1. ICS 252 Introduction to Computer Design
Fall 2007
Eli Bozorgzadeh
Computer Science Department-UCI

2. ICS 252-Intro to Computer Design
References
Lecture notes by Kia Bazargan on High level Synthesis [©Bazargan]
Lecture note by Rajesh Gupta on scheduling [©Gupta]
Chapter 4 of the textbook
Fall 2007
ICS 252-Intro to Computer Design

3. Architectural Synthesis
Abstract model  data path (structural) and control unit (logic-level)
Constructing macroscopic structure of digital circuit
Data path: interconnection of hardware resources (arithmetic or logic), steering logic circuits, registers/memory arrays
Objectives:
Area, latency, clock cycle time, throughput ( for pipelined data path)
Fall 2007
ICS 252-Intro to Computer Design

4. Design specification for architectural synthesis
Resources
Functional resources
Primitive
Application-specific
Memory
Examples: registers, ROM, RAM
Interface
Internal: buses
External: I/O pads
Fall 2007
ICS 252-Intro to Computer Design

5. Design specification for architectural synthesis
Constraints
Interface constraints
Format and timing of the I/O data transfers
Implementation constraints
Area
Latency, cycle time
Fall 2007
ICS 252-Intro to Computer Design

6. Fundamental architectural synthesis problems
Input:
sequencing graph
a set of functional units
a set of constraints
Assumption:
Storage by registers
Interconnection by wires
Non-hierarchical graph with operations having bounded and known execution delay
Two tasks:
Placing operations in time and space
Determining the detailed interconnection of data path and control unit
Fall 2007
ICS 252-Intro to Computer Design

7. Temporal Domain: Scheduling
Given
Sequencing graph
Operations D={di,i=0,1,2,…,n}
Start time at each operation T={ti, i=1,2…,n}
Scheduling: task of determining the start time of each operation
Subject to
Precedence constraint by sequencing graph
Fall 2007
ICS 252-Intro to Computer Design

8. ICS 252-Intro to Computer Design
Scheduling
Latency of a scheduled graph:
Difference between the start time of source and sink
NOP 10 1 2 6 8
Time 1 * * * * + 9 11 7
Time 2 3 * * +
<
Time 3 4 -
Time 4 5 -
NOP
Fall 2007
ICS 252-Intro to Computer Design

9. Scheduling under resource constraints
NOP 1 10
Time 1 * + 2
Time 2 *
<
Time 3 3 * 6
Time 4 4 - * 7
Time 5 *
Time 6 8 * - 5 9
Time 7 +
Fall 2007
ICS 252-Intro to Computer Design

10. Spatial Domain: Binding
A resource binding is a mapping such that operation vi with type T is implemented with rth resource
One-to-one function, many-to-one functions (resource sharing), one-to-many functions (module selection)
On a scheduled graph
Fall 2007
ICS 252-Intro to Computer Design

11. Resource binding on a scheduled graph
NOP 10 1 2 6 8
Time 1 * * * * + 7 9 11
Time 2 3 * * +
<
Time 3 4 -
Time 4 5 -
NOP
Fall 2007
ICS 252-Intro to Computer Design

12. Area and performance estimation
Latency
By scheduling
Area
Resource-dominated
General circuit
Registers
Steering logics
Wiring
Control unit
Fall 2007
ICS 252-Intro to Computer Design

13. ICS 252-Intro to Computer Design
Strategies
Area/latency optimization
Cycle-time/area optimization
cycle-time/latency optimization
Fall 2007
ICS 252-Intro to Computer Design

14. ICS 252-Intro to Computer Design
Scheduling
Without resource constraint
Unconstrained Scheduling: ASAP
Latency-constrained scheduling :ALAP
Scheduling under timing constraint
With resource constraint
Exact: ILP model
Heuristic: List scheduling, force-directed scheduling, etc.
Fall 2007
ICS 252-Intro to Computer Design

15. ICS 252-Intro to Computer Design
ASAP Scheduling
Schedule each operation as-soon-as-possible
NOP 10 1 2 6 8
Time 1 * * * * + 9 11 7
Time 2 3 * * +
<
Time 3 4 -
Time 4 5 -
NOP
Fall 2007
ICS 252-Intro to Computer Design

16. ICS 252-Intro to Computer Design
ALAP scheduling
Schedule each operation as late as possible under given latency
NOP 1 2
Time 1 * * 6
Time 2 3 * * 7 8 10
Time 3 4 - * * + 9 11
Time 4 5 - +
<
NOP
Fall 2007
ICS 252-Intro to Computer Design

