minimum routing cost spanning trees

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Minimum Routing Cost Spanning Trees Kun-Mao Chao ( 趙趙趙 ) Department of Computer Scienc e and Information Engineering National Taiwan University, T aiwan E-mail: [email protected] WWW: http://www.csie.ntu.edu.tw/~k mchao

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Minimum Routing Cost Spanning Trees. Kun-Mao Chao ( 趙坤茂 ) Department of Computer Science and Information Engineering National Taiwan University, Taiwan E-mail: [email protected] WWW: http://www.csie.ntu.edu.tw/~kmchao. An MST with large routing cost. - PowerPoint PPT Presentation

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  • Minimum Routing Cost Spanning TreesKun-Mao Chao ()Department of Computer Science and Information EngineeringNational Taiwan University, Taiwan

    E-mail: [email protected]: http://www.csie.ntu.edu.tw/~kmchao

  • An MST with large routing cost

    2

    ......

    2

    2

    2

    2

    2

    1

    1

    1

    1

    1

  • A small routing cost tree with large weight

    1

    1

    ....

    1

    1

    v0

    1

    1

    1

    1

    v3

    v2

    v1

    vi

    v-2

    v-1

    ....

    ....

  • Minimum routing cost spanning treesGiven a graph, find a spanning tree with the minimum all-to-all distance

    NP-hard

  • Routing cost C(T)=192

    v4

    v5

    v1

    v2

    v3

    10

    5

    3

    1

  • Routing load l(T,e)

  • Routing cost C(T)=192

    v4

    v5

    v1

    v2

    v3

    10

    5

    3

    1

  • The impact of the topology

    1

    1

    1

    ...

    vn

    v2

    v1

    v3

    v2

    v1

    v3

    5

    T2

    5

    T1

    5

    ...

    5

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    vn

  • bound on routing loadxn - x

  • bound on routing load>=n>=nWhy?

  • MedianLet r be the median of graph G=(V,E,W), i.e., the vertex with the minimum total distance to all vertices.In other words, r minimizes the function

  • A 2-approximationA shortest-paths tree rooted at the median of a graph is a 2-approximation of an MRCT of the graph. (Please refer to our discussions in class. A note on this has been posted in our course website.)

  • Centroid

    r

    r1

    r2

    (a)

    (b)

  • Some interesting verticesCentroidMedianCenter* a tree with positive edge lengths, the median coincides with the centroid.

  • 1/2, 1/3, 1/4-separators

    v1

    v2

    v3

    v4

    v5

    v1

    v2

    v3

    v4

    v5

    v1

    v2

    (a)

    (b)

    v3

    v4

    (d)

    v5

    (c)

    v1

    v2

    v3

    v4

    v5

  • A -separator

  • A 15/8-approximation algorithmUse a minimal 1/3-separator to estimate a lower of the routing cost of an MRCTThere exists a path P which is a minimal 1/3-separator

    The endpoints of P are useful in constructing a lower routing cost spanning tree

  • A 3/2-approximation algorithmBesides the two endpoints of P, a centroid is used to lower the upper bound.