1.5-side Boundary Labeling Chun-Cheng LinNational Chiao Tung University Sheung-Hung PoonNational Tsing Hua University Shigeo TakahashiThe University of Tokyo Hsiang-Yu WuThe University of Tokyo Hsu-Chun YenNational Taiwan University

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

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

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

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

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

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

Assumptions All the parameters are integers No two sites with the same x- or y- coordinate Map height = label height sum Legal leader pjpj pipi maplabel A left A right p i–1 j i p j+1 pjpj pipi maplabel A left A right p i–1 j i p j+1 #1 #2 #3 #4 #5 #6

Our Contributions (LabelSize,LabelPort,Objective)timereference (uniform,FR/sliding,TLLM)O(n log n)Thm 1 (uniform,FP,TLLM)O(n5)O(n5)Thm 2 (uniform,FR/FP/sliding,TBM)O(n5)O(n5)Thm 2 (nonuniform,FR/FP/sliding,TLLM)NP-complete*Thm 3 (nonuniform,FR/FP/sliding,TBM)NP-complete*Thm 4 * Pseudo-polynomial time algorithms and fixed-parameter algorithms are designed for those intractable problems. Solved all the problems of all the combinations of (LabelSize, LabelPort, Objective).

(LabelSize,LabelPort,Objective)timereference (uniform,FR/sliding,TLLM)O(n log n)Thm 1 (uniform,FP,TLLM)O(n5)O(n5)Thm 2 (uniform,FR/FP/sliding,TBM)O(n5)O(n5)Thm 2 (nonuniform,FR/FP/sliding,TLLM)NP-complete*Thm 3 (nonuniform,FR/FP/sliding,TBM)NP-complete*Thm 4

Lemma 1. All direct leaders are optimal for the above concerned case. (LabelSize,LabelPort,Objective)timereference (uniform,FR/sliding,TLLM)O(n log n)Thm 1 p U B p B U p B lhlh B lvlv B leader l p B |U||U|

(LabelSize,LabelPort,Objective)timereference (uniform,FR/sliding,TLLM)O(n log n)Thm 1 (uniform,FP,TLLM)O(n5)O(n5)Thm 2 (uniform,FR/FP/sliding,TBM)O(n5)O(n5)Thm 2 (nonuniform,FR/FP/sliding,TLLM)NP-complete*Thm 3 (nonuniform,FR/FP/sliding,TBM)NP-complete*Thm 4

Theorem 2. The above case can be solved by dynamic programming in O(n 5 ) time. (LabelSize,LabelPort,Objective)timereference (uniform,FP,TLLM)O(n5)O(n5)Thm 2 pbpb papa (c+b-a)-th c-th maplabel S(a, b, c) = // all direct leaders // downward indirect leader // upward indirect leader // the solution of the problem with p a, p a+1, …, p b connected to label positions c to c+(b-a)+1 # = (b+a)+1

Theorem 2. The above case can be solved by dynamic programming in O(n 5 ) time. (LabelSize,LabelPort,Objective)timereference (uniform,FP,TLLM)O(n5)O(n5)Thm 2 S(a, b, c) = // all direct leaders // downward indirect leader // upward indirect leader // the solution of the problem with p a, p a+1, …, p b connected to label positions c to c+(b-a)+1 pbpb papa (c+b-a)-th c-th maplabel

Theorem 2. The above case can be solved by dynamic programming in O(n 5 ) time. (LabelSize,LabelPort,Objective)timereference (uniform,FP,TLLM)O(n5)O(n5)Thm 2 pbpb p a+i+1 p a+i-1 pa+jpa+j p a+j-1 papa p a+i (c+b-a)-th (c+i)-th (c+j-1)-th c-th (c+j+1)-th (c+i+1)-th maplabel S(a+i+1, b, c+i+1) S(a+j, a+i-1, c+j+1) S(a, a+j-1, c) (c+j)-th S(a, b, c) = // all direct leaders // downward indirect leader // upward indirect leader // the solution of the problem with p a, p a+1, …, p b connected to label positions c to c+(b-a)+1

Theorem 2. The above case can be solved by dynamic programming in O(n 5 ) time. (LabelSize,LabelPort,Objective)timereference (uniform,FP,TLLM)O(n5)O(n5)Thm 2 pbpb p a+j+1 pa+jpa+j p a+i+1 p a+i-1 papa pa+ipa+i (c+b-a)-th (c+i)-th (c+j-1)-th c-th (c+i-1)-th (c+j+1)-th maplabel S(a+j+1, b, c+j+1) S(a, a+i-1, c) (c+j)-th S(a+i+1, a+j, c+i) pbpb p a+i+1 p a+i-1 pa+jpa+j p a+j-1 papa p a+i (c+b-a)-th (c+i)-th (c+j-1)-th c-th (c+j+1)-th (c+i+1)-th maplabel S(a+i+1, b, c+i+1) S(a+j, a+i-1, c+j+1) S(a, a+j-1, c) (c+j)-th S(a, b, c) = // all direct leaders // downward indirect leader // upward indirect leader // the solution of the problem with p a, p a+1, …, p b connected to label positions c to c+(b-a)+1

