SCHEDULING
SOURCESMark Manwaring
Kia Bazargan
Giovanni De Micheli
Gupta
Youn-Long Lin
M. Balakrishnan
Camposano,
J. Hofstede,
Knapp,
MacMillen
Lin
Overview of Hardware Synthesis
assign operations to
physical resources
under given
constraints
assign times to
operations under
given constraints
reduce the amount of hardware, optimize the design in
general.
May be done with the consideration of additional constraints.
Outline of scheduling
• The scheduling problem.
1. Scheduling without constraints.
2. Scheduling under timing constraints.
• Relative scheduling.
3. Scheduling under resource constraints.
• The ILP model (Integer Linear Programming).
• Heuristic methods (graph coloring, etc).
Timing constraints versus resource constraints
What is
Scheduling
Example of scheduling:
ASAP
This is As Soon as
Possible Scheduling
(ASAP).
It can be used as a
bound in other
methods like ILP or
when latency only is
important, not area.
Example of Scheduling
Adder number
1
Multiplier
number 2
Adder number
3
More complex example of
Scheduling
What is necessary to solve the
scheduling problem?
• Circuit model:
• Sequencing graph.
• Cycle-time is given.
• Operation delays expressed in cycles.
• Scheduling:
• Determine the start times for the operations.
• Satisfying all the sequencing (timing and resource) constraints.
• Goal:
• Determine area/latency trade-off.
Do you remember what is
latency?
• Scheduling affects
• Area: maximum number of concurrent
operations of same type is a lower
bound on required hardware resources.
• Performance: concurrency of resulting
implementation.
Taxonomy of scheduling
problems to solve
1. Unconstrained scheduling.
2. Scheduling with timing constraints:
•
•
Latency.
Detailed timing constraints.
3. Scheduling with resource constraints.
•
Related problems:
•
•
•
Chaining. What is chaining?
Synchronization. What is synchronization?
Pipeline scheduling.
• Simplest scheduling model
• All operations have bounded delays.
• All delays are in cycles.
• Cycle-time is given.
• No constraints - no bounds on area.
• Goal
• Minimize latency.
What areScheduling
types of Scheduling
2
Algorithms?
Scheduling problems are NP-hard, so all kind of heuristics are used
•
•
•
•
•
•
•
•
•
•
•
ASAP – As soon as possible
ALAP
List scheduling – Resource Constrained algorithms
Force directed algorithms – time constrained
Path based
Percolation algorithms
Simulated annealing
Tabu search and other heuristics
Simulated evolution
Linear Programming
Integer Linear Programming - time constrained
Types of Scheduling Processes
Assignment of operations to time (control steps)
within given constraints and minimizing a cost
function
• Time-constrained
• Resource-constrained
• With or without control (conditions)
• With or without iterations (infinite loops)
• Constructive
• Iterative
Classification of Scheduling
Algorithms
Scheduling
Algorithms
Simplest model of
scheduling
• All operations have bounded delays.
• All delays are expressed in numbers of cycles
of a single one-phase clock.
• Cycle-time is given.
• No constraints - no bounds on area.
• Goal:
• Minimize latency.
Minimum Latency
Unconstrained Scheduling
Minimum Latency Unconstrained
• Unconstrained
scheduling
used
when
Scheduling Problem
• Dedicated resources are used.
• Operations differ in type.
• Operations cost is marginal when compared to that of
steering logic, registers, wiring, and control logic.
• Binding is done before scheduling: resource conflicts
solved by serializing operations sharing same resource.
• Deriving bounds on latency for constrained problems.
Minimum-latency
unconstrained
scheduling problem
•
•
Given a set of operations V with set of
corresponding integer delays D and a partial order
on the operations E:
tj
ti
dj
Find an integer labeling of the operations
 : V --> Z + , such that:
titj +dj
•
t i =  (v i ),
•
t i  t j + d j  i, j such that (v j , v i )  E
•
and tn is minimum.
(v j , v i )
di
Input to
di must
be stable
• Operations with zero mobility:
•
•
{v 1, v 2, v 3, v 4, v 5 }.
•
They are on the critical path.
Operations with mobility one:
•
•
{v 6 , v 7 }.
mobility two:
Operations with mobility two:
•
{v 8 , v 9 , v
Example of using
mobility
10
,v
11
}.
