SCHEDULING

1
SCHEDULING

• SOURCES-
• Mark Manwaring
• Kia Bazargan
• Giovanni De Micheli
• Gupta
• Youn-Long Lin

M. Balakrishnan Camposano, J. Hofstede,
Knapp, MacMillenLin
2
Overview of Hardware Synthesis
assign times to operations under given constraints
assign operations to physical resources under
given constraints

• reduce the amount of hardware, optimize the
  design in general.
• May be done with the consideration of additional
  constraints.

3
Outline of scheduling

• The scheduling problem.
• 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
4
What is Scheduling
5
Example of schedulingASAP
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.
6
(No Transcript)
7
Example of Scheduling
Adder number 1
Multiplier number 2
Adder number 3
8
More complex example of Scheduling
9
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?
10

• Scheduling affects
• Area maximum number of concurrent operations of
  same type is a lower bound on required hardware
  resources.
• Performance concurrency of resulting
  implementation.

11
Taxonomy of scheduling problems to solve

• 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.

12

• 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.

13
Scheduling 2
What are types of Scheduling 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

• Time Resource Tradeoff
• Scheduling is temporal binding

14
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

15
Classification of Scheduling Algorithms
16

• Scheduling Algorithms

17
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.

18
Minimum Latency Unconstrained Scheduling
19
Minimum-Latency UnconstrainedScheduling Problem
Minimum Latency Unconstrained Scheduling Problem

• Unconstrained scheduling used when
• 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.

20
Minimum-latency unconstrainedscheduling problem
t j

• Given a set of operations V with set of
  corresponding integer delays D and a partial
  order on the operations E
• Find an integer labeling of the operations
• ? V --gt Z , such that
• t i ? (v i ),
• t i ? t j d j ? i, j such that (v j , v
  i ) ? E
• and tn is minimum.

t i
dj
(v j , v i )
di
t i ? t j d j
Input to di must be stable
21
Example of using mobility

• 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 .
• Operations with mobility two
• v 8 , v 9 , v 10 , v 11 .

mobility two
1. Start from ALAP 2.Use mobility to improve
22
Operation Scheduling Formalisation
Classes of scheduling algorithms
23
(No Transcript)
24

• Time Constrained Scheduling

25
Latency-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.

26
Time Constrained Scheduling
27
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.
• 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

28
Scheduling under Detailed TimingConstraints
Scheduling under Detailed Timing Constraints

• Example
• 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.

29
Constraint Graph Model
Mul delay 2 ADD delay 1
30
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.

31
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 gt t j ? t i
  lij .
• Add backward edges for maximum constraints.
• Edge (vi , vj) with weight
• - u ij gt t j ? t i uij
• because t j ? t i uij gt t i ? t j - uij

Add this edge for min constraint
Add this edge for max constraint
32
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.

33
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

34
Example of using constraint graph with minimum
and maximum constraints
So now we can calculate this table from
sequencing graph
explain
Sequencing graph
Constraint graph
35
Methods for Schedulingunder Detailed Timing
Constraints
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.

36
Methods for Scheduling Under Detailed Timing
Constraints
Shown in last slide
Will follow
37
Methods for schedulingunder 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.

38
Bellman-Fords 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 j1 i
  min s j i , (s j k w q, i ),
• if (s j1 i s j i
  ?i ) return (TRUE)
• return (FALSE)

k?i
39
Longest path problem

• Use shortest path algorithms
• by reversing signs on weights.
• Modify algorithms
• by changing min with max.
• Remarks
• Dijkstras algorithm is not relevant.
• Inconsistent problem
• Positive-weighted cycles.

40
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.

145
246
3 -12
41
Liao-Wongs algorithm

• LIAO WONG(G( V, E? F, W))
• for ( i 1 to N)
• l 1 i 0
• for ( j 1 to F 1)
• foreach vertex v i
• l j1 i
  longest path in G( V, E,W E )
• flag TRUE
• foreach edge ( v p, v q) ?F
  
• if ( l j1 q
  lt l j 1 p w p,q )
• flag
  FALSE
• E E ? (
  v 0 , v q) with weight ( l j 1 p w p,q)
• 
  
• 
  
• if ( flag ) return (TRUE)
  
• return (FALSE)

adjust
42
Example Liao-Wong
(b) adjusted by adding longest path from node 0
to node 2
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.

43
Method for schedulingwith 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.

44
Relative scheduling method
45
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.

46
Relative scheduling under timingconstraints

• Problem definition
• Detailed timing constraints.
• Unbounded delay operations.
• Solution
• May or may not exist.
• Problem may be ill-specified.

47
Relative scheduling under timingconstraints

• 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.

48
Example of Relative scheduling under timing
constraints
(a) (b) Ill-posed constraint
(c) well-posed constraint
49
Relative scheduling approach

• Analyze graph
• Detect anchors.
• Well-posedness test.
• Determine dependencies from anchors.
• Schedule operations with respect to relevant
  anchors
• Bellman-Ford, Liao-Wong, Ku algorithms.
• Combine schedules to determine start times

50
Example of Relative scheduling
51
Example of control-unit synthesized for Relative
scheduling
52
(No Transcript)
53
Operations must be serialized
54
Scheduling and Binding
55
(No Transcript)
56
binding
schedule
57
Scheduling Problem formalization
Scheduling Problem
58
Various Operator types in Scheduling
59
ASAP Scheduling
60
ASAP Scheduling
61
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
62
Example - ASAP

• Solution
• Multipliers 4
• ALUs 2

Latency Time4
63
ALAP Scheduling
64
ALAP Scheduling
successor
65
ALAP scheduling algorithm

• As Late as Possible - ALAP
• Similar to depth-first search

66
Example ALAP

• Solution
• multipliers 2
• ALUs 3

Latency Time4
67
Remarks on ALAP and mobility

• ALAP solves a latency-constrained problem.
• Latency bound can be set to latency computed by
  ASAP algorithm. lt-- 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.

