Optimizing Active Ranges [.5ex] for Consistent Dynamic Map ... · 1 _ x 2 _ x 3 x 1 _ x 3 _ x 4 x 1 _ x 2 _ x 4 x 2 _ x 3 _ x 4 x 1 x 2 x 3 x 4 planar 3SAT formula ' (set of labels,

ModelComplexity

Approximation

Optimizing Active Rangesfor Consistent Dynamic Map Labeling

Ken Been1 Martin Nöllenburg2 Sheung-Hung Poon3

Alexander Wolff4

1Yeshiva University

2Universität Karlsruhe

3National Tsing Hua University

4TU Eindhoven

Been, Nöllenburg, Poon, Wolff 1 19 Dynamic Map Labeling

ModelComplexity

Approximation

Outline

Model

Complexity

ApproximationTop-to-bottom sweep algorithmLevel-based algorithm

Been, Nöllenburg, Poon, Wolff 2 19 Dynamic Map Labeling

ModelComplexity

Approximation

Requirements in static map labeling

non-overlapping labels

proximity of feature and labelunambiguitymaximize number of labeled features

A-TownB-VillageC-City

Been, Nöllenburg, Poon, Wolff 3 19 Dynamic Map Labeling

ModelComplexity

Approximation

Requirements in static map labeling

non-overlapping labels

proximity of feature and labelunambiguitymaximize number of labeled features

A-TownB-Village

C-City

Been, Nöllenburg, Poon, Wolff 3 19 Dynamic Map Labeling

ModelComplexity

Approximation

Requirements in static map labeling

non-overlapping labelsproximity of feature and label

unambiguitymaximize number of labeled features

A-TownB-Village

C-City

airport

train station

Been, Nöllenburg, Poon, Wolff 3 19 Dynamic Map Labeling

ModelComplexity

Approximation

Requirements in static map labeling

non-overlapping labelsproximity of feature and label

unambiguitymaximize number of labeled features

A-TownB-Village

C-City

airport

train station

Been, Nöllenburg, Poon, Wolff 3 19 Dynamic Map Labeling

ModelComplexity

Approximation

Requirements in static map labeling

non-overlapping labelsproximity of feature and labelunambiguity

maximize number of labeled features

A-TownB-Village

C-City

airport

train station

hotelrestaurant

store

Been, Nöllenburg, Poon, Wolff 3 19 Dynamic Map Labeling

ModelComplexity

Approximation

Requirements in static map labeling

non-overlapping labelsproximity of feature and labelunambiguity

maximize number of labeled features

A-TownB-Village

C-City

airport

train station

hotelrestaurant

store

Been, Nöllenburg, Poon, Wolff 3 19 Dynamic Map Labeling

ModelComplexity

Approximation

Requirements in static map labeling

non-overlapping labelsproximity of feature and labelunambiguitymaximize number of labeled features

A-TownB-Village

C-City

airport

train station

hotelrestaurant

storecastle monumenthill

museum

Been, Nöllenburg, Poon, Wolff 3 19 Dynamic Map Labeling

ModelComplexity

Approximation

Requirements in static map labeling

non-overlapping labelsproximity of feature and labelunambiguitymaximize number of labeled features

A-TownB-Village

C-City

airport

train station

hotelrestaurant

storecastle monumenthill

museum opera

Been, Nöllenburg, Poon, Wolff 3 19 Dynamic Map Labeling

ModelComplexity

Approximation

Requirements in dynamic map labeling

interactive maps add morerequirements

static map at each scalenon-overlapping labelsfeature–label proximityunambiguitymaximize label number

during zooming & panningno popping of labelsno jumping of labelsmap independent ofnavigation history

Been, Nöllenburg, Poon, Wolff 4 19 Dynamic Map Labeling

ModelComplexity

Approximation

Static model

Static selectionBoolean function that selects subset of non-overlapping labels

Static placement1 transform label L to world coordinates

(by translation, rotation, dilation)2 transform world coordinates to screen coordinates with

dilation factor 1/s (define s as the scale of the map)

L L̂

y

x

Been, Nöllenburg, Poon, Wolff 5 19 Dynamic Map Labeling

ModelComplexity

Approximation

Our dynamic model ∼[Been, Daiches, Yap 2006]Dynamic selectionBoolean function of scale selects each label Li in at most onescale interval [ai , Ai ], its active range

