❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞ ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ▼❛r❝❤ ✻✱ ✷✵✶✷ ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
❋❡✇ r❡♠✐♥❞❡rs ❛❜♦✉t ❞✐str✐❜✉t❡❞ ❑✲▼❡❛♥s ❑▼❡❛♥s ❙❡q✉❡♥t✐❛❧ ❚✐♠❡ = ( ✸ ❑♥❞ + ❑♥ + ❑❞ + ♥❞ ) ∗ ■ ∗ ❚ ✢♦♣ ≃ ✸ ❑♥❞ ∗ ■ ∗ ❚ ✢♦♣ ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
❙▼P ❈♦st ❑▼❡❛♥s ❙▼P ❉✐str✐❜✉t❡❞ ❈♦st = ❚ ❝♦♠♣ P = ( ✸ ❑♥❞ + ❑♥ + ❑❞ + ♥❞ ) ∗ ■ ∗ ❚ ✢♦♣ P ≃ ✸ ❑♥❞ ∗ ■ ∗ ❚ ✢♦♣ P ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
❉▼▼ ❈♦st ❑▼❡❛♥s ❉▼▼ ❈♦st = ❚ ❝♦♠♣ + ❚ ❝♦♠♠ P P = ( ✸ ❑♥❞ + ❑♥ + ❑❞ + ♥❞ ) ∗ ■ ∗ ❚ ✢♦♣ + ❚ ❝♦♠♠ P P ≃ ✸ ❑♥❞ ∗ ■ ∗ ❚ ✢♦♣ + ❖ ( ❧♦❣ ( P )) P ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
Worker 1 Centroids local version merge into Worker 2 Worker 2 Centroids local Centroids partial version version merge Worker 3 into Centroids local version merge into Worker 4 Worker 4 Worker 4 Centroids aggregated Centroids local Centroids partial version version version ❋✐❣✉r❡✿ ▼❡r❣✐♥❣ ❝❡♥tr♦ï❞s ❧♦❣✐❝ ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
❝❧❛ss✐❝❛❧ ✐ss✉❡ ✿ ♣✐♣❡s ❚❤❡ s✐③❡ ♦❢ t❤❡ ❜✐❣❣❡st ♣✐♣❡ ✭♦♥ t♦♣ ♦❢ t❤❡ tr❡❡✮ ❛♥❞ t❤❡ ❛❣❣r❡❣❛t❡❞ ❜❛♥❞✇✐❞t❤✳ ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
♠♦❞❡❧✐s❛t✐♦♥ ♦❢ ❚ ❝♦♠♠ ✐♥ ❉▼▼ P ⌈ ❧♦❣ ✷ ( P ) ⌉ ( ✷ ■❑❞ ∗ ❙♦❉ � ❚ ❝♦♠♠ + ✺ ■❑❞ ∗ ❚ ✢♦♣ ) = P ❇ ❉▼▼ , ▼P■ ♣ = ✶ P ✷ ♣ ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
