Efficient Steiner Tree Construction Based on Spanning Graphs

Hai Zhou

Electrical and Computer Engineering

Northwestern University

Rectilinear Steiner tree

Steiner points

• NP-hard• but efficient heuristic are critical in modern VLSI design

Spanning tree as a starting point

• L(MST) <= 1.5L(SMT)• Shorten tree length by modification

Edge-based approach

• Borah, Owens, and Irwin (TCAD 94)

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Questions

• How to construct the initial spanning tree?• How to find the edge point pair candidates for

connection?• How to find the longest edge on the formed

cycle?

Answer: spanning graphs

Answer: spanning graphs

• A graph on a set of given points• A MST on the graph is a MST of the given

points– geometrical MST = spanning graph + graph MST

• Spanning graph also represent geometrical proximity information– it also provides candidates for point-edge

connection

• It is a sparse graph: O(n) number of edges

Spanning graphs in O(n log n) time

divide&conquer

(Shamos 78?)

sweep-line

(Fortune 87)

div.&con.

(Hwang 79)

sweep-line

(Shute et al 91)

div.&con.

(Guibas et al 83)

sweep-line

(Zhou et al 01)

Euclidean rectilinear

Delaunay Delaunay Non-Delaunay

Octal partition

s

R1

R2

R3

R4R5

R6

R7

R8

• ||pq||<max(||ps||, ||qs||)• Only the closest point in each region needs to be

connected to s

R1

s

p

q

Find closest points in R1

• Sweep points in increase x+y• Keep points waiting for

closest point in A (active set)• Checking current point with A

– make connections– delete connected points– add current point in A

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x+y

Active set

• Active set can be linearly order according to x

• Use a binary search tree– Finding insertion place O(log n)– Deleting each point O(log n)– Inserting a point O(log n)

• Each point is inserted and deleted at most once: O(n log n)

Spanning graph for point-edge candidates

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p

a

b

• (p,a) or (p,b) is usually in the spanning graph if nothing blocks p from (a,b)

Finding longest edge on cycle

• Use Tarjan’s least common ancestor algorithm

• Almost linear time• The bug appeared in Borah’s original O(n2)

algorithm does not exist here (proved)

Algorithm RST

• construct a rectilinear spanning graph• sort the edges in the graph• use Kruskal to build MST, at the same time

– build the merging binary tree and– possible point-edge candidates (by spanning

graph)

• use least common ancestors on merging binary tree to find the longest edges in cycles

• iteratively update point-edge connections

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(%)

GeoSteiner BI1S BOI RST

Performance

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GeoSteiner BI1S BOI RST

Running times

Correction

• The running times of RST reported in the paper are much worse because we accidentally switched the sorting method to bubble sort (O(n2)) instead of quick sort

BGA

• Mandoiu et al. (ASP-DAC 03)• Similar to Borah et al. but more general• O(nlog2n)

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RST BGA

Compared with BGA

Trees by RST less than 1% longer lengths

Summary

• Besides generating spanning trees, spanning graphs also provide proximity information

• Use for Steiner tree improvements• RST: efficient O(nlog n) heuristic• Future research: improving tree lengths