Dynamic placement

static placement L̂s for each scale scontinuous with stransforms label L to extended world coordinates (x , y , s)L̂s is cross section of extended world coordinates at scale s

Been, Nöllenburg, Poon, Wolff 6 19 Dynamic Map Labeling

ModelComplexity

Approximation

Our dynamic model ∼[Been, Daiches, Yap 2006]Dynamic selectionBoolean function of scale selects each label Li in at most onescale interval [ai , Ai ], its active range → no popping

Dynamic placement

static placement L̂s for each scale scontinuous with stransforms label L to extended world coordinates (x , y , s)L̂s is cross section of extended world coordinates at scale s

Been, Nöllenburg, Poon, Wolff 6 19 Dynamic Map Labeling

ModelComplexity

Approximation

Our dynamic model ∼[Been, Daiches, Yap 2006]Dynamic selectionBoolean function of scale selects each label Li in at most onescale interval [ai , Ai ], its active range → no popping

Dynamic placement

static placement L̂s for each scale scontinuous with stransforms label L to extended world coordinates (x , y , s)L̂s is cross section of extended world coordinates at scale s

Been, Nöllenburg, Poon, Wolff 6 19 Dynamic Map Labeling

ModelComplexity

Approximation

Our dynamic model ∼[Been, Daiches, Yap 2006]Dynamic selectionBoolean function of scale selects each label Li in at most onescale interval [ai , Ai ], its active range → no popping

Dynamic placement

static placement L̂s for each scale scontinuous with s → no jumpingtransforms label L to extended world coordinates (x , y , s)L̂s is cross section of extended world coordinates at scale s

Been, Nöllenburg, Poon, Wolff 6 19 Dynamic Map Labeling

ModelComplexity

Approximation

Extended world coordinates

scale as 3rd dimension

union of label shapesover scale: “extrusion”

restriction to active range:“truncated extrusion”

here:axis-aligned rectangular labelsinvariant-point placementproportional dilation

Been, Nöllenburg, Poon, Wolff 7 19 Dynamic Map Labeling

ModelComplexity

Approximation

Active-range optimization

Problem

IN: • labels L1, . . . , Ln with dynamic placement,• available ranges [si , Si ] for i = 1, . . . , n.

OUT: active ranges [ai , Ai ] ⊆ [si , Si ] such that

– total active range height H =∑

i(Ai − ai) is max,– truncated extrusions do not overlap.

s

x

Been, Nöllenburg, Poon, Wolff 8 19 Dynamic Map Labeling

ModelComplexity

Approximation

Active-range optimization

Problem

IN: • labels L1, . . . , Ln with dynamic placement,• available ranges [si , Si ] for i = 1, . . . , n.

OUT: active ranges [ai , Ai ] ⊆ [si , Si ] such that

– total active range height H =∑

i(Ai − ai) is max,– truncated extrusions do not overlap.

s

x

Been, Nöllenburg, Poon, Wolff 8 19 Dynamic Map Labeling

ModelComplexity

Approximation

Active-range optimization

Problem

IN: • labels L1, . . . , Ln with dynamic placement,• available ranges [si , Si ] for i = 1, . . . , n.

OUT: active ranges [ai , Ai ] ⊆ [si , Si ] such that

– total active range height H =∑

i(Ai − ai) is max,– truncated extrusions do not overlap.

Simple problem

All available ranges are [0, Smax].

Been, Nöllenburg, Poon, Wolff 8 19 Dynamic Map Labeling

ModelComplexity

Approximation

Outline

Model

Complexity

ApproximationTop-to-bottom sweep algorithmLevel-based algorithm

Been, Nöllenburg, Poon, Wolff 9 19 Dynamic Map Labeling

ModelComplexity

Approximation

NP-hardness

TheoremThe active-range optimization problem is NP-hard –even the simple variant.

Sketch of proof

By reduction from PLANAR 3SAT.

Been, Nöllenburg, Poon, Wolff 10 19 Dynamic Map Labeling

ModelComplexity

Approximation

NP-hardness

TheoremThe active-range optimization problem is NP-hard –even the simple variant.

Sketch of proof

By reduction from PLANAR 3SAT.

Been, Nöllenburg, Poon, Wolff 10 19 Dynamic Map Labeling

ModelComplexity

Approximation

NP-hardness

TheoremThe active-range optimization problem is NP-hard –even the simple variant.

