Feodor F. Dragan 1990 Ph.D. in Theoretical Computer Science Institute 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) ???

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

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.

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

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

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.

Median Points of Simple Rectilinear Polygons 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 Method

A link central point and the link diameter of a simple rectilinear polygon Chepoi & 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)

A link central point and the link diameter of a simple rectilinear polygon 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 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)

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.

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

Distance Approximating Trees in Graphs Brandstaedt & Chepoi & Dragan, ESA’97, J. of Algorithms ’99, European J. of Combinatorics 2000 A graph G=(V,E) Vertex set Edges Adjacent Incident Path Connected A tree T=(V,E’) connected and minimum number of edges

Distance Approximating Trees in Graphs Shortest path Topological distance in graphs The problem : approximate by a simpler distance (e.g., by ) applications in Communication networks Data analysis Motion planning Network design Phylogeny reconstruction Numerical taxonomy

Distance Approximating Trees in Graphs Case t-spanners Multiplicative Tree t-Spanners for any Additive Tree r-Spanners

Distance Approximating Trees in Graphs Tree Spanners Problem: Given G and integer t, decide whether G has a multiplicative tree t-Spanner. For all graphs (Cai & Corneil) NP-complete for t>3 Linear for t=1,2 Open for t=3 For special graph classes Multiplicative tree 3-spanners in linear time for interval and permutation graphs (Madanlal & Venkatesan & Rangan) Additive tree 2-spanners in linear time for interval and distance hereditary graphs (Prisner) Additive tree 4-spanner for cocomparability graphs (Prisner) For chordal graphs For every fixed integer t there is a chordal graph without tree t-spanners (additive as well as multiplicative) (McKee) Question: Whether strongly chordal graphs have tree t-spanners with small t (Prisner, STACS’97)

Distance Approximating Trees in Graphs Our result Every strongly chordal graph (even every dually chordal graph) has a multiplicative 4-spanner which is also an additive 3-spanner. Such a tree can be constructed in linear time. A graph G is chordal if it does not contain any chordless cycle of length at least four. A chordal graph is strongly chordal if it does not contain any induced sun.

Distance Approximating Trees in Graphs A strongly chordal graph and an additive tree 3-spanner of it, produced by our algorithm.

Distance Approximating Trees in Graphs Case (it is allowed to use new edges) distance -approximating trees A tree T=(V,E’) is a distance –approximating tree of a graph G=(V,E) if for any

Distance Approximating Trees in Graphs Tree Spanners Problem: Given G and integer t, decide whether G has a multiplicative tree t-Spanner. For all graphs (Cai & Corneil) NP-complete for t>3 Linear for t=1,2 Open for t=3 For special graph classes Multiplicative tree 3-spanners in linear time for interval and permutation graphs (Madanlal & Venkatesan & Rangan) Additive tree 2-spanners in linear time for interval and distance hereditary graphs (Prisner) Additive tree 4-spanner for cocomparability graphs (Prisner) For chordal graphs For every fixed integer t there is a chordal graph without tree t-spanners (additive as well as multiplicative) (McKee) Question: Whether strongly chordal graphs have tree t-spanners with small t (Prisner, STACS’97)