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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.

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slide1
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)
  • ???
research interests
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
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 PolygonsChepoi & Dragan,Location Science, 1996
median points of simple rectilinear polygons1
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 polygons2
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
slide9
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)
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
a link central point and the link diameter of a simple rectilinear polygon1
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 polygon2
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 polygon3
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
a link central point and the link diameter of a simple rectilinear polygon4
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
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