Many-to-One Boundary Labeling

Hao-Jen Kao, Chun-Cheng Lin, Hsu-Chun Yen

Dept. of Electrical EngineeringNational Taiwan University

2

Outline

Introduction

Motivations

Problem setting

Our results

Conclusion & Future work

3

Map labelingPoint features

e.g., city

Line features

e.g., river

Area features

e.g., mountain

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Boundary labeling (Bekos et al., GD 2004)

(Bekos & Symvonis, GD 2005)

Type-opo leaders Type-po leders Type-s leaders

Min (total leader length)s.t. #(leader crossing) = 0

1-side, 2-side, 4-side

sitelabel

leader

5

Variants

Polygons labeling (Bekos et. al, APVIS 2006)

Multi-stack boundary labeling (Bekos et. al, FSTTCS 2006)

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Motivations

In practice, it is not uncommon to see more than one site to be associated with the same label

Ex1: The language distribution of a countryEach city site

The main language used in a city label

Ex2: Religion distribution in each state of a country

Ex3: The association or organization composed of some countries in the world

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Many-site-to-one-label boundary labeling (a.k.a. Many-to-one boundary labeling)

Type-opo leaders Type-po leders Type-s leaders

Main aesthetic criteria:To minimize the leader crossings

To minimize the total leader length

Crossing problem

Leader length problem

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Our main results

objective # of sidesleader type

complexity solution

Min #(crossing)

1-side opo NP-complete 3-approx.

2-side opo NP-complete3(1+.301/c)-

approx.

1-side po NP-complete heuristic

Min

Total leader length

any any O(n2 log3n)

Note that c is a number depending on the sum of edge weights.

9

Main assumption

AssumptionThere are no two sites with the same x- or y- coordinates

When we consider the crossing problem for the labeling with type-opo leaders, only y-coordinates matter.

1

2

#(crossings) = 2 #(crossings) = 2downwardupward

2

1

10

1-side-opo crossing problem is NP-C

The Decision Crossing Problem (DCP)

DCP is NP-C. (Eades & Wormald, 1994)

DCP 1-side-opo crossing problem

Fixed ordering

Find an orderings.t. #(crossing) is minimized.

#(crossings) M #(crossings) 4M + #(self-contributed crossings)

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Median algorithm (Eades & Wormald, 1994)

Median algorithm is 3-approximation of 1-side-opo crossing problem(The correctness proof is along a similar line of that of [Eades & Wormald, 1994])

3-approximation

Arbitrary Median algorithm

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Brown booby

Taiwan hill partridge

Masked palm civet

Hawk

Melogale moschata

Bamboo partridge

Chinese pangolin

Mallard

Experimental resultDistribution of someanimals in Taiwan:

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2-side-opo crossing problem is NP-C even when n1 = n2

2-side-opo crossing problem even when n1 = n2

Legal operations:Swapping two nodes between the two sides

Change the node ordering in each side

1-side-opo crossing problem 2-side-opo crossing problem even when n1 = n2

2

N

2

N +1

l1

l2

l3

r1

ln

r2

r3

p1

p2

p3

pN

rnpn

+1

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Max-Bisection Problem

There exists a 1.431-approximiation algorithm for the Max-Bisection problem (Ye, 1999).

By using the approximation algorithm for the Max-Bisection problem, we can find a 3(1+.301/c)-approximation for the 2-side-opo crossing problem, where c is a number depending on the sum of edge weights.

3(1+.301/c)-approximation

weighted graph|V| = n

# = n/2 # = n/2

Max (edge weight sum on the cut)

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Algorithm

Median algorithm

1

3 1

111

Completeweighted graph

Step 1. Step 2. Step 3.

Max-Bisection

sites labels

Less crossings

sites labelslabels

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Brown booby

Masked palm civet

Hawk

Chinese pangolin

Taiwan hill partridge

Melogale moschata

Bamboo partridge

Mallard

Experimental result

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1-side-po crossing problem is NP-C

1-side-opo crossing problem 1-side-po crossing problem

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Greedy heuristic

Link the leftmost site and the sites with the same color

Experimental results

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Total leader length problem

For any number of sides and any type of leaders, minimizing the total leader length for many-to-one labeling can be solved in O(n2 log3n) time

3

4

1

2

1

4

2

3

complete weightedbipartite graph

edge weight= Manhattan distance

Find minimum weight matching

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Conclusion

objective # of sidesleader type

complexity solution

Min #(crossing)

1-side opo NP-complete 3-approx.

2-side opo NP-complete3(1+.301/c)-

approx.

1-side po NP-complete heuristic

Min

Total leader length

any any O(n2 log3n)

Note that c is a number depending on the sum of edge weights.

21

Future work

Is there an approximation algorithm for the 1-side-po crossing problem?

Is the 2-side-po crossing problem tractable?

Is the 4-side many-to-one labeling tractable?

Can we simultaneously achieve the objective to minimize #(crossing) as well as minimize the total leader length?