Download  Minimum Routing Cost Spanning Trees

Minimum Routing Cost Spanning TreesKunMao Chao ()Department of Computer Science and Information EngineeringNational Taiwan University, Taiwan
Email: [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 alltoall distance
NPhard

Routing cost C(T)=192
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Routing load l(T,e)

Routing cost C(T)=192
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The impact of the topology
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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 2approximationA shortestpaths tree rooted at the median of a graph is a 2approximation 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, 1/4separators
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A separator

A 15/8approximation algorithmUse a minimal 1/3separator to estimate a lower of the routing cost of an MRCTThere exists a path P which is a minimal 1/3separator
The endpoints of P are useful in constructing a lower routing cost spanning tree

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