Slope Propagation in Static Timing Analysis

1
Slope Propagation in Static Timing Analysis

• David Blaauw, Vladimir Zolotov, Savithri
  Sundareswaran, Chanhee Oh and Rajendran Panda
• Motorola Inc. Austin, TX
• Motorola India Electronics Ltd., Bangalore,
  India
• Presented by Boaz Ribak

2
Agenda

• Timing analysis.
• Timing Graph, Critical Path, and Delay Function.
• Latest Propagation Algorithm (LPA).
• Proposed Propagation Algorithm.
• Backward Propagation of Required Times.
• Results.
• Conclusions.

3
Timing Analysis

• Dynamic simulation.
• Exercises all possible paths in a circuit.
• Good only for small circuits.
• Tends to be error prone.
• Static timing analysis.
• Large designs timing verification.
• Circuits optimization tools.
• transistor and gate sizing tools.
• Logic synthesis tools.

4
Static Timing Analysis

• Arrival time / signal
• The latest time a signal can transition at a
  node due to a signal change at a circuit input.
• 
  from input to output
• Required times
• The latest time a signal is required to
  transition at a node in order to meet performance
  constraints.
• 
  from output to input

5
Arrival time / Signal

• Crossing time the time the signal reaches the ½
  Vdd.
• Slope the total transition time.
• As signals are propagated across a gate, their
  crossing times and slopes are updated with the
  gates delay.

6
Timing graph

• A directed graph, G N, E, ns,nf , having
  exactly 1 source node and 1 sink node.
• N n1,n2,...,nk nodes in the circuit.
• E e1,e2,...,el the connections in the
  circuit between gate inputs and outputs. eij
  exists iff there is a gate in the circuit such
  that ni is connected to its input and nj to its
  output.
• ns the source node.
• nf the sink node.

nf
ns
7
Timing graph (cont)

• Each edge e in G is assigned 2 functions
• de de(sI) the signal propagation delay from a
  gates input to its output.
• se se(sI) the slope of the signal at the
  gates output.
• Both de and se are functions of sI - the gate
  input slope.
• Different input slopes in the same edge will give
  different delay and output slopes.
• Property 1 If slope saltsb then delay de(sa
  )ltde(sb )
• 
  slope se(sa )ltse(sb ).
• For any given path P in graph G we can find the
  slopes of all the nodes in the path, if the slope
  of the first node is given. sjsij(si )

8
The critical path
dij(si)

• Path delay dP
• A path having the maximum delay among all paths
  with the same ending node is called critical.
• The critical path - The critical path of the sink
  node nf.
• The delay of the timing graph, d(G) the delay
  of the critical path.
• Finding the critical path and the delay of the
  timing graph is the goal in timing analysis.

9
Finding the critical path

• Obvious approach - In a given timing graph,
  enumerate all paths from its source to sink,
  compute their delays, and select the path with
  the worst delay.
• Worst-case number of paths in a circuit is
  exponential with circuit size.
• The traditional approach Use the propagations
  of signals from the source node to the sink node.
• A signal at a node Sn n, Tn, s, P
• n the node.
• Tn the signal crossing time.
• s the signal slope at node n.
• P the signal propagation path from ns to the
  node n.

10
Latest Propagation Algorithm (LPA)

• 1. Assign to the source node n0 the signal S0
  n0 ,T0 ,s0 ,P0 where T0 0, P0 (n0 ).
• 2. Visit each node, ni, in the graph in
  topological order doing the following
• 2.1. For each edge eki with signal Sk nk ,Tk
  ,sk ,Pk
• create a new signal Ski ni ,Tki
  ,ski ,Pki where
• Tki Tk dki(sk ), ski
  ski(sk ), Pki (Pk ,ni ).
• 2.2. From all signal Ski select Slatest
  ni,Tlatest ,slatest ,Platest
  where Tlatest max(Tki ).
• 2.3. Assign the computed signal Slatest
  to node ni.