1. Start from
ALAP
2.Use mobility to
improve
Operation Scheduling Formalisation
Classes of scheduling algorithms
Time
Constrained
Scheduling
Latency Constrained
Scheduling
• ALAP solves a latency-constrained problem.
• Latency bound can be set to latency computed by ASAP
algorithm.
• Mobility
• Defined for each operation.
• Difference between ALAP and ASAP schedule.
• Zero mobility implies that an operation can be started only at one
given time step.
• Mobility greater than 0 measures span of time interval in which
an operation may start.
• Slack on the start time.
Time Constrained
Scheduling
•
Scheduling under Detailed
Motivation Timing Constraints
• Interface design.
• Control over operation start time.
• Constraints
• Upper/lower bounds on start-time difference of any operation pair.
• Minimum timing constraints between two operations
• An operation follows another by at least a number of prescribed time
steps
• lij  0 requires tj  ti + lij
• Maximum timing constraints between two operations
• An operation follows another by at most a number of prescribed time
steps
• uij  0 requires tj  ti + uij
•
Scheduling under Detailed
Example Timing Constraints
• Circuit reads data from a bus, performs computation, writes
result back on the bus.
• Bus interface constraint: data written three cycles after read.
• Minimum and maximum constraint of 3 cycles between read and
write operations.
• Example
• Two circuits required to communicate simultaneously to external
circuits.
• Cycle in which data available is irrelevant.
• Minimum and maximum timing constraint of zero cycles between
two write operations.
Mul delay = 2
ADD delay =1
Scheduling under detailed
timing constraints
• Motivation:
• Interface design.
• Control over
operation start time.
•
Constraints:
• Upper/lower bounds on start-time
difference of any operation pair.
•
Feasibility of a solution.
Constraint graph model
•
Start from a sequencing graph.
•
Model delays as weights on edges.
•
Add forward edges for minimum constraints.
•
•
Edge (vi , vj) with weight lij => t j  t i +lij .
Add backward edges for maximum constraints.
•
Edge (vi , vj) with weight:
•
- u ij => t j  t i + uij
•
because t j  t i + uij => t i  t j - uij
Add this edge for
min constraint
Add this edge for
max constraint
Method for Scheduling with
Unbounded-Delay Operations
• Unbounded delays
• Synchronization.
• Unbounded-delay operations (e.g.
loops).
• Anchors.
• Unbounded-delay operations.
• Relative scheduling
• Schedule ops w.r. to the anchors.
• Combine schedules.
Method for Scheduling Under
Detailed Timing Constraints
• Presence of maximum timing constraints may
prevent existence of a consistent schedule.
• Required upper bound on time distance between
operations may be inconsistent with first
operation execution time.
• Minimum timing constraints may conflict with
maximum timing constraints.
• A criterion to determine existence of a schedule:
• For each maximum timing constraint uij
• Longest weighted path between vi and vj must
be  uij
So now we can calculate
this table from
sequencing graph
Example of using
constraint graph with
minimum and maximum
constraints
explain
Sequencing graph
Constraint graph
Method for Scheduling Under
Detailed Timing Constraints
• Weight of longest path from source to a vertex is the
minimum start time of a vertex.
• Bellman-Ford or Lia-Wong algorithm provides the
schedule.
• A necessary condition for existence of a schedule is
constraint graph has no positive cycles.
Methods for Scheduling
Under Detailed Timing
Constraints
Shown
in last
slide
Will
follow
Methods for scheduling
under detailed timing constraints
• Start from the Sequencing Graph. Assumption:
• All delays are fixed and known.
• Set of linear inequalities.
• Longest path problem.
• Algorithms for the longest path problem were
discussed in Chapter 2 of De Micheli:
• Bellman-Ford,
• Liao-Wong.
Bellman-Ford’s
algorithm
BELLMAN_FORD(G(V, E, W))
{
s 1 0 = 0;
for (i = 1 to N)
s
1
i
=w
0, i
;
for (j =1 to N){
for (i =1 to N){
s
j+1
i
= min { s
j
i
, (s
j
k
+w
q, i
}
if (s
j+1
i
== s j
}
return (FALSE)
}
i
i ) return (TRUE) ;
ki
)},
Longest path problem
• Use shortest path algorithms:
• by reversing signs on weights.
