Transcript
• 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

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• A small routing cost tree with large weight

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• 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

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• Routing cost C(T)=192

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• The impact of the topology

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• bound on routing loadxn - x

• 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

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

• 1/2, 1/3, 1/4-separators

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• 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.