11
Latest Propagation Algorithm (LPA)
Example
S7n7,0.6,0.4,(ns,n1,n2,n7)
n7
S7-13n13,1.1,0.7,(ns,n1,n2,n7,n13)
S10-13n13,1.5,0.3,(ns,n3,n10,n13)
d7-13(0.4)0.5 s7-13(0.4)0.7
S12-13n13,1.35,0.7,(ns,n4,n8,n12,n13)
d10-13(0.2)0.6 s10-13(0.2)0.3
n10
S10n10,0.9,0.2,(ns,n3,n10)
n13
d12-13(0.8)0.8 s12-13(0.8)0.7
S12n12,0.55,0.8,(ns,n4, n8,n12)
n12
12
Latest Propagation Algorithm (LPA)

• 1. Assign to the source node n0 the signal S0
  n0 ,T0 ,s0 ,P0 where T0 0, P0 (n0 ).
• 2. Visit each node, ni, in the graph in
  topological order doing the following
• 2.1. For each edge eki with signal Sk nk ,Tk
  ,sk ,Pk
• create a new signal Ski ni ,Tki
  ,ski ,Pki where
• Tki Tk dki(sk ), ski
  ski(sk ), Pki (Pk ,ni ).
• 2.2. From all signal Ski select Slatest
  ni,Tlatest ,slatest ,Platest
  where Tlatest max(Tki ).
• 2.3. Assign the computed signal Slatest
  to node ni.

13
Latest Propagation Algorithm (LPA)
Example
S7n7,0.6,0.4,(ns,n1,n2,n7)
n7
S7-13n13,1.1,0.7,(ns,n1,n2,n7,n13)
S10-13n13,1.5,0.3,(ns,n3,n10,n13)
d7-13(0.4)0.5 s7-13(0.4)0.7
S12-13n13,1.35,0.7,(ns,n4,n8,n12,n13)
d10-13(0.2)0.6 s10-13(0.2)0.3
n10
S10n10,0.9,0.2,(ns,n3,n10)
S13n13,1.5,0.3,(ns,n3,n10,n13)
n13
d12-13(0.8)0.8 s12-13(0.8)0.7
S12n12,0.55,0.8,(ns,n4, n8,n12)
n12
14
LPA (cont)

• Run time complexity is linear with the number of
  edges in the timing graph.
• Does LPA always find the correct critical path?
• LPA is a greedy algorithm, based on the
  assumption that at each node, the arriving signal
  with the latest crossing time results in the
  longest path delay and is therefore propagated
  forward, while all earlier arriving signals are
  not.
• Is this assumption always correct?

15
LPA (cont)

• Example
• LPA The critical path starts from B. Circuit
  delay 0.82ns.
• The correct critical path starts from A. Circuit
  delay 0.95ns.

at node D
at node E
(c)
(b)
16
LPA (cont)
Example LPA results in discontinuities in the
calculated worst circuit delay with respect to
transistor and gate sizes.
Worst circuit delay as a function of the size of
gate G1.

• Crossing time of signal B at node D becomes
  earlier than A.
• LPA switches from signal B to A.
• The slope used for G3 delay changes from 0.1ns to
  1.36
• A sudden increase in G3s delay, and in the worst
  circuit delay.

17
LPA - Conclusions

• LPA may identify the incorrect critical path and
  report an optimistic delay for the circuit.
• The circuit delay calculated by LPA is a
  discontinuous function with respect to transistor
  and gate sizes.
• severe problem for efficient gradient-based
  optimization methods.

18
Proposed Propagation Algorithm

• 1. Assign to the source node n0 the signal set
  C0 S0,
• where S0 n0 ,T0 ,s0 ,P0, T0 0, P0 (n0).
• 2. Visit each node, ni, in topological order and
  compute its
• signal set CkSk1 , Sk2 ,... as follows
• 2.1. For each edge eki with signal set
  Ck Sk1 , Sk2 ,...
• create a new signal set CkiSki1
  , Ski2 ,..., with
• Skij (ni ,Tkij ,skij ,Pki ),
  where
• Tkij Tkj dki(skj ), skij
  ski(skj ), Pki (Pk,ni ).
• 2.2. Assign to node ni signal set Ci
  the union of signal
• sets Cki.
• 2.3. Remove from Ci any signal Sij if Ci
  has another
• signal Sik such as TijltTik and
  sijltsik.

