A Minimum Cost Path Search Algorithm Through Tile Obstacles

Zhaoyun Xing and Russell Kao

Sun Microsystems Laboratories

4/3/2001 ISPD 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

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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

2( )O n S

T

n

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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

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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

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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

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• 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

ba

c d

s

x

E

G

1 1 1( , )P x y2 2 2( , )P x y

1 2 1 2 1 2( , ) | | | |c P P x x y y α β

,( )

( ) ,

h c x bc x

x b h b x d

β

α β

h

c d xb

c(x)

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• 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

ba

c d

s

x

E

( )f x

h[ , ]

( ) min{ ( ) | |}s a b

g x f s s x

α βh

G

( )g x

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Linear Minimum Convolution (LMC)

• Definition– and a piecewise linear function defined on [a, b], their LMC ( )f x

* )( ) min( ( ) | |), ( , )a s b

f x f s s x x

(α α

α 0

( * )( )f xα

a b x

( )f x

α

0 k α

0k α

f(x) is a line segment with slope k

α

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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

( * )( )f xα

a b x

( )f x

2( )O n

( )O n

( )O n

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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

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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!