Moldova State University (1988 – 1996) University of Duisburg (1994 – 1995)

1. Feodor F. Dragan1990 Ph.D. in Theoretical Computer ScienceInstitute of Mathematics of the Byelorussian Academy of Science, Minsk • Moldova State University (1988 – 1996) • University of Duisburg (1994 – 1995) • University of Rostock (1996 – 1999) • UCLA (1999 – 2000) • ???

2. Research interests • Design and analysis of algorithms • Algorithmic graph and hypergraph theory • Computational geometry • Facility location problems • Operations research • Combinatorial optimization • VLSI CAD • Data analysis • Computational biology • Discrete convexity and geometry of discrete metric spaces

3. Efficient algorithms for some optimization problems • Median Points of Simple Rectilinear Polygons • A Link Central Point and the Link Diameter of a Simple Rectilinear Polygon • Computational geometry • Facility location problems • Operations research • Design and analysis of algorithms • Discrete convexity and geometry of discrete metric spaces. • Distance Approximating Trees in Graphs • Algorithmic graph theory • Data analysis • Networks design • etc.

4. Simple rectilinear polygon, vertices, edges Rectilinear path in P Length of the path - metric d(x,y) in P Median Points of Simple Rectilinear PolygonsChepoi & Dragan,Location Science, 1996

5. number of users located at a point Weber Function is a median point if Med(P) Median Points of Simple Rectilinear Polygons

6. Median Points of Simple Rectilinear Polygons • Problem formulation (facility location problem) • Given P, • Find Med(P) • Algorithmic results • Med(P) can be found in O(nlogN + N) time. • If all users are located on vertices of P then in O(N + n) time.

7. Theoretical results used (P,d) is a median space Any convex compact subset of a median space is gated Med(P) is convex and forms a simple rectilinear polygon inside of P Majority role etc. etc. etc. Median Points of Simple Rectilinear Polygons

8. Median Points of Simple Rectilinear PolygonsMethod

9. A link central point and the link diameter of a simple rectilinear polygonChepoi & Dragan,Comput. Sci. J. of Moldova, `93; Russian J. of Oper. Res., `94 • Link-distance in general polygons (Suri. PhD th. `87, motivated by robot motion-planning and broadcasting problems) • Minimum number of line segments/ of turns the path makes • Rectilinear/orthogonal link-distance in rectilinear polygons (M. de Berg `91)

10. Eccentricity Function is a central point if is the minimum eccentricity of a point in P. is the maximum eccentricity of a point in P. C(P) A link central point and the link diameter of a simple rectilinear polygon

11. A link central point and the link diameter of a simple rectilinear polygon • Problem formulation (facility location problem) • Given P • Find C(P), rad(P), diam(P) • Previous results • In simple polygons • O(nlogn) for C(P) [Djidjev et al. `89],[Ke `89] • O(nlogn) for the diameter [Suri `87] • In simple rectilinear polygons • O(nlogn) for the diameter [de Berg `91] • Open for C(P)[de Berg `91] • Our algorithmic results • A link central point, the link radius, the link diameter of a simple rectilinear polygon can be found in O(n) time. (the same results were obtained independently by Nilsson & Schuierer in 1994 (1996); they used completely different approach)

12. A link central point and the link diameter of a simple rectilinear polygon • Theoretical results used • For any point x, the set of furthest points from x contains a vertex of P. • A pair of vertices with can be found in linear time.

13. Theoretical results used (c.) The center C(P) is not necessarily connected but forms an orthogonal convex set. diam(C(P)) <5 The Helly property for intervals, etc., etc., etc. A link central point and the link diameter of a simple rectilinear polygon

14. Method eccentricity of a cut visibility intervals let Case 1. Case 2. or find instaircase, or repeat all for A link central point and the link diameter of a simple rectilinear polygon