1. More Efficient Generation of Plane Triangulations Shin-ichi Nakano Takeaki Uno Gunma University National Institute of JAPAN Informatics, JAPAN 23/Sep/2003 GD2003

2. Outline of Our Research We consider the problem of enumerating Plane triangulations with n vertices having r vertices on the outerface having r vertices on the outerface We propose an algorithm, and reduce the time complexity per triangulation from O(r 2 n) to O(rn) from O(r 2 n) to O(rn)

3. Why Enumerate Graph Objects ? We can obtain complete lists of graph objects, used for - - proving properties - - finding counter examples - - making bench mark problems, etc. We can generate candidates for solutions of - - optimization problems - - clustering - - data mining, etc. Recent increase of computer power intensifies the advantage of enumeration

4. Plane triangulation: a plane graph ( planer graph with order of edges, embedding) all whose inner faces are triangles Plane Triangulation outer face Inner face biconnected plane triangulations Enumerate all biconnected plane triangulations with n vertices not biconnected

5. Two plane triangulations are the same if there is a one-to-one mapping of vertices and edges preserving the adjacency and order of edges Isomorphism Not output the same triangulation twice v1 u1 u2 v2 f1 e4 e3 e2 e1 f2 f3 f4

6. A simple way: A simple way: add a triangle one-by-one  so many duplications How We Enumerate ? To avoid duplication is difficult

7. 98, B.D.McKay O( n 2 ) ? per one 01, Nakano (ICALP 01) O( r 2 n ) per one ( generalized problem) Existing Research Generalized problem: Enumerating biconnected triangulations with n vertices having r vertices on the outer face We improve this We improve this  O( rn )

8. based Enumerate based biconnected triangulations Framework of Nakano 01 Output one of r same triangulations based  decide output or not, at each based triangulation  O(1) for each (r duplications) (based edge is like root of rooted tree)

9. unique string Define a unique string for given triangulationbase edge triangulation and base edge Compute the strings, and output if the string of current solution is the minimum How to Decide ? I'm minimum! I'm not Computing string: O(n) r strings: O(rn)  O(r 2 n) for each triangulation We reduce this to O(n)

10. Remove triangles adjacent to the outerface, iteratively Treat these one-by-one String for Given Triangulation and Edge 1st2nd3rd

11. "Sequence of triangles" in clockwise order from base edge Each triangle is a letter representing - - including an edge of outer face? - - how far to previous triangle including each its vertex Give String to Base Edge start from the base edge trace triangles in clockwise order the obtained string represents the structure, and unique

12. Connect the top and the end of the string  circular string Each string is obtained by cutting this at the edge Find Minimum String finding the edge minimizing the string = finding the position to be cut cut here

13. A version of "circular string linearization problem"  linear time in the length Circular String Linearization ABCEAGAFFA… Find the minimum string obtained by a cut minimize not

14. If two or more are the minimum we evaluate next triangles Tie Break 1st 2nd 3rd

15. Conclusion and Future work Reduce the time complexity for enumerating biconnected plane triangulations with n vertices having r vertices on outer face from O(r 2 n) to O(rn) for each triangulation We find a way to give an edge on the outer face for biconnected plane triangulations in O(n) time -------------- Can we generalize this technique for other plane graph objects ? Can we enumerate only based triangulations giving minimum strings ?