Algorithms for Labeling Focus Regions Martin Fink Lehrstuhl f ur - PowerPoint PPT Presentation

Algorithms for Labeling Focus Regions Martin Fink Lehrstuhl f¨ ur Informatik I Universit¨ at W¨ urzburg Joint work with Jan-Henrik Haunert, Andr´ e Schulz, Joachim Spoerhase, and Alexander Wolff 1 /14

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Paragon Restaurant & Bar Top Pot Doughnuts I Lola Vios Cafe Waterfront Seafood Grill Tulio Queen City Grill Daniel’s Broiler Circa Metropolitan Grill Maximilien Sodo Deli 2 /14

Paragon Restaurant & Bar Top Pot Doughnuts I Lola Top Pot Doughnuts II Vios Cafe Cascadia Restaurant Waterfront Seafood Grill Tulio Mama’s Mexican Kitchen Queen City Grill Daniel’s Broiler Etta’s Circa Metropolitan Grill Maximilien Sodo Deli 2 /14

Our models free leaders 3 /14

Our models radial leaders free leaders 3 /14

Previous Work Excentric Labeling [Fekete and Plaisant, 1999] 4 /14

Previous Work Excentric Labeling Boundary Labeling [Fekete and Plaisant, 1999] [Bekos et al., 2007] 4 /14

Previous Work Excentric Labeling Boundary Labeling Necklace Maps [Speckmann and Verbeek, [Fekete and Plaisant, 1999] [Bekos et al., 2007] 2010] 4 /14

The Radial Leader Model minimum allowed angle to avoid label collisions 5 /14

The Radial Leader Model minimum allowed angle to avoid label collisions maximize number of visible labels by a dynamic program 5 /14

The Radial Leader Model minimum allowed angle to avoid label collisions maximize number of visible labels by a dynamic program s 1 s 7 s 2 ≥ α c s 6 s 3 s 5 s 4 5 /14

The Radial Leader Model minimum allowed angle to avoid label collisions maximize number of visible labels by a dynamic program s 1 s 7 s 2 c s 6 s 3 s 5 s 4 5 /14

The Radial Leader Model minimum allowed angle to avoid label collisions maximize number of visible labels by a dynamic program s 1 s 7 s 2 c s 6 s 3 s 5 s 4 5 /14

The Radial Leader Model minimum allowed angle to avoid label collisions maximize number of visible labels by a dynamic program s 1 s 7 s 2 c s 6 s 3 s 5 s 4 5 /14

The Radial Leader Model minimum allowed angle to avoid label collisions maximize number of visible labels by a dynamic program s 1 s 7 s 2 c s 6 s 3 s 5 s 4 5 /14

The Radial Leader Model minimum allowed angle to avoid label collisions maximize number of visible labels by a dynamic program s 1 s 7 s 2 c s 6 s 3 s 5 s 4 5 /14

The Radial Leader Model minimum allowed angle to avoid label collisions maximize number of visible labels by a dynamic program s 1 s 7 s 2 O ( n log n ) time c s 6 s 3 s 5 s 4 5 /14

The Radial Leader Model minimum allowed angle to avoid label collisions maximize number of visible labels by a dynamic program s 1 s 7 s 2 O ( n log n ) time c weighted version: prefer higher s 6 rated points s 3 s 5 s 4 5 /14

The Radial Leader Model minimum allowed angle to avoid label collisions maximize number of visible labels by a dynamic program s 1 s 7 s 2 O ( n log n ) time c weighted version: prefer higher s 6 rated points s 3 s 5 s 4 O ( n 2 ) time 5 /14

The Radial Leader Model with Flexible Center Position find disk that respects minimum α angle α 6 /14

The Radial Leader Model with Flexible Center Position find disk that respects minimum α angle α p consider double disk D ( p , q ) of minimum angle α formed with p α and q D ( p , q ) q 6 /14

The Radial Leader Model with Flexible Center Position find disk that respects minimum α angle α consider double disk D ( p , q ) of minimum angle α formed with p and q build arrangement of all D ( · , · ) 6 /14

The Radial Leader Model with Flexible Center Position find disk that respects minimum α angle α consider double disk D ( p , q ) of minimum angle α formed with p and q build arrangement of all D ( · , · ) check for intersection (cell of � n � depht ) 2 6 /14

The Radial Leader Model with Flexible Center Position find disk that respects minimum α angle α consider double disk D ( p , q ) of minimum angle α formed with p and q build arrangement of all D ( · , · ) check for intersection (cell of � n � depht ) 2 choose center in intersection O ( n 4 log n ) time 6 /14

The Radial Leader Model with Flexible Center Position find disk that respects minimum α angle α consider double disk D ( p , q ) of minimum angle α formed with p and q build arrangement of all D ( · , · ) Variants check for intersection (cell of maximize labels � n � depht ) 2 O ( n 5 ) time choose center in intersection O ( n 4 log n ) time 6 /14

