Reach Aware Steiner Trees Stephan Held and Sophie Spirkl Research - PowerPoint PPT Presentation

Reach Aware Steiner Trees Stephan Held and Sophie Spirkl Research Institute for Discrete Mathematics, University of Bonn Aussois, January 6-10, 2014 Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 1 / 24

Steiner Trees in Chip Design distribution of electrical signals Here: � � 8 million nets/trees � 100 million wire segments � 1 kilometer total wire length � This talk: ℓ 1 -norm. Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 2 / 24

Buffer Insertion Steiner trees can be too long for a single root gate/transistor r . Repeaters are needed to speed up electrical signals. t 1 t 3 r t 2 Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 3 / 24

Buffer Insertion Steiner trees can be too long for a single root gate/transistor r . Repeaters are needed to speed up electrical signals. t 1 t 3 r t 2 Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 3 / 24

Reach Aware Steiner Trees Obstacles may inhibit proper repeater insertion. obstacle-unaware Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 4 / 24

Reach Aware Steiner Trees Obstacles may inhibit proper repeater insertion. obstacle-unaware obstacle-avoiding Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 4 / 24

Reach Aware Steiner Trees Obstacles may inhibit proper repeater insertion. obstacle-unaware obstacle-avoiding reach-aware Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 4 / 24

Reach Aware Steiner Trees Reach Aware Steiner Trees Input terminals T , rectilinear obstacles O , ≤ L a reach length L > = 0 . A Steiner tree Y connecting T is reach aware if the length of each component in the intersection of Y with the interior of the blocked area o ∈ O o ) ◦ is bounded by L . ( � Formulation does not depend on representation of blocked area. Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 5 / 24

Problem Formulation Reach Aware Steiner Tree Problem Find a reach aware Steiner tree of minimum length. Our goal 2-approximation with a running time of ˜ O ( n ) (w.r.t. ℓ 1 -norm). O ( n ) is shorthand for O ( n log k n ) for some k ) ( ˜ Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 6 / 24

1 . 5 -Approximation Without Obstacles Smith, Lee, and Liebman [1980] 1 Compute Voronoi diagram ( O ( | T | log | T | ) ). 2 Minimum spanning tree in dual graph (Delaunay triangulation) ( O ( | T | log | T | ) ). 3 Embed and post-optimize (flip L’s) . Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 7 / 24

1 . 5 -Approximation Without Obstacles Smith, Lee, and Liebman [1980] 1 Compute Voronoi diagram ( O ( | T | log | T | ) ). 2 Minimum spanning tree in dual graph (Delaunay triangulation) ( O ( | T | log | T | ) ). 3 Embed and post-optimize (flip L’s) . Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 7 / 24

1 . 5 -Approximation Without Obstacles Smith, Lee, and Liebman [1980] 1 Compute Voronoi diagram ( O ( | T | log | T | ) ). 2 Minimum spanning tree in dual graph (Delaunay triangulation) ( O ( | T | log | T | ) ). 3 Embed and post-optimize (flip L’s) . Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 7 / 24

1 . 5 -Approximation Without Obstacles Smith, Lee, and Liebman [1980] 1 Compute Voronoi diagram ( O ( | T | log | T | ) ). 2 Minimum spanning tree in dual graph (Delaunay triangulation) ( O ( | T | log | T | ) ). 3 Embed and post-optimize (flip L’s) . Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 7 / 24

Obstacle Avoiding Steiner Trees ( L = 0 ) Extensively studied, e.g.: PTAS: Liu et al. [2001]. 2-approximation: Lin et al. [2008], Long et al. [2008], Liu et al. [2009]. Exact: Huang et al. [2013], based on Warme, Winter, and Zachariasen [1999] for the ℓ 2 -norm. Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 8 / 24

General L ≥ 0 M¨ uller-Hannemann and Peyer [2003] Augmented Hanan-grid: 2-approximation with (super) quadratic running time and space requirement 2 k 2 k − 1 α - approximation for obstacles of bounded complexity and any k ≥ 4 , where α is the approximation ratio in graphs. 1 Compute Hanan grid 2 Remove blocked vertices and edges 3 Add edges that are shorter than L 4 Run Steiner tree algorithm for graphs Quadratic size of Hanan grid too big in practice! Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 9 / 24

