Minimum Routing Cost Spanning Trees Kun-Mao Chao ( 趙坤茂 ) Department of Computer Science and Information Engineering National Taiwan University, Taiwan.

Minimum Routing Cost Spanning Trees

Kun-Mao Chao (趙坤茂 )Department of Computer Science an

d Information EngineeringNational Taiwan University, Taiwan

E-mail: [email protected]

WWW: http://www.csie.ntu.edu.tw/~kmchao

2

An MST with large routing cost

2

......

2

2

2

2

2

1

1

11

1

3

A small routing cost tree with large weight

1

1

....

11 v0

1

1

1

1

v3

v2

v1

vi

v-2

v-1

.... ....

3/)2|(|),(

1),(

0

1

ivvw

vvw

i

ii

4

Minimum routing cost spanning trees

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

• NP-hardTvu

vud

vudTC

T

vuT

on and between

path)shortest the(of

distance theis ),( where

),()( Minimize,

5

Routing cost C(T)=192v1 v2

v3

10

v4 v5

5 3 1

6

Routing load l(T,e)

)(

)(),()(TEe

eweTlTC

7

Routing cost C(T)=192v1 v2

v3

10

v4 v5

5 3 1

1921838581012)(

8412)),(,(

8412)),(,(

8412)),(,(

12322)),(,(

52

42

31

21

TC

vvTl

vvTl

vvTl

vvTl

8

The impact of the topology

v1 vnv2 v3

T2

1 1 1...

T1

v2

v1v3

5

55

...5

5

5vn

21 )1(10)( nTC

3

)1)(1(

)(2)(11

2

nnn

iniTCni

9

bound on routing load

x n - x

)1(2Minimum21

Maximum

].1,1[in integer an is where

2 2

)(2load Routing

2

2

n

n

nx

xnx

xnx

10

bound on routing load

>=δn >=δn

)(2load Routing nnn

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

.on and between (distance)

length path -shortest theis ),( where

),()(

Guv

uvd

uvdvf

G

vuG

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

r r1 r2

(a) (b)

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-separatorsv1v2 v3v4

v5

v1v2 v3v4

v5

v1v2 v3v4

v5

v1v2 v3v4

v5

(a) (b)

(d)(c)

16

A δ-separator

<=kn<=kn

<=kn

<=kn

<=kn <=kn

<=kn<=kn

iS

brn(T,S,i)={B1,B2,B3}

B1

B2 B3

VB(T,S,i)

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

Vv

MRCT Pwn

Pvdn

MRCTC )(9

4),(

34

)(2

18

A 3/2-approximation algorithm

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