Planar Orientations

1
Planar Orientations

• Chapter 4 (4.1-4.6) in the book
• Written By Tomer Heber

2
Outline Goal

• In this lesson we will deal with planar st-graphs
  (which will be explained later on).
• We will review several algorithms for
  representing a st-planar graph.
• We will also learn that by using those
  representations we can turn a st-planar graph, to
  a more specific shape (such as a polyline).
• The goal of this lesson is simple Which is to
  learn of ways to represent a st-graph in a more
  useful and aesthetic representation.

3
Numbering of Digraphs

• A topological numbering of G is an assignment of
  numbers to the vertices of G, such that, for
  every edge (u,v) of G, the number assigned to v
  is greater then the one assigned to u.

5
2
3
4
2
1
4
Numbering of Digraphs (2)

• A topological sorting is a topological numbering
  of G, such that every vertex is assigned a
  distinct integer between 1 and n.
• A topological sorting is unique if G has a
  directed path that visits every vertex.

6
5
3
4
2
1
5
Numbering of Digraphs (3)

• The following statements are equivalent
• G is acyclic.
• G admits a topological numbering.
• G admits a topological sorting.
• In other words a topological numbering (sorting)
  can be done only on an acyclic graph.

6
Planar Graph

• A planar graph is a graph which can be embedded
  in the plane, i.e., it can be drawn on the plane
  in such a way that its edges intersect only at
  their endpoints.

Faces
7
ST-Graph

• An acyclic digraph with a single source s and a
  single sink t is called an st-graph.

Sink t has no outgoing edges!
t
s
Source s has no incoming edges!
8
ST-Graph (2)

• Let G be an st-graph. The following simple
  properties hold
• Given a topological numbering of G. every
  directed path of G visits with increasing
  numbers.
• For every vertex v of G, there exists a simple
  directed path from s (source) to t (sink) that
  contains v.

9
Planar ST-Graph

• A planar st-graph is an st-graph that is planar
  and embedded with vertices s and t on the
  boundary of the external face.

t
s
10
Planar ST-Graph (2)

• Let G be a planer st-graph and F be its set of
  faces. We conventionally assume that F contains
  two representatives for the external face
• The left external face s
• The right external face t

s
t
11
Planar ST-Graph (3)

• For each edge e (u,v), we define orig(e) u
  and dest(e) v.
• We define left(e) (resp. right(e)) to be the face
  to the left (resp. right) of e.

dest(e) v
v
orig(e) u
f1
left(e) f1
e
u
left(e) f2
f2
12
Planar ST-Graph Properties

• Lemma 1 Each face f of G consists of two
  directed paths with common origin called orig(f),
  and common destination called dest(f).
• Proof Let f be a face of G for which the lemma
  is not true.

t
dest(f)
There must be a path from vertex u to the sink t
w
Since the graph is planar there must be a node
where the passes intersect
We receive a cycle!
u
There must be a path from the source s to
vertex w
orig(f)
s
13
Planar ST-Graph Properties (2)

• Lemma 2 The incoming edges for each vertex of G
  appear consecutively around v, and so do the
  outgoing edges.
• Proof The lemma holds trivially for the vertices
  s and t. Let v be any other vertex, and suppose
  for a contradiction, that there are edges (v,w0),
  (w1,v), (v,w2), (w3,v).

t
w0
w1
We now have a cycle!
v
x
w2
w3
s
Because every vertex in a st-graph has a simple
path from s to t we add the following paths
Because the graph is planar we add the vertex x
to the graph.
14
Planar ST-Graph Properties (3)

• Since the incoming (outgoing) edges of each
  vertex of G appear consecutively we define the
  face separating the incoming edges from the
  outgoing edges in clockwise order, left(v), and
  the other separating face is called right(v).

15
Planar ST-Graph G

• We define a digraph G associated with planar
  st-graph G, as follows
• The vertex set of G is the set F of faces
  (recall that F has two representatives, s and
  t, of the external face.
• For every edge e ! (s,t) of G, G has an e
  (f,g) where f left(e) and g right(e).

e
left(e)
right(e)
Notice that G is a planar st-graph as well
16
Lemma 3

• An element of is called an
  object of planar st-graph G.
• For vertex v, we define orig(v) dest(v) v.
• For a face f, we define left(f) right(f) f.
• Reminder we have already defined previously
  left(v), right(v), left(e), right(e), orig(e),
  dest(e), orig(f) and dest(f).

17
Lemma 3 (2)

• Lemma 3 For any two objects o1 and o2 of a
  planar st-graph G, exactly one of the following
  holds
• G has a directed path from dest(o1) to orig(o2).
• G has a directed path from dest(o2) to orig(o1).
• G has a directed path from right(o1) to
  left(o2).
• G has a directed path from right(o2) to left(o2).

18
Tile

• A tile is a rectangle with sides parallel to the
  coordinate axes.
• A tile can be unbounded or can degenerate to a
  segment or a point.
• Two tiles are horizontally (vertically) adjacent
  if they share a portion of a vertical
  (horizontal) side.