• Modify algorithms:
• by changing min with max.
• Remarks:
• Dijkstra’s algorithm is not relevant.
• Inconsistent problem:
• Positive-weighted cycles.
Example – Bellman-Ford
Use shortest path algorithms:
by reversing signs on
weights.
source
• Iteration 1: l 0 =0, l 1 =3, l 2 = 1, l 3 =.
• Iteration 2: l 0 =0, l 1 =3, l 2 =2, l 3 =5.
• Iteration 3: l 0 =0, l 1 =3, l 2 =2, l 3 =6.
1+4=5
2+4=6
3 -1=2
LIAO WONG(G( V, E F, W))
{
Liao-Wong’s
algorithm
for ( i = 1 to N)
l
1
i
= 0;
for ( j = 1 to |F|+ 1) {
foreach vertex v i
l
j+1
i
= longest path in G( V, E,W
E
);
flag = TRUE;
foreach edge ( v p, v q) F {
if ( l
j+1
q
< l
j+ 1
p
+w
p,q
){
flag = FALSE;
E = E  ( v 0 , v q) with weight ( l
}
}
if ( flag ) return (TRUE) ;
}
return (FALSE)
}
j+ 1
p
+w
p,q)
adjust
Example – Liao-Wong
Looking for longest path
from node 0 to node 3
Only positive
edges from (a)
• Iteration 1: l 0 = 0, l 1 = 3, l 2 = 1, l 3 = 5.
• Adjust: add edge (v 0 , v 2 ) with weight 2.
• Iteration 2: l 0 = 0, l 1 = 3, l 2 = 2, l 3 = 6.
(b) adjusted by
adding longest
path from node
0 to node 2
Method for scheduling
with unbounded-delay operations
• Unbounded delays:
• Synchronization.
• Unbounded-delay operations (e.g. loops).
• Anchors.
• Unbounded-delay operations.
• Relative scheduling:
• Schedule operations with respect to the anchors.
• Combine schedules.
Relative
scheduling
method
Relative scheduling
method
• For each vertex:
• Determine relevant anchor set R(.).
• Anchors affecting start time.
• Determine time offset from anchors.
• Start-time:
• Expressed by:
• Computed only at run-time because delays of anchors are unknown.
timing
constraints
Relative scheduling under
• Problem definition:
• Detailed timing constraints.
• Unbounded delay operations.
• Solution:
• May or may not exist.
• Problem may be ill-specified.
Relative scheduling under timing
constraints
• Feasible problem:
• A solution exists when unknown delays are zero.
• Well-posed problem:
• A solution exists for any value of the unknown delays.
• Theorem:
• A constraint graph can be made well-posed iff there
are no cycles with unbounded weights.
Example of Relative scheduling under timing constraints
(a) & (b) Ill-posed constraint (c) well-posed constraint
Relative scheduling approach
1. Analyze graph:
1. Detect anchors.
2. Well-posedness test.
3. Determine dependencies from anchors.
2. Schedule operations with respect to relevant anchors:
•
Bellman-Ford, Liao-Wong, Ku algorithms.
3. Combine schedules to determine start times:
Example of Relative
scheduling
Example of control-unit synthesized for
Relative scheduling
Operations must be serialized
Scheduling and Binding
schedule
binding
Scheduling Problem
Scheduling
formalization
Problem
Various Operator
types in Scheduling
ASAP
Scheduling
ASAP Scheduling
ASAP scheduling algorithm
ASAP ( Gs(V, E)){
Schedule v0 by setting t S 0 = 1;
repeat {
Select a vertex vi whose predecessors are all scheduled;
Schedule vi by setting t
S
i
= max t
S
j
+ dj ;
}
until (vn is scheduled) ;
return (t S );
}
j:(vj,vi)E
Similar to breadth-first search
Example - ASAP
• Solution
• Multipliers = 4
• ALUs = 2
Latency Time=4
ALAP
Scheduling
ALAP Scheduling
successor
ALAP scheduling algorithm
• As Late as Possible - ALAP
• Similar to depth-first search
Example ALAP
• Solution
• multipliers =
2
• ALUs = 3
Latency Time=4
Remarks on ALAP and mobility
• ALAP solves a latency-constrained problem.
• Latency bound can be set to latency computed by
ASAP algorithm. <-- using bounds, also in other approaches
• Mobility:
• Mobility is defined for each operation.
