1.5-side Boundary Labeling

1. 1.5-side Boundary Labeling

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

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

4. #1 #6 #2 indirect leader #1 #3 #2 #3 #4 #4 #5 #5 #6 1.5-side Boundary Labeling • type-opo: direct leader vs. indirect leader • Annotation system for wordprocessing S/W direct leader

5. Problem Setting • (labelSize, labelPort, Objective) #1 #3 #2 #1 #2 Uniform label Nonuniform label

6. #1 #2 Problem Setting • (labelSize, labelPort, Objective) #1 #1  #2  #2 Fixed-ratio port (FR) Fixed-position port (FP) Sliding port

7. Problem Setting • (labelSize, labelPort, Objective) #1 #4 #1 #2 #3 #1 #2 #2 #3 #3 #3 #4 #1 #2 #(bends) = 6 #(bends) = 2 longer length shorter length Min (total leader length) (TLLM for short) Min (total bend num) (TBM for short)

8. #1 #6 #2 #3 #4 #5 pj+1 map label map label pj j pj+1 Aleft j pj Aright Aright pi Aleft i pi–1 pi i pi–1 Assumptions • All the parameters are integers • No two sites with the same x- or y- coordinate • Map height = label height sum • Legal leader

9. Our Contributions Solved all the problems of all the combinations of (LabelSize, LabelPort, Objective). * Pseudo-polynomial time algorithms and fixed-parameter algorithms are designed for those intractable problems.

11. B B p p B lh B lv • Lemma 1. All direct leaders are optimal for the above concerned case. leader l |U| U U B B p p

13. S(a, b, c) = // the solution of the problem with pa, pa+1, …, pb connected to label positions c to c+(b-a)+1 // all direct leaders // downward indirect leader // upward indirect leader • Theorem 2. The above case can be solved by dynamic programming in O(n5) time. map label (c+b-a)-th pb # = (b+a)+1 pa c-th

14. S(a, b, c) = // the solution of the problem with pa, pa+1, …, pb connected to label positions c to c+(b-a)+1 // all direct leaders // downward indirect leader // upward indirect leader • Theorem 2. The above case can be solved by dynamic programming in O(n5) time. map label (c+b-a)-th pb pa c-th

15. map label (c+b-a)-th S(a+i+1, b, c+i+1) pb (c+i+1)-th pa+i+1 (c+i)-th pa+i pa+i-1 S(a+j, a+i-1, c+j+1) (c+j+1)-th pa+j (c+j)-th pa+j-1 (c+j-1)-th S(a, a+j-1, c) pa c-th S(a, b, c) = // the solution of the problem with pa, pa+1, …, pb connected to label positions c to c+(b-a)+1 // all direct leaders // downward indirect leader // upward indirect leader • Theorem 2. The above case can be solved by dynamic programming in O(n5) time.

16. map label map label (c+b-a)-th (c+b-a)-th pb S(a+j+1, b, c+j+1) S(a+i+1, b, c+i+1) pb (c+i+1)-th (c+j+1)-th pa+j+1 pa+i+1 (c+i)-th (c+j)-th pa+i pa+j pa+i-1 S(a+j, a+i-1, c+j+1) (c+j-1)-th S(a+i+1, a+j, c+i) pa+i+1 (c+j+1)-th pa+j pa+i (c+i)-th (c+j)-th pa+j-1 pa+i-1 (c+j-1)-th (c+i-1)-th S(a, a+i-1, c) pa S(a, a+j-1, c) pa c-th c-th S(a, b, c) = // the solution of the problem with pa, pa+1, …, pb connected to label positions c to c+(b-a)+1 // all direct leaders // downward indirect leader // upward indirect leader • Theorem 2. The above case can be solved by dynamic programming in O(n5) time.

18. J3 J1 J0 J2 J4 0 M Total Discrepancy Problem is NP-complete J0 J1 J2 J3 J4 •  job Ji {J0, J1, …, J2n} • Execution time length li , where I0 < I1 < … < l2n • Preferred midtimeM = (l0 + l1 + … + l2n) /2 • For a planned schedule • Actual midtime of Ji = mi() • Min ( |m0() – M| + |m1() – M| + … + |m2n() – M| + |m2n+1() – M’|) • Properties for the optimal scheduleopt • No gaps between two jobs • m0(opt) = M • | {Ji : mi < M } | = | {Ji : mi > M } | • opt= An, An-1, …, A1, J0, B1, B2, …, Bn where {Ai, Bi} = {J2i-1, J2i}

19. Theorem 3. Total Discrepancy Problem L(nonuniform, FR/FP/sliding, TLLM). J3 J1 J0 J2 J4 0 M

21. Theorem 4. Subset Sum ProblemL(nonuniform, FR/FP/sliding, TBM). Subset Sum Problem Input:A = {a1, …, an} and a num B = (a1 + … + an)/2 Question: find a subset A’A such that sum(elements in A’) = B pn+2 < hmin < hmin pn+1 h/2 hmin

22. * Pseudo-polynomial time algorithms and fixed-parameter algorithms are designed for those intractable problems.

23. Theorem 5 (pseudo-polynomial algorithm).The above two cases can be solved in O(n4h) time, where h is the map height. S(a, b,t ) = // the solution of the problem with pa, pa+1, …, pb connected to y-coordinate t S(a, b, c) = // the solution of the problem with pa, pa+1, …, pb connected to label positions c to c+(b-a)+1 // all direct leaders (uniform label case) // downward indirect leader // upward indirect leader

24. type-1, …, type-(k-1) (i –1) labels using type-1, type-2, …, type-k type-k • Theorem 6 (fixed-parameter algorithm).The above two cases using k different label heights can be solved in O(nk+4) time. • Theorem 5. The above two cases can be solved in O(n4h) time. • Lemma 2. num( positions of each label using k different label heights ) = O(nk). pf. • Induction on k • Assume num(…(k-1) …) = O(nk-1) • Consider each label, which is the i-th label from the bottom h = nk O(nk-1) positions at most O(n)

25. Conclusion Solved all the problems of all the combinations of (LabelSize, LabelPort, Objective). * Pseudo-polynomial algorithms and fixed-parameter algorithms are designed for those intractable problems.