The Radial Leader Model with Flexible Center Position find disk that respects minimum α angle α consider double disk D ( p , q ) of minimum angle α formed with p and q build arrangement of all D ( · , · ) Variants check for intersection (cell of maximize labels � n � depht ) 2 O ( n 5 ) time choose center in intersection maximize angle O ( n 4 log n ) time O ( n 6 ) time 6 /14

The free leader model labels vertically distributed with unit distances 7 /14

The free leader model labels vertically distributed with unit distances compute non-crossing leaders 7 /14

The free leader model labels vertically distributed with unit distances compute non-crossing leaders minimize total leader length: weighted bipartite matching [Bekos et al., 2007] 7 /14

The free leader model labels vertically distributed with unit distances compute non-crossing leaders minimize total leader length: weighted bipartite matching no crossings � 7 /14

The free leader model labels vertically distributed with unit distances compute non-crossing leaders minimize total leader length: weighted bipartite matching no crossings � 7 /14

The free leader model labels vertically distributed with unit distances compute non-crossing leaders minimize total leader length: weighted bipartite matching no crossings � � O ( n 2+ ε ) fast 7 /14

Selecting labeled sites not all sites can be labeled Paragon Restaurant & Bar Top Pot Doughnuts I Lola Vios Cafe Waterfront Seafood Grill Tulio Queen City Grill Daniel’s Broiler Circa Metropolitan Grill Maximilien Sodo Deli 8 /14

Selecting labeled sites not all sites can be labeled label good subset Paragon Restaurant & Bar Top Pot Doughnuts I Lola Vios Cafe Waterfront Seafood Grill Tulio Queen City Grill Daniel’s Broiler Circa Metropolitan Grill Maximilien Sodo Deli 8 /14

Selecting labeled sites not all sites can be labeled label good subset Paragon Restaurant & Bar Top Pot Doughnuts I Lola Vios Cafe short leaders ✗ Waterfront Seafood Grill Tulio Queen City Grill Daniel’s Broiler Circa Metropolitan Grill Maximilien Sodo Deli 8 /14

Selecting labeled sites not all sites can be labeled label good subset Paragon Restaurant & Bar Top Pot Doughnuts I Lola Vios Cafe high weight ✗ Waterfront Seafood Grill Tulio Queen City Grill Daniel’s Broiler Circa Metropolitan Grill Maximilien Sodo Deli 8 /14

Selecting labeled sites not all sites can be labeled label good subset Paragon Restaurant & Bar Top Pot Doughnuts I – nice distribution – represent all sites Lola Vios Cafe Waterfront Seafood Grill Tulio Queen City Grill Daniel’s Broiler Circa Metropolitan Grill Maximilien Sodo Deli 8 /14

Clustered Labeling 1 labeled site k unlabeled sites Paragon Restaurant & Bar Top Pot Doughnuts I Lola Top Pot Doughnuts II Vios Cafe Cascadia Restaurant Waterfront Seafood Grill Tulio Mama’s Mexican Kitchen Queen City Grill Daniel’s Broiler Etta’s Circa Metropolitan Grill Maximilien Sodo Deli 9 /14

Clustered Labeling 1 labeled site k unlabeled sites minimize leader length + distance to attached sites Paragon Restaurant & Bar Top Pot Doughnuts I Lola Top Pot Doughnuts II Vios Cafe Cascadia Restaurant Waterfront Seafood Grill Tulio Mama’s Mexican Kitchen Queen City Grill Daniel’s Broiler Etta’s Circa Metropolitan Grill Maximilien Sodo Deli 9 /14

Clustered Labeling 1 labeled site k unlabeled sites minimize leader length + distance to attached sites Paragon Restaurant & Bar Top Pot Doughnuts I Facility Location model: Lola Top Pot Doughnuts II Vios Cafe Cascadia Restaurant solved by ILP Waterfront Seafood Grill Tulio Mama’s Mexican Kitchen Queen City Grill Daniel’s Broiler Etta’s Circa Metropolitan Grill Maximilien Sodo Deli 9 /14

Clustered Labeling 1 labeled site k unlabeled sites minimize leader length + distance to attached sites Paragon Restaurant & Bar Top Pot Doughnuts I Facility Location model: Lola Top Pot Doughnuts II Vios Cafe Cascadia Restaurant solved by ILP Waterfront Seafood Grill Tulio Mama’s Mexican Kitchen 95 sites, 20 labels: Queen City Grill Daniel’s Broiler Etta’s 124s Circa Metropolitan Grill Maximilien Sodo Deli 9 /14

A Heuristic for Clustered Labeling Randomized initialization heuristic for k -median/ k -means [Arthur and Vassilvitski, 2007] 10 /14

A Heuristic for Clustered Labeling Randomized initialization heuristic for k -median/ k -means [Arthur and Vassilvitski, 2007] 10 /14