19
Proposed Algorithm (cont)
Example
S7-1n7,0.6,0.4,(ns,n1,n7)
C7
S7-2n7,0.3,0.6,(ns,n2,n7)
n7
S7-13-1n13,1.1,0.7,(ns,n1,n7,n13)
d7-13 (0.4)0.5
C7-13
S7-13-2n13,0.9,1,(ns,n2,n7,n13)
d7-13 (0.6)0.6
s7-13 (0.4)0.7
S12-13-1n13,1.6,0.1,(ns,n3,n10,n13)
s7-13 (0.6)1
C12-13
S12-13-2n13,1.15,0.8,(ns,n12,n13)
d12-13 (0.2)0.7
d12-13 (0.8)0.6
s12-13 (0.2)0.1
n13
s12-13 (0.8)0.8
S12-1n12,0.9,0.2,(ns,n3,n10)
C12
n12
S12-2n12,0.55,0.8,(ns,n12)
20
Proposed Propagation Algorithm

• 1. Assign to the source node n0 the signal set
  C0 S0,
• where S0 n0 ,T0 ,s0 ,P0, T0 0, P0 (n0).
• 2. Visit each node, ni, in topological order and
  compute its
• signal set CkSk1 , Sk2 ,... as follows
• 2.1. For each edge eki with signal set
  Ck Sk1 , Sk2 ,...
• create a new signal set CkiSki1
  , Ski2 ,..., with
• Skij (ni ,Tkij ,skij ,Pki ),
  where
• Tkij Tkj dki(skj ), skij
  ski(skj ), Pki (Pk,ni ).
• 2.2. Assign to node ni signal set Ci
  the union of signal
• sets Cki.
• 2.3. Remove from Ci any signal Sij if Ci
  has another
• signal Sik such as TijltTik and
  sijltsik.

21
Proposed Algorithm (cont)
Example
S7-1n7,0.6,0.4,(ns,n1,n7)
C7
S7-2n7,0.3,0.6,(ns,n2,n7)
n7
d7-13 (0.4)0.5
S13-1n13,1.1,0.7,(ns,n1,n7,n13)
d7-13 (0.6)0.6
S13-2n13,0.9,1,(ns,n2,n7,n13)
s7-13 (0.4)0.7
C13
S13-3n13,1.6,0.1,(ns,n3,n10,n13)
s7-13 (0.6)1
S13-4n13,1.15,0.8,(ns,n12,n13)
d12-13 (0.2)0.7
d12-13 (0.8)0.6
s12-13 (0.2)0.1
n13
s12-13 (0.8)0.8
S12-1n12,0.9,0.2,(ns,n3,n10)
C12
n12
S12-2n12,0.55,0.8,(ns,n12)
22
Proposed Propagation Algorithm

• 1. Assign to the source node n0 the signal set
  C0 S0,
• where S0 n0 ,T0 ,s0 ,P0, T0 0, P0 (n0).
• 2. Visit each node, ni, in topological order and
  compute its
• signal set CkSk1 , Sk2 ,... as follows
• 2.1. For each edge eki with signal set
  Ck Sk1 , Sk2 ,...
• create a new signal set CkiSki1
  , Ski2 ,..., with
• Skij (ni ,Tkij ,skij ,Pki ),
  where
• Tkij Tkj dki(skj ), skij
  ski(skj ), Pki (Pk,ni ).
• 2.2. Assign to node ni signal set Ci
  the union of signal
• sets Cki.
• 2.3. Remove from Ci any signal Sij if Ci
  has another
• signal Sik such as TijltTik and
  sijltsik.

