Tabu searching for high-level synthesis

TS for HLS

M. SevauxK. Sorensen

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Scheduling

Implementation

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Experiments

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Tabu Searching for High Level Synthesis

Marc Sevaux Kenneth Sorensen

University of South-BrittanyLorient – France

[email protected]

Centre for Industrial Management, KULeuven – Belgium

[email protected]

FRANCORO – ROADeFGrenoble – France

January 20-23, 2007

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High-level synthesis

Problem

A source code (C, VHDL) has to be executed on anelectronic component in a certain amount of time and thedesign of the electronic component is “left” to the decisionmaker

A field programmable gate array(FPGA) is a semiconductor devicecontaining programmable logic compo-nents. A library combines these logiccomponents into basic operators

An experimental platform transforms the source code into agraph of tasks to be executed by the basic operators

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A small graph

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A long term strategy

HLS Scheduling Simplifed Problem

◮ Release dates

◮ Due dates

◮ Minimize the numberof operator used

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HLS Scheduling Simplifed Problem

◮ Release dates

◮ Due dates

◮ Minimize the numberof operator used

HLS Scheduling Central Problem

◮ Precedences

◮ Multi-purposeoperators

◮ Memory allocation

◮ Cost function

◮ Cmax

◮ Total flow

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HLS SchedulingCentral Problem

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HLS SchedulingSimplified Problem

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MIP Formulation

See A. Rossi’s talk

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MIP Formulation

See A. Rossi’s talkTabu Search

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MIP Formulation

See A. Rossi’s talkTabu Search

PPC

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MIP Formulation


IROCOI

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MIP Formulation


IROCOI

HLS SchedulingCentral Problem

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“Scheduling” problem

A list of tasks (jobs) with release dates and due dates has tobe scheduled on a minimum number of processors.

A simplified version of the problem considers

◮ only one class of operators → identical processors

◮ precedence constraints can be ignored → simpletransformation

A similar problem to the one presented before is

◮ Minimising the number of late jobs on parallelprocessors

◮ Noted Pm|rj |∑

Uj in the classical notation

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A first approach

Based on previous results on Pm|rj |∑

wjUj , we can

◮ Compute a lower bound of the number of machines

◮ Solve the problem using previous work

◮ Increase the number of machines until all jobs aresequenced

Work in progress

◮ Computing lower bounds → Bin packing or VRPTW

◮ Solver when the number of machine is known◮ fully operational for small size instances◮ need to be improved for handling large instances

◮ Iterative steps → change the strategy

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A parallel machine scheduling problem

Scheduling problem definition

◮ m identical machines in parallel

◮ n jobs to be scheduled

pj processing timerj release datedj due date

◮ Objective: minimise the number of late jobs

◮ Uj = 1 if job j is late and 0 otherwise

◮ Classical notation: Pm|rj |∑

Uj

◮ NP-hard in a strong sense

Important: Late jobs do not need to be scheduled

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Specifications for implementation

A special encoding

◮ m lists of early jobs with starting times = machine lists

◮ 1 list of late jobs = late list

Job 1 2 3 4 5

rj 0 0 2 3 4pj 3 6 3 4 1dj 5 7 8 9 7

M1

M2

t950

J

J J J1 5 3

2J4

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Elastic placement

General assertion

◮ if Jj is early → tj ∈ [rj , dj − pj ]

Dynamic structure

◮ tj is never fixed but computed on-demand depending onthe partial sequence of its machine

r d

J

j j

j

t j

J Ji k

remark: similarities with constraint programming

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Inserting a job

Inserting a job Jj between two consecutive jobs Ji and Jk

requires the computation of a maximum available interval[hmin, hmax]:

◮ hmin: minimum date at which Ji can terminate(by pushing Ji and its predecessors)

◮ hmax: maximum date at which Jk can start(by pushing Jk and its successors)

Checking the size of the interval:

◮ max(rj , hmin) + pj ≤ min(dj , hmax)

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Pushing jobs

Limits of the interval are computed recursively

eeti = max(ri , eetpred(i)) + pi

JkJi

eeti

lstk

lstk = min(dk , lstsucc(k)) − pk

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Best insertion

Condition for insertion

◮ if max(rj , eetpred(j)) + pj ≤ min(dj , lstsucc(j))

