Chapter 3 Restriction (1) Steiner Tree Ding-Zhu Du

y* f(x*) = min f(x) x in Ω Ω x* f(y*) = min f(y) y in Γ Γ y restriction

y* f(x*) = max f(x) x in Ω Ω x* f(y*) = max f(y) y in Γ Γ y restriction

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 120 A C B S D E >120

p.r. of MST in Euclidean plane AB S 120 AB p.r. <

Gilbert-Pollak Conjecture The Steiner ratio = p.r. = proved by Du and Hwang in

p.r. of MST in rectilinear plane = proved by F.K Hwang

How do we get a better approximation? Euclidean SMT (PTAS) Rectilinear SMT (PTAS) Network SMT: k-restricted Steiner tree Guillotine cut 1.55

Full Components size 4 size 3

size 4 size 3 Every full component has size < k. k-restricted Steiner Tree This is a 4-restricted Steiner tree.

p.r. of k-restricted Steiner tree 0 Consider a full component of SMT. 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

There are t ways to do the t-level tree partition. By Shafting, we know One of them gives a k-restricted Steiner tree with total length < (1 + 1/t) SMT where

The k-Steiner ratio for network Steiner trees (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