CS137: Today Electronic Design Automation Scheduling - - PDF document

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CS137: Electronic Design Automation

Day 20: November 23, 2005 Scheduling Variants and Approaches

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Today

• Scheduling

– Force-Directed – SAT/ILP – Branch-and-Bound

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Last Time

• Resources aren’t free
• Share to reduce costs
• Schedule operations on resources
• Greedy approximation algorithm

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Force-Directed

• Problem: how exploit schedule freedom

(slack) to minimize instantaneous resources

– Directly solve time constrained – Trying to minimize resources

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Force-Directed

• Given a node, can schedule anywhere

between ASAP and ALAP schedule time

– Between latest schedule predecessor and ALAP – Between ASAP and already scheduled successors

• N.b.: Scheduling node will limit freedom
• f nodes in path

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Single Resource Challenge

A7 A8 B11 A9 B2 B3 B4 A1 A2 A3 A4 A5 A6 A10 A11 A13 A12 B5 B1 B6 B7 B8 B9 B10

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Force-Directed

• If everything where scheduled, except

for the target node, we would:

– examine resource usage in all timeslots allowed by precedence – place in timeslot which has least increase maximum resources

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Force-Directed

• Problem: don’t know resource

utilization during scheduling

• Strategy: estimate resource utilization

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Force-Directed Estimate

• Assume a node is uniformly distributed

within slack region

– between earliest and latest possible schedule time

• Use this estimate to identify most used

timeslots

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 2 3/9 2 1/9

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Force-Directed

• Scheduling a node will shift distribution

– all of scheduled node’s cost goes into one timeslot – predecessor/successors may have freedom limited so shift their contributions

• Want to shift distribution to minimize

maximum resource utilization (estimate)

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 Repeat

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 2 3/9 2 1/9 Repeat

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 3 4/9

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 3 2/9

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 2 3/9 2 1/9

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 2 3/9

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 2 3/9

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 3

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 2 13/18

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 3 2/9

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 2 13/18

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 2 13/18

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 2 13/18

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 2 13/18

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 2 13/18

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8

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Single Resource Challenge

2 2 8 2 8 8 8 2 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 Many steps…

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Force-Directed Algorithm

1. ASAP/ALAP schedule to determine range of times for each node 2. Compute estimated resource usage 3. Pick most constrained node (in largest time slot…)

– Evaluate effects of placing in feasible time slots (compute forces) – Place in minimum cost slot and update estimates – Repeat until done

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Time

• Evaluate force of putting in timeslot

O(NT)

– Potentially perturbing slack on net prefix/postfix for this node N – Each node potentially in T slots

• Evaluate all timeslots can put in O(NT2)
• N nodes to place
• O(N2T2)

– Loose bound--don’t get both T slots and N perturbations

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SAT/ILP (Integer-Linear Programming)

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Two Constraint Challenge

• Processing elements have limited

memory

– Instruction memory (data memory)

• Tasks have different requirements for

compute and instruction memory

– i.e. Run length not correlated to code length

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Task

• Task: schedule tasks onto PEs obeying

both memory and compute capacity limits

Example and ILP solution From Plishker et al. NSCD2004

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Task

• Task: schedule tasks onto PEs obeying

both memory and compute capacities

• two capacity partitioning problem

– …actually, didn’t say anything about communication…

• two capacity bin packing problem
• Task: i <Ci,Ii>

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SAT Packing

• Ai,j – task i assigned to resource j

Constraints

• Coverage constraints
• Uniqueness constraints
• Cardinality constraints

– PE compute – PE memory

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Allow Code Sharing

• Two tasks of same type can share code
• Instead of memory capacity

– Vector of memory usage

• Compute PE Imem vector

– As OR of task vectors assigned to it

• Compute mem space as sum of non-

zero vector entries

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Allow Code Sharing

• Two tasks of same type can share code
• Task has vector of memory uage

– Task i needs set of instructions k: Ti,k

• Compute PE Imem vector

– OR (all i): PE.Imemj,k+=Ai,j * Ti,k

• PE Mem space

– PE.Total_Imemj= Σ(PE.Imemj,k*Instrs(k))

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Symmetries

• As with partitioning, many symmetries
• Speedup with symmetry breaking

– Tasks in same class are equivalent – PEs indistinguishable – Total ordering on tasks and PEs – Add constraints to force tasks to be assigned to PEs by ordering – Plishker claims “significant runtime speedup” – Using GALENA [DAC 2003] psuedo-Boolean SAT solver

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Plishker Task Example

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Results

SAT/ILP Solve Greedy (first-fit) binpack Solutions in < 1 second

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Why can they do this?

• Ignore precedence?
• Ignore Interconnect?

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Why can they do this?

• Ignore precedence?

– feed forward, buffered

• Ignore Interconnect?

– Through shared memory, not dominant?

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Interconnect Buffers

• Allow “Software Pipelining”

Each data item Spatial we would pipeline, running all three at once Think of each schedule instance as one timestep in spatial pipeline.

