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Many-to-One Boundary Labeling

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.

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Many-to-One Boundary Labeling

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

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

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

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

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

  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. Crossing problem Leader length problem 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

  8. Our main results Note that c is a number depending on the sum of edge weights.

  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 1 2 2 upward downward #(crossings) = 2 #(crossings) = 2

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

  11. 3-approximation • 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]) Arbitrary Median algorithm

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

  13. 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 +1 +1 2-side-opo crossing problem is NP-C even when n1 = n2 l1 r1 p1 l2 p2 r2 p3 r3 l3 pn ln rn pN

  14. 3(1+.301/c)-approximation • 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. # = n/2 # = n/2 weighted graph |V| = n Max (edge weight sum on the cut)

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

  16. Brown booby Taiwan hill partridge Masked palm civet Melogale moschata Hawk Bamboo partridge Chinese pangolin 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 edge weight = Manhattan distance 3 1 1 2 4 2 3 4 complete weighted bipartite graph Find minimum weight matching

  20. Conclusion 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?

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