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  1. ▼♦❞❡❧❧✐♥❣ ❛ ▼❛✐♥t❡♥❛♥❝❡ ❙❝❤❡❞✉❧✐♥❣ Pr♦❜❧❡♠ ✇✐t❤ ❆❧t❡r♥❛t✐✈❡ ❘❡s♦✉r❝❡s ❆❧✐❛❛ ▼✳ ❇❛❞r ✶ ❆r♥❛✉❞ ▼❛❧❛♣❡rt ✷ ❑❡♥♥❡t❤ ◆✳ ❇r♦✇♥ ✶ ✶ ❈♦r❦ ❈♦♥str❛✐♥t ❈♦♠♣✉t❛t✐♦♥ ❈❡♥tr❡✱ ❯♥✐✈❡rs✐t② ❈♦❧❧❡❣❡ ❈♦r❦✳ ✷ ➱❝♦❧❡ ❞❡s ▼✐♥❡s ❞❡ ◆❛♥t❡s ❈♦♥str❛✐♥t ▼♦❞❡❧❧✐♥❣ ❛♥❞ ❘❡❢♦r♠✉❧❛t✐♦♥ ❲♦r❦s❤♦♣✱ ✷✵✶✵ ❇❛❞r ✱ ▼❛❧❛♣❡rt✱ ❇r♦✇♥ ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❆❧t❡r♥❛t✐✈❡ ❘❡s♦✉r❝❡s ✭❈♦r❦ ❈♦♥str❛✐♥t ❈♦♠♣✉t❛t✐♦♥ ❈❡♥tr❡✱ ❯♥✐✈❡rs✐t② ❈♦❧❧❡❣❡ ❈♦r❦✳✱ ➱❝♦❧❡ ❞❡s ▼✐♥❡s ❞❡ ◆❛♥t❡s ✮ ▼❖❉❘❊❋ ✷✵✶✵ ✶ ✴ ✷✶

  2. ❆❜str❛❝t ❇✉✐❧❞✐♥❣ ♠❛✐♥t❡♥❛♥❝❡ ✐♠♣❛❝t ♦♥ ❜✉✐❧❞✐♥❣ ♦♣❡r❛t✐♦♥ ❝♦sts✱ ❡♥❡r❣② ✉s❡✱ ❛♥❞ ♣r♦❞✉❝t✐✈✐t② ♦❢ ♦❝❝✉♣❛♥ts✳ ❇✉✐❧❞✐♥❣ ♠❛✐♥t❡♥❛♥❝❡ s❝❤❡❞✉❧✐♥❣ ♠♦❞❡❧s Pr✐♠✐t✐✈❡ s❝❤❡❞✉❧✐♥❣ ❝♦♥str❛✐♥ts ●❧♦❜❛❧ ❝♦♥str❛✐♥ts✿ ❢♦r ❛❧t❡r♥❛t✐✈❡ ❞✐s❥✉♥❝t✐✈❡ r❡s♦✉r❝❡s✳ ❊✈❛❧✉❛t✐♦♥ s❤♦✇s t❤❡ ❜❛s✐❝ ♠♦❞❡❧ ✐s ❢❛st❡r ✐♥ s♠❛❧❧❡r ♣r♦❜❧❡♠s❀ ❤♦✇❡✈❡r✱ t❤❡ ❣❧♦❜❛❧ ❝♦♥str❛✐♥t ♠♦❞❡❧ s❝❛❧❡s ❜❡tt❡r✳ ❇❛❞r ✱ ▼❛❧❛♣❡rt✱ ❇r♦✇♥ ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❆❧t❡r♥❛t✐✈❡ ❘❡s♦✉r❝❡s ✭❈♦r❦ ❈♦♥str❛✐♥t ❈♦♠♣✉t❛t✐♦♥ ❈❡♥tr❡✱ ❯♥✐✈❡rs✐t② ❈♦❧❧❡❣❡ ❈♦r❦✳✱ ➱❝♦❧❡ ❞❡s ▼✐♥❡s ❞❡ ◆❛♥t❡s ✮ ▼❖❉❘❊❋ ✷✵✶✵ ✷ ✴ ✷✶