♠♦❞❡❧✐s❛t✐♦♥ ♦❢ ❚ ❝♦♠♠ ✐♥ ❉▼▼ ✭✷✮ P ■❑❞ ∗ ❙♦❉ ❚ ❝♦♠♠ ❇ P , ❛✈❣ ❉▼▼ , ▼P■ + ✺ ■❑❞ ∗ ❚ ✢♦♣ ) ∗ ⌈ ❧♦❣ ✷ ( P ) ⌉ = ( ✷ P ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
❙♣❡❡❞❯♣ ❑▼❡❛♥s ❙❡q✉❡♥t✐❛❧ ❈♦st ❙♣❡❡❞✉♣ = ❑▼❡❛♥s❉▼▼❉✐str✐❜✉t❡❞❈♦st = ❑▼❡❛♥s❙❡q✉❡♥t✐❛❧❈♦st ❚ ❝♦♠♣ + ❚ ❝♦♠♠ P P ✸ ♥❑❞■❚ ✢♦♣ = � � ✸ ♥❑❞■❚ ✢♦♣ ✷ ■❑❞ ∗ ❙♦❉ + ✺ ■❑❞❚ ✢♦♣ + ⌈ ❧♦❣ ✷ ( P ) ⌉ P ❇ ❉▼▼ , ▼P■ P ✸ ♥❚ ✢♦♣ = � � ✸ ♥❚ ✢♦♣ ❙♦❉ + ✺ ❚ ✢♦♣ + ✷ ⌈ ❧♦❣ ✷ ( P ) ⌉ P ❇ ❉▼▼ , ▼P■ P ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
❙♦♠❡ ❝♦♠♠❡♥ts ◮ ❘❆▼ ✐s ♥♦t ♠♦❞❡❧✐③❡ ⇒ s♣❡❡❞ ✉♣ ♠✐❣❤t ❜❡ ❣r❡❛t❡r ◮ s♣❡❡❞✉♣ ✐♥❞❡♣❡♥❞❛♥t ♦❢ ❑✱ ■ ❛♥❞ ❞ ◮ ✇✐♥♥❡r ♣❛♣❡r ♠♦❞❡❧ ✦ ♥ → + ∞ ❙♣❡❡❞❯♣ = P ❧✐♠ ✭✶✮ ✭✷✮ ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
❋✐rst ❝❛s❡ st✉❞② ✿ ❊❉❋ ◮ ♥ ❂ ✷✵ ✵✵✵ ✵✵✵ t✐♠❡ s❡r✐❡s ◮ ❞ ❂ ✷✹ ❤♦✉rs ✯ ✸✻✺ ❞❛②s ✯ ✶✵ ②❡❛rs ❂ ✽✼✻✵✵ ✈❛❧✉❡s ♣❡r s❡r✐❡s ◮ ❑ ❂ √ ♥ ❂ ✹✹✼✷ ❝❧✉st❡rs ◮ ◆♦t❡ ✿ ✈♦❧✉♠❡ ♦❢ t✐♠❡s❡r✐❡s ✐s ✶✷✳✸ ❚❡r❛❇②t❡s✳ ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
✶ ♠♦♥♦❝♦r❡ ✸✼ ②❡❛rs ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
❨❛❤♦♦ t❡r❛s♦rt ✇✐♥♥❡r ◮ ✾✶✵ ♥♦❞❡s ◮ ✹ ❞✉❛❧ ❝♦r❡ ❳❡♦♥s ❅ ✷✳✵❣❤③ ♣❡r ❛ ♥♦❞❡ ◮ ✶ ❣✐❣❛❜✐t ❡t❤❡r♥❡t ♦♥ ❡❛❝❤ ♥♦❞❡ ◮ ✹✵ ♥♦❞❡s ♣❡r ❛ r❛❝❦ ◮ ✽ ❣✐❣❛❜✐t ❡t❤❡r♥❡t ✉♣❧✐♥❦s ❢r♦♠ ❡❛❝❤ r❛❝❦ t♦ t❤❡ ❝♦r❡ ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
