Center and Diameter Problems

1
Center and Diameter Problems in Plane
Triangulations and Quadrangulations
Victor D. Chepoi Universite Aix-Marseille II,
France Feodor F. Dragan Kent State University,
Ohio Yann Vaxes Universite Aix-Marseille II,
France

2
The Center and Diameter Problems

• G (V,E) is a connected, finite, and undirected
  graph
• The length of a path from a vertex v to a vertex
  u is the number of edges in the path
• The distance d(u,v) is the length of a shortest
  (u,v)-path
• The eccentricity e(v) of a vertex v is the
  maximum distance from v to a vertex in G

• The radius r(G) is the minimum eccentricity of a
  vertex in G and the diameter d(G) is the maximum
  eccentricity
• The center C(G) of G is the subgraph induced by
  the set of all central vertices, i.e., vertices
  whose eccentricities are equal to r(G)
• The diameter problem find d(G) and x,y such
  that d(x,y)d(G)
• The center problem find a central vertex v of G
  or whole C(G)

3
Graphs Considered

• Trigraphs plane triangulations with inner
  vertices of degree at least six
• Squaregraphs plane quadrangulations with inner
  vertices of degree at least four
• Triangular and Square Systems the subgraphs of
  the regular triangular and square grids which are
  induced by the vertices lying on a simple circuit
  and inside the region bounded by this circuit.
• Benzenoids, alias, Hexagonal Systems subgraphs
  of the regular hexagonal grid bounded by a simple
  circuit
• Kinggraphs the graphs resulting from
  squaregraphs by transforming each inner face
  into a 4-clique (includes all subgraphs of the
  King grid bounded by a simple circuit).

4
Motivation

• The diameter and center problems are basic
  problems in algorithmic graph theory and
  computational geometry.
• They naturally arise in communication and
  transportation networks, robot-motion planning
  and also in several other areas.
• in distributed systems, centers are ideal
  locations for placing resources that need to be
  shared among different processes in a network,
• if a graph represents a road network with its
  vertices representing communities, one may have
  the problem of locating optimally a hospital,
  police station, fire station, or any other
  "emergency service facility.
• the diameter of a communication network gives a
  lower bound on the time needed to transmit a
  message from an arbitrary source node to all
  other nodes.

5
Motivation (cont.)

• The starting point of our investigations of
  these subclasses of planar graphs was a
  question how to find the center (i.e., all
  central vertices) of a benzenoid system
  efficiently A. Balaban in his H. Skolnik award
  lecture.
• Benzenoids represent a significant class of
  chemical graphs and their encoding constitutes an
  important subject of research in computational
  chemistry.
• One canonical way of such an encoding is to
  label the carbon atoms level-wise starting from
  the center.
• Trying to find an efficient algorithm for the
  center problem on benzenoids, we noticed that its
  solution can be obtained from the solution of the
  same problem on two triangular systems inferred
  from the initial benzenoid

6
Motivation (cont.)

• Squaregraphs can be used to model a road
  network (streets) in a city.
• Squaregraphs and Kinggraphs contain as
  particular cases two important classes of
  discrete metric spaces, extensively studied in
  digital geometry
• simply connected sets of lattice points in the
  plane under the graph structure defined by
  4-neighbor adjacency (city block distance) or
  8-neighbor adjacency (chessboard distance)
• Such sets of lattice points ("pixels") arise
  when planar regions are digitized they can be
  regarded as discrete approximations of these
  regions.

7
Our Results

• We designed a general approach for computing
  the centers and diameters which can be applied
  not only to triangular systems but also to all
  trigraphs, squaregraphs and kinggraphs.
• It gives linear time algorithms for computing
  the centers and diameters in all those graph
  classes as well as in all benzenoids.
• Additionally, we characterized centers of
  trigraphs and kinggraphs (answering a question
  posed by Khuller et al. KhRoWu00).
• The centers of squaregraphs were characterized
  by Postaru Po84 and, independently, by Khuller
  et al. in KhRoWu00 for the particular case of
  the square systems.
• Some properties of centers of square systems
  have been given by Metivier and Saheb in
  MeSa96.
• No linear time algorithms were known for
  computing the centers and diameters of those
  graphs.

