Creating and Exploiting Flexibility in Steiner Trees

1
Creating and Exploiting Flexibility in Steiner
Trees

• Elaheh Bozorgzadeh, Ryan Kastner, Majid
  Sarrafzadeh
• Embedded and Reconfigurable System Design (ER
  LAB)
• Computer Science Department
• UCLA

2
Outline

• Introduction
• Definition and Preliminaries
• Flexibility in Rectilinear Steiner Tree
• Our Approach to Create and Exploit Flexibility
• Experimental Results
• Conclusion and Future Work

3
Global Routing

• Routability
• Important factor in global routing solution
• Satisfied if detailed router is able to find
  feasible solution from global router.
• Depends on highly constrained regions in global
  routing solution like congested regions.
• Delay
• Wire delay is becoming increasingly important
• Only 10 of the nets are timing critical.

Routability can be emphasized more when routing
non-critical nets.
4
Flexibility under Routing

• Flexibility
• Geometrical property of RST
• Related to routability of steiner tree
• Flexible Edge
• Non-horizontal/vertical edge route
• Has more than one shortest path
• Exploited as soft edge in the routing algorithm
  proposed
• by Hu and Sepatnekar (ICCAD2000)

Flexible edges
Steiner Node
5
Pattern Routing and Flexibility

• Pattern routing the flexible edges
• More than one patterns defined
• More ability to maneuver the congested region

Congested Area
6
Flexibility under Routing

• Two given RSTs with same topology
• Study impact of flexibility in congestion

Flexible RST
Non-flexible RST
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Flexibility under Routing

• Two given RSTs with same topology
• Study impact of flexibility in congestion

Flexible RST
Non-flexible RST
8
Flexibility Function

• Flexibility of an edge Two
  possible functions
• Flexibility of an edge increases if w or l
  increases.

w
l
9
Rectilinear Steiner Tree Constraints

• Given RST is stable (introduced by Ho, et.al DAC
  89)
• Topology of RST remains unchanged

stable
unstable
Two RSTs with same topology
10
Generating Flexibility in RST

• Problem Formulation
• Given a stable Rectilinear Steiner Tree,
• Maximize the flexibility of the RST
• Subject to
• ? Topology remains unchanged (and thus
  if we do min-length edge connection, total length
  remains unchanged)
• ? No initial flexible edge is degraded in
  flexibility.

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Flexible Edges

• Flexible edges can be generated by moving
  themovable edges in RST.
• Movable Edge
• Steiner-to-steiner edge.
• Edge degree of each steiner point is 3.
• parallel edges exists at both ends.
• Flexible candidate exists at least at one end.

parallel edges
flexible candidate
movable edge
12
Flexible Edges

• Flexible edges can be generated by moving
  themovable edges in RST.
• Movable Edge
• Steiner-to-steiner edge.
• Edge degree of each steiner point is 3.
• parallel edges exists at both ends.
• Flexible candidate exists at least at one end.

parallel edges
flexible candidate
movable edge
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Pseudo code

• Input Edge Set of an RST S
• Output RST R
• Algorithm Generate Flexible Tree
• Begin
• For Each edge e
• If e and its adjacent edges are a movable set
• Create Movable Set
• Check Overlap
• For each movable set M
• If M has no overlap
• Move edge M
• Move Overlapped edges
• End

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Example of Flexible RST Construction
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Example of Flexible RST Construction
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Example of Flexible RST Construction
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Complexity of Our Algorithm

• Algorithm GenerateFlexibleTree generates the most
  flexible RST from a given stable RST under the
  constraints of wirelength and topology and
  stability remaining unchanged.
• Our method solves the problem optimally.
• If a linear flexibility function is used, the
  time complexity of the algorithm is O(E).
• If quadratic flexibility function is used, the
  time complexity of the algorithm is
  O(E2k)(pseudo-polynomial), where E is the edges
  in RST and k in the number of overlapping movable
  set pairs (k is normally small).

18
Preliminary Experiments

• Preliminary Experiments to show relationship
  between routability and flexibility (many other
  flows are possible)

Maze route the nets other than nets in C
C 4 terminal Nets
Route nets in C in nonflexible pattern
Route nets in C in flexible pattern
Compare Congestion!
Pattern-route flexible edges (L-shape, Z-shape)
19
Experiments

• MCNC standard cell Benchmarks and ISPD98
  benchmark
• Circuits placed by placer DRAGON

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Conclusions and Future Work

• Introduced flexibility , a geometrical property
  of Steiner trees related to routability.
• Proposed an algorithm to generate optimally a
  flexible RST from given stable RST, which can be
  applied in early stages to assign the location of
  Steiner points in order to deal with congestion
  better.
• Preliminary experimental results show that
  flexible Steiner tree cause less congestion on
  routing resources
• Developing constructive Steiner tree algorithms
  which generate and exploit the flexibility in
  routing is a suggested future work.

21
Introduction

• Global Routing
• Finding approximate path (route) for each net
• Generating steiner tree for each net
• Steiner tree construction with minimum cost is
  NP-hard.
• Objectives
• Minimizing wirelength
• ?Like Maze router and extended versions
• Minimizing the required number of vias
• Minimizing delay
• ?Buffer insertion, wiresizing
• Minimizing Congestion
• Our cost Total excess demand of routing edges in
  grid graph

Route edge