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Interconnect Buffer

A B C 50 100 50 A B C PE0 PE1 A B C A B C A B C A B C A B C A B C A B C A B C

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Add Precedence to SAT/ILP?

• Assign start time to each task
• Precedence: constrain start of each

task to be greater than start+run of each predecessor

• Time Exclusivity: constrain non-
• verlap of startstart+run-1 on nodes
• n same PE

– Maybe formulate as order on PE – And make PE order predecessor like a task predecessor?

Untested conjecture

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Memory Schedule Variants

• Persistent: holds memory whole time

– E.g. task state, instructions

• Task temporary: only uses memory

space while task running

• Intra-Task: use memory between point
• f production and consumption

– E.g. Def-Use chains

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Memory Schedule Variants

• Persistent:

– Binpacking in memory

• Task temporary:

– Co-schedule memory slot with execution

• Intra-Task:

– Lifetime in memory depends on scheduling def and last use – Phase Ordered: Register coloring

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Branch-and-Bound

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Brute-Force

• Try all schedules
• Branching/Backtracking Search
• Start w/ nothing scheduled (ready

queue)

• At each move (branch) pick:

– available resource time slot – ready task (predecessors completed) – schedule task on resource

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Example

T1 T2 T3 T4 T5 T6 T1→time 1 T3→time 1 T2→time 1 idle→time 1 T3→time2 T4→time 2 idle→time 2 Target: 2 FUs

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Branching Search

• Explores entire state space

– finds optimum schedule

• Exponential work

– O (N(resources*time-slots) )

• Many schedules completely

uninteresting

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Reducing Work

• 1. Canonicalize “equivalent” schedule

configurations

• 2. Identify “dominating” schedule

configurations

• 3. Prune partial configurations which will

lead to worse (or unacceptable results)

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“Equivalent” Schedules

• If multiple resources of same type

– assignment of task to particular resource at a particular timeslot is not distinguishing

T1 T2 T3 T2 T1 T3

Keep track of resource usage by capacity at time-slot.

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“Equivalent” Schedule Prefixes

T1 T3 T2 T4 T1 T3 T2 T4 T1 T2 T3 T4 T5 T6

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“Non-Equivalent” Schedule Prefixes

T1 T2 T3 T2 T3 T1 T1 T2 T3 T4 T5 T6

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Pruning Prefixes?

• I’m not sure there is an efficient way

(general)?

• Keep track of schedule set

– walk through state-graph of scheduled prefixes – unfortunately, set is power-set so 2N – …but not all feasible, so shape of graph may simplify

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Dominant Schedules

• A strictly shorter schedule

– scheduling the same or more tasks – will always be superior to the longer schedule

T1 T2 T1 T2 T5 T4 T4 T5 T3 T3 T3 T2 T1 T5 T4

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Pruning

• If can establish a particular schedule

path will be worse than one we’ve already seen

– we can discard it w/out further exploration

• In particular:

– LB=current schedule time + lower_bound_estimate – if LB greater than existing solution, prune

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Pruning Techniques

Establish Lower Bound on schedule time

• Critical Path (ASAP schedule)
• Resource Bound
• Critical Chain

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“Critical Chain” Lower Bound

• Bottleneck resource present coupled

resource and latency bound

Single red resource

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“Critical Chain” Lower Bound

• Bottleneck resource present coupled

resource and latency bound

Single red resource

Critical path 5 Resource Bound (1,1) 4 Critical Chain (1,1) 7

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Alpha-Beta Search

• Generalization

– keep both upper and lower bound estimates on partial schedule

• Lower bounds from CP, RB, CC
• Upper bounds with List Scheduling

– expand most promising paths

• (least upper bound, least lower bound)

– prune based on lower bounds exceeding known upper bound – (technique typically used in games/Chess)

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Alpha-Beta

• Each scheduling decision will tighten

– lower/upper bound estimates

• Can choose to expand

– least current time (breadth first) – least lower bound remaining (depth first) – least lower bound estimate – least upper bound estimate

• Can control greediness

– weighting lower/upper bound – selecting “most promising”

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Note

• Aggressive pruning and ordering

– can sometimes make polynomial time in practice – often cannot prove will be polynomial time – usually represents problem structure we still need to understand

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Multiple Resources

• Works for multiple resource case
• Computing lower-bounds per resource

– resource constrained

• Sometimes deal with resource coupling

– e.g. must have 1 A and 1 B simultaneously

• r in fixed time slot relation
• e.g. bus and memory port

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Summary

• Resource estimates and Refinement
• SAT/ILP Schedule
• Software Pipelining
• Branch-and-bound search

– “equivalent” states – dominators – estimates/pruning

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Admin

• Class

– Friday (no holiday) – Next week:

• Monday, Friday
• No Wed.
• Next week’s reading all online

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Big Ideas:

• Estimate Resource Usage
• Use dominators to reduce work
• Techniques:

– Force-Directed – SAT/ILP – Coloring – Search

• Branch-and-Bound
• Alpha-Beta