Download - Minimum Routing Cost Spanning Trees
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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
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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
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Routing cost C(T)=192
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Routing load l(T,e)
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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
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bound on routing load>=n>=nWhy?
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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
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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.)
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Centroid
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Some interesting verticesCentroidMedianCenter* a tree with positive edge lengths, the median coincides with the centroid.
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1/2, 1/3, 1/4-separators
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A -separator
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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
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A 3/2-approximation algorithmBesides the two endpoints of P, a centroid is used to lower the upper bound.
