Combining Traditional Map Labeling with Boundary Labeling

Combining Traditional Map Labeling with Boundary Labeling

[Sofsem 2011]

M. A. Bekos1, M. Kaufmann2, D. Papadopoulos1, A. Symvonis1

1 School of Applied Mathematical & Physical Sciences,

National Technical University of Athens, Greece

{mikebekos,dpapadopoulos,symvonis}@math.ntua.gr

2 University of Tubingen, Institute for Informatics, Germany

[email protected]

Sofsem 11 (2011/01/27) Mixed Labeling 1 / 29

Map Labeling


Map Labeling


Known Map Labeling Models

1 Fixed Position Models:

1-position 2-position 4-position

2 Sliding Models:

1-slider 2-slider 4-slider


Boundary Labeling: A Different Labeling Approach

• Large Labels → Overlaps.

• The map contains information, which must not be obscured.


Boundary Labeling: A Different Labeling Approach

• Large Labels → Overlaps.

• The map contains information, which must not be obscured.


The Boundary Labeling Model.

According to Boundary Labeling Model:

• Labels are placed on the boundary of a rectangle R (enclosing rectangle).

• Each site is connected to its label by non-intersected polylines (leaders).

• The labels can be attached to one, two or all four sides of R.

Drawing

R


Boundary Labeling: Leader Types

Different Types of Leaders:

• Straight-line type-s.

• Rectilinear type-po and type-opo.

• Octilinear type-do/od/pd.

R

Type-s leaders

R

Type-po leaders

RTrack Routing Area

Type-opo leaders

R

Type-do/od/pd leaders


Can we Combine Both Approaches?


Can we Combine Both Approaches?


Mixed Labeling

Input:

• Sites: P = {s1, s2, . . . sn}, si = (xi, yi)

• Labels: L = {`1, `2, . . . `n}• Internal

• External

• Enclosing rectangle R

• Model: (m, k, t)

• m ∈ {1P, 2P, 4P, 1S, 2S, 4S}• k ∈ {Left-Sided, Right-Sided, Two-Sided}• t ∈ {s, po, opo, do, od, pd}


Mixed Labeling

Output:

• Legal Labeling:

• Non-overlapping Internal Labels.

• Non-intersecting Leaders.

• No intersections among Leaders and Internal Labels

• Optimization: Maximize the no of internal labels


Previous Work

Formann and Wagner [1991]:

• ∃ overlap-free labeling under 4-position? → NP-complete.

Iturriaga and Lubiw [1997]:

• ∃ overlap-free labeling under 1-slider? → NP-complete.

⇒ Any mixed labeling problem with 4P or any slider model is difficult.

Fowler, Paterson, Tanimoto [1981]:

• Maximize the no. of non-overlapping labels under 1-position → NP-complete.

∃ variants where we can give an algorithm that maximizes the no. of internal labels


Previous Work










Previous Work










Previous Work










The (1P, Right-Sided, opo) Mixed Labeling Model

• Internal labels: 1P

• External labels: Right

• Leaders: opo



Property:

ab

ab

ab



1 Start with boundary labeling.

2 Identify overlaps and mark leaders

permanent.

3 If no permanent leader crosses the

internal label of a site, then remove

the leader and place an internal

label.

Complexity: O(n log2 n)





permanent.




label.






permanent.




label.






permanent.




label.






permanent.




label.



The (1P, Left-Sided, opo) Mixed Labeling Model


• External labels: Left

• Leaders: opo



Previous property is not applied:

ab

ab

ab



Idea: Use Recursion

1 Fix an internal label.

2 Split into half-sized subproblems.

3 Recursively solve the subproblems

4 Keep the solution which maximizes number of internal labels.



1 Splitting Line: L

2 Fix s: leftmost with internal label.

3 Recurse.

L

R

T (k) =

{2kT (k/2) + n2 log2 n, k > 1

O(1), k = 1

Complexity: O(nlogn+3)



1 Splitting Line: L


3 Recurse.

Ls

R

Rs

T (k) =

{2kT (k/2) + n2 log2 n, k > 1

O(1), k = 1




1 Splitting Line: L


3 Recurse.

Ls

R

Rs

T (k) =

{2kT (k/2) + n2 log2 n, k > 1

O(1), k = 1




1 Splitting Line: L


3 Recurse.

Ls

R

Rs

T (k) =

{2kT (k/2) + n2 log2 n, k > 1

O(1), k = 1




• Can we obtain a more efficient algorithm?

