1. Many-to-One Boundary Labeling Hao-Jen Kao, Chun-Cheng Lin, Hsu-Chun Yen Dept. of Electrical Engineering National Taiwan University

2. 2 Outline Introduction Motivations Problem setting Our results Conclusion & Future work

3. 3 Map labeling Point features e.g., city Line features e.g., river Area features e.g., mountain

4. 4 Boundary labeling (Bekos et al., GD 2004) (Bekos & Symvonis, GD 2005) Type-opo leadersType-po ledersType-s leaders Min (total leader length) s.t. #(leader crossing) = 0 1-side, 2-side, 4-side site label leader

5. 5 Variants Polygons labeling (Bekos et. al, APVIS 2006) Multi-stack boundary labeling (Bekos et. al, FSTTCS 2006)

6. 6 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 country Each 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

7. 7 Many-site-to-one-label boundary labeling (a.k.a. Many-to-one boundary labeling) Type-opo leadersType-po ledersType-s leaders Main aesthetic criteria: To minimize the leader crossings To minimize the total leader length Crossing problemLeader length problem

8. 8 Our main results objective# of sides leader type complexitysolution Min #(crossing) 1-sideopoNP-complete3-approx. 2-sideopoNP-complete 3(1+.301/c)- approx. 1-sidepoNP-completeheuristic Min Total leader length any O(n 2 log 3 n) Note that c is a number depending on the sum of edge weights.

9. 9 Main assumption Assumption There 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 downwardupward 2 1

10. 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 ordering s.t. #(crossing) is minimized. #(crossings)  M#(crossings)  4M + #(self-contributed crossings)

11. 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 ArbitraryMedian algorithm

12. 12 Brown booby Taiwan hill partridge Masked palm civet Hawk Melogale moschata Bamboo partridge Chinese pangolin Mallard Experimental result Distribution of some animals in Taiwan:

13. 13 2-side-opo crossing problem is NP-C even when n 1 = n 2 2-side-opo crossing problem even when n 1 = n 2 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 n 1 = n 2 +1 l1l1 l2l2 l3l3 r1r1 lnln r2r2 r3r3 p1p1 p2p2 p3p3 pNpN rnrn pnpn

14. 14 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 Max (edge weight sum on the cut)

15. 15 Algorithm Median algorithm 1 3 1 1 1 1 Complete weighted graph Step 1.Step 2.Step 3. Max-Bisection sites labels Less crossings sites labels

16. 16 Brown booby Masked palm civet Hawk Chinese pangolin Taiwan hill partridge Melogale moschata Bamboo partridge Mallard Experimental result

17. 17 1-side-po crossing problem is NP-C 1-side-opo crossing problem  1-side-po crossing problem

18. 18 Greedy heuristic Link the leftmost site and the sites with the same color Experimental results

19. 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(n 2 log 3 n) time 3 4 1 2 1 4 2 3 complete weighted bipartite graph edge weight = Manhattan distance Find minimum weight matching

20. 20 Conclusion objective# of sides leader type complexitysolution Min #(crossing) 1-sideopoNP-complete3-approx. 2-sideopoNP-complete 3(1+.301/c)- approx. 1-sidepoNP-completeheuristic Min Total leader length any O(n 2 log 3 n) Note that c is a number depending on the sum of edge weights.

21. 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?