A'B' Kahng, Ion I' Mandoiu

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Highly Scalable Algorithms for Rectilinear and
Octilinear Steiner Trees

• A.B. Kahng, Ion I. Mandoiu
• University of California at San Diego, USA
• A.Z. Zelikovsky
• Georgia State University, USA
• Supported in part by MARCO GSRC and Cadence
  Design Systems, Inc.

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Outline

• Single net routing problem
• Problem definition
• Previous work
• Motivation for highly scalable heuristics
• The batched greedy algorithm
• High-level algorithm
• Efficient generation of triples
• Efficient bottleneck-edge computation
• Experimental results and conclusions

• Single net routing problem
• Problem definition
• Previous work
• Motivation for highly scalable heuristics
• The batched greedy algorithm
• High-level algorithm
• Efficient generation of triples
• Efficient bottleneck-edge computation
• Experimental results and conclusions

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Single Net Routing Problem
Given set of terminals in the plane Find
minimum length interconnection
? Rectilinear/Octilinear Minimum Steiner Trees
(RMST/OMST)
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Why Minimum Steiner Tree Routing?

• Advantages
• Minimum routing area
• Minimum total capacitance
• Reduced power consumption
• Steiner tree routing appropriate for
• Non-critical nets
• Physically small instances

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Previous Work on Steiner Tree Problem

• Long history
• Euclidean version Gauss 1836
• Rectilinear version Hanan 1966
• Octilinear version SarrafzadehWong 1992
• Steiner tree problem in graphs Hakimi/Levin
  1971
• Fundamental results
• All versions are NP-hard Karp 1972, GJ77
• Minimum Spanning Tree (MST) gives good
  approximation
• Always within factor 2 of optimum 3 papers,
  1979-1981
• Within factor 1.5 in rectilinear plane Hwang
  1976

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Previous RSMT/OSMT Algorithms

• Exact algorithms
• GeoSteiner Warme,WinterZachariasen
• Approximation algorithms
• Zelikovsky 1993, BermanRamaier 1994,
  HougardyPromel 1999, RajagopalanVazirani 1999,
  RobinsZelikovsky 2000,
• Best-performing RSMT heuristics (within 0.5 of
  optimum)
• Iterated 1-Steiner heuristic KahngRobins 1992
• Edge-based heuristic Borah,OwensIrwin 1999
• Iterated Rajagopalan-Vazirani Mandoiu,VaziraniGa
  nley 2000
• Not practical for tens of thousands of terminals!

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Motivation for Highly Scalable Heuristics

• Most nets are small (lt20 terminals)
• But nets with gt104 terminals become increasingly
  common
• Scan-enable nets in large designs using full-scan
  test
• All flip-flops need to receive the scan-enable
  signal
• Nets with pre-routes and non-zero terminal
  dimensions
• Each terminal represented by set of electrically
  equivalent pins
• ?RSMT/OSMT instances with up to 105 pins

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Requirements for Highly Scalable RSMT/OSMT
Heuristics

• Linear memory
• Sub-quadratic runtime
• Solutions within 0.5 of optimum
• Previous Steiner tree heuristics do not meet
  first two requirements

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Outline

• Single net routing problem
• Problem definition
• Previous work
• Motivation for highly scalable heuristics
• The batched greedy algorithm
• High-level algorithm
• Efficient generation of triples
• Efficient bottleneck-edge computation
• Experimental results and conclusions

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Triple Contraction

• Connect 3 terminals (triple) using shortest
  connection

• Remove longest edge on each of the 2 formed
  cycles

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High-level Algorithm

• Greedy Triple-Contraction Algorithm Zelikovsky
  1993
• Compute MST of terminals
• While there exist triples with positive gain, do
• Find triple with maximum gain
• Contract triple remove longest edges, replace
  triple with 2 zero-cost edges
• Output MST of terminals and centers of contracted
  triples

• Expensive to compute max-gain triple in Step 2
• Best implementation uses complex dynamic MST
  datastructures
• We use a batched implementation
• Find positive-gain triples
• Contract triples in descending order of gain
  without recomputing gains
• A triple is selected if 2 longest edges not used
  by previous triples

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Efficient Generation of Triples

• O(n3) triples overall
• Use geometry to avoid generating all of them!
• Fossmeier,KaufmannZelikovsky 1997 sufficient
  to consider a set of O(n) triples with
• Empty interior (? no other terminal in bounding
  box)
• Positive gain
• We use a practical O(nlogn) divide-and-conquer
  algorithm to compute a superset of size O(n logn)
• Some triples may have negative gain
• Eliminated after gain computation
• Some triples may be non-empty
• Can be removed, but too few to justify the extra
  testing time

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Divide-and-conquer for Empty Triples
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Divide-and-conquer for Empty Triples
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Divide-and-conquer for Empty Triples
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Divide-and-conquer for Empty Triples
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Efficient Bottleneck-edge Computation
Bottleneck edges needed for computing triple gains
Given tree T, vertices u,v Find most expensive
edge on path between u and v

• We give a new data structure
• O(logn) time per bottleneck-edge query
• O(n logn) pre-processing (O(n) if edges already
  sorted)
• Much more practical than methods based on
  least-common-ancestors
• ? Gain evaluation in O(logn) time per triple
• ? O(n log2n) total time for the batched greedy
  algorithm

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Outline

• Single net routing problem
• Problem definition
• Previous work
• Motivation for highly scalable heuristics
• The batched greedy algorithm
• High-level algorithm
• Efficient generation of triples
• Efficient bottleneck-edge computation
• Experimental results and conclusions

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Experimental Setup

• Testcases
• Random nets with 100 to 500,000 terminals
• 100 samples for each size
• Nets extracted from recent designs (330 to 34,000
  terminals)
• Compared algorithms
• Batched greedy
  O(n log2n)
• MST GuibasStolfi 1983
  O(n logn)
• Prim-based heuristic Rohe 2001
  O(nlog2n)
• Edge-based heuristic of Borah,OwensIrwin 1999
  O(n2)
• GeoSteiner 4.0, beta version Nielsen,WinterZacha
  riasen 2002

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Quality Random Rectilinear Tests

• BatchGreedy quality slightly better than
  Edge-based, 1 better than Prim-based
• Within 0.7 of optimum computed by GeoSteiner

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CPU Time Random Rectilinear Tests

• BatchGreedy highly scalable, practical runtime
  up to 105 terminals
• Edge-Based impractical for gt104 terminals

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CPU Time Rectilinear Industry Tests

• 34k terminals 24 seconds BatchGreedy vs. 86
  minutes Edge-based

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Quality Rectilinear Industry Tests

• BatchGreedy up to 1 better than Prim-Based,
  within 0.5 of GeoSteiner
• Slightly better than Edge-Based in half of the
  cases

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CPU Time Octilinear Industry Tests

• 34k terminals 25 seconds BatchGreedy vs. 17.5
  hours Edge-based
• Octilinear Prim-Based not available

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Quality Octilinear Industry Testcases

• BatchGreedy slightly better than Edge-Based,
  within 0.5 of GeoSteiner

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Conclusions

• Presented new rectilinear/octilinear Steiner tree
  heuristic
• Highly-scalable
• O(n) memory, O(nlog2n) runtime, with small hidden
  constants
• High-quality
• Better quality than O(n2) edge-based heuristic
• Within 0.7 of optimum computed by GeoSteiner
• Ongoing extensions
• Via costs
• Preferred directions
• Routing obstacles

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Thank You for Your Attention!

• Further details on our work are available on the
  groups website, http//vlsicad.ucsd.edu