• Difference between ALAP and ASAP schedule.
• What is mobility?number of cycles that I can move
upwards or downwards the operation
• Slack on the start time.
Resource
Constrained
Scheduling under
resource constraints
• Classical scheduling problem.
• Fix area bound - minimize latency.
• The amount of available resources affects the
achievable latency.
• Dual problem:
• Fix latency bound - minimize resources.
• Assumption:
• All delays bounded and known.
Minimum latency resource-constrained
scheduling problem
• Given a set of operations V with integer delays D a partial
order on the operations E, and upper bounds {ak ; k = 1,
2,…,nres}:
• Find an integer labeling of the operations  : V --> Z+ such that :
• t i = '(v i ),
• t i t j +d j 8 i; j s:t: (v j ; v i ) 2 E,
• jfv i jT (v i ) = k and t i l < t i +d i gj a k
• and tn is minimum.
• Number of operations of any given type in
any schedule step does not exceed bound.
:V {1,2, …nres}
Scheduling under resource
constraints
• Intractable problem.
• Algorithms:
• Exact:
• Integer linear program.
• Hu (restrictive assumptions).
• Approximate:
• List scheduling.
• Force-directed scheduling.
Resource Constraint Scheduling
ML-RCS:
minimize latency,
bound on resources
MR-LCS:
minimize resources,
bound on latency
Algorithm of Hu for
Resource Constraint Scheduling
ML-RCS:
minimize latency,
bound on resources
Hu's algorithm
•
Additional assumptions:
•
•
Graph is a forest.
Algorithm:
•
Label vertices with distance
from sink.
• Greedy strategy.
• Exact solution.
Distance
from sink
• Assumptions:
• One resource type only.
• All operations have unit delay.
Example of using
Hu’s algorithm
Algorithm
Hu's schedule with a resources
• Set step, l = 1.
• Repeat until all operations are
scheduled:
• Select s  a resources with:
• All predecessors scheduled.
• Maximal labels.
• Schedule the s operations at step l.
• Increment step l = l +1.
•All
operations
have unit
delay.
•All
operations
have the
same type.
• Minimum latency with a = 3
resources.
•
Step 1:
• Select {v 1 , v 2 , v 6 }.
•
Step 2:
• Select {v 3 , v 7 , v 8 }.
•
Step 3:
• Select {v 4 , v 9 , v
•
10
Step 4:
• Select {v 5 , v
11
}.
We always select 3
resources
}.
Algorithm
Hu's schedule with a resources
• Minimum latency with a = 3
resources.
•
Step 1:
• Select {v 1 , v 2 , v 6 }.
•
Step 2:
• Select {v 3 , v 7 , v 8 }.
•
Step 3:
• Select {v 4 , v 9 , v
•
10
Step 4:
• Select {v 5 , v
11
}.
We always select 3
resources
}.
Algorithm
Hu's schedule with a resources
Example of Hu’s Algorithm
Distance
from sink
Example
of Hu’s
Algorithm
Exactness of Hu's algorithm
• Theorem1:
• Given a DAG with operations of the same type.
• a is a lower bound on the number of resources to complete a
schedule with latency  .
•  is a positive integer.
• Theorem2:
•Hu's algorithm applied to a tree with unit-cycle resources
achieves latency  with a resources.
• Corollary:
• Since a is a lower bound on the number of resources for
achieving , then  is minimum.
LIST
scheduling
List scheduling idea
Example
DFG
List Schedule
• Assumptions
• a1 = 2 multipliers with delay 1.
• a2 = 2 ALUs with delay 1.
• First Step
• U1,1 = {v1, v2, v6, v8}
• Select {v1, v2}
• U1,2 = {v10}; selected
• Second step
• U2,1 = {v3, v6, v8}
• select {v3, v6}
• U2,2 = {v11}; selected
• Third step
• U3,1 = {v7, v8}
• Select {v7, v8}
• U3,2 = {v4}; selected
• Fourth step
• U4,2 = {v5, v9}; selected
List Scheduling
Algorithms
• Priority list heuristics.
• Assign a weight to each vertex indicating
its scheduling priority
•Longest path to sink.
•Longest path to timing constraint.
List scheduling
algorithms
• Heuristic method for:
• 1. Minimum latency subject to resource bound.