  3. ❖✉t❧✐♥❡ ❇✉✐❧❞✐♥❣ ▼❛✐♥t❡♥❛♥❝❡ ❙❝❤❡❞✉❧✐♥❣ ✶ Pr♦❜❧❡♠ ❉❡s❝r✐♣t✐♦♥ ❙❝❤❡❞✉❧✐♥❣ ▼♦❞❡❧s ✷ ❇❛s✐❝ ▼♦❞❡❧ ●❧♦❜❛❧ ❈♦♥str❛✐♥t ▼♦❞❡❧ ✸ ●❧♦❜❛❧ ❈♦♥str❛✐♥t ❆❧t❡r♥❛t✐✈❡ ❉✐s❥✉♥❝t✐✈❡ ❈♦♥str❛✐♥t ✉s❡❘❡s♦✉r❝❡s ❈♦♥str❛✐♥t ✹ P❡r❢♦r♠❛♥❝❡ ❊✈❛❧✉❛t✐♦♥ ❊①♣❡r✐♠❡♥t❛❧ ❘❡s✉❧ts ✺ ❈♦♥❝❧✉s✐♦♥ ❇❛❞r ✱ ▼❛❧❛♣❡rt✱ ❇r♦✇♥ ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❆❧t❡r♥❛t✐✈❡ ❘❡s♦✉r❝❡s ✭❈♦r❦ ❈♦♥str❛✐♥t ❈♦♠♣✉t❛t✐♦♥ ❈❡♥tr❡✱ ❯♥✐✈❡rs✐t② ❈♦❧❧❡❣❡ ❈♦r❦✳✱ ➱❝♦❧❡ ❞❡s ▼✐♥❡s ❞❡ ◆❛♥t❡s ✮ ▼❖❉❘❊❋ ✷✵✶✵ ✸ ✴ ✷✶

  4. ❖✉t❧✐♥❡ ❇✉✐❧❞✐♥❣ ▼❛✐♥t❡♥❛♥❝❡ ❙❝❤❡❞✉❧✐♥❣ ✶ Pr♦❜❧❡♠ ❉❡s❝r✐♣t✐♦♥ ❙❝❤❡❞✉❧✐♥❣ ▼♦❞❡❧s ✷ ❇❛s✐❝ ▼♦❞❡❧ ●❧♦❜❛❧ ❈♦♥str❛✐♥t ▼♦❞❡❧ ✸ ●❧♦❜❛❧ ❈♦♥str❛✐♥t ❆❧t❡r♥❛t✐✈❡ ❉✐s❥✉♥❝t✐✈❡ ❈♦♥str❛✐♥t ✉s❡❘❡s♦✉r❝❡s ❈♦♥str❛✐♥t ✹ P❡r❢♦r♠❛♥❝❡ ❊✈❛❧✉❛t✐♦♥ ❊①♣❡r✐♠❡♥t❛❧ ❘❡s✉❧ts ✺ ❈♦♥❝❧✉s✐♦♥ ❇❛❞r ✱ ▼❛❧❛♣❡rt✱ ❇r♦✇♥ ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❆❧t❡r♥❛t✐✈❡ ❘❡s♦✉r❝❡s ✭❈♦r❦ ❈♦♥str❛✐♥t ❈♦♠♣✉t❛t✐♦♥ ❈❡♥tr❡✱ ❯♥✐✈❡rs✐t② ❈♦❧❧❡❣❡ ❈♦r❦✳✱ ➱❝♦❧❡ ❞❡s ▼✐♥❡s ❞❡ ◆❛♥t❡s ✮ ▼❖❉❘❊❋ ✷✵✶✵ ✹ ✴ ✷✶

  5. Pr♦❜❧❡♠ ❉❡s❝r✐♣t✐♦♥ ❇❛❞r ✱ ▼❛❧❛♣❡rt✱ ❇r♦✇♥ ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❆❧t❡r♥❛t✐✈❡ ❘❡s♦✉r❝❡s ✭❈♦r❦ ❈♦♥str❛✐♥t ❈♦♠♣✉t❛t✐♦♥ ❈❡♥tr❡✱ ❯♥✐✈❡rs✐t② ❈♦❧❧❡❣❡ ❈♦r❦✳✱ ➱❝♦❧❡ ❞❡s ▼✐♥❡s ❞❡ ◆❛♥t❡s ✮ ▼❖❉❘❊❋ ✷✵✶✵ ✺ ✴ ✷✶

