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Metric Space Revisited

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... question is when are two metric spaces (M,d) and (M',d') considered isomorphic. ... How many such non-isomorphic drawings are there? ( Hint: there are six) ... – PowerPoint PPT presentation

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Title: Metric Space Revisited


1
Metric Space - Revisited
  • If (M,d) is a metric space, then for any A
    µ M with the induced metric (A,d) is also a
    metric space, a subspace.
  • A natural question is when are two metric spaces
    (M,d) and (M,d) considered isomorphic.
  • There are two types of mappings that are
    candiates for isomorphism.

2
Isometries
  • Let (M,d) and (M,d) be two metric spaces. A
    bijective mapping s M ! M is called is
    isometry, if for every pair of points u,v 2 M we
    have
  • d(u,v) d(s(u),s(v)).
  • Clearly, isometric spaces are indistingushable as
    far as metric properties are concerned.

3
Euclidean metric in Rn.
  • The set of real n-tuples
  • Rn x (x1,x2,...,xn)xi 2 R, 1 i n
  • carries a number of important mathematical
    structures. The mapping
  • dp(x, y) (x1 y1)p (x2 y2)p ... (xn
    yn)p1/p.
  • makes (Rn,dp) a metric space for 1 p 1.
  • For p 2 the usual Euclidean metric is obtained.

4
Metric in C.
  • Let z a bi and w c di be two complex
    numbers.
  • Define d(z,w) z w. Then (C,d) is a metric
    space.
  • Note that (C,d) is isometric to the Euclidean
    plane (R,d2).

5
Similarity I
  • Let (M,d) and (M,d) be two metric spaces. A
    mapping hM ! M with the property that for any
    four points a,b,c,d 2 M we have
  • If d(a,b) d(c,d) then d(h(s),h(b))
    d(h(c),h(d)) is called similarity (of type I).

6
Similarity II
  • Let (M,d) and (M,d) be two metric spaces and r
    2 R\0. A mapping hM ! M with the property
    that for any pair of points a,b, 2 M we have
  • If d(a,b) r d(h(a),h(b)) then h is called
    similarity (of type II) and r is called the
    dilation factor.

7
Type I vs. Type II
  • Clearly each similarity of type II is also a
    similarity of type I. In general, the converse is
    false.
  • Theorem. A similarity on (Rn,d2) of type I is
    also of type II. (Proof can be found in Paul B.
    Yale Geomerty and Symmetry, Dover, 1988 (reprint
    from 1968))

8
Finite Metric Space
  • In a finite metric space (M,d) we may assume that
    min d(u,v) 1. Max d(u,v) is called the diameter
    of M. Quotient Max d(u,v)/Min d(u,v) is called
    dilation coefficient.

9
Representation of Graphs
  • Let G be a graph and let V be a set. A pair of
    mappings rVV(G) ! V and rEV(G) ! P(V) is called
    a V-representation of graph G if for any edge e
    uv 2 E(G) we have rV(u),rV(v) µ rE(uv). If
    there is no danger of confusion we will drop the
    subscripts and denote both mappings simply by r.
  • Usually we require V to be a vector space (this
    is what C. Godsil and G. Royle do in their book
    Algebraic Graph Theory, Springer, 2001). But that
    is not always the case. In their definition
    Godsil and Royle use a single mapping defined on
    the vertices. Insuch a case we may extend the
    mapping on the edge set in an arbitrary way, for
    instance by taking rE(uv) rV(u),rV(v).

10
Representation of Graphs in Metric Space
  • There are important and deep results by László
    Lovász et al.
  • Sometimes we may take V to be a metric space,
    projective space or some other structure.
  • If (V,d) is a metric space we may define the
    energy of representation.

11
Point Configuration
  • A point configuration S µ V is a collection of
    elements of some space V. Later we will consider
    point configurations in R2.
  • If r is a V-representation of G then the image S
    r(V(G)) is a point configuration.
  • We say that r is vertex faithful is rV(G) ! S is
    a bijection. We are mostly interested in vertex
    faithful representations.

