Divide and Conquer Algorithms for Trees and Series

Transcription

Algorithms for Graph Visualization
Summer Semester 2016
Lecture #4
Divide-and-Conquer Algorithms:
Trees and Series-Parallel Graphs
(based on slides from Martin Nöllenburg and Robert Görke, KIT)
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Steven Chaplick
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Lehrstuhl für Informatik I
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Universität Würzburg
Uses of Divide & Conquer
Well suited for inductively/recursively defined Graph Classes
1. draw the left subtree
2. draw the right subtree
3. combine together + draw root
Terminology
Rooted Binary Trees:
r
root
Subtree T (v)
v
right
child
left
child
depth(v): distance from the root
traversal
• preorder
• inorder
• postorder
left subtree Tl (v)
right subtree Tr (v)
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Steven Chaplick
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Overview
Grid Size
balanced drawings of binary trees
O(nh)
radial drawings of trees
O(nh)
compact drawings of trees
O(n log n)
upward drawings of series parallel graphs
exp.
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Steven Chaplick
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Algorithm of Reingold and Tilford (’81)
Trivial:
Divide:
If T = {v}, draw v (e.g., as a small disk).
Run Alg. recursively on the left and right subtrees.
Conquer: shift the partial drawings to up to 2 units apart
place the root r one above and centrally btw. children.
or 3!
Grid Drawing?
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4-7
Steven Chaplick
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Implementation in 2 Phases:
1. postorder (bottom-up):
contours and x-offsets
gather the predecessors
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2. preorder (top-down):
calculate absolute coordinates
Contour: linked list of vertices (-coordinates)
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Steven Chaplick
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Phase 1:
1. compute T` (v) und Tr (v)
2. trace the right contour of T` (v) and left of Tr (v)
3. Find dv = min. horiz. distance between v` und vr
4. x-offset(v` ) = −ddv /2e, x-offset(vr ) = ddv /2e
v
5. Build left contour of Tv from:
v,
v`
left contour of T` (v),
left contour of any low
hanging part of Tr (v)
6. Symmetrically for right contour.
vr
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Steven Chaplick
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Phase 1:
1. compute T` (v) und Tr (v)
2. trace the right contour of T` (v) and left of Tr (v)
3. Find dv = min. horiz. distance between v` und vr
4. x-offset(v` ) = −ddv /2e, x-offset(vr ) = ddv /2e
v
5. Build left contour of Tv from:
v,
v`
left contour of T` (v),
left contour of any low
hanging part of Tr (v)
6. Symmetrically for right contour.
vr
5-10
Runtime?
Steven Chaplick
P
v (1
+ min{h` (v), hr (v)}) = n +
·
P
·
v
min{. . . } ≤ n + n
Algorithm of Reingold und Tilford (’81)
Phase 2:
Set y-coordinate y(v) = −depth(v) for each vertex v.
Set x(w) := 0 for the root w, then in preorder for v ∈ V :
x(v` ) := x(v) + x-offset(v` ) and
x(vr ) := x(v) + x-offset(vr ).
Runtime? O(n)
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Steven Chaplick
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Summary for Balanced Drawings of Binary Trees
Theorem
[Reingold & Tilford ’81]
For a binary tree with n vertices, in O(n) time we can produce
a drawing Γ such that:
Easily generalizes to
arbitrary trees!
Γ is layered, i.e., y ≡ − depth,
Γ is planar, straightline, and strictly downward,
Γ matches the embedding (i.e., right children on the right),
all vertices: horiz. & vert. distances ≥ 1, and on the grid,
example?
the area is O(n2 ),
parent always centered above children.
Min. width (but without the grid): by LP!
Min. width and on the grid: NP-hard! [Supowit & Reingold ’83]
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Steven Chaplick
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Example of width variation
Output of the
Algorithm:
Optimal
Drawing:
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Steven Chaplick
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2. Radial Drawings of Trees
Grid Size
O(nh)
O(nh)
O(n log n)
exp.
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Steven Chaplick