• 2. Minimum resource subject to latency bound.
• Greedy strategy (like Hu's).
• General graphs (unlike Hu's).
LIST scheduling
List scheduling algorithm
for minimum resource Usage
List scheduling algorithm
for minimum latency for resource bound
• Candidate Operations Ul,k
• Operations of type k whose predecessors are scheduled and completed at
time step before l
U l ,k  {vi  V : Τype(vi )  k and t j  d j  l j : (v j , vi )  E}
• Unfinished operations Tl,k are operations of type k that started at earlier cycles
and whose execution is not finished at time l
Tl ,k  {vi  V : Τype(vi )  k and t j  d j  l j : (v j , vi )  E}
• Note that when execution delays are 1, Tl,k is empty.
List scheduling algorithm
for minimum latency for resource bound
LIST_L ( G(V, E), a ) {
l = 1;
repeat {
for each resource type k = 1, 2,…,nres {
Determine candidate operations U
l,k
;
Determine unfinished operations T l,k ;
Select Sk  U l,k vertices, such that |Sk| + |T l,k|  a k ;
Schedule the S k operations at step l;
}
l = l +1;
}
until (vn is scheduled) ;
return (t );
}
List scheduling
algorithm
1. List
scheduling
for minimum latency
algorithm
for minimum latency for resource bound
• Assumptions:
• a 1 = 3 multipliers with
delay 2.
• a 2 = 1 ALUs with delay 1.
Now we need
two time units for multiplier
• Assumptions
Example
• a1 = 3 multipliers with delay 2.
• a2 = 1 ALU with delay 1.
•
•
•
•
•
•
Assume =4
Let a = [1, 1]T
First Step
• U1,1 = {v1, v2, v6, v8}
• Operations with zero slack {v1, v2}
• a = [2, 1]T
• U1,2 = {v10}
Second step
• U2,1 = {v3, v6, v8}
• Operations with zero slack {v3, v6}
• U2,2 = {v11}
Third step
• U3,1 = {v7, v8}
• Operations with zero slack {v7, v8}
• U3,2 = {v4}
Fourth step
• U4,2 = {v5, v9}
• Both have zero slack; a = [2, 2]T
1. List scheduling algorithm
for minimum latency for resource bound
•
Assumptions:
•
•
a 1 = 3 multipliers with delay 2.
a 2 = 1 ALUs with delay 1.
•
•
•
•
Solution
3 multipliers as assumed
1 ALU as assumed
LATENCY 7
Compute the latest possible start times
tL by ALAP ( G(V,E), );
2. List scheduling algorithm
for minimum resource usage
2. Example: List scheduling algorithm for
minimum resource usage
We can move
them
Now we assume
the same time of
each operation
From ALAP
• overlap
•
•
•
•
Solution
2 multipliers
1 ALU
LATENCY 4
List Scheduling Algorithm example
Assume 1
ML-RCS
ALU
Assume 3
multipliers
*
*
*
-
*
*
+
*
+
<
Other way of explanation
of the same as in last
slide
t=0mul 3m 1 ALU
t=1
t=2 3mul
t=3 1 ALU
t=4 1 ALU
t=5 1 ALU
Now we assume the same time of
each operation
1 ALU
List
Scheduling
Algorithm
example
ML-RCS
Generic List
Scheduling
List Scheduling Example
•
DFG with mobility labeling (inside <>)
•
ready operation list/resource constraint
•
The scheduled DFG
Static-List Scheduling
•
DFG
•
Partial schedule of five nodes
•
Priority list
The final
schedule
Classification of
Scheduling Approaches
Classification of
Scheduling Approaches
Classification of Scheduling
Approaches
Scheduling with chaining
•
Consider propagation delays of resources not in terms of cycles.
•
Use scheduling to chain multiple operations in the same control step.
•
Useful technique to explore effect of cycle-time on area/latency trade-off.
•
Algorithms:
• ILP,
• ALAP/ASAP,
• List scheduling.
Example of Scheduling with chaining
• Cycle-time: 60.
Summary
• Scheduling determines area/latency trade-off.
• Intractable problem in general:
• Heuristic algorithms.
• ILP formulation (small-case problems).
• Chaining:
• Incorporate cycle-time considerations.
