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ELEN 468 Advanced Logic Design

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ELEN 468 Advanced Logic Design Lecture 28 Interconnect Timing Optimization III Dependence on Steiner Tree Rectilinear Steiner Minimum Tree Given a signal net, find ... – PowerPoint PPT presentation

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Title: ELEN 468 Advanced Logic Design


1
ELEN 468Advanced Logic Design
  • Lecture 28
  • Interconnect Timing Optimization III

2
Dependence on Steiner Tree
Timing critical
Timing critical
3
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
4
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

5
Iterative 1-Steiner Algorithm
  • In each step, add one Steiner node such that the
    spanning tree is minimized

6
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

7
Area Radius Trade-off
  • Find a solution in middle
  • Not too much area
  • Not too long radius
  • How to find an ideal point?

8
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
9
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

10
Spanning Tree ? Steiner Tree
11
Rectilinear Steiner Arborescence (RSA)
  • Every source-sink path is the shortest
  • Minimum total wire length

12
RSA Heuristic
  • Assume all sinks in first quadrant
  • Initially, each sink is a subtree
  • Iteratively merge or grow subtrees toward the
    source

13
RSA Example
14
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))

15
RSA Diagonal Line Sweep
16
Buffered A-Tree
17
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

18
SERT Connecting Sink
19
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

20
Exploit Non-Hanan Points
Delay is dominated by capacitance Or timing
constraint is loose Minimize total wirelength
21
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
22
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
23
P-Tree Abstract Tree
g
d
c
f
a
e
g
b
f
24
P-Tree Abstract Tree Generation
25
P-Tree Embedding
Hanan grid
j
i
d
c
a
h
b
26
Buffered P-Tree
Optimal tree Topta,b,c,d
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