12
Graph Representation An Example
  • For the cube graph Q3 there are several useful
    representations
  • 3 dimensional real representation In R3 the
    eight vertices are mapped to the eight points of
    0,13.
  • The two drawings of Q3 in the Euclidean plane can
    be interpreted as representations in
  • 2 dimensional real representation R2 or in
  • 1 dimensional complex representation C.
  • In the latter case, the points in the complex
    plane are given by eikp0 L k L 7.

13
Extending Representation to Edges
  • Usually we try to extend r mapping to the edges.
    In the case V R2 or V R3 finding a
    representation means actually drawing graph G in
    V R2 or V R3 .
  • Each edge euv is then represented as the line
    segment connecting r(u) and r(v).
  • Hence r(e) conv(r(u),r(v)).
  • In general we extend r to the edges r E(G) !
    P(V) and require that for e uv, r(u),r(v) µ
    r(e).
  • If nothing is said about edge extension, we
    assume r(e) r(u),r(v).

14
Edge Extensions
  • Let e uv 2 E(G).
  • There are several possible edge extensions
  • r(e) r(u),r(v).
  • r(e) r(u),r,r(v).
  • r (r(u)r(v))/2.
  • r(e) conv(r(u),r(v)).
  • r(e) aff(r(u),r(v))
  • We may speak of barycentric, convex and affine
    edge extensions, respectively.
  • But there are several other interpretations of r
    and a variety of possible edge extensions.

r(u)
r(u)
r(u)
r(u)
r(u)
r
r
r(v)
r(v)
r(v)
r(v)
r(v)
15
Three Classical Results
  • Steinitz Theorem, Fary Theorm and Tutte Theorem
    can be interpreted as graph representations.

16
Graph Representation vs. Graph Drawing
  • There is some overlap but there are many
    differences.
  • In graph drawing (in the broad sense of the word)
    the object is to find algorithms to draw a graph
    (usually in the plane) with certain restrictions
    or with some optimization criterion. Computer
    Science Approach. See for example Annotated
    bibliography on graph drawing algorithms, by Di
    Battista, Eades, Tamassia and Tollis.
  • In graph representation we label vertices ( add
    coordinates). We may look at this as a functor
    from the category of graphs to the category of
    coordinatized graphs. Mathematical Approach. We
    will use the word graph drawing in a narrow sense
    of the word.

17
The Energy
  • Usually we try to find among the representations
    of certain type the one that is optimal in
    cetrain sense.
  • To this end we may define an energy function E(r)
    and then seek for representation that minimizes
    the energy.
  • There are several such energy functions used in
    various problem areas.

18
Some Energy Models
  • Spring embedders
  • Molecular mechanics
  • Tutte drawing
  • Schlegel diagram drawing (B. Plestenjak).
  • Connection to Markov Chains
  • ...
  • Laplace Representation

19
The Laplace Representation
  • Let r be a representation in Rk. Define
  • E(r) Suv 2 E(G) r(u)-r(v)2
  • It turns out that the minimum (under some
    reasonable conditions) is achieved as follows.
  • Take the Laplace matrix of G.
  • Q(G) D(G)-A(G)
  • Find the eigenvalues 0 l1 l2 ...
    ln.
  • Find the corresponding orthonormal eigenvectors
    x1, x2, ..., xn.
  • Form a matrix R x2x3 ... xk1
  • Let r(vi) rowi(R).

An R3 Laplace representation of a fullerene
(skeleton of a trivalent polyhedron with
pentagonal and hexagonal faces)
20
Nodal Domains
  • One dimensional representation defines partition
    of the vertex set in three classes V, V-, V0.
  • A nodal domain is a connected component of the
    graph induced by Vor V-. Weak nodal domain V
    V0.

21
Nodal Domains
  • The Example on the left represents nodal domains
    obtained from the Laplace representation of
    G(10,4).

22
Congruence and Similarity
  • A representation in any metric sapce, in
    particular in Rn, can be scaled without being
    changed too much, If r is injective on vertices,
    we may scale it in such a way that Min d(u,v)
    1, for u v. Each vertex faithful representation
    is similar to a standard one.

