minimum routing cost spanning trees
DESCRIPTION
Minimum Routing Cost Spanning Trees. Kun-Mao Chao ( 趙坤茂 ) Department of Computer Science and Information Engineering National Taiwan University, Taiwan E-mail: [email protected] WWW: http://www.csie.ntu.edu.tw/~kmchao. An MST with large routing cost. - PowerPoint PPT PresentationTRANSCRIPT
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)
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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)
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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)
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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
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A 3/2-approximation algorithm
• Besides the two endpoints of P, a centroid is used to lower the upper bound.