❨❛❤♦♦ t❡r❛s♦rt ✇✐♥♥❡r✭✷✮ = ( ✸ ♥❑❞ + ♥❑ + ❞❑ + ♥❞ ) ∗ ■ ∗ ❚ ✢♦♣ ❚ ❝♦♠♣ P P = ✸✷✷✽✾✸ s❡❝♦♥❞s ❚ ❝♦♠♠ = ( ✷ ∗ ❇r♦❛❞❝❛st t✐♠❡ ❜❡t✇❡❡♥ ✷ ♣r♦❝❡ss♦rs + ▼❡r❣✐♥❣ t✐♠❡ ) ∗ ⌈ ❧♦❣ ✷ P P = ( ✷ ■❑❞ ∗ ❙♦❉ + ✺ ■❑❞ ∗ ❚ ✢♦♣ ) ∗ ⌈ ❧♦❣ ✷ ( P ) ⌉ ❇ ❉▼▼ , ▼P■ P , ❛✈❣ = ✷✷✺✽✻✻ s❡❝♦♥❞s ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
❨❛❤♦♦ t❡r❛s♦rt ✇✐♥♥❡r✭✷✮ ❜♦✉♥❞❡❞ ❜② ❛❣❣r❡❣❛t❡❞ ❜❛♥❞✇✐❞t❤ ❇ ❉▼▼ , ▼P■ = ✶✵✾ ▼❜✐ts P , ❛✈❣ ❚ ❝♦♠♠ = ( ✷ ∗ ❇r♦❛❞❝❛st t✐♠❡ ❜❡t✇❡❡♥ ✷ ♣r♦❝❡ss♦rs + ▼❡r❣✐♥❣ t✐♠❡ ) ∗ ⌈ ❧♦❣ ✷ P P = ( ✷ ■❑❞ ∗ ❙♦❉ + ✺ ■❑❞ ∗ ❚ ✢♦♣ ) ∗ ⌈ ❧♦❣ ✷ ( P ) ⌉ ❇ ❉▼▼ , ▼P■ P , ❛✈❣ = ✹✸✹✶✺✺ s❡❝♦♥❞s ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
Pr✐❝✐♥❣ ♦♥ ❆③✉r❡ ◮ ❙t♦r❛❣❡ ♣r✐❝❡ ✿ ✶✽✹✺ ❞♦❧❧❛rs✴▼♦♥t❤ ✭✵✱✶✺ ❞♦❧❧❛rs ✴●❇✴▼♦♥t❤✮ ◮ ◆♦ ❝❤❛r❣❡ ❢♦r ✐♥t❡r♥❛❧ tr❛♥s❢❡rs ❄ ◮ ✉♣❞❛t❡ ✿ ✶✵ ●❇②t❡s✴▼♦♥t❤ ✿ ✸ ❞♦❧❧❛rs ✭✵✳✸❞♦❧❧❛rs ✴●❇✮ ◮ ❝♦st ♦❢ r✉♥♥✐♥❣ t❤❡ ❑▼❡❛♥s ✿ ✭✈❡r② r♦✉❣❤✮ ✵✳✶✷❞♦❧❧❛rs✴❤♦✉r ✯ ✸✷✹✶✷✵ ❂ ✹✵ ✵✵✵❞♦❧❧❛rs ◮ st♦r❛❣❡ r❡q✉❡sts ✿ ❄ ❢r♦♠ ✵✳✵✶ ❞♦❧❧❛rs t♦ ✶✶ ♠✐❧❧✐♦♥s ❞♦❧❧❛rs ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
❋✐❣✉r❡✿ P❛❛s✱ ■❛❛s✱ ❙❛❛❙ ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
P❛❛s ❛♥❞ ■❛❛s ✿ s♦♠❡ tr❛❞❡♦✛s ◮ ❛❜str❛❝t✐♦♥✴❝♦♥tr♦❧ tr❛❞❡♦✛ ◮ s❝❛❧❛❜✐❧✐t②✴❝♦st t♦ ❞❡✈❡❧♦♣ tr❛❞❡♦✛ ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
■♥tr♦❞✉❝✐♥❣ ▲♦❦❛❞✳❈❧♦✉❞ ◮ ❤✐❣❤❡r ❧❡✈❡❧ ❛❜str❛❝t✐♦♥s ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
❆③✉r❡ ❆❜str❛❝t✐♦♥s ◮ ❆③✉r❡ ◗✉❡✉❡s ◮ ❆③✉r❡ ❇❧♦❜❙t♦r❛❣❡ ◮ ❆③✉r❡ ❚❛❜❧❡❙t♦r❛❣❡ ◮ ❆③✉r❡ ❙◗▲❙t♦r❛❣❡ ▼❛tt❤✐❡✉ ❉✉r✉t✱ ❋❛❜r✐❝❡ ❘♦ss✐ ❉❡s✐❣♥✐♥❣ ❑✲▼❡❛♥s ❢♦r t❤❡ ❝❧♦✉❞
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