Sketch of proof

By reduction from PLANAR 3SAT.

x1 ∨ x2 ∨ x3

x1 ∨ x3 ∨ x4

x1 ∨ x2 ∨ x4

x2 ∨ x3 ∨ x4

x1 x2 x3 x4

planar 3SAT formula ϕ

Been, Nöllenburg, Poon, Wolff 10 19 Dynamic Map Labeling

ModelComplexity

Approximation

NP-hardness

TheoremThe active-range optimization problem is NP-hard –even the simple variant.

Sketch of proof

By reduction from PLANAR 3SAT.

x1 ∨ x2 ∨ x3

x1 ∨ x3 ∨ x4

x1 ∨ x2 ∨ x4

x2 ∨ x3 ∨ x4

x1 x2 x3 x4

planar 3SAT formula ϕ

(set of labels, int k ) s.t.H ≥ k ⇔ ϕ satisfiable

Been, Nöllenburg, Poon, Wolff 10 19 Dynamic Map Labeling

ModelComplexity

Approximation

NP-hardness

TheoremThe active-range optimization problem is NP-hard –even the simple variant.

Sketch of proof

By reduction from PLANAR 3SAT.

x1 ∨ x2 ∨ x3

x1 ∨ x3 ∨ x4

x1 ∨ x2 ∨ x4

x2 ∨ x3 ∨ x4

x1 x2 x3 x4

planar 3SAT formula ϕ

(set of labels, int k ) s.t.H ≥ k ⇔ ϕ satisfiable

Been, Nöllenburg, Poon, Wolff 10 19 Dynamic Map Labeling

ModelComplexity

Approximation

NP-hardness

TheoremThe active-range optimization problem is NP-hard –even the simple variant.

Sketch of proof

By reduction from PLANAR 3SAT.

x1 ∨ x2 ∨ x3

x1 ∨ x3 ∨ x4

x1 ∨ x2 ∨ x4

x2 ∨ x3 ∨ x4

x1 x2 x3 x4

planar 3SAT formula ϕ

(set of labels, int k ) s.t.H ≥ k ⇔ ϕ satisfiable

Been, Nöllenburg, Poon, Wolff 10 19 Dynamic Map Labeling

ModelComplexity

Approximation

Variable gadget

variable is true

Been, Nöllenburg, Poon, Wolff 11 19 Dynamic Map Labeling

ModelComplexity

Approximation

Variable gadget

variable is false

Been, Nöllenburg, Poon, Wolff 11 19 Dynamic Map Labeling

ModelComplexity

Approximation

Clause gadget

3 literals are true

Been, Nöllenburg, Poon, Wolff 12 19 Dynamic Map Labeling

ModelComplexity

Approximation

Clause gadget

3 literals are true → contribution to H: 2 · Smax

Been, Nöllenburg, Poon, Wolff 12 19 Dynamic Map Labeling

ModelComplexity

Approximation

Clause gadget

2 literals are true → contribution to H: 2 · Smax

Been, Nöllenburg, Poon, Wolff 12 19 Dynamic Map Labeling

ModelComplexity

Approximation

Clause gadget

1 literal is true → contribution to H: 2 · Smax

Been, Nöllenburg, Poon, Wolff 12 19 Dynamic Map Labeling

ModelComplexity

Approximation

Clause gadget

0 literals are true → contribution to H: ?

Been, Nöllenburg, Poon, Wolff 12 19 Dynamic Map Labeling

ModelComplexity

Approximation

Clause gadget

0 literals are true → contribution to H: 1.5 · Smax

Been, Nöllenburg, Poon, Wolff 12 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Outline

Model

Complexity

ApproximationTop-to-bottom sweep algorithmLevel-based algorithm

Been, Nöllenburg, Poon, Wolff 13 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

x

Theorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Top-to-bottom fill-down sweep

Algorithm1 sweep from top to bottom2 at each event try to fill

available but inactive extru.

Subroutine try to fill extrusion Ei

If Ei doesn’t intersect any activeextrusion at current scale s,

then set [ai , Ai ] = [si , s].

s

xTheorem

For segments of congruent triangles, this is a 12 -approximation.

Been, Nöllenburg, Poon, Wolff 14 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Approximation

Blocking Lemma

If an extrusion E never blocksmore than c pairwise independent extrusions,

then our algorithm computes a 1/c-approximation.