23
Unit Distance Graphs
  • Let r be a representation in Rk. Define
  • Ep (r) (Suv 2 E(G) r(u)-r(v)p) (1/p)
  • We assume that Min uv 2 E(G) r(u)-r(v) 1
  • In the limit when p ! 1 we get
  • E1 (r) Maxuv 2 E(G) r(u)-r(v)
  • The number E1 (r) is called dilation
    coefficient.
  • Hence E1 (r) 1. In the special case E1 (r) 1
    we call this representation a unit distance
    graph.

24
Homework
  • H1. It is easy to verify that K4 is not a unit
    distance graph in the plane. Consider a drawing
    of K4 in the plane with only two distinct edge
    lengths. How many such non-isomorphic drawings
    are there? (Hint there are six). Compute the
    dilation coefficient for all such drawings.

25
Flat Torus
  • Take a unit square and identify two opposite
    pairs of sides. The resulting topological space
    is a torus.In order to make it metric sapce we
    can extend the usual Euclidean distance .
  • dT(r,s) Mind(r,s(0,1)), d(r,s(1,0)),
    d(r,s(1,1)), d(r,s(0,-1)). d(r,s(-1,0)),
    d(r,s(-1,1)), d(r,s(1,-1)), d(r,s(-1,-1)).

r (rx,ry)
s (sx,sy)
26
Exercises
  • N1. Given a standard drawing of G(n,r) with inner
    radius r and outer radius R, determine the
    dialation coefficient of this planar
    representation.
  • N2. Select the optimal quotient R/r in the
    previous exercise.
  • N3. In the unit flat torus draw the circle of
    radius ½ centered in the point (¼, ¼).

27
Embeddings vs. Representations
  • Let S be a nice topological space such as
    metric space and G be a general graph. A mapping
    ?G ? S is defined as follows
  • Injective mapping ?V(G) ? S
  • Family of continuous mappings ?e0,1 ? S, for
    each edge e uv so that ?e( 0) ?(u) and ?e(1)
    ?(v).
  • In the interior of the interval ?e is
    injective.
  • Each embedding would qualify.
  • Note that ? defines a representation of G in S.

28
Embeddings are Representations
  • Think of K3 K3 embedded in torus, that, in
    turn, is embedded in R3. We obtain a
    representation of this graph in torus and another
    one in R3.

29
Stereographic Projection
  • There is a homeomorphic mapping of a sphere
    without the north pole N to the Euclidean plane
    R2. It is called a stereographic projection.
  • Take the unit sphere x2 y2 z2 1 and
    the plane z 0.
  • The mapping p T0(x0,y0,z0) a
    T1(x1,y1) is shown on the left.

N
T0
T1
30
Stereographic Projection
  • The mapping p T0(x0,y0,z0) a
    T1(x1,y1) is shown on the left.
  • r1 r0/(1-z0)
  • x1 x0/(1-z0)
  • y1 y0/(1-z0)

N
T0
T1
31
Stereographic projection and representations
  • We may use stereographic projection to get a R2
    drawing from a R3 drawing.
  • Note that the representation of edges is computed
    anew!

32
Example
  • Take the Dodecahedron and a random point N on a
    sphere.
  • Stereographic projection is depicted below.
  • A better strategy is to take N to be a face
    center.

33
Example
  • A better strategy is to take N to be a face
    center as shown on the left.
  • Only vertices are projected. The edges are
    re-computed.

34
Schlegel Diagram
  • Usually Schlegel diagram is defined for a
    (convex) polyhedron. Normally, it is definded as
    a projection of the polyhedron on one of its
    faces.
  • We understand this notion in a broader sense.
    This is a drawing of a graph G within the convex
    region defined by some of its vertices S ½ V(G).

35
Exercises
  • N1. Use Laplace representation followed by
    stereographic projection to get schlegel diagrams
    of platonic graphs.
  • N2. Use a generalization of Tuttes method to
    slove the same problem.
  • N3. Repeat the the two exercises for some of the
    archimedean solids and their duals.
  • N4. Is there a unit distance representation for
    the subdivision graph S(K4)?
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