68
Resource Constrained
69
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.

70
Minimum latency resource-constrainedscheduling
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
  --gt Z such that
• t i '(v i ),
• t i t j d j 8 i j st (v j v i ) 2 E,
• jfv i jT (v i ) k and t i l lt t i d i gj a k
• and tn is minimum.

71

• Number of operations of any given type in any
  schedule step does not exceed bound.

V? 1,2, nres
72
Scheduling under resource constraints

• Intractable problem.
• Algorithms
• Exact
• Integer linear program.
• Hu (restrictive assumptions).
• Approximate
• List scheduling.
• Force-directed scheduling.

73
Resource Constraint Scheduling
ML-RCS minimize latency, bound on resources
MR-LCS minimize resources, bound on latency
74
Algorithm of Hu for Resource Constraint Scheduling
ML-RCS minimize latency, bound on resources
75
Example of using Hus algorithm
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.

76
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.

77

• 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 .

Algorithm Hu's schedule with a resources
We always select 3 resources
78

• 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 .

Algorithm Hu's schedule with a resources
We always select 3 resources
79
Example of Hus Algorithm
Distance from sink
80
Example of Hus Algorithm
81
(No Transcript)
82
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.

83
LIST scheduling
84
List scheduling idea
85
Example
DFG
List Schedule
86
Example

• 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

87
List Scheduling Algorithms
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.

88
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).

89
LIST scheduling
90
List scheduling algorithmfor minimum resource
Usage
91
List Scheduling Algorithm for Minimum Latency
List scheduling algorithmfor minimum latency
for resource bound

• Candidate Operations Ul,k
• Operations of type k whose predecessors are
  scheduled and completed at time step before l
• Unfinished operations Tl,k are operations of type
  k that started at earlier cycles and whose
  execution is not finished at time l
• Note that when execution delays are 1, Tl,k is
  empty.

92
List scheduling algorithmfor 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 )

93
1. List scheduling algorithmfor minimum latency
for resource bound
List scheduling algorithmfor minimum latency

• Assumptions
• a 1 3 multipliers with delay 2.
• a 2 1 ALUs with delay 1.

Now we need two time units for multiplier
94
Example

• Assumptions
• a1 3 multipliers with delay 2.
• a2 1 ALU with delay 1.

95

• Assume ?4
• Let a 1, 1T
• First Step
• U1,1 v1, v2, v6, v8
• Operations with zero slack v1, v2
• a 2, 1T
• 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, 2T

96
1. List scheduling algorithmfor 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

97
Compute the latest possible start times tL by
ALAP ( G(V,E), ?)
2. List scheduling algorithm for minimum resource
usage
98
2. Example List scheduling algorithm for minimum
resource usage

• Solution
• 2 multipliers
• 1 ALU
• LATENCY 4

We can move them
Now we assume the same time of each operation
From ALAP

• overlap

99
List Scheduling Algorithm example ML-RCS
Assume 3 multipliers
Assume 1 ALU







lt

-
Other way of explanation of the same as in last
slide
-
t0mul 3m 1 ALU
t1 1 ALU
t2 3mul
t3 1 ALU
t4 1 ALU
t5 1 ALU
Now we assume the same time of each operation
100
List Scheduling Algorithm example ML-RCS
101
Generic List Scheduling
102
List Scheduling Example

• The scheduled DFG

• DFG with mobility labeling (inside ltgt)
• ready operation list/resource constraint

103
Static-List Scheduling

• DFG
• Partial schedule of five nodes

• Priority list

The final schedule
104
Classification of Scheduling Approaches
105
Classification of Scheduling Approaches
106
Classification of Scheduling Approaches
107
(No Transcript)
108
(No Transcript)
109
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.

110
Example of Scheduling with chaining

• Cycle-time 60.

111
Summary

• Scheduling determines area/latency trade-off.
• Intractable problem in general
• Heuristic algorithms.
• ILP formulation (small-case problems).
• Chaining
• Incorporate cycle-time considerations.

112
(No Transcript)
113
Other scheduling approaches
114
Three variants of Scheduling
115
Finite Impulse Response Filter
Example 1
This can be directly used for synthesis with 11
multipliers, 10 adders and 10 registers. But the
latency would be 1 multiplier 10 adders
116
FIR Scheduling
2 multipliers
1 adder, 1 multiplier
This is also bad as we use both multipliers only
in stage 1
There are many ways to solve this problem,
transform the tree, schedule, allocate, pipeline,
retime
117
Example Cascade Filter optimization
Example 2
Look to ci coefficients and compare with two
previous slides.
This is not FIR, six coefficients
118
Cascade Filter Scheduling, cont example 2
2 mul, 2 add
119
Infinite Impulse Response Filter
Example 3
IIR Filter, 10 coefficients
120
IIR Filter Scheduling continued
IIR Filter, 10 coefficients
Two adders two multipliers
121
(No Transcript)
122
Minimize Resources Subject to bound on Latency
MR-LCS