Ion I. Mandoiu

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Approximation Algorithms for VLSI Routing

• Ion I. Mandoiu
• Ph.D. Defense of Research
• August 11, 2000

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VLSI Routing
VLSI Physical Design Electrical description ?
Geometrical layout
VLSI Global Routing Given locations for net
terminals Find tree interconnection for each net

• Minimizing
• total length (RSMT problem)
• skew (ZST problem)
• number of buffers (MSPT problem)

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Overview of Results
Single-net routing

• New RSMT heuristic
• runs ?10 times faster, and gives higher-quality
  solutions than previous best RSMT heuristic

• Improved ZST approximation algorithms
• very fast O(n log n) running time

• Tight analysis of the MST heuristic for MSPT

Multi-net routing

• MCF-based approximation algorithms for global
  buffering via buffer blocks

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• A New RSMT Heuristic

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The RSMT problem
MST gives 3/2 approximation H76
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Why RSMT?

• Minimum wire length gives
• Minimum area
• Minimum resistance/capacitance

• RSMT used for
• Non-critical nets
• Physically small instances

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Key Results on RSMT Problem

• NP-hard GJ77

• Iterated 1-Steiner heuristic KR90
• Greedily adds Steiner points to the tree
• Almost 11 improvement over MST on average
• Fast batched implementation (BI1S)

• Exact algorithm GeoSteiner 3.0 WWZ98
• Branch-and-cut
• 11.5 improvement over MST on average
• Average speed comparable to BI1S!!!

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The IRV Algorithm High-Level Idea

• Iterative method in each step add/remove one
  Steiner point to/from tree

• Unlike Iterated 1-Steiner heuristic, do not
  insist on choosing best Steiner point in each step

• Steiner point to be added is chosen using a
  powerful LP formulation of the Steiner tree
  problem in graphs, called the bidirected cut
  formulation

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The Bidirected Cut Formulation
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The Bidirected Cut Formulation
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The Bidirected Cut Formulation
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The Bidirected Cut Formulation
C
C
Valid cut
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The Bidirected Cut Formulation (cont.)
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The Bidirected Cut Formulation (cont.)
LP relaxation
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The Bidirected Cut Formulation (cont.)
Dual LP
LP relaxation
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The Bidirected Cut Formulation (cont.)
Dual LP
LP relaxation

• Gives optimum integer solution if all vertices
  are terminals, i.e., for the MST problem (E 66)
• Integrality gap believed to be very close to 1

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The RV Algorithm

• RV99 3/2 approximation algorithm for Steiner
  tree problem in graphs based on bidirected cut
  formulation

• Applies only to quasi-bipartite graphs, i.e.,
  graphs with no edges connecting pairs of Steiner
  vertices

• Uses the primal-dual method for approximation
  algorithms

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Adaptation to General Graphs

• Remove Steiner-Steiner edges from graph G
• Run RV algorithm on remaining graph
• Repeat, treating Steiner vertices picked by the
  RV algorithm as terminals
• Stop when no new Steiner vertices are picked

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Implementation Issues

• Size of graphs resulting from reduction to grid
  is a potential bottleneck
• Use efficiently computable reductions
• Vertex reduction based on the empty rectangle
  test
• Edge reductions based on bound on max. degree of
  geometric MSTs (RS 95)

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Experimental Setup

• Test bed for experiments
• Random instances ranging in size between 10 and
  250 terminals, 1000 instances/size
• Instances extracted from industrial designs
• Measure of quality percent improvement over MST

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Average Improvement over MST
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Average CPU Time
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Results on Industrial Instances
No Term
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• Improved ZST and BST Approximation Algorithms

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Zero-Skew Trees
Zero-Skew Tree rooted tree in which all
root-to-leaf paths have the same length
Used in VLSI clock routing network multicasting
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The Zero-Skew Tree Problem
Zero-Skew Tree Problem Given set of terminals in
rectilinear plane Find zero-skew tree with
minimum total length

• Previous results CKKRST99
• NP-hard for general metric spaces
• factor 2e 5.44 approximation

• Our results
• factor 4 approximation for general metric spaces
• factor 3 approximation for rectilinear plane

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Overview

• Constructive lower-bound on optimum ZST length
• Converting spanning tree to zero-skew trees
• Finding spanning trees with small conversion
  cost
• Improved conversion using Steiner points
• Approximation algorithms for bounded-skew trees
• Conclusions and open problems

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ZST Lower-Bound
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ZST Lower-Bound
(CKKRST 99)
N(r)min. of balls of radius r that cover all
sinks
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ZST Lower-Bound
(CKKRST 99)
N(r)min. of balls of radius r that cover all
sinks
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ZST Lower-Bound
(CKKRST 99)
N(r)min. of balls of radius r that cover all
sinks
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ZST Lower-Bound
(CKKRST 99)
N(r)min. of balls of radius r that cover all
sinks
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Constructive Lower-Bound
Computing N(r) is NP-hard, but
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Constructive Lower-Bound
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Stretching Rooted Spanning Trees

