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Minimum Routing Cost Spanning Trees

This paper discusses the challenge of finding a minimum routing cost spanning tree (MRCT) for a given graph, which is known to be NP-hard. It introduces various concepts such as the median vertex minimizing total distance, and methods including 2-approximation algorithms based on shortest-paths trees and centroid definitions. The research highlights the importance of topology bounds on routing load and presents approximate algorithms for estimating lower and upper bounds on routing costs, utilizing 1/3-separators and additional tree vertices to enhance estimation accuracy.

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Minimum Routing Cost Spanning Trees

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  1. Minimum Routing Cost Spanning Trees Kun-Mao Chao (趙坤茂) Department of Computer Science and Information Engineering National Taiwan University, Taiwan E-mail: kmchao@csie.ntu.edu.tw WWW: http://www.csie.ntu.edu.tw/~kmchao

  2. An MST with large routing cost

  3. A small routing cost tree with large weight

  4. Minimum routing cost spanning trees • Given a graph, find a spanning tree with the minimum all-to-all distance • NP-hard

  5. Routing cost C(T)=192

  6. Routing load l(T,e)

  7. Routing cost C(T)=192

  8. The impact of the topology

  9. bound on routing load x n - x

  10. bound on routing load >=δn >=δn Why?

  11. Median • Let 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

  12. A 2-approximation • A 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.)

  13. Centroid

  14. Some interesting vertices • Centroid • Median • Center * a tree with positive edge lengths, the median coincides with the centroid.

  15. 1/2, 1/3, 1/4-separators

  16. A δ-separator

  17. A 15/8-approximation algorithm • Use a minimal 1/3-separator to estimate a lower of the routing cost of an MRCT • There 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

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

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