1. مرتضي صاحب الزماني 1 Maze Routing

2. Homework 4 مهلت تحویل : 23 اردیبهشت پروژه 1 : انتخاب طرح : امروز مرتضي صاحب الزماني 2

3. 3 Maze Routing Problem Given: –A planar rectangular grid graph. –Two points on the graph, S and T. –Obstacles modeled as blocked vertices. Objective: –Find the shortest path connecting S and T. Can be used to solve global or detailed routing (switchbox) problems.

4. مرتضي صاحب الزماني 4 Grid Graph Area Routing S T X S T X Simplified Representation X Grid Graph (Maze) S T X

5. مرتضي صاحب الزماني 5 Maze Routing S T

6. مرتضي صاحب الزماني 6 Lee’s Algorithm “ An Algorithm for Path Connection and its Application ”, C.Y. Lee, IRE Transactions on Electronic Computers, 1961.

7. مرتضي صاحب الزماني 7 Basic Idea A Breadth-First Search (BFS) of the grid graph. Always find the shortest path possible. Consists of 2 phases: –Wave Propagation –Retrace

8. مرتضي صاحب الزماني 8 An Illustration S T 012 12 345 456 3 3 5

9. مرتضي صاحب الزماني 9 Wave Propagation At step i, all vertices at Manhattan-distance i from S are labeled with i. A Propagation List (FIFO) is used to keep track of the vertices to consider in next step. S T 0 S T 012 12 345 456 3 3 S T 012 12 3 3 3 5 After Step 0After Step 3 After Step 6

10. مرتضي صاحب الزماني 10 Retrace Trace back the actual route. Starting from T. At vertex with i, goto any vertex with label i-1. S T 012 12 345 456 3 3 5 Final Labeling

11. مرتضي صاحب الزماني 11 Example A B 1 2 2 2

12. مرتضي صاحب الزماني 12 Example (continued) A B 1 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 10 11 12

13. مرتضي صاحب الزماني 13 Retrace the Path A B 1 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 10 11 12 11109876 5 4 3 2 1

14. مرتضي صاحب الزماني 14 Alternative Paths Guideline: do not change direction unless you must

15. مرتضي صاحب الزماني 15 Maze Routing Summary Four Main Steps: 1.Expansion – Breadth first search to find all paths from source to target 2.Backtrace – Walk the shortest path back to the source and mark the path cells used 3.Clean Up – Erase all distance marks from other grid cells before the next net is routed 4.Reduce Capacities.

16. مرتضي صاحب الزماني 16 How many grids visited using Lee’s algorithm? S T

17. مرتضي صاحب الزماني 17 Time and Space Complexity For a grid of size w  h: Time per net = O(wh) Space = O(wh log wh) (Since O(log wh) bits are needed to store each label.) For a 40000 x 40000 grid: 32 bits per label Total 4.8*4/3 Gbytes of memory!

18. مرتضي صاحب الزماني 18 Lee Algorithm

19. References David Pan, VLSI Physical Design Automation, Lecture Slides, University of Texas, 2009. مرتضي صاحب الزماني 19