Steiner Tree Problem

1. Steiner Tree Problem Shortest Networks Junction Points Steiner Points Mathematics in Management Science

2. Problem You want to locate a distribution warehouse which will service Akron, Cincinnati, and Toledo. Your company has a small fleet of planes to use for shipping. Where should the warehouse be built?

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7. 136 203 241 What is MST? Is there a shorter network?

8. An Easier Problem What is the shortest underground fiber-optic cable network connecting the three towns?

9. Start with the minimum spanning tree; MST found using Kruskal’s Algorithm. MST pix’d in (a); length is 1000 miles. T-network in (b) is shorter having total length approximately 933 miles. But we can do better still!

10. The Y-network shown in (c) is even shorter than the T-network in (b). In this network there is a “Y”- junction at S, with equal branches connecting S to each of A, B, C. This Y-network is approximately 866 miles long.

11. Steiner Points A key feature of the Y-network above is that three branches come together at the junction pointS forming equal 120 angles. Such a junction pointS is called a Steiner point. What is a junction point??

12. Shortest Network Problem Start with a physical network – edges weighted with actual distances. Free to add new vertices & new edges. Want to find a minimal length (shortest) network which joins all the vertices. Call such a network (which will be a tree, right?) a shortest network. Note Can add vertices and edges!

13. Junction Points Suppose have constructed some network joining all vertices. A vertex of degree 2 or more is called a junction point; this any point where two or more segments of the network come together. There are several types of junction pts.

14. Example MST in (a) has a junction point at A T-network in (b) has junction point at J Y-network in (c) has junction point at S

15. Types of Junction Points Start with a physical network; add new vertices & new edges to find network which joins all the vertices. A junction point is called • anative junction point if it is one of the original vertices • an interior junction point if not.

16. Steiner Points An interior junction point of degree 3 whose three edges come together at equal 120 angles is called a Steiner pointof the network. Steiner points are where the network forms a “perfect” Y-junction. Steiner points tell us how to find SNs!

17. 6.4 5.4 S 6 Example How to find S ?

18. Shortest Network Connecting 3 Pts If all three angles of a triangle are less than 120, then the SN linking the three vertices is obtained by finding a Steiner point S inside the triangle and joining S to each of the vertices. If one of the angles of the triangle is 120 or more, the SN consists of the two shortest sides of the triangle (i.e., the MST).

19. Examples In (a) the SN is the MST. In (b) the SN goes thru the Steiner pt S.

20. 794 794 750 600 300 300 520 Isosceles Example What is SN here? MST has length 1314. T-network has length 1270. Y-network has length 1200. Y-networkis the SN.

21. 6.4 5.4 S 6 Example How to find S ?

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