◮ Jj can be inserted at that position

Best insertion

◮ All insertion positions are scanned

◮ Jj is inserted where the idle time between pred(j) andsucc(j) is minimum

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Tabu Search Algorithm

TS specifications

◮ use a standard TS template

◮ take advantage of special encoding

◮ implement tenure refinements

Neighbourhood structure

◮ Only feasible solutions allowed

◮ Move 1 job from the late list to one of the early lists

◮ move 0 to k early jobs on a same machine → late list

◮ Cost = difference in objective function value

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TS parameters

Initial solutions

◮ Greedy approach: shortest processing time

◮ Sort the list of all jobs in increasing order of pj

◮ Fill machine lists sequentially with best insertion

Tabu search parameters

◮ itmax iterations

◮ Tabu criterion: sum of processing times of late jobs

The tabu tenure should be dynamically adjusted

◮ large tenure → all moves might be tabu

◮ small tenure → cycle around a local optimum

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Dynamic tenure and cycle detection

Tenure adjustments

◮ all moves tabu → decrease the tabu tenure by 1

◮ cycle detection → increase the tabu tenure by thelength of the cycle and set the next move tabu (allmoves of the cycle can be set tabu)

Cycle detection

◮ cycle greater than tabu tenure → keep history of“visited solutions”

◮ history stores only tabu criterion

◮ cycle detection = two identical substrings in history(same values, same length) and consecutives

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Numerical experiments

Objectives

◮ use real-life instances

◮ compare with other aproaches

◮ estimate the quality of the solution

Current status

◮ modify the real-life instances

◮ other approaches are not adapted to our goals

◮ quality compared to optimal solution or lower bounds

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Problem instances (1)

Functions are designed for Filtering operations in DSP

Simple functions

Functions Nodes Arcs Cmax (ns)

FFT 4 75 137 70FIR 16 96 172 170FIR 32 192 348 330FFT 8 227 433 110FIR 64 384 700 650DCT 4x4 482 944 100FFT 16 611 1185 150

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Problem instances (2)

More complex functions

Functions Nodes Arcs Cmax (ns)

LMS 128 1287 2556 1350FFT 32 1539 3009 190FFT 64 3715 7297 230DCT 8x8 3970 7872 180FFT 128 8707 17153 270FFT 256 19971 39425 310FFT 512 45059 89089 350

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Current approaches

Exact solution methods

◮ Constraint Programming-based B&B approachBaptiste Ph., Jouglet A., Le Pape C., Nuijten W. (2001)50 jobs in 10 min., no solution for 100 jobs

◮ Time-indexed ILP formulationM. Sevaux and A. Rossi (2007)up to 1000 jobs in 1 min.

Approximate methods

◮ Heuristics and basic version of SA and TSM. Sevaux and Ph. Thomin (2001)

◮ VNS/TS with specific neighbourhoodM. Sevaux and K. Sorensen (2006)50 jobs → 15% optimal sol., 100 jobs → ≈ 3 sec.

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Results

# of MILP Tabu Search methodjobs Av. m Av. m Dev. % # opt. time (s)

20 9.3 9.3 0.0 20 2.240 17.7 17.8 0.6 16 8.860 24.9 25.8 3.6 7 15.580 32.9 34.4 4.6 0 37.6

100 40.4 42.6 5.5 2 49.9

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Tabu tenure evolution

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tabu

tenu

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iterations (log scale)

tenure value

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Conclusion

Conclusion

◮ TS easy approach to tackle “large” problems

◮ Uses a specialised encoding

◮ Fast and flexible approach(unrelated machines, precedence constraints)

Potential problems / interests

◮ large size instances (45 000 jobs)Data Flow Graph (DFG) → Control Data Flow Graph45 000 jobs → 200∼500 jobs

◮ tabu tenure evolution → decrease too slowly

◮ short time horizon

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Perspectives

Ongoing work

◮ Precedences

◮ Multi-purposeoperators

◮ Cost function

◮ Cmax

Next steps of the study

◮ Work on tight lower bounds

◮ Exploit “real” data

◮ Insert resulting algorithm in GAUT

http://web.univ-ubs.fr/gaut/

http://web.univ-ubs.fr/gaut/

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Tabu Searching for High Level Synthesis

Marc Sevaux Kenneth Sorensen

University of South-BrittanyLorient – France

[email protected]

Centre for Industrial Management, KULeuven – Belgium

[email protected]

FRANCORO – ROADeFGrenoble – France

January 20-23, 2007

Tabu searching for high-level synthesis

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