General L ≥ 0 M¨ uller-Hannemann and Peyer [2003] Augmented Hanan-grid: 2-approximation with (super) quadratic running time and space requirement 2 k 2 k − 1 α - approximation for obstacles of bounded complexity and any k ≥ 4 , where α is the approximation ratio in graphs. 1 Compute Hanan grid 2 Remove blocked vertices and edges 3 Add edges that are shorter than L 4 Run Steiner tree algorithm for graphs Quadratic size of Hanan grid too big in practice! Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 9 / 24

General L ≥ 0 M¨ uller-Hannemann and Peyer [2003] Augmented Hanan-grid: 2-approximation with (super) quadratic running time and space requirement 2 k 2 k − 1 α - approximation for obstacles of bounded complexity and any k ≥ 4 , where α is the approximation ratio in graphs. 1 Compute Hanan grid 2 Remove blocked vertices and edges 3 Add edges that are shorter than L 4 Run Steiner tree algorithm for graphs Quadratic size of Hanan grid too big in practice! Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 9 / 24

General L ≥ 0 M¨ uller-Hannemann and Peyer [2003] Augmented Hanan-grid: 2-approximation with (super) quadratic running time and space requirement 2 k 2 k − 1 α - approximation for obstacles of bounded complexity and any k ≥ 4 , where α is the approximation ratio in graphs. 1 Compute Hanan grid 2 Remove blocked vertices and edges 3 Add edges that are shorter than L 4 Run Steiner tree algorithm for graphs Quadratic size of Hanan grid too big in practice! Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 9 / 24

General L ≥ 0 M¨ uller-Hannemann and Peyer [2003] Augmented Hanan-grid: 2-approximation with (super) quadratic running time and space requirement 2 k 2 k − 1 α - approximation for obstacles of bounded complexity and any k ≥ 4 , where α is the approximation ratio in graphs. 1 Compute Hanan grid 2 Remove blocked vertices and edges 3 Add edges that are shorter than L 4 Run Steiner tree algorithm for graphs Quadratic size of Hanan grid too big in practice! Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 9 / 24

General L ≥ 0 M¨ uller-Hannemann and Peyer [2003] Augmented Hanan-grid: 2-approximation with (super) quadratic running time and space requirement 2 k 2 k − 1 α - approximation for obstacles of bounded complexity and any k ≥ 4 , where α is the approximation ratio in graphs. 1 Compute Hanan grid 2 Remove blocked vertices and edges 3 Add edges that are shorter than L 4 Run Steiner tree algorithm for graphs Quadratic size of Hanan grid too big in practice! Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 9 / 24

General L ≥ 0 M¨ uller-Hannemann and Peyer [2003] Augmented Hanan-grid: 2-approximation with (super) quadratic running time and space requirement 2 k 2 k − 1 α - approximation for obstacles of bounded complexity and any k ≥ 4 , where α is the approximation ratio in graphs. 1 Compute Hanan grid 2 Remove blocked vertices and edges 3 Add edges that are shorter than L 4 Run Steiner tree algorithm for graphs Quadratic size of Hanan grid too big in practice! Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 9 / 24

Shortest path preserving graph for L = 0 Clarkson et al. [1987] Idea: Spanning tree approximations do not require Hanan grid. sufficient: graph contains a shortest path for each pair of terminals. For L = 0 a shortest path can be decomposed into into subpaths between terminals and obstacle corners that are ℓ 1 -shortest. 1 Set endpoints E := set of terminals and obstacle corners. 2 Insert vertical median line M that partitions E , resp. its unblocked parts. 3 Connect every endpoint to the median if obstacles allow 4 Recursively proceed in left and right partition Theorem: The graph size is O ( l log l ) and the construction takes O ( l (log l ) 2 ) time, where l = | T | + | O | . Stephan Held and Sophie Spirkl Reach Aware Steiner Trees Aussois, January 6-10, 2014 10 / 24