E blocks F at scale s if E is active and overlaps F at s.F and F ′ are independent at s if they do not overlap at s.

ProofIntegrate if-condition over all scales ⇒ then-statement.

Been, Nöllenburg, Poon, Wolff 15 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Approximation

Blocking Lemma

If an extrusion E never blocksmore than c pairwise independent extrusions,

then our algorithm computes a 1/c-approximation.

E blocks F at scale s if E is active and overlaps F at s.

F and F ′ are independent at s if they do not overlap at s.

s

E

F F ′

ProofIntegrate if-condition over all scales ⇒ then-statement.

Been, Nöllenburg, Poon, Wolff 15 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Approximation

Blocking Lemma

If an extrusion E never blocksmore than c pairwise independent extrusions,

then our algorithm computes a 1/c-approximation.

E blocks F at scale s if E is active and overlaps F at s.F and F ′ are independent at s if they do not overlap at s.

s

E

F F ′

ProofIntegrate if-condition over all scales ⇒ then-statement.

Been, Nöllenburg, Poon, Wolff 15 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Approximation

Blocking Lemma

If an extrusion E never blocksmore than c pairwise independent extrusions,

then our algorithm computes a 1/c-approximation.

E blocks F at scale s if E is active and overlaps F at s.F and F ′ are independent at s if they do not overlap at s.

ProofIntegrate if-condition over all scales ⇒ then-statement.

Been, Nöllenburg, Poon, Wolff 15 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Example: frustal segments of congruent cones

each label at each scale has the same shapeblocking lemma ⇒ sweep yields 1/4-approximation

Been, Nöllenburg, Poon, Wolff 16 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Level-based algorithm (sketch)

Settingn arbitrary square conesavailable ranges [0, Smax]

use horizontal planes at scales Smax/2i for i = 0, . . . , log n

Been, Nöllenburg, Poon, Wolff 17 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Level-based algorithm

Been, Nöllenburg, Poon, Wolff 18 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Level-based algorithm

Been, Nöllenburg, Poon, Wolff 18 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Level-based algorithm

Been, Nöllenburg, Poon, Wolff 18 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Level-based algorithm

Been, Nöllenburg, Poon, Wolff 18 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Level-based algorithm

Been, Nöllenburg, Poon, Wolff 18 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Level-based algorithm

Been, Nöllenburg, Poon, Wolff 18 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Summaryextrusions approx. running time

frustal segm. of congr. cones 1/4 O((n + k) log2 n)

congruent frusta 1/(4W ) O(n4)

arbitrary square cones (simple) 1/24 O(n log3 n)

congruent square cones (simple) 1/8 O(n log3 n)

Open Problems

better approximations, also in 1d(non-) existence of a PTASmore realistic extrusion shapes

Been, Nöllenburg, Poon, Wolff 19 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Summaryextrusions approx. running time

frustal segm. of congr. cones 1/4 O((n + k) log2 n)

congruent frusta 1/(4W ) O(n4)

arbitrary square cones (simple) 1/24 O(n log3 n)

congruent square cones (simple) 1/8 O(n log3 n)

Open Problems

better approximations, also in 1d(non-) existence of a PTASmore realistic extrusion shapes

Been, Nöllenburg, Poon, Wolff 19 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Summaryextrusions approx. running time

frustal segm. of congr. cones 1/4 O((n + k) log2 n)

congruent frusta 1/(4W ) O(n4)

arbitrary square cones (simple) 1/24 O(n log3 n)

congruent square cones (simple) 1/8 O(n log3 n)

Open Problems

better approximations, also in 1d(non-) existence of a PTASmore realistic extrusion shapes

Been, Nöllenburg, Poon, Wolff 19 19 Dynamic Map Labeling

ModelComplexity

Approximation

Top-to-bottom sweep algorithmLevel-based algorithm

Summaryextrusions approx. running time

frustal segm. of congr. cones 1/4 O((n + k) log2 n)

congruent frusta 1/(4W ) O(n4)

arbitrary square cones (simple) 1/24 O(n log3 n)

congruent square cones (simple) 1/8 O(n log3 n)

Open Problems

better approximations, also in 1d(non-) existence of a PTASmore realistic extrusion shapes ε

Been, Nöllenburg, Poon, Wolff 19 19 Dynamic Map Labeling