CAD for Physical Design

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CAD for Physical Design

• Tutorial 4
• Feb 25th, 2009

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Outline

• H. Zhou, Efficient Steiner Tree Construction
  Based on Spanning Graphs. ACM International
  Symposium on Physical Design, Monterey, CA, 2003.

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Rectilinear Steiner tree
Steiner points

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

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Spanning tree as a starting point

• L(MST)
• Shorten tree length by modification

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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?

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Answer spanning graphs
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Answer spanning graphs

• Definition Given a set of points on the plane, a
  spanning graph is a graph on the points that
  contains at least one minimal spanning tree
• 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

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Construct Spanning Graph - Octal partition
R1
p
R1
R8
q
R2
R7
s
R3
R6
R4
R5
s

• pq
• Only the closest point in each region needs to be
  connected to s
• So, connections will form a spanning graph of
  cardinality O(n)

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Find closest points in R1

• In order to find the closest points in regions R1
  and R2 for all points, the points will be sorted
  in nondecreasing order of x y.
• Sweep points in increase xy. In this way, after
  each point p is swept, the next point seen in its
  R1 or R2 region will be the closest point in that
  region
• 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

4
2
3
1
xy
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Active set

• Active set can be linearly order
• 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)

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Rectilinear Spanning Graph Alg.
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Spanning graph for point-edge candidates

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

a
e
p
b

• For each edge e in the current tree, all points
  that are neighbors of either of the end points of
  e will be considered to form point-edge pairs
  with e

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Spanning graph for point-edge candidates

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

a
e
p
b

• For each edge e in the current tree, all points
  that are neighbors of either of the end points of
  e will be considered to form point-edge pairs
  with e

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Finding longest edge on cycle

• When connecting a point to an edge, the longest
  edge on the formed cycle needs to be deleted
• Review Kruskals algorithm to construct MST
• Given G(V, E), find MST of G
• Sort the edges
• T ?
• Consider the edges in ascending order of their
  weights until T contains n-1 edges
• Add an edge e to T iff adding e to T does not
  create cycle in T

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Finding longest edge on cycle

• The structure of these connecting operations in
  Kruskals algorithm can be represented by a
  merging binary tree
• Leaves represent the points
• Internal nodes represent edges
• When an edge is added, a node is created
  representing the edge and has its two children
  representing the two components connected by this
  edge

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Finding longest edge on cycle

• The longest edge between two points is the least
  common ancestor of the two points in the binary
  tree
• To find the longest edge on the cycle formed by a
  point-edge pair (p, e), we need to find the
  longest edge between p and the one end point of e
  that is in the same component before adding e

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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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RST Algorithm
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Spanning Graph
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Minimal Spanning Tree
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Steiner Tree
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Performance
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Running times
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Summary

• Besides generating spanning trees, spanning
  graphs also provide proximity information
• Use for Steiner tree improvements
• RST efficient O(nlog n) heuristic