Analytic Placement

Layout Project:  Sending the RTL file: −Thursday, 27 Farvardin  Final deadline: −Tuesday, 22 Ordibehesht  New Project: −Soon 2

Analytic Placement Analytic placement:  Minimizes an objective function using a mathematical technique  Examples: −Numerical methods −Linear programming  Often requires assumptions like: −Differentiable obj. func. −Dimensionless objects 3

Analytic Placement Quadratic placement:  Cost function: −to facilitate the calculation of partial derivatives 4 n: total number of cells c(i,j): connectivity between cells i and j  Squared Euclidean distance  Only two-point-connections

Analytic Placement  Each dimension can be considered independently: 5

Analytic Placement 6  Placement problem converted to “convex quadratic optimization problem”  Convexity   local minimum = global minimum

7 Analytic Placement – Quadratic Placement  System of linear equations  Iterative numerical methods find a solution −Conjugate gradient (CG) −Successive over-relaxation (SOR)

Analytic Placement – Quadratic Placement A[i][j] = -c(i,j) when i ≠ j, A[i][i] = the sum of incident connection weights of cell i X = vector of all the x-coordinates of the non-fixed cells b x [i] = the sum of x-coordinates of all fixed cells attached to i Y = vector of all the y-coordinates of the non-fixed cells b y [i] = the sum of y-coordinates of all fixed cells attached to i 8

Formulation in Vector Notation 9

Quadratic Placement: Example Given:  Fixed blocks: p1 (100,175) p2 (200,225)  Free blocks: a, b, c  nets N1-N4. −N1 (P1,a) −N2 (a,b) −N3 (b,c) −N4 (c,P2) Task:  Find (xa, ya), (xb, yb) and (xc, yc). 10

Quadratic Placement: Example Solution for x-coordinates: 11 AX = b x

Quadratic Placement: Example  Solve for X: 12

Quadratic Placement: Example Solution for y-coordinates: 13 AY = b y

Quadratic Placement: Example  Solve for Y: 14  May need detailed placement and legalization

15 Second stage of quadratic placers:  Cells are spread out to remove overlaps Methods:  Adding fake nets that pull cells away from dense regions toward anchors  Geometric sorting and scaling  Repulsion forces, etc. Quadratic Placement

Advantages:  Captures the placement problem concisely in mathematical terms  Leverages efficient algorithms from numerical analysis and available software  Can be applied to large circuits without netlist clustering (flat)  Stability: small changes in the input do not lead to large changes in the output Disadvantages:  Connections to fixed objects are necessary: I/O pads, pins of fixed macros, etc. 16

GORDIAN Global Optimization and Rectangle Dissection A Min-Cut Placement tool:  Places the cells by performing global optimization −Formulated as a quadratic program  Uses FM to improve partitioning 17

Partitioning  Finds a good cut direction and position  Improves the cut value using FM 18

 Before the next level of partitioning, does the global optimization with additional constraints −the center of gravity of each partition is at the center of the region.  Always solves a single QP, i.e., global Applying the Idea Recursively Center of Gravity 19

Force-Directed Placement 20

Force-Directed Placement Reducing the placement problem to solving a set of simultaneous linear equations to determine equilibrium locations for cells. Analogy to Hooke's law: F = kd, F: force, k: spring constant, d: distance Goal: Map cells to the layout surface. 21

Zero-Force Target Location Cell i connects to several cells j's at distances d ij 's by wires of weights w ij 's. Total force: F i =  j w ij d ij The zero-force target (ZFT) locations (x o,y o ): 22

Zero-Force Target Location 23

Force-Directed Placement 24 Can be iterative:

Force-Directed Placement 25 Constructive Iterative

26 Advantages:  Conceptually simple, easy to implement  Primarily intended for global placement, but can also be adapted to detailed placement Disadvantages:  Does not scale to large placement instances  Is not very effective in spreading cells in densest regions  Poor trade-off between solution quality and runtime  In practice, FDP is extended by specialized techniques for cell spreading Force-Directed Placement

Branch and Bound Technique 27

Branch and Bound Method Select a valid solution: Branch. Stop searching a branch if the cost so far > the previously searched branches: Bound (pruning the decision tree). 28

Branch and Bound Method Gate array placement: Given a netlist, map blocks {B1, B2, B3}  slots {S1, S2, S3}. B 1 -S 1 B 1 -S 2 B 1 -S 3 B 2 -S 2 B 2 -S 3 B 2 -S 1 B 2 -S 3 B 3 -S B 3 -S Tree pruning B 3 -S

Branch and Bound Method 30