1. A linear-time algorithm to compute a MAD tree of an interval graph Elias Dahlhaus, Peter Dankelmann, R.Ravi

2. What is MAD tree? ► Average distance:, edge is unweighted, edge is unweighted ► A minimum average distance (MAD) spanning tree of G is a spanning tree of G which has minimum average distance.

3. In general case, finding a MAD tree is NP-hard ► B.Y. Wu, G. Lancia, V. Bafna, K.M. Chao, R. Ravi, C.Y. Tang, 1999, polynomial-time approximation ► E. Dahlhaus, 2003, distance-hereditary graph in linear time ► P. Dankelmann, outerplanar graph in polynomial time ► This paper represents an algorithm to compute a MAD tree of an interval graph in linear time

4. Definition ► Distance of a vertex v in G: ► Median vertex: vertex c for which is minimum

5. Lemma 1 a) If T is a MAD tree of G and c is a median vertex of T then every T-path starting at c is an induced path in G (i.e. no diagonals in G). b) T and c can be chosen such that there is no vertex c ’ ≠ c such that is strictly contained in v1 viVj-1vj diagonal

6. Proof a V1= c viVj-1vj T1T2

7. Proof b c c’c’

8. Interval graph ► Induced path L-path R-path

9. Definition ► h(v) is a neighbor x of v such that r(x) is maximum, h i (v)=h(h i- 1 (v)) ► k(v) is a neighbor y of v such that l(y) is minimum, k i (v)=h(k i- 1 (v))

10. Definition (cont.) ► :set of all vertices at distance i from v whose interval is on the right of v ► : set of all vertices at distance i from v whose interval is on the left of v

11. Theorem 1 ► If G is an interval graph then there is a MAD tree T of G with a median vertex c, such that

12. Proof ► To show that only one vertex in has T- neighbors in. Assume

13. Linear time algorithm ► Assumption: interval representation of a graph is given and l(v), r(v) are sorted. ► Evaluating following quantities: ► num R (v): number of neighbors of h(v) that are not neighbors of v ► :number of vertices of the tree ► :total distance of ► :distance of h(v) in

14. Evaluating num R (v) ► Number of vertices w, such that Determined in overall linear time.

15. Evaluating others by recursion

16. Determine the total distance of T v v

17. Remained Problem ► Vertex weighted interval graph ► Edge weighted interval graph