Tiles
Vertically adjacent
19
Tessellation Representation

• Let G be a planar st-graph. A tessellation
  representation T for G maps each object o of G
  into a tile T(o) such that
• The interiors of tiles T(o1) and T(o2) are
  disjoint whenever o1 ! o2.
• The union of all tiles T(o),
  is a rectangle.
• Tiles T(o1) and T(o2) are horizontally adjacent
  if and only if o1 left(o2) or o1 right(o2) or
  o2 left(o1) or o2 right(o1).
• Tiles T(o1) and T(o2) are vertically adjacent if
  and only if o1 orig(o2) or o1 dest(o2) or o2
  orig(o1) or o2 dest(o1).

20
Tessellation Representation (2)
4
3
1
4
f
3
3
2
4
X(left(e)) 0
X(left(f)) 3
2
2
0
X(right(e)) 1
X(right(f)) 3
2
1
Y(orig(e)) 0
Y(orig(f)) 2
Y(dest(e)) 2
Y(dest(f)) 4
1
3
1
e
1
0
0
0
1
2
3
4
21
Tessellation Representation (3)

• The correctness of the algorithm is based on
  Lemma 3
• Let there be tile t1 and tile t2, from Lemma 3 t1
  is either above t2, below t2, left of t2 or
  right of t2. And only one of this directions is
  true.
• Since each line of the algorithm is O(n), the
  total runtime of the algorithm is O(n).
• The size of the Tessellation Representation can
  be modified by modifying the topological
  numbering (e.g. increasing the numbering to be
  0..2..4 instead of 0..1..2 will make a
  Tessellation Representation twice bigger).

22
Visibility Representation

• Let G be a planar st-graph. A visibility
  representation of G draws each vertex v as a
  horizontal segment, called vertex segment
  , and each edge (u,v) as vertical segment, called
  edge segment such that
• The vertex segments do not overlap.
• The edge segments do not overlap.
• Edge-segment has its bottom end
  point on , its top end-point on
  , and does not intersect any other vertex
  segment.

23
Visibility Representation (2)
24
Visibility Representation (3)
4
3
1
4
3
3
2
4
2
2
0
2
1
1
3
1
0
1
0
0
1
2
3
4
25
Visibility Representation (4)
4
3
1
4
3
3
2
4
2
2
0
2
1
1
3
1
0
1
0
0
1
2
3
4
26
Visibility Representation (5)

• The correctness of the algorithm By lemma 3 and
  the construction of the algorithm
• Any two vertex segments are separated by a
  horizontal or vertical strip of at least unit
  width (The vertex segments do not overlap).
• Any two edge segments on opposite sides of a face
  are separated by a vertical strip of at least a
  unit width (The edge segments do not overlap).
• Each edge segments (u,v) has its bottom point
  intersecting with u vertex segment, and his upper
  point intersecting with v vertex segment
  (sufficing the 3rd condition).
• The runtime of the algorithm is O(n) since each
  step is O(n).

27
Constrained Visibility Representation

• Let G be a planar st-graph with n vertices. Two
  paths of p1 and p2 of G are said to be
  nonintersecting if
• They are edge disjoint.
• And do not cross at common vertices.

p1
p2
p3
28
Constrained Visibility Representation (2)

• Given a collection ? of nonintersecting paths of
  G, we consider the problem of constructing a
  visibility representation G of G such that for
  every path p in ? we have the following
  constraint
• Any two edges e and e of p, the edge segments
  G(e) and G(e) have the same x-coordinate.
• Note We assume that ? covers all the edges in
  graph G (Such that any edge who is not in any of
  the nonintersecting paths in the collection ?, is
  a nonintersecting path himself in ?).

29
Constrained Visibility Representation (3)
30
Constrained Visibility Representation (4)
?p1, p2,p3,p4,p5, p6, p7, p8
4
p8
3
p8
0
p7
0.5
2.5
p7
f5
f6
f5
f6
p6
p1
p6
2
3
1.5
1
-0.5
2
3.5
p1
f4
f7
2
p2
f4
f7
p2
f1
2
f1
1
f3
2.5
f3
p3
p5
f2
p4
f2
0.5
p3
0
1
p4
p5
3
1
0
31
Constrained Visibility Representation (5)
4
3
2
1
0
0
1
2
3
3
p8
0
4
p7
0.5
2.5
p8
Total Runtime O(n)
p7
f5
f6
f5
f6
p6
p1
p6
2
3
1.5
1
-0.5
2
3.5
p1
f4
f7
2
p2
f4
f7
p2
f1
2
f1
1
f3
2.5
f3
p3
p5
f2
p4
f2
0.5
p3
0
1
p4
p5
3
1
0
32
Polyline Drawing

• We can construct a planar upward polyline drawing
  of a planar st-graph G using its visibility
  representation.
• We draw each vertex in an arbitrary point inside
  its vertex segment.
• We draw each edge (u,v) of G as a three segment
  polygonal chain.

33
Polyline Drawing (2)
4
3
2
2
1
1
0
34
Polyline Drawing (3)

• Since every step in the algorithm is O(n), the
  total complexity runtime is O(n).
• We can reduce the number of bends if we put
  each place vertex on intersections between an
  edge segment and vertex segment.

35
Polyline Drawing (4)

• The technique can be extended for constrained
  visibility representation of a planar st-graph
  which is called constrained-polyline.
• In a constrained-polyline all the internal
  vertices in a path p in ? are vertically aligned.

The algorithm for Constrained polyline
36
Questions???