Minimum Routing Cost Spanning Trees

Minimum Routing Cost Spanning

Trees

Kun-Mao Chao ( 趙坤茂 )

Department of Computer Science an d Information Engineering

National Taiwan University, Taiwan

E-mail: [email protected]

A small routing cost tree with large

weight

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Minimum routing cost spanning

trees

• Given a graph, find a spanning tree with the minimum all-to-all distance

• NP-hard T v u v u d v u d T C T v u T on and between path) shortest the (of distance the is ) , ( where ) , ( ) ( Minimize ,  

Routing load l(T,e)

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Routing cost C(T)=192

192 1 8 3 8 5 8 10 12 ) ( 8 4 1 2 )) , ( , ( 8 4 1 2 )) , ( , ( 8 4 1 2 )) , ( , ( 12 3 2 2 )) , ( , ( 5 2 4 2 3 1 2 1                          T C v v T l v v T l v v T l v v T l

The impact of the topology

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bound on routing load

x n - x ) 1 ( 2 Minimum 2 1 Maximum ]. 1 , 1 [ in integer an is where 2 2 ) ( 2 load Routing 2 2         n n n x x nx x n x

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Why?

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

. on and between (distance) length path -shortest the is ) , ( where ) , ( ) ( G u v u v d u v d v f G v u G  

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

Some interesting vertices

• Centroid • Median • Center

* a tree with positive edge lengths, the medi an coincides with the centroid.

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    V v MRCT w P n P v d n MRCT C ( ) 9 4 ) , ( 3 4 ) ( 2

A 3/2-approximation algorithm

• Besides the two endpoints of P, a centroid is used to lower the upper bound.