1. Feodor F. Dragan 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)
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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. 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

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

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. 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.

8. Median Points of Simple Rectilinear Polygons Method

9. 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)

10. 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)

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. 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.

14. 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

15. 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

16. 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

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

18. 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)

19. 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.

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

21. 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

22. 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)