17. ICS 252-Intro to Computer Design
Quick overview
Tasks in high-level synthesis
Scheduling √
Allocation
Binding
Scheduling
Without resource constraint √
With resource constraint (next)
ILP model: exact solution
List scheduling, force-directed scheduling: heuristic
Fall 2007
ICS 252-Intro to Computer Design

18. Resource Constraint Scheduling
Constrained scheduling
General case NP-complete
Minimize latency given constraints on area or the resources (ML-RCS)
Minimize resources subject to bound on latency (MR-LCS)
Exact solution methods:
Integer linear programming (ILP) model
Hu’s algorithm
Heuristics:
List scheduling
Force-directed scheduling
[©Bazargan]
Fall 2007
ICS 252-Intro to Computer Design

19. ILP model for scheduling problem
Binary variable xil
xil=1 if start time of operation i is at time l
xil=0 otherwise
Constraints: set of linear inequalities
Uniqueness of start time for each operation
How to compute ti , start time of operation i?
Sequencing relation
Fall 2007
ICS 252-Intro to Computer Design

20. Start Time vs. Execution Time
For each operation vi , only one start time
If di=1, then the following questions are the same:
Does operation vi start at step l?
Is operation vi running at step l?
But if di>1, then the two questions should be formulated as:
Does xil = 1 hold?
Does the following hold? ?
[©Bazargan]
Fall 2007
ICS 252-Intro to Computer Design

21. Operation vi Still Running at Step l ?
Is v9 running at step 6?
Is x9,6 + x9,5 + x9,4 = 1 ?
Note:
Only one (if any) of the above three cases can happen
To meet resource constraints, we have to ask the same question for ALL steps, and ALL operations of that type v9 4 5 6
x9,6=1 v9 4 5 6
x9,5=1 v9 4 5 6
x9,4=1
[©Bazargan]
Fall 2007
ICS 252-Intro to Computer Design

22. Operation vi Still Running at Step l ?
Is vi running at step l ?
Is xi,l + xi,l xi,l-di+1 = 1 ? vi l
l-1
l-di+1
...
xi,l=1 vi l
l-1
l-di+1
...
xi,l-1=1 vi l
l-1
l-di+1
...
xi,l-di+1=1
. . .
[©Bazargan]
Fall 2007
ICS 252-Intro to Computer Design

23. ICS 252-Intro to Computer Design
ILP Model (cont’d)
Constraints (cont’d)
Resource bound at each time step of scheduling
Objective
Minimize cTt
t = vector of start time of operations
Fall 2007
ICS 252-Intro to Computer Design

24. ICS 252-Intro to Computer Design
ILP Example
Assume l = 4
First, perform ASAP and ALAP
(we can write the ILP without ASAP and ALAP, but using ASAP and ALAP will simplify the inequalities) +
NOP 
< - 1 2 3 4 +
NOP 
< - 1 2 3 4 v1 v1 v2 v6 v8
v10 v2 v3 v7 v9
v11 v6 v3 v4 v4 v8 v7
v10 v5 v5 v9
v11 vn
[©Bazargan] vn
Fall 2007
ICS 252-Intro to Computer Design

25. ILP Example: Unique Start Times Constraint
Without using ASAP and ALAP values:
Using ASAP and ALAP:
[©Bazargan]
Fall 2007
ICS 252-Intro to Computer Design

26. ILP Example: Dependency Constraints
Using ASAP and ALAP, the non-trivial inequalities are: (assuming unit delay for + and *)
[©Bazargan]
Fall 2007
ICS 252-Intro to Computer Design

27. ILP Example: Resource Constraints
Resource constraints (assuming 2 adders and 2 multipliers)
Objective:
Since l=4 and sink has no mobility, any feasible solution is optimum, but we can use the following anyway:
[©Bazargan]
Fall 2007
ICS 252-Intro to Computer Design

28. ILP Formulation of MR-LCS
Dual problem to ML-RCS
Objective:
Goal is to optimize total resource usage, a.
Objective function is cTa , where entries in c are respective area costs of resources
Constraints:
Same as ML-RCS constraints, plus:
Latency constraint added:
Note: unknown ak appears in constraints.
[©Gupta]
Fall 2007
ICS 252-Intro to Computer Design