(LabelSize,LabelPort,Objective)timereference (uniform,FR/sliding,TLLM)O(n log n)Thm 1 (uniform,FP,TLLM)O(n5)O(n5)Thm 2 (uniform,FR/FP/sliding,TBM)O(n5)O(n5)Thm 2 (nonuniform,FR/FP/sliding,TLLM)NP-complete*Thm 3 (nonuniform,FR/FP/sliding,TBM)NP-complete*Thm 4

Total Discrepancy Problem is NP-complete  job J i  {J 0, J 1, …, J 2n } Execution time length l i, where I 0 < I 1 < … < l 2n Preferred midtime M = (l 0 + l 1 + … + l 2n ) /2 For a planned schedule  Actual midtime of J i = m i (  ) Min ( |m 0 (  ) – M| + |m 1 (  ) – M| + … + |m 2n (  ) – M| + |m 2n+1 (  ) – M’|) Properties for the optimal schedule  opt No gaps between two jobs m 0 (  opt ) = M | {J i : m i M } |  opt = A n, A n-1, …, A 1, J 0, B 1, B 2, …, B n where {A i, B i } = {J 2i-1, J 2i } 0M J0J0 J1J1 J2J2 J3J3 J4J4 J0J0 J1J1 J2J2 J3J3 J4J4

Theorem 3. Total Discrepancy Problem  L(nonuniform, FR/FP/sliding, TLLM). (LabelSize,LabelPort,Objective)timereference (nonuniform,FR/FP/sliding,TLLM)NP-complete*Thm 3 0M J0J0 J1J1 J2J2 J3J3 J4J4

(LabelSize,LabelPort,Objective)timereference (uniform,FR/sliding,TLLM)O(n log n)Thm 1 (uniform,FP,TLLM)O(n5)O(n5)Thm 2 (uniform,FR/FP/sliding,TBM)O(n5)O(n5)Thm 2 (nonuniform,FR/FP/sliding,TLLM)NP-complete*Thm 3 (nonuniform,FR/FP/sliding,TBM)NP-complete*Thm 4

Theorem 4. Subset Sum Problem  L(nonuniform, FR/FP/sliding, TBM). (LabelSize,LabelPort,Objective)timereference (nonuniform,FR/FP/sliding,TBM)NP-complete*Thm 4 p n+1 p n+2 h min Subset Sum Problem Input: A = {a 1, …, a n } and a num B = (a 1 + … + a n )/2 Question: find a subset A’  A such that sum(elements in A’) = B < h min h/2

(LabelSize,LabelPort,Objective)timereference (uniform,FR/sliding,TLLM)O(n log n)Thm 1 (uniform,FP,TLLM)O(n5)O(n5)Thm 2 (uniform,FR/FP/sliding,TBM)O(n5)O(n5)Thm 2 (nonuniform,FR/FP/sliding,TLLM)NP-complete*Thm 3 (nonuniform,FR/FP/sliding,TBM)NP-complete*Thm 4 * Pseudo-polynomial time algorithms and fixed-parameter algorithms are designed for those intractable problems.

Theorem 5 (pseudo-polynomial algorithm). The above two cases can be solved in O(n 4 h) time, where h is the map height. (LabelSize,LabelPort,Objective)timereference (nonuniform,FR/FP/sliding,TLLM)NP-complete*Thm 3 (nonuniform,FR/FP/sliding,TBM)NP-complete*Thm 4 S(a, b, t ) = S(a, b, c ) = // all direct leaders // downward indirect leader // upward indirect leader // the solution of the problem with p a, p a+1, …, p b connected to label positions c to c+(b-a)+1 // the solution of the problem with p a, p a+1, …, p b connected to y-coordinate t (uniform label case)

Theorem 6 (fixed-parameter algorithm). The above two cases using k different label heights can be solved in O(n k+4 ) time. Theorem 5. The above two cases can be solved in O(n 4 h) time. Lemma 2. num( positions of each label using k different label heights ) = O(n k ). pf. Induction on k Assume num(…(k-1) …) = O(n k-1 ) Consider each label, which is the i-th label from the bottom (LabelSize,LabelPort,Objective)timereference (nonuniform,FR/FP/sliding,TLLM)NP-complete*Thm 3 (nonuniform,FR/FP/sliding,TBM)NP-complete*Thm 4 h = n k type-k type-1, …, type-(k-1) (i –1) labels using type-1, type-2, …, type-k O(n k-1 ) positions at most O(n)

Conclusion (LabelSize,LabelPort,Objective)timereference (uniform,FR/sliding,TLLM)O(n log n)Thm 1 (uniform,FP,TLLM)O(n5)O(n5)Thm 2 (uniform,FR/FP/sliding,TBM)O(n5)O(n5)Thm 2 (nonuniform,FR/FP/sliding,TLLM)NP-complete*Thm 3 (nonuniform,FR/FP/sliding,TBM)NP-complete*Thm 4 * Pseudo-polynomial algorithms and fixed-parameter algorithms are designed for those intractable problems. Solved all the problems of all the combinations of (LabelSize, LabelPort, Objective).