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**Group: Wilber L. Duran Duo (Steve) Liu**

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

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**Multilevel Approach to Full-Chip Gridless Routing**

Jason Cong, Jie Fang, and Yan Zhang Computer Science Department, UCLA

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**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

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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

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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

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**Multilevel vs. Hierarchical Approaches**

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**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

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**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

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**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

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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

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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

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**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

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**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

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**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

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**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

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**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

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**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

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Experiments Results

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**Experiment Results (cont.)**

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**Experiment Results (cont.)**

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**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

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**Full-chip Multilevel Routing for Power and Signal Integrity**

Authors: Jinjun Xiong and Lei He EE Department University of California, Los Angeles

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**Overview Introduction Design Constraints Problem derivation**

Power Net Estimation Formula Algorithm Description Experimental Results

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**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

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**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

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**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

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**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.

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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

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**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.

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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.

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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.

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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

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**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

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**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:

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**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

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**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

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GSPR Algorithm

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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

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**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.

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**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.

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**GSPR Vs. 3-step Algorithm**

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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.

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