29. ICS 252-Intro to Computer Design
Hu’s Algorithm
Simple case of the scheduling problem
Operations of unit delay
Operations (and resources) of the same type
Hu’s algorithm
Greedy
Polynomial AND optimal
Computes lower bound on number of resources for a given latency OR: computes lower bound on latency subject to resource constraints
Basic idea:
Label operations based on their distances from the sink
Try to schedule nodes with higher labels first (i.e., most “critical” operations have priority)
[©Gupta]
Fall 2007
ICS 252-Intro to Computer Design

30. ICS 252-Intro to Computer Design
Hu’s Algorithm
HU (G(V,E), a) {
Label the vertices // label = length of longest path passing through the vertex
l = 1
repeat {
U = unscheduled vertices in V whose predecessors have been scheduled (or have no predecessors)
Select S  U such that |S|  a and labels in S are maximal
Schedule the S operations at step l by setting ti=l, i: vi  S.
l = l + 1
} until vn is scheduled. }
[©Bazargan]
Fall 2007
ICS 252-Intro to Computer Design

31. Hu’s Algorithm: Example
[©Gupta]
Fall 2007
ICS 252-Intro to Computer Design

32. ICS 252-Intro to Computer Design
List Scheduling
Greedy algorithm for ML-RCS and MR-LCS
Does NOT guarantee optimum solution
Similar to Hu’s algorithm
Operation selection decided by criticality
O(n) time complexity
More general input
Resource constraints on different resource types
[©Bazargan]
Fall 2007
ICS 252-Intro to Computer Design

33. List Scheduling Algorithm: ML-RCS
LIST_L (G(V,E), a) {
l = 1
repeat {
for each resource type k {
Ul,k = available vertices in V.
Tl,k = operations in progress.
Select Sk  Ul,k such that |Sk| + |Tl,k|  ak
Schedule the Sk operations at step l }
l = l + 1
} until vn is scheduled.
[©Bazargan]
Fall 2007
ICS 252-Intro to Computer Design

34. List Scheduling Example
[©Gupta]
Fall 2007
ICS 252-Intro to Computer Design

35. List Scheduling Algorithm: MR-LCS
LIST_R (G(V,E), l’) {
a = 1, l = 1
Compute the ALAP times tL.
if t0L < 0
return (not feasible)
repeat {
for each resource type k {
Ul,k = available vertices in V.
Compute the slacks { si = tiL - l,  vi Ul,k }.
Schedule operations with zero slack, update a
Schedule additional Sk  Ul,k under a constraints }
l = l + 1
} until vn is scheduled.
[©Bazargan]
Fall 2007
ICS 252-Intro to Computer Design

36. Force-Directed Scheduling
Similar to list scheduling
Can handle ML-RCS and MR-LCS
For ML-RCS, schedules step-by-step
BUT, selection of the operations tries to find the globally best set of operations
Idea:
Find the mobility mi = tiL – tiS of operations
Look at the operation type probability distributions
Try to flatten the operation type distributions
Definition: operation probability density
pi ( l ) = Pr { vi starts at step l }.
Assume uniform distribution:
[©Gupta]
Fall 2007
ICS 252-Intro to Computer Design

37. Force-Directed Scheduling: Definitions
Operation-type distribution (NOT normalized to 1)
Operation probabilities over control steps:
Distribution graph of type k over all steps:
qk ( l ) can be thought of as expected operator cost for implementing operations of type k at step l.
[©Bazargan]
Fall 2007
ICS 252-Intro to Computer Design

38. ICS 252-Intro to Computer Design
Example +
NOP 
< - 1 2 3 4 1 2
1.66
0.33
2.83
2.33
.83
[©Bazargan]
Fall 2007
ICS 252-Intro to Computer Design

39. Force-Directed Scheduling Algorithm: Idea
Very similar to LIST_L(G(V,E), a)
Compute mobility of operations using ASAP and ALAP
Computer operation probabilities and type distributions
Select and schedule operations
Update operation probabilities and type distributions
Go to next control step
Difference with list scheduling in selecting operations
Select operations with least force
Consider the effect on the type distribution
Consider the effect on successor nodes and their type distributions
[©Bazargan]
Fall 2007
ICS 252-Intro to Computer Design

40. ICS 252-Intro to Computer Design
Summary
Architectural Synthesis
Scheduling
Binding and resource allocation
Challenge: Area/performance trade-off
Without resource constraints
With resource constraints
Exact solution
Heuristics
Next: Resource sharing and binding
Fall 2007
ICS 252-Intro to Computer Design