1. Group: Wilber L. Duran Duo (Steve) Liu
EE201 Spring 2004
Multilevel Routing
Group: Wilber L. Duran
Duo (Steve) Liu

2. Multilevel Approach to Full-Chip Gridless Routing
Jason Cong, Jie Fang, and Yan Zhang
Computer Science Department, UCLA

3. Traditional Routing System
Global routing
Partitions the entire routing region into tiles or channels and a rough route for each net is determined among these tiles to minimize the overall congestion
Detailed routing
Performed at each tile, where the exact implementation of each net is determined
Uses flat approaches or two level approaches
Maze searching algorithm, line-probe algorithm
All flat approaches have a scaling problem when it comes to large designs

4. Proposed Solution
As the designs grow, more levels of routing are needed for larger designs
Rather than a predetermined, manual partition of levels which may have discontinuity between levels, an automated flow is needed to enable seamless transitions between the levels
Propose a novel multilevel routing framework for the gridless routing problem

5. Overview
The multilevel framework features an iterative coarsening algorithm and an iterative refinement algorithm in a “V-shaped” flow
On the downward pass, the design is recursively coarsened and an estimation of routing resources is calculated at each level
At the coarsest level, a multicommodity flow algorithm is used to generate an initial routing result
On the upward pass, a modified maze searching algorithm is carried out iteratively to refine the results from level to level

6. Multilevel vs. Hierarchical Approaches

7. Multilevel vs. Hierarchical Approaches
In Multilevel approach, the uncoarsening pass allows the fine level router to refine the coarse level result and the coarse level solution only provides a guide to fine level path searching
Provides the flexibility to deviate from the coarse level path when more detailed information about local resource and congestion is considered
This feature makes the multilevel method converge to better solutions with higher efficiency

8. Build Multilevel Routing Region
The routing region is first partitioned into an array of fine tiles, each with the same height and width. This level is denoted as level 0
Then build a three-dimensional routing graph, denoted as G0
The edge capacity represents the routing resources at the common boundary of two tiles

9. Line-sweeping Algorithm
Boundary capacity is computed by the following formula:
The inter-layer edge capacity is computed as the sum of empty slices intersections between the two tiles connected by the edge

10. Coarsening Process
The grid graph G0 stores accurate routing capacity estimation at the finest level
At a coarser level (level i+1), the tiles are built from the finer level tiles (level i) by merging neighboring tiles
Gi+1 can be derived from the fine level graph Gi directly
C(ui+1,vi+1) on Gi+1 is the sum of the capacities of the edges in Gi that connect the tiles merged into ui+1 and the tiles merged into vi+1
Iteratively coarsen the tiles and the routing graphs until the size of the graph falls below a predetermined threshold

11. Initial Routing
A set of tile-to-tile paths are computed for the nets crossing the coarsest tile boundaries.
It is quite important to the final result of multilevel routing
Capability of handling performance issues caused by long interconnects
A bad initial routing solution can slow down the refinement process and may even degrade the final solution

12. Initial Routing (cont.)
Use multicommodity flow based algorithm
It is fast enough for a relatively big grid size
It considers all the nets at the same time
It can be integrated with other optimization algorithms to consider special requirements of certain critical nets
The objective is to minimize the congestion on the routing graph G0
Current implementation does not consider delay minimization and focuses mainly on routability and wire length optimization
Use only the shortest paths as candidates for each net

13. Multicommodity Flow Algorithm
Pi = {Pi,1, … , Pi,li} be the set of possible paths of given net i
C(e) is the capacity of each edge on the routing graph
Wi,e is the cost for net i to go through edge e
Xi,j is an integer variable with possible values 1 or 0 indicating if path Pi,j is chosen or not

14. Multicommodity Flow Algorithm (cont.)
Relax Xi,j >= 0 to convert the problem to a linear programming problem
A maximum flow approximation algorithm is used to compute the fraction value of Xi,j
After picking a path, increase the flow along the path as much as possible to saturate the minimum capacity edge along the path
After the fractional result for each path are computed, map the fractional results to integer results
Use a randomized rounding algorithm
Does not guarantee that there is no overflow at the tile boundaries

15. Upward Pass of Multilevel
Paths computed by the initial flow-based algorithm are refined from level to level until the finest tiles are finally reached
Multilevel framework allows the finer level to change coarser-level routing solutions

16. Constrained Maze Refinement
Local Nets are the nets that are relatively short and do not cross coarser tile boundaries. Finding paths for them is relatively easy
Another set of nets are those carried over from the previous coarser-level routing

17. Constrained Maze Refinement (cont.)
A preferred region is defined as the set of tiles that the coarse level path goes through
Weights and penalties associated with each routing graph edge are computed
Additional penalties are assigned to graph edges linking to and going between the graph nodes corresponding to tiles that are not located within the preferred region
Dijkstra’s shortest path algorithm is used to find a weighted shortest path for each net

18. Experiments Results

19. Experiment Results (cont.)

20. Experiment Results (cont.)

21. Summary Present a novel routing system using a multilevel method
It scales well on larger designs and provides a good framework for integrating different algorithms and allows different algorithms to be used on different levels
A flow-based algorithm is used to compute the initial routing results
A modified maze-searching algorithm is used to iteratively refine the results

22. Full-chip Multilevel Routing for Power and Signal Integrity
Authors: Jinjun Xiong and Lei He
EE Department
University of California, Los Angeles

