▼♦❞❡❧✐♥❣ ❛♥❞ ▼❡❛s✉r✐♥❣ ●r❛♣❤ ❙✐♠✐❧❛r✐t②✿ ❚❤❡ ❈❛s❡ ❢♦r ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡ ❚❤❡♦r❡t✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥❛❧ ❋❧✉✐❞ ❉②♥❛♠✐❝s ▲❛❜♦r❛t♦r② ✕ ❇❧❛✐r P❡r♦t ▼❛tt❤✐❡✉ ❘♦② ✶ ✱ ❙t❡❢❛♥ ❙❝❤♠✐❞ ✷ ✱ ●✐❧❧❡s ❚r❡❞❛♥ ✶ ✶ − ▲❛❜♦r❛t♦r② ❢♦r ❆♥❛❧②s✐s ❛♥❞ ❆r❝❤✐t❡❝t✉r❡ ♦❢ ❙②st❡♠s ✷ − ❚❯✲❇❡r❧✐♥✴❉❡✉ts❝❤❡ ❚❡❧❡❦♦♠ ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
❉②♥❛♠✐❝ ●r❛♣❤s ❊✈❡r②✇❤❡r❡ ❍✉❣❡ ❲❡ ❝❛♥✬t ♠❡❛s✉r❡ t❤❡♠ ❡❛s✐❧② ❉✐st❛♥❝❡ ❂❘♦♦t ♦❢ ❉②♥❛♠✐s♠ ❈❤❛r❛❝t❡r✐s❛t✐♦♥ ■♥t❡r♣♦❧❛t✐♦♥ ❊①tr❛♣♦❧❛t✐♦♥ ❈♦♦r❞✐♥❛t❡ ❙②st❡♠✱✳✳✳✳ ❲❤✐❝❤ ❞✐st❛♥❝❡s ❢♦r ❣r❛♣❤s ❄ ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
▼♦❞❡❧ ◆❛♠❡❞ ●r❛♣❤s✿ ❆❧✐❝❡✕❊✈❡✕❇♦❜ ! = ❆❧✐❝❡✕❇♦❜✕❊✈❡ ❯♥❞✐r❡❝t❡❞ ❣r❛♣❤s ✭s❤♦✉❧❞ ❜❡ ♦❦ ❢♦r ❞✐r❡❝t❡❞ ♦♥❡s✮ G 0 G 1 ❆❧❧ ♥♦❞❡s ♣r❡s❡♥t ❛t t = ✵ ✧❉②♥❛♠✐❝✐t②✧✿ str❡❛♠ ♦❢ ❡❞❣❡ G 2 ❛❞❞✐t✐♦♥s✴❞❡❧❡t✐♦♥s✿ ● ✵ = ( ❱ , ❊ ✵ ) , ● ✶ = ( ❱ , ❊ ✶ ) , . . . ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
●r❛♣❤ ❊❞✐t ❉✐st❛♥❝❡ ●r❛♣❤ ❊❞✐t ❉✐st❛♥❝❡ ❂ ❞ ●❊❉ ❖♥❧② ❦♥♦✇♥ ♣r♦♣❡r ❣r❛♣❤ ❞✐st❛♥❝❡ ❞ ●❊❉ ( ❆ , ❇ ) = ♥✉♠❜❡r ♦❢ ❣r❛♣❤ ❡❞✐t ♦♣❡r❛t✐♦♥s ❢r♦♠ ❆ t♦ ❇ ◆❛♠❡❞ ❣r❛♣❤s ⇔ ❈❤❡❛♣ ❚♦♦ ✧❜❧✉♥t✧ � � ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
❈❡♥tr❛❧✐t✐❡s ❇♦r❣❛tt✐✱❊✈❡r❡tt ✭✷✵✵✻✮✿ ✧t❤❡ ♦♥❧② t❤✐♥❣ ♣❡♦♣❧❡ ❛❣r❡❡ ❛❜♦✉t ❛ ❝❡♥tr❛❧✐t② ✐s t❤❛t ✐t ✐s ❛ ♥♦❞❡✲❧❡✈❡❧ ♠❡❛s✉r❡✧✳ ▲♦✈❡❞ ❜② ❙◆❆♥❛❧②sts ❈❛♠✐❧❧❡ ❏♦r❞❛♥ ❉❡✜♥✐t✐♦♥ ✭❈❡♥tr❛❧✐t②✮ ❆ ❝❡♥tr❛❧✐t② ❈ ✐s ❛ ❢✉♥❝t✐♦♥ ❈ : ( ● , ✈ ) → R + t❤❛t t❛❦❡s ❛ ❣r❛♣❤ ● = ( ❱ , ❊ ) ❛♥❞ ❛ ✈❡rt❡① ✈ ∈ ❱ ( ● ) ❛♥❞ r❡t✉r♥s ❛ ♣♦s✐t✐✈❡ ✈❛❧✉❡ ❈ ( ● , ✈ ) ✳ ❉❡❣r❡❡ ❝ ❞ ( ● , ✐ ) = ❞❡❣r❡❡ ( ✐ ) ❈❧♦s❡♥❡ss ❝ ❝ ( ● , ✐ ) = � ❥ � = ✐ ❞ ● ( ✐ , ❥ ) ❇❡t✇❡❡♥♥❡ss ❝ ❝ ( ● , ✐ ) = � ❥ � = ✐ , ❦ � = ✐ δ ✐ ∈ s♣ ( ❥ , ❦ ) ✧●r❛♣❤ ❜❡t✇❡❡♥♥❡ss✧ ✲ ❈❧❛✉❞✐♦ ❘♦❝❝❤✐♥✐ ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
❲❤❛t ❆❜♦✉t ❖t❤❡r ❉✐st❛♥❝❡s ❄ ■♠❛❣✐♥❡ ❛ ❣r❛♣❤ ❞✐st❛♥❝❡ ❞ : G ✷ → R + G 1 d d(G ,G ) 1 2 ■ ❝❛♥ ❝♦♥str✉❝t ②♦✉ ❛ ❝❡♥tr❛❧✐t② ♦✉t ♦❢ G 2 ✐t ✦ ❝ ❞ ( ● , ✈ ) = ❞ ( ● , ● − { ✐ } ) G G C(G,i) d(G ,G ) i G-{i} 1 2 ❱❡r② ✐♥tr✐❣✉✐♥❣✿ ❞ ●❊❉ ( ● , ● − { ✐ } ) = ❞❡❣r❡❡ ● ( ✐ ) ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
❆ ❈♦♥♥❡❝t✐♦♥ ❇❡t✇❡❡♥ ❇♦t❤ ❈♦♥❝❡♣ts ❄ ❲❡ ❤❛✈❡ ♠❛♥② ❝❡♥tr❛❧✐t✐❡s✱ ❢❡✇ ❞✐st❛♥❝❡s✳ ❈❛♥ ✇❡ ❝♦♥str✉❝t t❤❡ ♦t❤❡r ✇❛② r♦✉♥❞ ❄ ❨❡s✦ ❉❡✜♥✐t✐♦♥ ✭❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡✮ ●✐✈❡♥ ❛ ❝❡♥tr❛❧✐t② ❈ ✱ ✇❡ ❞❡✜♥❡ t❤❡ ❝❡♥tr❛❧✐t② ❞✐st❛♥❝❡ ❞ ❈ ( ● ✶ , ● ✷ ) ❜❡t✇❡❡♥ t✇♦ ♥❡✐❣❤❜♦r✐♥❣ ❣r❛♣❤s ❛s t❤❡ ❝♦♠♣♦♥❡♥t✲✇✐s❡ ❞✐✛❡r❡♥❝❡✿ � ∀ ( ● ✶ , ● ✷ ) ∈ ❊ ( G ) , ❞ ❈ ( ● ✶ , ● ✷ ) = | ❈ ( ● ✶ , ✈ ) − ❈ ( ● ✷ , ✈ ) | . ✈ ∈ ❱ ◆❛t✉r❛❧ ❡①t❡♥s✐♦♥ ❢♦r ♥♦♥✲♥❡✐❣❤❜♦r✐♥❣ ❣r❛♣❤ ❝♦✉♣❧❡s✿ ❞ ❈ ( ● ✶ , ● ✷ ) ❂ ❣r❛♣❤✲✐♥❞✉❝❡❞ ❞✐st❛♥❝❡ ♦♥ t❤❡ ✈❛❧✉❡❞ ❣r❛♣❤ G ✳ ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
6.3 2.6 8.4 1.2 ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
G 0 G 1 G 2 ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
G 0 G 1 1.0 8.4 2.6 G 2 6.3 3.1 1.2 4.6 ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