• ZST root spanning tree root

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Stretching Rooted Spanning Trees
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Stretching Rooted Spanning Trees
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Zero-Skew Spanning Tree Problem
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How good are the MST and Min-Star?
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The Rooted-Kruskal Algorithm

• While ? 2 roots remain

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The Rooted-Kruskal Algorithm
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How good is Rooted-Kruskal?
Lemma delay(T) ? length(T)
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How good is Rooted-Kruskal?
Lemma length(T) ? 2 OPT
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Factor 4 Approximation
Algorithm Rooted-Kruskal Stretching

• Length after stretching length(T) delay(T)
• delay(T) ? length(T)
• length(T) ? 2 OPT

? ZST length ? 4 OPT
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Stretching Using Steiner Points
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Factor 3 Approximation
Algorithm Rooted-Kruskal Improved Stretching

• Length after stretching length(T) ½ delay(T)
• delay(T) ? length(T)
• length(T) ? 2 OPT

? ZST length ? 3 OPT
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Practical Considerations

• For a fixed topology, minimum length ZST can be
  found in linear time using the Deferred Merge
  Embedding (DME) algorithm Eda91, BK92, CHH92
• Practical algo Rooted-Kruskal Stretching DME

Theorem Both stretching algorithms lead to the
same ZST topology when applied to the
Rooted-Kruskal tree
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Running Time

• Stretching O(N logN)
• Rooted-Kruskal O(N logN) using the dynamic
  closest-pair data structure of B98
• DME O(N) Eda91, BK92, CHH92

? O(N logN) overall
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Extension to Other Metric Spaces
Everything works as in rectilinear plane, except

• No equivalent of DME known for other spaces

• The space must be metrically convex to apply
  second stretching algorithm

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Bounded-Skew Trees
b-bounded-skew tree difference between length of
any two root-to-leaf paths is at most b
Bounded-Skew Tree Problem given a set of
terminals and bound bgt0, find a b-bounded-skew
tree with minimum total length

• Previous approximation guarantees CKKRST 99
• factor 16.11 for arbitrary metrics
• factor 12.53 for rectilinear plane

Our results factor 14, resp. 9 approximation
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BST construction idea lower bound
Two stage BST construction

• Cover terminals by disjoint b-bounded-skew trees
• Connect roots via a zero-skew tree

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Constructing the tree cover
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BST Approximation
Algorithm Output tree cover ? approximate ZST on
W
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BST Approximation
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Summary of Results on ZST/BST
Problem Zero-Skew Zero-Skew Bounded-skew Bounded-skew
Metric General Rectilinear General Rectilinear

Previous factor 5.44 5.44 16.11 12.53
New factor 4 3 14 9
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Open Problems

• Complexity of ZST problem in rectilinear plane
• Complexity of finding the spanning tree with
  minimum lengthdelay?
• Zero-skew Steiner ratio supremum, over all
  sets of terminals, of the ratio between minimum
  ZST length and minimum spanning tree lengthdelay
• What is the ratio for rectilinear plane?
• What is the ratio for arbitrary spaces? ( ?4,
  ?3)
• Planar ZST / BST

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• Tight Analysis of the MST Heuristic for MSPT

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The MSPT Problem
MSPT Problem find bounded edge-length Steiner
tree with min. number of Steiner points
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The MST Heuristic for MSPT

• Find MST, subdivide edges to meet edge-length
  constraints

LX99 Approximation factor ?5 in Euclidean plane
Our result Approximation factor of MST heuristic
is D-1, where D is the MST number of the space
(maximum possible degree of a minimum-degree MST)
? Factor 3 in rectilinear plane, 4 in Euclidean
plane
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• Provably Good Global Buffering by MTMCF
  Approximation

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Global Routing via Buffer Blocks
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Global Buffering Problem

• Given
• L/U bounds on edge lengths
• buffer block locations and capacities
• list of nets, each net has
• upper-bound and parity requirement on buffers
  for each source-sink path
• non-negative weight (criticality coefficient)
• Find routing of a max. weight set of nets
  s.t.
• each edge length is between L and U
• buffers for any source-sink pair satisfy given
  constraints
• of nets passing through buffer block b ?
  capacity(b)

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Integer MTMCF Formulation
Approach solve LP relaxation randomized
rounding
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MTMCF Approximation

• Garg/Konemann Fleisher (edge-cap. MCF)
• ?-MCF algorithm
• w(v) ?, f 0
• For i 1 to N do
• For k 1,,K do
• Find min. weight valid tree T for net k
• While w(T) lt min 1, ?(12?)i do
• f(T) f(T)1
• For every v, w(v) ? (1 ??(T,v)/c(v))w(v)
• End while
• End for
• End for
• Output f/N

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Rounding fractional MCF

• Raghavan-Thompson random walk from source
• probability of choosing an arc/node proport. node
  flow
• Probability of routing net proportional net flow
• Algorithm
• decrease flow by (1-?)
• route nets with randomized rounding
• With high probability no node capacity violations

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Experimental results
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Conclusions

• Improved approximation algorithms and heuristics
  for NP-hard problems arising in VLSI routing
• Experimentally validated, theoretical
  guarantees doubled by good practical results