Other scheduling
approaches
Three variants of
Scheduling
Finite Impulse
Response Filter
Example 1
D
IN[i]
*
c0
c1
D
D
D
D
D
D
D
D
D
IN[i-1]
IN[i-2]
IN[i-3]
IN[i-4]
IN[i-5]
IN[i-6]
IN[i-7]
IN[i-8]
IN[i-9]
*
A0
c2
+
c3
B1
B0
A0
*
A1
+
*
c4
B2
A2
+
*
B1
c5
B3
A3
+
*
c6
B4
A4
+
*
c7
B5
A5
+
*
c8
B6
A6
+
*
c9
B7
A7
+
*
c10
B8
A8
+
IN[i-10]
*
B9
A9
+
OUT
This can be directly used for synthesis with 11 multipliers, 10 adders and 10
registers. But the latency would be 1 multiplier + 10 adders
IN[i]
c0
1
IN[i-1]
c1
*
*
B0
c2
2 multipliers
IN[i-2]
A0
2
+
*
B1
A1
3
+
B2
A2
4
c3 IN[i-3]
*
+
B3
A3
5
c4 IN[i-4]
*
+
B4
A4
6
1 adder, 1 multiplier
c5 IN[i-5]
*
This is also bad
as we use both
multipliers only
in stage 1
c6 IN[i-6]
+
*
B5
A5
7
FIR
Scheduling
+
c7 IN[i-7]
*
B6
c8 IN[i-8]
A6
8
+
B7
A7
9
*
+
B8
10
c9 IN[i-9]
*
A8
+
B9
A9
11
c10 IN[i-10]
+
OU
*
There are many ways
to solve this problem,
transform the tree,
schedule, allocate,
pipeline, retime
Example: Cascade Filter
optimization
Example 2
c2
*
A0[i-2]
A4
+
c1
A2
*
A5
IN[i]
c5
+
This is not FIR,
six coefficients
D
A0[i-1]
D
A0[i]
C0[i-2]
C4
c0
*
*
A1
+
+
A3
+
C5
B
+
c4
C2
D
*
c3
*
C0[i-1]
D
C0[i]
C1
+
C3
+
OUT
Look to ci coefficients and compare with two
previous slides.
c2
*
A4
+
A5
IN[i]
+
c5
A0[i-2]
c1
A2
*
D
A0[i-1]
A1
+
+
A3
D
A0[i]
C0[i-2]
C4
c0
*
*
+
C5
B
+
c4
C2
D
*
c3
*
C0[i-1]
C1
+
C3
D
C0[i]
+
OUT
c1
1
2
A0[i-1]
c2
A0[i-2]
*
*
A2
A4
+
c0 A0[i-2]
*
IN
c5C0[i-2] c4
C0[i-1]
A5
2 mul, 2 add
Cascade
Filter
Scheduling,
cont example
2
3
4
+
+
A0[i]
A3
* *
C4
+
B
5
6
c3
C0[i-1
C2
+
C0[i-2]
**
C1
C5
+
+
c0[i]
C3
+
OUT
Example 3
Infinite Impulse
Response Filter
c3
c2
*
c1
+
*
A0[i-1]
D
A4
IN
A6
D
+
A0[i]
c0
*
*
A1
A5
*
C2
+
+
A3
+
c6
C0[i-2]
*
A0[i2]
A2
c7
C4
B
c5
D
C1
c4
c9
c8
D0[i-1]
C6
*
C0[i-1]
D
+
IIR Filter, 10 coefficients
*
C5
*
+
D2
C3
C0[i]
+
D
+
*
D
D0[i]
D1
+
OUT
IIR Filter Scheduling continued
c3A0[i-2]c1
1
2
3
4
5
6
7
8
A0[i-1]
* *
A2
c2
IIR Filter, 10 coefficients
A0[i-2] c0 A0[i-1]
A6
+
A4
IN
+
A0[i]
* *
A1
c7
C0[i-2]
c5
Two adders two
multipliers
C0[i-1]
A5
+ * *
+
+ * *
+
+
+
*
+
*
+
C0[i-1]
A3
B
C2
C6
C0[i-2]
c4
C5
C4
c6
C1
c9
C3
D0[i-1]
C0[i]
D2
c8
D
D0[i]
D1
OUT
D0[i-1]
Minimize Resources Subject to
bound on Latency = MR-LCS