A Fast Algorithm for Context Aware Buffer Insertion

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A Fast Algorithm for ContextAware Buffer
Insertion

• Ashok Jagannathan,
• Sung-Woo Hur,
• John Lillis
• University of Illinois, Chicago.
• June, 2000

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Outline

• Motivation
• Problem Formulation
• Algorithm
• Results
• Properties
• Conclusion

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Motivation

• Buffer insertion
• Restrictions on buffer locations (Macros, IPs..)
• We may have to detour to pick up a buffer

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Motivation

• Given
• Candidate buffer locations
• Buffer library
• Source/sink terminals (2 pins)
• Objectives
• Min cost path s.t. delay lt Dspec
• Weakly NP-hard
• Shortest weight-constrained path problem GJ
• Min-delay path
• Polynomial-time solvable

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Motivation

• Pseudo-polynomial algorithm
• Paths characterized by (g,d) pair
• P(v) set of all non-dominated s-v paths
• Propagate solutions from source to sink

• Problems
• P(v) Not polynomially-bounded in E,V
• Comparatively slow in practice

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Motivation

• Finding min delay path
• Fast-path algorithm Zhou et. al., DAC99
• Reasonably fast

• Problem
• May not be good use of resources

delay
min delay
cost
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Motivation

• Pseudo-polynomial algorithm
• Aware of cost/delay tradeoff
• Relatively slow
• Fast-path algorithm
• Unaware of tradeoff
• Polynomial time

• Need something in between
• Composite objective (cost vs. delay)
• Systematically explore tradeoff curve
• Should be fast

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Problem Formulation

• We propose
• Max Delay Reduction to Cost Ratio (DRCR)
• Given
• Candidate locations buffer library
• Source, sink
• Reference delay, Dref
• Objective
• A buffered source/sink path p s.t.
• (Dref - Dp)
• Gp
• is maximized.
• Dp delay of p
• Gp cost of p

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Buffer Graph

• Directed
• Nodes Buffers source,sink
• Edges Candidate connections
• Edges
• Not detailed routes
• buffer-buffer connections
• ge estimated cost of the edge
• routing cost
• cost of destination buffer
• anything you want
• de estimated interconnect delay (input to input)

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Buffer Graph - Illustration
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Buffer Graph - Illustration
With cascading
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Buffer Graph - Illustration
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Buffer Graph

• Properties
• Independent of cost model
• Almost independent of delay estimation model
• Path topological route buffering
• Nodes determine type of buffer
• Allows cascaded-buffers
• Observation
• Need not be a complete graph
• Dont want lengthy unbuffered wires
• Sufficient to connect buffers within distance m
• Depends on tech parameters

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DRCR Algorithm

• Similar to the ratio-cycle problem by Lawler et
  al.
• Objective
• Given Dref, find a path p in G with max

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DRCR Algorithm

• Rmax Gp Dp Dref
• If we find Rmax, we have found p
• Observation
• Rmax ?ge ?de Dref
• Edge weights we Rmax ge de
• LHS length(p) ?we
• How to find Rmax ?
• Start with conjecture I
• Iteratively refine I to obtain Rmax
• Use binary search

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DRCR Algorithm

• Re-weight edges with we I ge de
• Find shortest s-t path p (Dijkstra)
• length(p) I Gp Dp
• Three cases
• I Gp Dp lt Dref
• I Gp Dp gt Dref
• I Gp Dp Dref

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DRCR Algorithm

• I Gp Dp lt Dref

Ratio for path p is better than I
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DRCR Algorithm

• I Gp Dp gt Dref

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DRCR Algorithm

• I Gp Dp Dref

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DRCR Algorithm

• Re-weight edges with we I ge de
• Find shortest s-t path p (Dijkstra)
• length(p) I Gp Dp
• Three cases
• I Gp Dp lt Dref
• I Gp Dp gt Dref
• I Gp Dp Dref

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DRCR Algorithm

• Polynomial time
• O(logDmax x E log V)

• What do we really sacrifice ?
• Only solutions on Lower Convex hull(LCH)
• 30-40 lie on LCH

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Properties

• Theorem
• For any Dref
• (Gp,Dp) is optimal iff its on LCH
• As Dref decreases, cost increases
• Decrease Dref -- move right on the curve

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Results

• Goal Evaluate computational feasibility
• Experiments done on randomly generated graphs
• 0.18µm tech from literature

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Discussion

• Dref is artificial
• Observation
• For any I, shortest path is optimum for some Dref
• Use I to probe tradeoff curve
• Increase I -- move left
• Decrease I -- move right

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Conclusion

• Study context-aware buffer insertion
• New problem efficient solution
• Potential application floorplan level
  wire-planning
• Results show viability of the approach
• Variant to probe tradeoff curve
• Fits applications with largely convex tradeoffs

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Thank you!