  6. ❖✉t❧✐♥❡ ❇✉✐❧❞✐♥❣ ▼❛✐♥t❡♥❛♥❝❡ ❙❝❤❡❞✉❧✐♥❣ ✶ Pr♦❜❧❡♠ ❉❡s❝r✐♣t✐♦♥ ❙❝❤❡❞✉❧✐♥❣ ▼♦❞❡❧s ✷ ❇❛s✐❝ ▼♦❞❡❧ ●❧♦❜❛❧ ❈♦♥str❛✐♥t ▼♦❞❡❧ ✸ ●❧♦❜❛❧ ❈♦♥str❛✐♥t ❆❧t❡r♥❛t✐✈❡ ❉✐s❥✉♥❝t✐✈❡ ❈♦♥str❛✐♥t ✉s❡❘❡s♦✉r❝❡s ❈♦♥str❛✐♥t ✹ P❡r❢♦r♠❛♥❝❡ ❊✈❛❧✉❛t✐♦♥ ❊①♣❡r✐♠❡♥t❛❧ ❘❡s✉❧ts ✺ ❈♦♥❝❧✉s✐♦♥ ❇❛❞r ✱ ▼❛❧❛♣❡rt✱ ❇r♦✇♥ ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❆❧t❡r♥❛t✐✈❡ ❘❡s♦✉r❝❡s ✭❈♦r❦ ❈♦♥str❛✐♥t ❈♦♠♣✉t❛t✐♦♥ ❈❡♥tr❡✱ ❯♥✐✈❡rs✐t② ❈♦❧❧❡❣❡ ❈♦r❦✳✱ ➱❝♦❧❡ ❞❡s ▼✐♥❡s ❞❡ ◆❛♥t❡s ✮ ▼❖❉❘❊❋ ✷✵✶✵ ✻ ✴ ✷✶

  7. ❇❛s✐❝ ▼♦❞❡❧ ✭✶✴✷✮ ❱❛r✐❛❜❧❡s ❆ s❡t ❚ = { ✶ ,... ♥ } ♦❢ ❛❝t✐✈✐t✐❡s ❛♥❞ ❛ s❡t ❘ = { ✶ ,... ♠ } ♦❢ ❡♥❣✐♥❡❡rs✳ ❉♦♠❛✐♥s ♦❢ ✈❛r✐❛❜❧❡s ❛r❡ ✐♥t❡❣❡rs✳ ❋♦r ❡❛❝❤ ❛❝t✐✈✐t② ✐ ∈ ❚ ✿ st❛rt ✐ ✿ st❛rt t✐♠❡ ❡♥❞ ✐ ✿ ❡♥❞ t✐♠❡ ❞ ✐ ✿ ❡①♣❡❝t❡❞ ❞✉r❛t✐♦♥ ❝ ✐ ✿ ❛ss✐❣♥❡❞ ❡♥❣✐♥❡❡r ① ✐ ✿ s❝❤❡❞✉❧✐♥❣ ❝♦st✳ s ✐ ✿ ❧✐♥❦s st❛rt t✐♠❡ t♦ ❝♦sts ♣ ✿ s❝❤❡❞✉❧❡ t♦t❛❧ ❝♦st✳ ❈♦♥st❛♥ts ❢♦r ❡❛❝❤ ❛❝t✐✈✐t② ✐ ∈ ❚ ❞t ✐ ✿ ❛rr❛② ♦❢ st❛rt t✐♠❡s t❝♦st ✐ ✿ ❛rr❛② r❡♣r❡s❡♥t✐♥❣ t❤❡ ❝♦st ♦❢ ❡❛❝❤ st❛rt t✐♠❡✳ ❇❛❞r ✱ ▼❛❧❛♣❡rt✱ ❇r♦✇♥ ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❆❧t❡r♥❛t✐✈❡ ❘❡s♦✉r❝❡s ✭❈♦r❦ ❈♦♥str❛✐♥t ❈♦♠♣✉t❛t✐♦♥ ❈❡♥tr❡✱ ❯♥✐✈❡rs✐t② ❈♦❧❧❡❣❡ ❈♦r❦✳✱ ➱❝♦❧❡ ❞❡s ▼✐♥❡s ❞❡ ◆❛♥t❡s ✮ ▼❖❉❘❊❋ ✷✵✶✵ ✼ ✴ ✷✶