• Idea: Relax the optimality constraint.



• ∀ site si → variable zi

• zi = true ⇔ si is labeled external

• zi = true ⇔ si is labeled internal

(a) Internal Labels Overlap: zi ∨ zj(b) Site - Internal Label: zi

(c) Internal Label - Leader: zi ∨ zj

sisj

sisj

sj

si

(a)

(b)

(c)

2-Sat with n variables, O(n2) clauses: Need a solution with min no of true variables.

⇒ 2-Approximation of O(n3) time [Gusfield and Pitt 1992]



• ∀ site si → variable zi

• zi = true ⇔ si is labeled external

• zi = true ⇔ si is labeled internal

(a) Internal Labels Overlap: zi ∨ zj(b) Site - Internal Label: zi

(c) Internal Label - Leader: zi ∨ zj

sisj

sisj

sj

si

(a)

(b)

(c)

2-Sat with n variables, O(n2) clauses: Need a solution with min no of true variables.

⇒ 2-Approximation of O(n3) time [Gusfield and Pitt 1992]


The (1P, Two-Sided, opo) Mixed Labeling Model


• External labels: Left-Right

• Leaders: opo




• External labels: Left-Right

• Leaders: opo

• Extra assumptions:

1 Uniform Label height: h

2 Each slice of height h

contains at most λ sites.



Idea: Use Recursion (similar to previous quasi-polynomial time algorithm)

1 Fix two labels.

2 Split into “half-sized” subproblems.

3 Recursively solve the subproblems

4 Keep the solution which maximizes number of internal labels.



1 Splitting Line: L


3 Fix t: leftmost with external label.

4 Recurse.

Ls

Ltop

Lbottom

R

Rs

R1

R2

R3

t

T (n) ≤

{λ2(2T (n/2 + λ) + n log2 n), n > 3λ

33λλ log2 λ, n ≤ 3λ

Complexity: (n/λ)O(log λ) · 3O(λ)



1 Splitting Line: L


3 Fix t: leftmost with external label.

4 Recurse.

Ls

Ltop

Lbottom

R

Rs

R1

R2

R3

t

T (n) ≤

{λ2(2T (n/2 + λ) + n log2 n), n > 3λ

33λλ log2 λ, n ≤ 3λ

Complexity: (n/λ)O(log λ) · 3O(λ)


Generalizations

A λ-height restricted instance of the (m, k, opo) problem, where m ∈ {2P, 4P},k ∈ {∅, Left-Sided, Right-Sided, Two-Sided} can be solved in time

(n/λ)O(λ log κ) · (κ+ µ)λ+1


Experimental Results

We have implemented:

1 Polynomial time algorithm: (1P, Right-Sided, opo)

2 Quasi-polynomial time algorithm: (1P, Left-Sided, opo)

3 λ-parameterized algorithm: (1P, Two-Sided, opo)

Evaluation Parameter: Density = total area of all labelsarea of enclosing rectangle


Experimental Results: Time Complexity

Low Density 1/10 High Density 1/6

• X-axis: Number of points

• Y -axis: Running time measured in sec

Note: Two-side algorithm is much slower

70 sites → ≈ 1 min.


Experimental Results: Quality of produced illustrations

Low Density 1/10 High Density 1/6

• X-axis: Number of points

• Y -axis: Quality in terms of the number of internal labels


Open Problems

• Our algorithms are mostly subexponential.

[ There is large space for improvements ]

Future Directions:

• Labels occupy two adjacent or four sides of R.

• Extension from type-opo leaders to more appealing type-po (or octilinear)

• Leaders only when they are bend-less.

• Leader length restrictions.


The End

Thank you...


Combining Traditional Map Labeling with Boundary Labeling

Documents