Mat 3770 Steiner Trees Overview Spanning Mat 3770 Trees Grid Steiner Trees Kruskal Na¨ ıve B&B Theorems Pretaxial B&B Epitaxial B&B Spring 2014 MDFDA Annealing Evolution Results
Topics Mat 3770 Steiner Trees Overview Background, Definitions, and Prior Theorems Spanning Trees Initial Algorithms Grid Kruskal Na¨ ıve B&B Theoretical Work Theorems Pretaxial Improved Performance Algorithms B&B Epitaxial B&B Conclusions MDFDA Annealing Evolution Results
Motivation for Studying Steiner Trees Mat 3770 Applications Steiner Trees Communications networks Overview Mechanical & Electrical systems in buildings and along Spanning Trees streets Grid Wire layout in VLSI chip design Kruskal Na¨ ıve B&B Theorems VLSI Chip Design Pretaxial B&B Epitaxial Given a collection of cells and a collection of nets, find a way B&B to position the cells ( placement ) and run the wires for net MDFDA Annealing connections ( routing ) so the wires are short with as few vias as Evolution possible, and the whole layout uses a minimum amount of area. Results
Gate Arrays Custom vs Semi–custom chip design Mat 3770 Steiner Trees A two dimensional array of replicated transistors fabricated just short of the interconnection phase, allowing customized Overview connections to define the overall circuits for semi–custom Spanning design chips. Trees Grid The interconnections are implemented on a rectangular grid Kruskal in the channels between the cells. Na¨ ıve B&B Theorems Pretaxial B&B Epitaxial B&B MDFDA Annealing Evolution Results Placement Interconnect
Fermat’s Problem Mat 3770 Steiner Trees In the early 1600’s, Pierre Fermat posed the problem: Overview Given a triangle, find the point in the plane such that Spanning Trees the sum of the distances to the vertices is minimized. Grid Kruskal Na¨ ıve B&B Theorems Evangelista Torricelli solved this problem in 1659: Pretaxial B&B If all angles are less than 120 o , then P is the point from Epitaxial which each side of the triangle subtends an angle of B&B 120 o , else it is the vertex of largest measure. MDFDA Annealing Evolution Results
Steiner’s Problem Mat 3770 Steiner Trees Overview In the early 1800’s, Jacob Steiner formalized the problem Spanning Trees mathematically and generalized it to n points: Grid Kruskal Na¨ ıve B&B Given n points in a plane, find a connected system of Theorems straight line segments of shortest total length such Pretaxial that any two of the given points can be joined by a B&B Epitaxial path consisting of segments of the system. B&B MDFDA Annealing Evolution Results
Spanning Trees Mat 3770 Related Problem: Minimal Spanning Trees Steiner Trees E A Overview F Spanning Trees B Grid Kruskal G C D Na¨ ıve B&B Theorems Steiner Minimal Trees : shorten trees by adding points Pretaxial B&B E A Epitaxial B&B F MDFDA B Annealing Evolution C G D Results
Shortest Rectilinear Steiner Spanning Tree Problem Mat 3770 Steiner Trees Connecting a set of points in the plane with a connected collection of vertical and horizontal lines with minimal Overview overall length. Spanning Trees Grid Kruskal Na¨ ıve B&B Theorems Pretaxial B&B Epitaxial B&B MDFDA Rectilinear Minimal Shortest Rectilinear Annealing Spanning Tree Steiner Spanning Tree Evolution flip connections and take out overlaps Results
Upper and Lower Bounds Theorem. ( Hwang ) An SRSST over a set of points is no Mat 3770 Steiner Trees smaller in length than two thirds the length of the RMST over the same set of points. Overview Spanning Trees Grid Kruskal Na¨ ıve B&B Theorems Pretaxial B&B Epitaxial B&B SRSST RMST MDFDA Annealing This gives us nice upper (Length(RMST)) Evolution and lower ( 2 3 Length(RMST)) Results bounds on the length of Steiner trees.
Grid & Enclosing Rectangle Theorem. ( Hannan ) An SRSST over a set of points exists Mat 3770 Steiner Trees on the grid induced by the points. Overview Spanning Trees Grid Kruskal Na¨ ıve B&B Theorems Pretaxial Note: The shortest path between two points is not unique. B&B Thus a solution to the SRSST Problem in general is not Epitaxial B&B unique. MDFDA The Enclosing Rectangle is the smallest rectangle Annealing containing the point set. Any SRSST must be at least half Evolution the perimeter of the enclosing rectangle. (Providing another Results lower bound.)