  8. ❇❛s✐❝ ▼♦❞❡❧ ✭✷✴✷✮ ❈♦♥str❛✐♥ts ∀ ✐ ∈ ❚ ✿ st❛rt ✐ + ❞ ✐ = ❡♥❞ ✐ ❉✉❡ t♦ ✐rr❡❣✉❧❛r t✐♠❡ ✇✐♥❞♦✇s ❛♥❞ ❝♦sts✱ ✇❡ ❛❞❞ ∀ ✐ ∈ ❚ ✿ ❞t ✐ [ s ✐ ] = st❛rt ✐ ∀ ✐ ∈ ❚ : t❝♦st ✐ [ s ✐ ] = ① ✐ ∀ ✐ , ∀ ❥ � = ✐ : ❝ ✐ = ❝ ❥ → ( ❡♥❞ ✐ ≤ st❛rt ❥ ) ∨ ( ❡♥❞ ❥ ≤ st❛rt ✐ ) ❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ♣r♦❜❧❡♠ ✐♥st❛♥❝❡✱ ✇❡ ❛❞❞ ❝♦♥str❛✐♥ts Concurrency : st❛rt ✐ = st❛rt ❥ Precedence : ❡♥❞ ✐ ≤ st❛rt ❥ Non-overlapping : ( ❡♥❞ ✐ ≤ st❛rt ❥ ) ∨ ( ❡♥❞ ❥ ≤ st❛rt ✐ ) ∀ ❙ , AllDifferent ( { ❝ ✐ : ✐ ∈ ❙ } ) ❚♦t❛❧ ❝♦st ♦❢ t❤❡ s❝❤❡❞✉❧❡✿ ♣ = ∑ ✐ ∈ ❚ ① ✐ ❖❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✐s t♦ ♠✐♥✐♠✐③❡ ♣ ✳ ❇❛❞r ✱ ▼❛❧❛♣❡rt✱ ❇r♦✇♥ ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❆❧t❡r♥❛t✐✈❡ ❘❡s♦✉r❝❡s ✭❈♦r❦ ❈♦♥str❛✐♥t ❈♦♠♣✉t❛t✐♦♥ ❈❡♥tr❡✱ ❯♥✐✈❡rs✐t② ❈♦❧❧❡❣❡ ❈♦r❦✳✱ ➱❝♦❧❡ ❞❡s ▼✐♥❡s ❞❡ ◆❛♥t❡s ✮ ▼❖❉❘❊❋ ✷✵✶✵ ✽ ✴ ✷✶

  9. ❖✉t❧✐♥❡ ❇✉✐❧❞✐♥❣ ▼❛✐♥t❡♥❛♥❝❡ ❙❝❤❡❞✉❧✐♥❣ ✶ Pr♦❜❧❡♠ ❉❡s❝r✐♣t✐♦♥ ❙❝❤❡❞✉❧✐♥❣ ▼♦❞❡❧s ✷ ❇❛s✐❝ ▼♦❞❡❧ ●❧♦❜❛❧ ❈♦♥str❛✐♥t ▼♦❞❡❧ ✸ ●❧♦❜❛❧ ❈♦♥str❛✐♥t ❆❧t❡r♥❛t✐✈❡ ❉✐s❥✉♥❝t✐✈❡ ❈♦♥str❛✐♥t ✉s❡❘❡s♦✉r❝❡s ❈♦♥str❛✐♥t ✹ P❡r❢♦r♠❛♥❝❡ ❊✈❛❧✉❛t✐♦♥ ❊①♣❡r✐♠❡♥t❛❧ ❘❡s✉❧ts ✺ ❈♦♥❝❧✉s✐♦♥ ❇❛❞r ✱ ▼❛❧❛♣❡rt✱ ❇r♦✇♥ ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ❆❧t❡r♥❛t✐✈❡ ❘❡s♦✉r❝❡s ✭❈♦r❦ ❈♦♥str❛✐♥t ❈♦♠♣✉t❛t✐♦♥ ❈❡♥tr❡✱ ❯♥✐✈❡rs✐t② ❈♦❧❧❡❣❡ ❈♦r❦✳✱ ➱❝♦❧❡ ❞❡s ▼✐♥❡s ❞❡ ◆❛♥t❡s ✮ ▼❖❉❘❊❋ ✷✵✶✵ ✾ ✴ ✷✶

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