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Example: Radial Tree Layouts
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Steven Chaplick
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An Algorithm for Radial Layout?
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10
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7
8
1
6
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1
9
1
9
10
1
8
1
7
1
·
1
10
5
1
1
3
1
11
9
1
1
7
5
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Steven Chaplick
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Restricting to Smaller Sectors
αv
ρi
αmin
τ
ρi+1
v
cos τ =
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⇒
αmax
Steven Chaplick
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ρi
ρi+1
i
αmin = αv − arccos ρρi+1
i
αmax = αv + arccos ρρi+1
·
Pseudocode for radial tree layout
RadialTreeLayout(tree T , root r ∈ T , radii ρ1 < · · · < ρk )
preorder(vertex v, t, αmin , αmax )
begin
postorder (r)
d v ← ρt
{ output }
preorder (r, 0, 0, 2π)
αv ← (αmin + αmax )/2
return (dv , αv )v∈V (T )
if t > 0 then
t
{vertex pos./ polar coord.}
αmin ← max{αmin , αv −arccos ρρt+1
}
ρt
α
←
min{α
,
α
+arccos
}
max
max
v
postorder(vertex v)
ρt+1
nv ← 1
foreach child w von v do
postorder (w)
nv ← nv + nw
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size of the subtree T (v)
Steven Chaplick
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left ← αmin
foreach child w von v do
·(αmax − αmin )
right ← left + nnvw
−1
preorder (w, t + 1, left, right)
left ← right
Runtime? O(n).
·
Correctness?
X
Overview
Grid Size
O(nh)
O(nh)
O(n log n)
exp.
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Steven Chaplick
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hv-Drawings
Definition.
An hv-drawing of a binary tree is a straight line drawing, so that
for each vertex v:
each child of v is either directly right or directly below v.
the smallest axis-parallel rectangle enclosing the subtrees of
the children of v are disjoint.
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Steven Chaplick
horizontal
combination
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vertical
combination
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Algorithm RightHeavyHVTreeDraw
Recursively construct drawings of the left and right
subtrees from the root.
Place the larger subtree on the right using the horizontal
combination, and the smaller on the left
Size of a subtree := number of vertices
at least ·2
at least ·2
at least ·2
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Obs.
The drawing has width ≤ n ,
Steven Chaplick
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height ≤ log2 n.
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Overview
Grid Size
O(nh)
O(nh)
O(n log n)
exp.
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Steven Chaplick
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Series Parallel Graphs
t
simple series parallel graph
s
Induction: combining two series parallel graphs G1 , G2 . . .
t1
t2
G2
G1
s2
s1
- . . . series . . .
t1 = s2
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s1
- . . . or parallel.
G2
G1
Steven Chaplick
t1 = t2
t2
G1
s1 = s2
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G2
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Decomposition Tree for SP-graphs
P
P
S
S
S
Q
Q
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Q
P
S
Q
·
Q
Q
S
Q
Generalization:
SPQR-Tree
Steven Chaplick
S
Q
Q
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Q
SP-Graphs: applications
PERT-Diagrams
Flow Charts
Provides:
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Steven Chaplick
(Program Evaluation and Review Technique)
Linear time algorithms for NP-complete problems
(e.g., Maximum Independent Set)
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Grid Size
Theorem
[Bertolazzi et al. ’92]
There is a family (Gn )n∈N of embedded SP-graphs where Gn
has 2n vertices and every upward planar drawing of Gn requires
tn+1
Ω(4n ) area.
Proof:
tn+1
∆2
tn
t0
tn
Gn
Gn
sn
sn+1
s0
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G0
sn−1
∆1
sn+1
Π
Gn+1
sn
a(Gn+1 ) ≥ a(Π) + a(∆1 ) + a(∆2 ) ≥ 2 · a(Π) ≥ 4 · a(Gn )
Steven Chaplick
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Divide and Conquer Algorithms for Trees and Series

Transcription

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