23
Proposed Algorithm (cont)
Example
S7-1n7,0.6,0.4,(ns,n1,n7)
C7
S7-2n7,0.3,0.6,(ns,n2,n7)
n7
T13-1ltT13-4 s13-1lts13-4 Signal S13-1 is
removed from C13.
d7-13 (0.4)0.5
d7-13 (0.6)0.6
s7-13 (0.4)0.7
s7-13 (0.6)1
S13-1n13,1.1,0.7,(ns,n1,n7,n13)
S13-2n13,0.9,1,(ns,n2,n7,n13)
C13
S13-3n13,1.6,0.1,(ns,n3,n10,n13)
d12-13 (0.2)0.7
S13-4n13,1.15,0.8,(ns,n12,n13)
d12-13 (0.8)0.6
s12-13 (0.2)0.1
n13
s12-13 (0.8)0.8
S12-1n12,0.9,0.2,(ns,n3,n10)
C12
n12
S12-2n12,0.55,0.8,(ns,n12)
24
Proposed Algorithm (cont)

• Differences from LPA
• All the signals that might have the longest delay
  are propagated (in LPA just 1 signal is
  propagated).
• The algorithm also checks signals slopes (in LPA
  just crossing times)
• Only signals that are clearly not the latest are
  pruned (in LPA all signals except the chosen
  one are pruned).

25
Proposed Algorithm (cont)

• The proposed algorithm can be shown to be
  correct, since it has 2 properties
• A signal that could be critical is never pruned.
• Since signal Sij is pruned only if there exists
  another signal Sik such that TijltTik and sijltsik,
  and since all signals propagate through the same
  edges after the node ni ,it is clear that Sij
  will always be earlier than Sik
• All potentially critical signals are propagated
  to the sink node, where the critical path and
  graph delay are determined by identifying the
  latest signal.

26
Proposed Algorithm (cont)

• The timing graph delay of the proposed algorithm
  is a continuous function if the edge delay or
  slope function are continuous functions of the
  chosen parameters.
• If there is no information available regarding
  the timing graph beyond node ni ,it is necessary
  for the algorithm to propagate all selected
  signals. It is impossible to reduce the number of
  propagated signals (except for in digital
  circuits).

27
Backward Propagation of Required Times

• Optimization algorithms require node slacks (the
  difference between the required time and the
  crossing time of a signal) for all nodes.
• As required times are propagated backward across
  edges, their crossing times are decreased by the
  edge delay.
• The arrival signal and associated require times
  are updated with the same edge delays as they are
  propagated, and remain consistent.

28
Backward Propagation (cont)

• Slack along the sub-critical path must always be
  higher than the slack of the critical path at
  node ni.
• If a signal Sia is pruned at node ni, a new
  required time must be created.
• Solution select from among the propagated
  signals at node ni, the signal Sip with a slope
  that has the least slope greater than Sia, and
  propagate backward its required time as Sias
  required time.

• The new required time will overestimate the
  delay between ni and nf (the new required time
  will be earlier than the exact required time).

i

• The over-estimated slack on
• sias path will guarantee the
• higher slack rule.

29
Results
Average error in LPA 19
30
Results (cont)
31
Results (cont)

• The area/delay trade-off produced during the
  optimization of circuit mux.
• The trade-off curve produced by LPA suffers
  discotinuances.
• The trade-off curve produced by the proposed
  algorithm is free of such discotinuances.

32
Conclusions

• LPA can significantly underestimate the delay of
  a circuit (in the results - as much as 38).
• LPA can lead to discontinuities in the circuit
  delay as a function of its transistor sizes.
• The presented algorithm proved to correctly
  calculate the critical path in a circuit and its
  circuit delay.
• This algorithm propagates the minimum number of
  possible arrival signals for a general timing
  graph.
• In digital logic gates, this number can be
  reduced.
• The algorithm increases the run time of timing
  analysis.
• An algorithm was presented for propagating the
  required times in a manner consistent with the
  arrival signal propagation for calculation of
  node slacks at all circuit nodes.

33
Backup
34
The continuous function of the delay

• Theorem 1 If the edge delay and slope functions
  are continuous with respect to some parameters,
  then the timing graph delay is also continuous
  with respect to these parameters.
• Proof The delay of each path in a graph is a
  finite sum of finite compositions of delay and
  slope functions of individual edges.
• The path delay is a continuous function with
  respect to the parameters of the slope and delay
  functions.
• The graph delay is the maximum of all path delays
  in the timing graph and is therefore also
  continuous, since the maximum operation is a
  continuous function.