23. Overview Introduction Design Constraints Problem derivation
Power Net Estimation Formula
Algorithm Description
Experimental Results

24. Introduction Major Concern in wire-limited deep sub-micron designs
- Power Distribution Networks
- Signal networks
Designed Separately
- PDN First
- SN second
Problem
- Iteration between both in order to find best design

25. Existing Approaches Feedback between Power Network and Signal Network
- Design Convergence is very slow
- Results in small benchmarks reported
2) Three Step design:
Signal Routing->Power Network->Signal Routing
- Requires iteration
- Is applied to real industrial practices

26. Design Constraints Power Network
- Designed as a mesh to provide a low impedance current return path for signals
- Power Pitch (max. separation between 2 adj. Power lines in a mesh structure)
Signal Integrity
- Crosstalk reduction via shielding
- Assumes shielding requirements for nets are inputs
- Signal nets that require:
I – 2 adj. Shields: S2_nets
II - 1 adj. Shield : S1_nets
III – 0 adj. Shield : S0_nets
- I & II are critical nets

27. Design Constraints Tessellate routing area into routing tiles
- Formulated into an undirected graph G(V,E).
- Each Vertex νєV = 1 routing tile
- Each edge e є E = routing area between 2 adj. Tiles. Capacity = # tracks available
In Multi-layer design an edge consists of more than 1 layer.
- Each layer is composed of eq. Spaced tracks.
- Each track is used by one net segment.

28. Design Constraints
Assuming uniform wire sizing for all power nets and uniform lengths for all finest routing tiles:
Model for total power network area:
(1)
St = #power nets in Rt
Rt = Routing region
Routing Density:
Ct = routing capacity and Gt = # signal nets
But, if Rt > 1 then overflows in Rt exists

29. Problem Formulation Shields inserted after Power Network Design
Typically during or after signal routing
 shields consume the already tight routing budget left for signal routing
If no solution possible then,
- Go back to modify power network design to min. area and allocate routing resources for shielding purposes.

30. Problem Formulation
Apply co-design to the power and signal Networks simultaneously.
Co-design is formulated as follows:
* Given a power pitch PGP, a placement solution, a netlist and the shielding requirements for all signal nets.
* GSPR synthesizes a Power Network and an extended global routing solution such that power pitch < PGP.
* It satisfies the shielding constraints for all nets and total Power Network area defined in ( 1 ) is minimized.

31. Design Methodology
GSPR synthesizes a global routing solution with power net estimation considering Power Pitch and shielding requirements
Then the Power Network is synthesized to satisfy the Power Pitch constraint.
Goal: Provide a simple & accurate Power estimation formula that calculates min. # Power nets that satisfies power pitch and net shielding constraints without knowing the Power Network solution.

32. Power Net Estimation
A valid track in Rt = solution that meets Power Pitch and signal shielding constraint.
The exact # of Power Nets is only known after we have fixed track assign. Solution.
- At this point is too late to correct bad routing solutions
 A formula is developed to estimate # power nets
Lemma 1:
- Given Rt with Capacity Ct, Min. # of Power Nets in Rt must be:
Pt =  Ct/ PGP  in order to satisfy PGP

33. Power Net Estimation Need to satisfy shielding requirements Lemma 2:
- Given Rt with m2 (#s2_nets), m1, m0
* Min. # of Power Nets:
St = ( m1/2 - b2) + (m2 + 1)*b2
Where, b2=1 , for m2>0
b2=0 Otherwise

34. Power Net Estimation In order to satisfy PGP and Shield constraints:
Theorem 1:
* Given routed nets and shielding requirements for signal integrity, Min # Power Nets as (Pt –1)^2
Then, upper bound on Min. # of Power Nets is:

35. GSPR Algorithm Power Integrity aware multi-level signal routing
Power Network synthesis and track assignment to satisfy both Power and Signal integrity constraints.
Multi-level routing framework consists in 2 parts:
- Coarsening Process
- un-coarsening Process

36. GSPR Algorithm Coarsening Process:
- Fine routing tiles merged recursively into coarser tiles.
- Stops when # of tiles in coarsest level is < threshold
2) Un-coarsening Process:
- Determines tile to tile solution for un-routed nets left by coarsening stage.
- Refines the routed solution

37. GSPR Algorithm

38. GSPR Algorithm
For each determined path, its cost function is defined as:
Gt = # of nets
St= # of Power Nets
Ct = Capacity of Rt
t = Dynamically factor to penalize for paths that tend to cause overflow

39. GSPR Algorithm Power Network Synthesis & track assignments:
* 2 step hierchical procedure
1) Synthesize a global Power Network -> 2 power nets
along the 2 edges of every routing region.
2) Synthesize local Power Networks & track assignment simultaneously.
Optimal local power network and track assignment in each routing region is decided by Theorem 1.
Results in no iteration.

40. Implementation Results
GSPR is implemented in C++ on Linux.
10 Industrial benchmarks are involved for testing.
It is assumed that the required power pitch PGP = 10 for the benchmarks shown in table 1.

41. GSPR Vs. 3-step Algorithm

42. Summary
Problem: Iterative process between Power Distribution and Signal Networks.
Formulation of Theorem 1 used in GSPR
Novel Design Methodology to co-design of Power and Signal Networks under integrity constraints
Algorithm Flow
Results: reduction of power network area of 19.4% compared to 3 step approach.