Chapter 3 Restriction
(1) Steiner Tree
Ding-Zhu Du
restriction
f(y*) = min f(y)
y in Γ
f(x*) = min f(x)
x in Ω
Ω
Γ
y*
x*
y
f ( y*)
f ( y)
f ( y )  f ( x*)

 1
f ( x*) f ( x*)
f ( x*)
restriction
f(y*) = max f(y)
y in Γ
f(x*) = max f(x)
x in Ω
Ω
Γ
y*
x*
y
f ( x*) f ( x*)
f ( x*)  f ( y )

 1 /(1 
)
f ( y*)
f ( y)
f ( x*)
Steiner Tree Problem
Given a set of points (called terminals) in a metric space,
find a shortest network interconnecting all terminals.
Steiner point
SMT: Steiner minimum tree
Spanning Tree – A restricted Steiner tree
No Steiner point exists!
MST: minimum spanning tree
What is the p.r. of MST ?
p.r. = 2
Euclidean Steiner Tree
A
120
C
>120
S
E
D
B
ABS  ADE
ES  AS
DE  BS
p.r. of MST in Euclidean plane
A
B
A
B
120
S
Assume SA  SB.
AB 2  SA2  SB 2  2 SA  SB cos 120 o
 SA2  SB 2  SA  SB
 3SB 2
AB  3SB
p.r. <
3
Gilbert-Pollak Conjecture
smt ( P)
The Steiner ratio = min
P mst ( P )
3

2
1
3
p.r. = 2 / 3
proved by Du and Hwang in 1990.
p.r. of MST in rectilinear plane
= 1.5
2
1
proved by F.K Hwang
How do we get a better approximation?
Euclidean SMT (PTAS)
Rectilinear SMT (PTAS)
Guillotine cut
Network SMT: k-restricted Steiner tree
1.55
Full Components
size 4
size 3
k-restricted Steiner Tree
Every full component has size < k.
size 4
size 3
This is a 4-restricted Steiner tree.
p.r. of k-restricted Steiner tree
Consider a full component of SMT.
0
Modify it into a rooted binary tree with adding
some edges of length 0.
A property of binary tree
There is a mapping f from internal nodes to leaves:
All paths (x,f(x)) are edge-disjoint.
t-level Tree Partition
< t+1
t+1
2 k
t
By Shafting, we know
There are t ways to do
the t-level tree partition.
One of them gives a k-restricted Steiner tree
with total length
< (1 + 1/t) SMT
where
t  log 2 k 
The k-Steiner ratio for network Steiner trees
smt ( P )
 k  inf
P smt ( P )
k
r2  s
r
k 
for k  2  s
r
(r  1)2  s
r
(Borchers & Du, 1995)
How do we compute k-restricted SMT?
What is the complexity for computing k-restricted SMT?
For k > 4, it is NP-hard !
For k=3, it is an open problem !
go to Greedy k-restricted Steiner tree