35
LPA (cont)

• Lemma Given two signals Sa, Sb at the same node
  n, where salt sa then for any signal path Q from
  node n to nf the path delay da(Q) of signal Sa is
  always less then the path delay db(Q) of signal
  Sb.
• Proof Follows directly from Property 1 of delay
  and slope functions, the recurrent dependence of
  gate output slopes on gate input and the additive
  property of path delay.

36
LPA (cont)

• Theorem 2 If, for some node ni of timing graph
  G, the slope slatest of LPAs selected signal
  Slatest is less than the slope sk of another
  signal Sik ni ,Tk ,sk ,Pk propagated to ni,
  then we can construct a new graph H, containing
  all nodes and edges in G that have already been
  visited by LPA, such that in H Sik is critical
  but Slatest latest is not.
• Proof Construct H such that it contains only
  the nodes from G that have been visited by LPA.
• Add a sink node nf and an edge ek (nk, nf) for
  each node nk of H that does not have an outgoing
  edge. To all edges ek assign delay functions
  di(s)0, except for ei (ni, nf).
• dmax - the maximum path delay of the set of all
  paths that dont pass through node ni. For ei
  (ni, nf) assign di(slatest ) dmax and di(sk)
  dmax Tlatest - Tk D , D gt 0.
• It is clear that signal Sik will arrive at nf
  later than signal Slatest.

37
LPA (cont)

• Corollary 1 There exist timing graphs for which
  LPA computes an incorrect critical path and
  delay.
• Theorem 3 It is possible to construct a Timing
  Graph G with edge delay functions de depending
  continuously on a certain edge parameter xe ,but
  such that the timing graph delay d(xe) computed
  by LPA is a discontinuous with respect to xe.
• Proof We use the timing graph H constructed in
  the proof of Theorem 2. Assuming that the edge
  delay function de of subpath Platest depends
  strongly and monotonically on parameter xe, we
  set xx0 such that the path delays of subpath
  Slatest and Pki are equal. Then, a small
  variation of x around x0 will result in a
  variation of the graph delay by D.

38
Proposed Algorithm (cont)

• Theorem 4 For any G and any of its nodes ni,
  any signal Sij ni ,Tij ,sij ,Pij that is
  pruned by the algorithm doesnt have the latest
  crossing time at any node nl following node ni.
• Proof Signal Sij is pruned only if there
  exists another signal Sik such that TijltTik and
  sijltsik. Since both signals propagate through the
  same edges after ni, and from the property of the
  slope function, it follows that at any node nl
  after ni the slope slj will be less than the
  slope slk. From this and the property of the edge
  delay function it follows that the edge delay
  along the path of signal Slj from ni to nl will
  always be less than the edge delay along the same
  path for signal Slk . Since the crossing time is
  the summation of edge delays from ni to nl, and
  since TijltTik at node ni , it follows that Sij
  will always be earlier than Sik .

39
Reduction in Propagated Signals in Digital
Circuits

• Property 2 For a digital gate connecting input
  node ni with output node nj , if two input signal
  waveforms Sia and Sib are related such that at
  any point along their transition Sia is earlier
  than Sib, then for all time points along the
  outputs Sja and Sjb, waveform Sja will be earlier
  than Sjb.

a digital gate can only produce an output signal
waveform Sjb that is earlier than signal Sja, if
the input signal Sib is earlier than signal Sia
on at least one time instance along its
transition.
40
Reduction in Propagated Signals in Digital
Circuits (cont)

• Finding a bound on the difference in crossing
  times at nf.
• (sib sia)/2

41
Reduction in Propagated Signals in Digital
Circuits (cont)

• Sia ni ,Tia ,sia ,Pia , Sib ni ,Tib ,sib
  ,Pib
• If Tib-Tia gt (sia-sib)/2 , Sia can be pruned from
  the propagation.
• Using this condition in step 2.3 of the algorithm
  limits the propagated signals.