Lower and upper bounds for the resource-constrained modulo - PowerPoint PPT Presentation

Lower and upper bounds for the resource-constrained modulo scheduling problem Christian Artigues 1 Maria Alejandra Ayala 2 Abir Benabid 3 Claire Hanen 4 1 LAAS - CNRS & Université de Toulouse, France 2 Universidad de los Andes, Mérida, Venezuela 3 King Saud University, Saudi Arabia 4 LIP6 & Université Paris Ouest Nanterre, France artigues@laas.fr, marialej@ula.ve, ben_abid_abir@yahoo.fr, Claire.Hanen@lip6.fr PMS 2012 - Leuven Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 1 / 28

Outline 1 Problem definition 2 Typical application : instruction scheduling for VLIW processors 3 Solution methods 4 Computational experiments Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 2 / 28

Outline Problem definition 1 Typical application : instruction scheduling for VLIW processors 2 Solution methods 3 Integer Linear programming (ILP) for the RCMSP Decomposed Software Pipelining An hybrid method Computational experiments 4 Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 3 / 28

Periodic scheduling Set V of unit-duration tasks with | V | = n . Each task i ∈ V has an infinite number of occurrences < i ; q > that are scheduled periodically A start time σ q i ∈ N has to be assigned to each task occurrence < i ; q > such that σ q i = σ 0 i + q λ where λ is the period (to be minimized). λ = 2 < i ; 0 > < i ; 1 > < i ; 2 > < i ; 3 > A periodic schedule is defined by σ i ≡ σ 0 , ∀ i with σ i ∈ { 0 , . . . , λ − 1 } Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 4 / 28

Uniform precedence constraints Set E of precedence constraints such that ( i , j ) ∈ E is defined by a a latency θ j i and a distance ω j i q + ω j ≥ σ q i + θ j σ i i , ∀ ( i , j ) ∈ E , ∀ q ∈ N j < i ; q > < i ; q + 1 > < i ; q + 2 > < i ; q + 3 > < j ; q > < j ; q + 1 > < j ; q + 2 > < j ; q + 3 > < k ; q > < k ; q + 1 > < k ; q + 2 >< k ; q + 3 > N λ = 1 Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 5 / 28

Uniform precedence constraints Set E of precedence constraints such that ( i , j ) ∈ E is defined by a a latency θ j i and a distance ω j i q + ω j ≥ σ q i + θ j σ i i , ∀ ( i , j ) ∈ E , ∀ q ∈ N j < i ; q > < i ; q + 1 > < i ; q + 2 > < i ; q + 3 > 0 0 0 0 i θ k i = 0 , ω k i = 0 < j ; q > < j ; q + 1 > < j ; q + 2 > < j ; q + 3 > k < k ; q > < k ; q + 1 > < k ; q + 2 >< k ; q + 3 > N λ = 1 Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 5 / 28

Uniform precedence constraints Set E of precedence constraints such that ( i , j ) ∈ E is defined by a a latency θ j i and a distance ω j i q + ω j ≥ σ q i + θ j σ i i , ∀ ( i , j ) ∈ E , ∀ q ∈ N j < i ; q > < i ; q + 1 > < i ; q + 2 > < i ; q + 3 > i θ k i = 0 , ω k i = 0 < j ; q > < j ; q + 1 > < j ; q + 2 > < j ; q + 3 > k θ i k = 1 , ω i k = 2 1 1 < k ; q > < k ; q + 1 > < k ; q + 2 >< k ; q + 3 > N λ = 1 Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 5 / 28

Uniform precedence constraints Set E of precedence constraints such that ( i , j ) ∈ E is defined by a a latency θ j i and a distance ω j i q + ω j ≥ σ q i + θ j σ i i , ∀ ( i , j ) ∈ E , ∀ q ∈ N j 1 , 1 < i ; q > < i ; q + 1 > < i ; q + 2 > < i ; q + 3 > j i 1 1 1 0 , 0 < j ; q > < j ; q + 1 > < j ; q + 2 > < j ; q + 3 > k 1 , 2 < k ; q > < k ; q + 1 > < k ; q + 2 >< k ; q + 3 > N λ = 1 Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 5 / 28

Uniform precedence constraints Set E of precedence constraints such that ( i , j ) ∈ E is defined by a a latency θ j i and a distance ω j i q + ω j ≥ σ q i + θ j σ i i , ∀ ( i , j ) ∈ E , ∀ q ∈ N j 1 , 1 < i ; q > < i ; q + 1 > < i ; q + 2 > < i ; q + 3 > j i 2 , 1 0 , 0 < j ; q > < j ; q + 1 > < j ; q + 2 > < j ; q + 3 > k 1 , 2 2 2 2 < k ; q > < k ; q + 1 > < k ; q + 2 >< k ; q + 3 > N λ = 1 Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 5 / 28

Uniform precedence constraints Set E of precedence constraints such that ( i , j ) ∈ E is defined by a a latency θ j i and a distance ω j i q + ω j ≥ σ q i + θ j σ i i , ∀ ( i , j ) ∈ E , ∀ q ∈ N j 1 , 1 < i ; q > < i ; q + 1 > < i ; q + 2 > < i ; q + 3 > j i 2 , 1 0 , 0 < j ; q > < j ; q + 1 > < j ; q + 2 > < j ; q + 3 > k 1 , 2 2 2 2 < k ; q − 1 > < k ; q > < k ; q + 1 > < k ; q + 2 >< k ; q + 3 > N λ = 1 Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 5 / 28

