016052023445.47 schema:affiliation grid-institutes:grid.6936.a
    118 schema:familyName Pyrga
    119 schema:givenName Evangelia
    120 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016052023445.47
    121 rdf:type schema:Person
    122 sg:pub.10.1007/11527954_2 schema:sameAs https://app.dimensions.ai/details/publication/pub.1037617251
    123 https://doi.org/10.1007/11527954_2
    124 rdf:type schema:CreativeWork
    125 sg:pub.10.1007/11600930_7 schema:sameAs https://app.dimensions.ai/details/publication/pub.1036428577
    126 https://doi.org/10.1007/11600930_7
    127 rdf:type schema:CreativeWork
    128 sg:pub.10.1007/978-3-540-69355-0_17 schema:sameAs https://app.dimensions.ai/details/publication/pub.1022553650
    129 https://doi.org/10.1007/978-3-540-69355-0_17
    130 rdf:type schema:CreativeWork
    131 sg:pub.10.1007/978-3-540-73545-8_20 schema:sameAs https://app.dimensions.ai/details/publication/pub.1017293670
    132 https://doi.org/10.1007/978-3-540-73545-8_20
    133 rdf:type schema:CreativeWork
    134 sg:pub.10.1007/978-3-642-23719-5_11 schema:sameAs https://app.dimensions.ai/details/publication/pub.1035402197
    135 https://doi.org/10.1007/978-3-642-23719-5_11
    136 rdf:type schema:CreativeWork
    137 sg:pub.10.1007/s10951-009-0118-8 schema:sameAs https://app.dimensions.ai/details/publication/pub.1044398600
    138 https://doi.org/10.1007/s10951-009-0118-8
    139 rdf:type schema:CreativeWork
    140 grid-institutes:grid.10025.36 schema:alternateName University of Liverpool, Ashton Building, Ashton Street, L69 3BX, Liverpool, UK
    141 schema:name University of Liverpool, Ashton Building, Ashton Street, L69 3BX, Liverpool, UK
    142 rdf:type schema:Organization
    143 grid-institutes:grid.419528.3 schema:alternateName Max-Planck-Institut für Informatik, Saarbrücken, Germany
    144 schema:name Max-Planck-Institut für Informatik, Saarbrücken, Germany
    145 rdf:type schema:Organization
    146 grid-institutes:grid.6936.a schema:alternateName Technische Universität München, Munich, Germany
    147 schema:name Technische Universität München, Munich, Germany
    148 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...