ELEN 468 Advanced Logic Design

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ELEN 468Advanced Logic Design

• Lecture 28
• Interconnect Timing Optimization III

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Dependence on Steiner Tree
Timing critical
Timing critical
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Rectilinear Steiner Minimum Tree

• Given a signal net, find the best tree connecting
  them
• Minimize wire area
• Wire area implies
• Cost
• Capacitive load ? delay
• Find Steiner minimum tree

Spanning tree
Steiner node
Steiner tree
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Hanan Grid and Hanan Theorem

• Hanan grid
• Draw vertical and horizontal lines through all
  pins
• Hanan Theorem
• There is always a Steiner minimum tree on Hanan
  grid

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Iterative 1-Steiner Algorithm

• In each step, add one Steiner node such that the
  spanning tree is minimized

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Area or Radius?
Radius the longest source-sink path length

• Dijkstras shortest path tree
• Short path to sinks
• Large total wire length

• Prims minimum spanning tree
• Small total wire length
• Long path to sinks

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Area Radius Trade-off

• Find a solution in middle
• Not too much area
• Not too long radius
• How to find an ideal point?

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Prims and Dijkstras Algorithms

• d(i,j) length of edge (i, j)
• p(i) length of path from source to i
• Prim min d(i,j) Dijkstra min d(i,j) p(i)

p(i)
i
j
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The Prim-Dijkstra Trade-off

• Prim add edge minimizing d(i,j)
• Dijkstra add edge minimizing p(i) d(i,j)
• Trade-off c?p(i) d(i,j) for 0 c 1
• When c0, trade-off Prim
• When c1, trade-off Dijkstra

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Spanning Tree ? Steiner Tree
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Rectilinear Steiner Arborescence (RSA)

• Every source-sink path is the shortest
• Minimum total wire length

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RSA Heuristic

• Assume all sinks in first quadrant
• Initially, each sink is a subtree
• Iteratively merge or grow subtrees toward the
  source

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RSA Example
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Merging Rule In RSA Heuristic

• Iteratively
• Find subtrees rooted at p and q maximizing
  min(xp, xq) min (yp, yq)
• Merge them to a new subtree rooted at r
  (min(xp, xq), min (yp, yq))

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RSA Diagonal Line Sweep
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Buffered A-Tree
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SERT Steiner Elmore Routing Tree

• Similar to Prims minimum spanning tree algorithm
• Connect one sink to partial tree in each step
• Find the sink such that the max sink delay is
  minimized
• First Elmore delay based Steiner tree algorithm

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SERT Connecting Sink
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SERT-C

• Steiner Elmore Routing Tree with identified
  critical sink
• Connect source with the critical sink directly
• Connect one sink to tree each step
• Do the connecting such that delay to the critical
  sink is minimized

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Exploit Non-Hanan Points
Delay is dominated by capacitance Or timing
constraint is loose Minimize total wirelength
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Non-Hanan Optimization Problem Formulation

• Minimize total wirelength subject to min slack ?
  0
• Maximize x subject to min slack ? 0
• Slack is convex function of x

x
L
Slack
Sink 1
Sink 3
x
Sink 2
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Non-Hanan Optimization
Slack

• Given a Steiner tree, disconnect each sink from
  the tree and reconnect it back
• Binary search is performed when a sink is
  reconnected to an edge
• If min slack lt 0 at both ends with same critical
  sink, no feasible solution
• If min slack gt 0 at one end, lt 0 at the other
  end, there is solution in between

x
Slack
x
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P-Tree Abstract Tree
g
d
c
f
a
e
g
b
f
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P-Tree Abstract Tree Generation
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P-Tree Embedding
Hanan grid
j
i
d
c
a
h
b
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Buffered P-Tree
Optimal tree Topta,b,c,d