Uniform precedence constraints Set E of precedence constraints such that ( i , j ) ∈ E is defined by a a latency θ j i and a distance ω j i q + ω j ≥ σ q i + θ j σ i i , ∀ ( i , j ) ∈ E , ∀ q ∈ N j 1 , 1 < i ; q > < i ; q + 1 > < i ; q + 2 > < i ; q + 3 > j i 3 , 1 0 , 0 < j ; q > < j ; q + 1 > < j ; q + 2 > < j ; q + 3 > k 1 , 2 3 3 3 < k ; q − 1 > < k ; q > < k ; q + 1 > < k ; q + 2 >< k ; q + 3 > N λ = 1 Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 5 / 28

Uniform precedence constraints Set E of precedence constraints such that ( i , j ) ∈ E is defined by a a latency θ j i and a distance ω j i q + ω j ≥ σ q i + θ j σ i i , ∀ ( i , j ) ∈ E , ∀ q ∈ N j 1 , 1 < i ; q > < i ; q + 1 > < i ; q + 2 > < i ; q + 3 > j 1 1 1 1 i 3 , 1 0 , 0 < j ; q > < j ; q + 1 > < j ; q + 2 > < j ; q + 3 > k 1 , 2 3 3 3 < k ; q − 2 > < k ; q − 1 > < k ; q > < k ; q + 1 > < k ; q + 2 >< k ; q + 3 > N λ = 1 Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 5 / 28

Uniform precedence constraints Set E of precedence constraints such that ( i , j ) ∈ E is defined by a a latency θ j i and a distance ω j i q + ω j ≥ σ q i + θ j σ i i , ∀ ( i , j ) ∈ E , ∀ q ∈ N j 1 , 1 < i ; q − 1 > < i ; q > < i ; q + 1 > < i ; q + 2 > < i ; q + 3 > j 1 1 1 1 i 3 , 1 0 , 0 < j ; q > < j ; q + 1 > < j ; q + 2 > < j ; q + 3 > k 1 , 2 3 3 3 < k ; q − 2 > < k ; q − 1 > < k ; q > < k ; q + 1 > < k ; q + 2 >< k ; q + 3 > N λ = 1 Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 5 / 28

Uniform precedence constraints Set E of precedence constraints such that ( i , j ) ∈ E is defined by a a latency θ j i and a distance ω j i q + ω j ≥ σ q i + θ j σ i i , ∀ ( i , j ) ∈ E , ∀ q ∈ N j 1 , 1 < i ; q − 1 > < i ; q > < i ; q + 1 > < i ; q + 2 > < i ; q + 3 > j 1 1 1 1 i 3 , 1 1 1 1 1 0 , 0 < j ; q > < j ; q + 1 > < j ; q + 2 > < j ; q + 3 > k 1 , 2 3 3 3 < k ; q − 2 > < k ; q − 1 > < k ; q > < k ; q + 1 > < k ; q + 2 >< k ; q + 3 > N λ = 1 Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 5 / 28

Uniform precedence constraints Set E of precedence constraints such that ( i , j ) ∈ E is defined by a a latency θ j i and a distance ω j i q + ω j ≥ σ q i + θ j σ i i , ∀ ( i , j ) ∈ E , ∀ q ∈ N j 1 , 1 < i ; q − 1 > < i ; q > < i ; q + 1 > < i ; q + 2 > j 1 1 1 i 1 1 1 1 3 , 1 0 , 0 < j ; q > < j ; q + 1 > < j ; q + 2 > < j ; q + 3 > k 1 , 2 3 3 < k ; q − 2 > < k ; q − 1 > < k ; q > < k ; q + 1 > < k ; q + 2 > N λ = 2 Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 5 / 28

Uniform precedence constraints Set E of precedence constraints such that ( i , j ) ∈ E is defined by a a latency θ j i and a distance ω j i q + ω j ≥ σ q i + θ j σ i , ∀ ( i , j ) ∈ E , ∀ q ∈ N i j 1 , 1 < i ; q − 1 > < i ; q > < i ; q + 1 > < i ; q + 2 > j i 3 , 1 0 , 0 < j ; q > < j ; q + 1 > < j ; q + 2 > < j ; q + 3 > k 1 , 2 < k ; q − 2 > < k ; q − 1 > < k ; q > < k ; q + 1 > < k ; q + 2 > N λ = 2, pattern σ i = 1, σ j = 0, σ k = 3, C max = 4 Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 5 / 28

Basic Cyclic scheduling problem Precedence constraints can be expressed using only σ i : q + ω j ≥ σ q i + θ j σ j + λ ( q + ω j i ) ≥ σ i + λ q + θ j σ ⇔ i j i i σ j ≥ σ i + θ j i − λω j ⇔ i The Basic Cyclic Scheduling Problem (BCSP) min λ σ j ≥ σ i + θ j i − λω j ∀ ( i , j ) ∈ E i σ i ∈ N ∀ i ∈ { 1 , ..., n } Remark : for a fixed λ we obtain a project schedu- − 1 j ling problem with minimum and maximum time i 0 1 lags that can be solved by the Bellman-Ford al- k − 3 gorithm. Artigues, Ayala, Benabid, Hanen LB & UB for the RCMSP PMS 2012 - Leuven 6 / 28