Grid Segments Mat 3770 a b c Steiner Trees g d f e j h i Overview Spanning k l m n Trees q Grid p o Kruskal s r t u Na¨ ıve B&B v w x Theorems Pretaxial Points and Their Induced Grid B&B Epitaxial B&B Number of Segments MDFDA (Horizontal & Vertical) × n nodes × ( n − 1) edges Annealing 4 points: 2 × 4 × 3 = 24 grid segments Evolution 5 points: 2 × 5 × 4 = 40 grid segments Results 6 points: 2 × 6 × 5 = 60 grid segments
NP–Complete Problem Mat 3770 Theorem. ( Garey and Johnson ) The problem of determining Steiner Trees the minimum length of an optimal rectilinear Steiner tree for a set of points in the plane is NP–Complete. Overview Spanning Trees Grid This implies it is probable that finding an optimal solution will Kruskal take worse than polynomial time. Na¨ ıve B&B Theorems That may be alright for small problems, but heuristic Pretaxial algorithms are still needed. B&B Epitaxial B&B Heuristic Algorithms MDFDA Annealing Non–optimal, but usually close Evolution & have performance guarantees Results
Research Sequence Mat 3770 Steiner Trees Extended Kruskal Overview Spanning Na¨ ıve Branch and Bound (optimal) Trees Grid Structural Theorems Kruskal Na¨ ıve B&B Pretaxial & Epitaxial Branch and Bound (optimal) Theorems Pretaxial MST–based Direction Assignment B&B Epitaxial Simulated Annealing B&B MDFDA Stochastic Evolution Annealing Evolution Results
Extended Kruskal Based on Kruskal’s MST algorithm: Mat 3770 Steiner Trees Place all points in individual subtrees Repeat Overview Connect closest subtrees Spanning Until a single tree is formed Trees Grid Two versions with different connection schemes: Kruskal Na¨ ıve B&B Theorems Pretaxial B&B Epitaxial B&B All grid intersection Only corner MDFDA points considered points considered Annealing Theorem. The EK algorithm is correct and produces an Evolution RSST no larger than the RMST. Results Guarantees result is no worse than 150% of optimal .
Na¨ ıve Branch and Bound Mat 3770 Steiner Trees Idea : look at all combinations of grid edges. Overview Spanning Trees General Branch and Bound Method Grid Examine all feasible solutions Kruskal in an orderly manner: Na¨ ıve B&B Theorems Pretaxial B&B Organize the search space as a tree Epitaxial B&B MDFDA Search the tree using a depth first approach Annealing Using lower and upper bounds Evolution to decrease the search Results
Grid and Search Tree a b Mat 3770 Steiner Trees e c d g f Overview Spanning h i Trees j Grid k l Kruskal Na¨ ıve B&B Induced Grid over three points Theorems Pretaxial B&B Epitaxial a B&B ab a b MDFDA Annealing abc ab ac a bc c b Evolution . . . etc . . . Results Na¨ ıve Search Tree
Pruning the Search Tree Mat 3770 Steiner Trees A small search space is needed to feasibly complete the tree traversal, so we prune the tree: Overview lower bound : larger of 2 3 RMST or 1 2 perimeter Spanning upper bound : RMST or result of heuristic Trees Grid Kruskal If collection of grid edges is Na¨ ıve B&B too large — discard Theorems too short — discard Pretaxial if they form a tree and it’s shorter than current known tree, B&B keep it Epitaxial B&B MDFDA Unfortunately, there were still too many combinations to Annealing check Evolution Results
Implementation All combinations of grid edges considered Mat 3770 Steiner Trees Search Space: O (2 2 n ( n − 1) ) 3 points yield 2 12 or 4096 combinations Overview Spanning Implementation: essentially used a binary odometer where Trees 1 = IN and 0 = OUT . Grid Kruskal Order the grid edges: Na¨ ıve B&B l k j i h g f e d c b a Theorems Pretaxial 0 0 0 0 0 0 0 1 1 0 1 0 B&B The 1’s at e , d , and b represent a forest containing those Epitaxial B&B edges and no others. MDFDA Annealing Determining whether the edges formed a tree (i.e., if they Evolution are connected) was very time consuming and the Search Results Space was too large.
Structural Theorems — Definitions Mat 3770 Idea : reduce the number of configurations must consider Steiner Trees Overview A line is a connected collection of one or more grid Spanning Trees segments, all with the same orientation (either horizontal or Grid vertical). Kruskal Na¨ ıve B&B A via occurs the intersection of a horizontal and a vertical Theorems grid segment. Pretaxial B&B Epitaxial B&B A non–repetitive set of points contains no duplicate x or y MDFDA coordinates. I.e., all x and y coordinates are unique. Annealing Evolution A point set can be perturbed to meet this requirement. Results
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