1. A Minimum Cost Path Search Algorithm Through Tile Obstacles Zhaoyun Xing and Russell Kao Sun Microsystems Laboratories

2. 4/3/2001ISPD 2001 Search Through Tile Obstacles A classical problem –Find a path for two points through some obstacle tiles in a rectangular area Many applications –Robotics arm path searching –VLSI routing Previous approaches –Line probe based algorithms –Graph based algorithms S T

3. 4/3/2001ISPD 2001 Graph Based Search Algorithm All try to build a graph that contains the shortest path –Maze uniform graph –Non-uniform graph –Connection graph Connection graph –Wu et al, 1987 –Extend S/T and obstacle boundary lines. –Until hit an obstacle or a boundary edge –Graph size is – is obstacle number S T

4. 4/3/2001ISPD 2001 Reduce The Maze Graph Non-uniform graph –Cong et al, 1999 –Extend the boundary lines of all obstacle tiles and H/V lines passing sources and destinations –Graph size is is the number of obstacles S T Connection graph –Wu et al, 1987 –Created similar to the non- uniform graph –Lines extended until they hit an obstacle or a boundary edge –Graph size is S T

5. 4/3/2001ISPD 2001 11 10 3 1 Tile Graph S T 13 12 9 6 8 7 5 2 4 1 2 3 4 5 6 7 8 9 10 11 12 13 T S Extend only in one direction Unoccupied space is fractured into maximal tiles Nodes: space tiles and S/T Edges: adjacency

6. 4/3/2001ISPD 2001 Motivations Tile graph is small –Size is linear in the number of obstacle tiles Previous tile expansion approaches –Not accurate edge cost –The search is guided by the estimated cost Our approach –Guide the search using an accurate cost propagation

7. 4/3/2001ISPD 2001 Outline of the Rest of the Talk Cost propagation through a tile –Cost definition –Propagation formulation Linear Minimum Convolution (LMC) The minimum cost path search algorithm Conclusion and future works

8. 4/3/2001ISPD 2001 Recti-linear cost inside a tile –Cost between two points and Focus –Cost propagation from [a, b] to [c, d] Example –Min cost is 0 on [a, b] –Min cost on [c, d] is –It is a piecewise linear function Cost Propagation through A Tile F b a c d s x E G h c d x b c(x)

9. 4/3/2001ISPD 2001 Assumptions –The minimum cost function on interval [a, b] is a piecewise linear function. The min cost function on [c, d] –Is still a piecewise linear function Bottom line –How to compute efficiently? Approach –Use a notation, we call it, Linear Minimum Convolution (LMC) Cost Propagation through A Tile F b a c d s x E h G

10. 4/3/2001ISPD 2001 Linear Minimum Convolution (LMC) Definition – and a piecewise linear function defined on [a, b], their LMC a b f(x) is a line segment with slope k

11. 4/3/2001ISPD 2001 LMC Observations –Easy to compute LMC of each line segment –Still need to compute the minimum function of line segments – Brute force approach is Compare f(x) with 2n legs (beam lines) Our clipping algorithm is linear –See Proceedings for detail a b

12. 4/3/2001ISPD 2001 Clipping Algorithm Two sweep algorithm –Forward and backward Key –Each beam contributes to LMC only up to the first clipping point Computation of LMC is efficient –The complexity is linear in number of segments in f(x)

13. 4/3/2001ISPD 2001 Search Algorithm Get tile list first –Use A* search –Get a tile list containing the shortest path Retrieve the point path –Build a connection grid graph based on tile list –This graph is small

14. 4/3/2001ISPD 2001 Conclusion and Future Work A new minimum cost path search algorithm –Tile graph based –Accurate cost propagation from tile to tile –Linear Minimum Convolution Future works –Explore the applicability of this algorithm to the VLSI routing –Experiments Thank you!