### On Timing Closure: Buffer Insertion for Hold

```On Timing Closure: Buffer
Insertion for Hold-Violation
Removal
Pei-Ci Wu
Martin D. F. Wong
DAC’14
Outline






Introduction
Preliminaries
Linear Programming Based
Optimization
Bottom-up Buffer Insertion
Experimental Results
Concluding Remarks
Introduction


Timing closure, which is to satisfy the timing
constraints, is a key problem in the physical
design
Setup (long-path) constraints


ensure that the signal transitions do not arrive
too late
hold-time (short-path) constraints

ensure that the signal transitions do not arrive
too early

after setup optimization has been performed.

Discrete cell sizes (i.e. discrete buffer sizes
for hold optimization) in modern industrial
designs

Cell libraries specified for the setup
constraints and the hold-time constraints are
usually different in modern industrial
designs
Preliminaries

Negative setup slacks and negative hold
slacks indicate setup violations and hold
violations

TNS


THS


the absolute value of the total negative setup
slacks of all the pins in PO
the absolute value of the total negative hold
slacks of all the pins in PO
TNS must not be worsen during holdviolation removal

Given:


a design and a buffer library,
find a buffering solution such that:

THS and the cost of buffering (i.e. area and
power consumption) are both minimized while
TNS is not worsen.
Linear Programming Based
Optimization

Inserting delay into wires to remove hold
violations



A linear programming formulation
Extend such formulation for the complex timing
constraints
Graph-reduction approach

Input


Combinational circuit C* s.t. for any pin p of C*,
hold_slackp < 0 and setup_slackp > 0
C* can then be represented as a directed
acyclic graph G(V,E)


V is the pins of C*
(i, j) ∈ E represents an edge



I : the zero in-degree pins
O : the zero out-degree pins
for each pin i in V

three real-value variables, xi(delays inserted at
pin i for hold-time constraints), hai, and sai

Hold-time constraints

For buffer library characterization is
necessary in order to get an empirical ratio
such that

we assume that the buffer only affects the
driver cell and the sink cells of the buffer

Delays introduced by inserting the buffer is


(a)
(b)


Setup constraints

Objective :

The setup constraints limit the delays that
can be inserted

ri is only necessary when there is no
feasible solution

Some pins with positive setup slacks and
positive hold slacks that are not included
Graph Reduction

Bottom-up Buffer Insertion

Given:


a pin i, hold delay DH and setup delay DS
Find a buffering solution at pin i from a
buffer library B:



hold delays introduced by the chosen buffers are
as close to DH
setup delays introduced by the chosen buffers
are not larger than DS
Minimize the area of the chosen buffers



DP based algorithm
A set of buffering candidates C(L, dh, ds, A)
is kept during the process
For each buffer in B, we insert it to any of
the existing candidates

New buffering candidates


(1) if d′s > DS, C′ is removed immediately
(2) if d′h <= dh, C′ is removed as well



d′h > DH + margin where margin is a parameter, then C′
is removed too
(3) C′ is dominated by any existing candidate
C*(I*, d*h, d*s, A*) if d′h < d*h and A′ > A*
Chose the candidate that has the largest
ratio of dh/A as the buffering solution

Bottom-up Methodology


process the pins by the bottom-up topological
ordering (i.e. from PO to PI)
DP algorithm cannot realize the exact
amount of hold delays/setup delays by
inserting buffers(extra delays)

Suppose now we are processing pin p,
collected extra delays




cur_setup_reqp = setup_reqp − ds_delay
extra delays = cur_setup_reqp – sap
Ds = xp + cur_setup_reqp − sap
Similarly to get Dh
Optimization Flow

Experimental Results


Concluding Remarks


First propose a linear programming based
approach that minimizes the number of
inserted delays
A bottom-up buffer insertion and the flow of
optimizing are presented
```