G 1 G 0 1.0 8.4 2.6 G 2 6.3 3.1 1.0 1.2 4.6 1.7 1.5 0.8 0.3 ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
❈♦♥♥❡❝t✐♦♥ ❈♦♥t✳ ❉❡✜♥✐t✐♦♥ ✭❙❡♥s✐t✐✈❡ ❈❡♥tr❛❧✐t②✮ ❈❡♥tr❛❧✐t② ❈ ✐s s❡♥s✐t✐✈❡ ✐✛ ∀ ● ∈ G , ∀ ❡ ∈ ❊ ( ● ) , ∃ ✈ ∈ ❱ ( ● ) s✳t✳ ❈ ( ● , ✈ ) � = ❈ ( ● \ { ❡ } , ✈ ) , ❞ ❈ ✐s ❛ ❞✐st❛♥❝❡ ✐✛ ❈ ✐s s❡♥s✐t✐✈❡ ◆♦t ❛❧❧ s❡♥s✐t✐✈❡ ✦ ❊①✿ ❊①❝❡♥tr✐❝✐t② ❙♦♠❡ ❝❡♥tr❛❧✐t✐❡s ♥❡❡❞ ❛❞❛♣t❛t✐♦♥s ❆♣♣r♦①✐♠❛t❡ ✭❝❤❡❛♣✮ ✈❡rs✐♦♥✿ � � ∀ ( ● ✶ , ● ✷ ) , ❞ ❈ ( ● ✶ , ● ✷ ) = | ❈ ( ● ✶ , ✈ ) − ❈ ✷ ( ● ✷ , ✈ ) | . ✈ ∈ ❱ ❝♦✲❝❡♥tr❛❧ ❣r❛♣❤s✦ ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
❊①♣❡r✐♠❡♥ts ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
❚♦♣♦❧♦❣② ❡✈♦❧✉t✐♦♥ ❉✐✛❡r❡♥t✐❛t❡ ❜❡t✇❡❡♥ ♣❛t❤s ✉s✐♥❣ ❝❡♥tr❛❧✐t②✲✐♥❞✉❝❡❞ ❞✐st❛♥❝❡s ❚❤❡s❡ ♣❛t❤s ❛r❡ ❡q✉✐✈❛❧❡♥t ✇rt ❞ ●❊❉ ❨❡t t❤❡② ✇♦✉❧❞♥✬t ✐♠♣❛❝t ♥❡t✇♦r❦s t❤❡ s❛♠❡ ✇❛②✳ ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
❚♦♣♦❧♦❣② ❡✈♦❧✉t✐♦♥ ❈♦♥t✳ Order: Dichotomic Incremental 0.3 0.2 BC 0.1 0.0 d c 0.08 0.06 0.04 CC 0.02 0.00 0 10 20 30 Round ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
❉②♥❛♠✐❝ ❚♦♣♦❧♦❣✐❡s ✷ ❉❛t❛s❡ts ❋❛❝❡❜♦♦❦ ❧✐❦❡ ❖❙◆ ❖♥❧✐♥❡ ♠❡ss❛❣❡s 4000 ❡①❝❤❛♥❣❡ Degree Distance FB ≈ ✷✵❦ ✉s❡rs 2000 ✶✽✼ s♥❛♣s❤♦ts✱ ✶ ❞❛② 0 s❛♠♣❧✐♥❣ 60 50 ❙♦✉❦ ♠♦❜✐❧✐t② ❞❛t❛s❡t 40 Souk ❙♦❝✐❛❧ ❝♦♥t❛❝ts ✇✐t❤✐♥ 30 ❛ ❝r♦✇❞ 20 ✹✺ ✐♥❞✐✈✐❞✉❛❧s 0 100 200 300 ✸✵✵ s♥❛♣s❤♦ts✱ ✸ s❡❝✳ Round s❛♠♣❧✐♥❣ ▼❛tt❤✐❡✉ ❘♦②✱ ❙t❡❢❛♥ ❙❝❤♠✐❞✱ ●✐❧❧❡s ❚r❡❞❛♥ ❈❡♥tr❛❧✐t② ❉✐st❛♥❝❡
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