Nontree Routing for Reliability

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Non-tree Routing for Reliability Yield
Improvement
A.B. Kahng UCSD B. Liu Incentia I.I. Mandoiu
UCSD
Work supported by Cadence, MARCO GSRC, and NSF
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Outline

• Motivation for non-tree routing
• Problem formulation
• Exact solution by integer programming
• Greedy heuristic
• Experimental results

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Motivation for Redundant Interconnect

• Manufacturing defects increasingly difficult to
  control in nanometer processes
• Cannot expect continued decreases in defect
  density
• Defects occur at
• Front end of the line (FEOL), i.e., devices
• Back end of the line (BEOL), i.e. interconnect
  and vias
• In nanometer processes BEOL defects are
  increasingly dominant
• Aluminum interconnects etched ? defect modality
  short faults
• Copper interconnects deposited ? defect modality
  open faults

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Catastrophic Interconnect Faults
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Opens vs. Shorts - Probability of Failure

• Open faults are significantly more likely to occur

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Opens vs. Shorts - Critical Area (CA)
Open fault CA larger than short fault CA
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Reliability Improvement Approaches

• Reduction of short critical area
• Conservative design rules
• Decompaction
• Effective in practice!
• Reduction of open critical area
• Wider wires
• Non-tree interconnect
• How effective? What are the tradeoffs involved?
• Related work
• McCoy-Robins 1995, Xue-Kuh 1995 non-tree
  interconnect for delay and skew reduction
• 2-Edge-Connectivity Augmentation (E2AUG)

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Our Contributions

• Post-processing approach to non-tree routing for
  reliability improvement
• One net at a time
• Easy to integrate in current flows
• Most appropriate for large non-critical nets
• Compact integer program, practical up to 100
  terminals
• Faster, near-optimal greedy heuristic
• Experimental study including comparison with best
  E2AUG heuristics and SPICE verification of delay
  and process variability

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Problem Formulation

• Manhattan Routed Tree Augmentation (MRTA) Problem
• Given
• Tree T routed in the Manhattan plane
• Feasible routing region FRR
• Wirelength increase budget W
• Find
• Augmenting paths A within FRR
• Such that
• Total length of augmenting paths is less than W
• Total length of biconnected edges in T?A is
  maximum

• Wirelength increase budget used to balance open
  CA decrease with short CA increase

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Allowed Augmenting Paths
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Hanan Grid Theorem
Theorem MRTA has an optimum solution on the
Hanan grid defined by tree nodes and FRR corners.
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Hanan Grid Theorem
Theorem MRTA has an optimum solution on the
Hanan grid defined by tree nodes and FRR corners.
Sliding in at least one direction is not
decreasing biconnectivity
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MRTA vs. 2-Edge-Connectivity Augmentation

• 2-Edge-Connectivity Augmentation (E2AUG) Problem
• Given weighted graph G(V,E) and spanning tree
  T, find minimum weight A ? E s.t. T?A is
  2-edge-connected, i.e., cannot be disconnected by
  removal of a single edge

• E2AUG can be solved by performing binary search
  on WL increase budget of MRTA ? MRTA is NP-hard
• Differences between MRTA and E2AUG
• WL increase budget
• Geometric context (Manhattan plane with
  obstacles)
• Partial parallel edges
• Steiner points (paths of type C and D)

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Integer Linear Program (type A-C paths)

• 
  Total biconnected length
• Subject to
• 
  Wirelength budget
• 
  e biconnected if ?p connecting Tu Tv
• 
  exe1 gives augmenting paths
• 
  eye1 gives biconnected tree edges

• P set of -- at most O(n2) -- augmenting paths
• WL budget is fully utilized by (implicit)
  parallel paths

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Integer Linear Program (type D paths)

• Subject to

• H Hanan grid defined tree nodes and FRR corners
• Exponentially many cut constraints
• Fractional relaxation can still be solved using
  the ellipsoid algorithm

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Greedy MRTA Algorithm

• Input Routed tree T, wirelength budget W,
  feasible routing region, set V of
• allowed augmenting path endpoints
• Output Augmented routing T ? A, with l(A) W
• 1. A mark all edges of T as bridges
• 2. Compute augmenting path lengths between every
  u,v ? V by V Dijkstra calls
• 3. Compute length of bridges on tree path between
  every u,v? V by V DFS calls
• 4. Find path p with l(p) W and max ratio
  between length of bridges on the tree path
  between ends of p and l(p)
• 5. If ratio ? 1 then
• Add p to A
• Mark all edges on the tree path between ends of p
  as biconnected
• Update V and compute lengths for newly allowed
  paths (C type augmentation)
• Go to step 3
• 6. Else exit

Runtime O(ND KN2), reduced to O(KN2) w/o
obstacles where N allowed endpoints, K
added paths, D Dijkstra runtime
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Experimental Setup

• Compared algorithms
• Greedy
• Integer program solved with CPLEX 7.0
• Best-drop E2AUG heuristic Khuller-Raghavachari-Zh
  u 99
• Uses min-weight branching to select best path to
  add and multiple restarts
• Modified to observe WL budget
• Recent E2AUG genetic algorithm Raidl-Ljubic
  2002
• Features compact edge-set representation
  stochastic local improvement for solution space
  reduction
• Test cases
• WL increase budget 1, 2, 5, 10, 20, no
  limit
• Net size between 5 and 1000 terminals
• Random nets routed using BOI heuristic
• Min-area and timing driven nets extracted from
  real designs
• No routing obstacles

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Extra wirelength () and runtime (sec.) for
Unlimited WL Increase Budget

• CPLEX finds optimum (least) wirelength increase
  with practical runtime for up to 100 sinks
• Greedy always within 3.5 of optimum runtime
  practical for up to 1000 sinks

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Biconnectivity () and runtime (sec.) for 10 WL
Increase

• Augmenting paths of type C (allowing node
  projections as augmenting path endpoints) give
  extra 1-5 biconnectivity
• Biconnectivity grows with net size
• Greedy within 1-2 of optimum (max)
  biconnectivity computed by CPLEX

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Biconnectivity-Wirelength Tradeoff for Type C
Augmentation, 20-terminals
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SPICE Max-Delay (ns) Improvement

• 52-56 terminal nets, routed for min-area
• 28.26 average and 62.15 maximum improvement in
  max-delay for 20 WL increase
• Smaller improvements for timing driven initial
  routings

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Process Variability Robustness

• Width ww0, w06.67
• Delay variation computed as (maxw d(w) minw
  d(w)) / d(w0)
• 13.79 average and 28.86 maximum reduction in
  delay variation for 20 WL increase

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Summary

• Post-processing tree augmentation approach to
  reliability and manufacturing yield improvement
• Results show significant biconnectivity increase
  with small increase in wirelength, especially for
  large nets

• Future work includes
• Multiple net augmentation
• Simultaneous non-tree augmentation decompaction
• Consideration of defect-size distribution
• Reliability with timing constraints

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Thank You!