. . .
8
Method The Diameter Problem

• For any vertex v of G, the set F(v) u V
  d(v,u)e(v) of furthest neighbors belong to
  . (Lyndon67 and Baues and Peyerimhoff01)

• To find
  we use row-wise maxima search of Aggarwal
  et al. AKMSW87 in totally monotone matrix.
• matrix D is totally monotone if
  D(i,k)ltD(i,l) implies D(j,k)ltD(j,l) for any
• The matrix is defined implicitly -- an entry is
  evaluated only when needed by the algorithm. If
  evaluating an entry takes O(f(n,m)) time, then
  the complexity of the algorithm is O((nm)f(n,m)).

is totally monotone matrix
9
d(p,q) in constant time after linear time
preprocessing

• Get metric interval I(v,w)
  x V d(v,x)d(x,w)d(v,w). It is a convex set
  and induces a triangular system.

• Get distances and projections of vertices from P
  and Q to I(v,w).

• Embed I(v,w) isometrically into the Cartesian
  product of three trees in linear time. Then, for
  any p,q, the distance can
  be computed in constant time.

10
Diameters of Squaregraphs and Kinggraphs

• Squaregraphs in a similar way as for trigraphs,
  even easier.

• Kinggraphs we reduce the problem to the
  squaregraphs

K
Q(K)

• Theorem 1 Diameters of trigraphs,
  squaregraphs, kinggraphs and benzenoids can be
  computed in linear time.
• The idea to use matrix-searching to compute the
  diameter of a simple polygon was employed by
  Hershberger and Suri in HeSu97.

11
Method The Center Problem

• Having a diametral pair, a simple region
  containing at least one central vertex will be
  located and preprocessed in such a way that all
  vertices of minimum eccentricity in this region
  can be found in linear time.
• Then, using the established structure of the
  center, the remaining part of the center can
  build up.

Get histogram H(c)
Get convex cut c of trigraph T
A histogram H(c) of a convex cut c is the union
of all metric triangles having one side on c. It
is an isometric subgraph of T (and of triangular
grid).
12
Method The Center Problem

• Get distances and projections of vertices from
  to H(c)
• harder, since H(c) is not necessarily convex it
  is convex if T does not contain inner vertices of
  degree 7.
• the case with inner vertices of degree 7 is
  handled separately.

• Embed H(c) into triangular grid and consider
  quadrangles.
• Find vertices of least eccentricity in each
  quadrangle (reduced to analyzing a system of at
  most 6 inequalities with two variables and one
  unknown parameter).
• Thus, can be
  found in linear time.

13
Centers of Trigraphs

• Structure of the center The center of a
  trigraph T is a 3-sun, a convex path, or a
  convex strip.
• Using the established structure of the center,
  one can get the entire center C(T) from the set
  in linear time.

• Here is the idea how we do this for a triangular
  system T

• Let
• Consider disk B(w,2) (3-sun case)
• Consider
  or intersections of C(T) with 9 convex cuts.

14
Centers of Squaregraphs, Kinggraphs and Benzenoids

• Squaregraphs in a similar way as for trigraphs.
  Even easier since

• Kinggraphs and Benzenoids The centers of
  hexagonal systems and kinggraphs are obtained by
  simply employing their relation to triangular
  systems and squaregraphs, respectively.
• The center of a kinggraph K is an isometric path
  or an isometric chain of .
• Theorem 2 Centers of trigraphs, squaregraphs,
  kinggraphs and benzenoids can be found in linear
  time.

15
Concluding Remarks and Open Problems

• We presented a general approach for computing
  in linear time the centers and diameters of
  trigraphs, benzenoid systems, squaregraphs and
  kinggraphs.
• We characterized centers of trigraphs and
  kinggraphs (answering a question posed by Khuller
  et al. KhRoWu00).
• Few interesting open problems remain
• Characterize the centers of benzenoid systems.
• To which other face regular planar graphs can
  this method be applied?
• Can it be extended to 3-dimensional variants of
  squaregraphs, kinggraphs and trigraphs?
• Can the p-center problem (p2,3,) be solved
  efficiently on those classes of graphs?

. . .