The Steiner problem with edge length 1 and

The Steiner problem with edge length 1 and 2 Source: Information Process Letter 32 (1989)171 -176 Author: Marshall Bern and Paul. Plassmann Reporter: Chih-Ying Lin (林知瑩) 1

Outline 1. 2. 3. 4. 5. 6. Introduction Problem Definition Preview Algorithm Example Ratio 2

Introduction • G=(V, E) with edge length≧ 0，and a set N V of distinguished vertices. • The Steiner problem asks for a minimum length tree within G that spans all members of N. • NP-complete problem G=(9, 11) 3

• Rayward-Smith’s average distance heuristic (ADH) is a 4/3 -approximation algorithm for this problem. • It is the first proof that a polynomial-time heuristic for an NP-complete Steiner problem achieves an approximation bound better than that given by a minimum spanning tree. 4

Problem Definition • Steiner(1, 2) 1. In complete graph 2. All length 1 or 2 The Steiner(1, 2) asks for a minimum length tree within G that spans all members of N. 5

Preview 6

Algorithm 1. Find a vertex v (optional or terminal) and a set S of terminals (possibly containing v) that minimize the average distance over all choices of v and S v=C S={C, F} minimize the average distance =[d(C, C)+d(C, F)]/(2 -1)=1 7

2. Replace S∪{v} by a single terminal vs and for each vertex u, let d(vs, u) be the minimum distance from u to a vertex of S∪{v}. 8

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Example G=(9, 36) and terminal node {C, D, H, I} Bold edge length =1 and unseen edge length=2 10

V=H, S={D, H, I} minimize the average distance ={d(H, D)+d(I, H))}/(3 -1) =2/2=1 11

V=C, S={C, VS 1} minimize the average distance ={d(C, VS 1)}/(2 -1) =2 12

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Instance • For an instance I of STEINER(1, 2) • |I|: =the number of vertices (terminal and optional) • |N|: =n • ratio(I)= HEU(I)／OPT(I) • HEU(I) and OPT(I) mean tree length. 14

4/3 -approximation algorithm for STEINER (1, 2) G(V, H) complete graph Square: =terminal node Circle: =optional Bold edge length is 1, other unseen is 2. 15

ADH=(1+1)/1=2 16

ADH=2/1=2 17

OPT(I) 18

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OPT(I)=12+5=17 21

HEU(I) 22

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HEU(I)=22 Ratio(I)=22/17=1. 29… 25

Ratio= HEU(I)/OPT(I)=[1/2(3 n-2)]/(2 n-2)=4/3 26

4/3 -approximation algorithm for STEINER (1, 2) For an instance I of STEINER(1, 2) |I|: =the number of vertices(terminal and optional) n: =|N| HEU(I): =the length of the tree found by ADH assuming a worst possible order of breaking ties OPT(I): = the length of an optimal Steiner tree ratio(I)=HEU(I)/OPT(I) 27

Lemma 1 • A worst-case instance I contains no pair of terminals 1 apart. 28

Lemma 2 • For instance I, the average distance in each reduction is greater than 1. 29

Lemma 3 • If I contains a K-star for K≧ 3, then ratio (I) ≦ 4/3. 30

Lemma 4 • OPT(I)≧n +|P| +1 • P: =A minimum-cardinality set of vertices (optional or terminal ) that dominates all terminals in instance I. 31

Lemma 5 • OPT(I) ≧ 3/2 n- 1/2 q-1 • q: = The number of equivalence classes in this partition that contain three terminals. 32

Theorem • ADH is a 4/3 -approximation algorithm for STEINER(1, 2). HEU(I) ≦ 2 n-s-2 OPT(I) ≧ 3/2 n- 1/2 q-1 33

Thank you 34