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Many-to-One Boundary Labeling. Hao-Jen Kao, Chun-Cheng Lin, Hsu-Chun Yen Dept. of Electrical Engineering National Taiwan University. Outline. Introduction Motivations Problem setting Our results Conclusion & Future work. Point features e.g., city. Line features e.g., river. - PowerPoint PPT Presentation
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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
4
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)
11
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
17
1-side-po crossing problem is NP-C
1-side-opo crossing problem 1-side-po crossing problem
18
Greedy heuristic
Link the leftmost site and the sites with the same color
Experimental results
19
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
20
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?