❉▼✽✻✺ ✖ ❍❡✉r✐st✐❝s ❛♥❞ ❆♣♣r♦①✐♠❛t✐♦♥ ❆❧❣♦r✐t❤♠s ❚❡❛❝❤❡rs✿ ▼❛r❝♦ ❈❤✐❛r❛♥❞✐♥✐ ✫ ▲❡♥❡ ❋❛✈r❤♦❧❞t ❈r❡❞✐ts✿ ✶✵ ❊❈❚❙ Pr❡r❡q✉✐s✐t❡s✿ � Pr♦❣r❛♠♠✐♥❣ ✭❉▼✺✵✷✱ ❉▼✺✵✸✱ ❉▼✺✺✵✮ � ❆❧❣♦r✐t❤♠s ❛♥❞ ❉❛t❛str✉❝t✉r❡s ✭❉▼✺✵✼✮ � ❈♦♠♣❧❡①✐t② ❛♥❞ ❈♦♠♣✉t❛❜✐❧✐t② ✭❉▼✺✵✽✱ ❉▼✺✺✸✮ � ▲✐♥❡❛r ❛♥❞ ■♥t❡❣❡r Pr♦❣r❛♠♠✐♥❣ ✭❉▼✺✺✾✱ ❉▼✺✹✺✱ ❉▼✺✺✹✱ ❉▼✽✼✶✮ ▲❛♥❣✉❛❣❡✿ ❊♥❣❧✐s❤ ❛♥❞✴♦r ❉❛♥✐s❤ ❈❧❛ss❡s✿ ■♥tr♦✿ ✷ h × ✷✵❀ tr❛✐♥✐♥❣✿ ✷ h × ✶✺ ▼❛t❡r✐❛❧✿ ❙❧✐❞❡s ✰ t❡①t❜♦♦❦ ✰ ❛rt✐❝❧❡s ✰ st❛rt✐♥❣ ❝♦❞❡ ❆ss❡ss♠❡♥t✿ Pr❛❝t✐❝❛❧ ♣r♦❥❡❝ts ✰ ♦r❛❧ ❡①❛♠
❆♣♣r♦①✐♠❛t✐♦♥ ❆❧❣♦r✐t❤♠s ❆ ✷ ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r ▼❡tr✐❝ ❚❙P✿ ❉♦✉❜❧❡✲❚r❡❡ ❉♦✉❜❧❡ tr❡❡ ❛❧❣♦r✐t❤♠✿ ✶✳ T ← ▼❙❚ ✷✳ ❉♦✉❜❧❡ ❛❧❧ ❡❞❣❡s ✐♥ T ✸✳ E tour ← ❊✉❧❡r t♦✉r ✹✳ ❚♦✉r ← ✈❡rt✐❝❡s ✐♥ ♦r❞❡r ♦❢ ❛♣♣❡❛r❛♥❝❡ ✐♥ E tour
❆♣♣r♦①✐♠❛t✐♦♥ ❆❧❣♦r✐t❤♠s ❆ ✷ ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r ▼❡tr✐❝ ❚❙P✿ ❉♦✉❜❧❡✲❚r❡❡ ▼❙❚ ❚♦✉r c ( ▼❙❚ ) ≤ c ( ❖P❚ ) : c ( ❚♦✉r ) ≤ ✷ · c ( MST ) ❖P❚ → ❙❚
❉♦✉❜❧❡ tr❡❡ ❛❧❣♦r✐t❤♠✿ ✶✳ T ← ▼❙❚ ✷✳ ❉♦✉❜❧❡ ❛❧❧ ❡❞❣❡s ✐♥ T ✸✳ E tour ← ❊✉❧❡r t♦✉r ✹✳ ❚♦✉r ← ✈❡rt✐❝❡s ✐♥ ♦r❞❡r ♦❢ ❛♣♣❡❛r❛♥❝❡ ✐♥ E tour ❈❤r✐st♦✜❞❡✬s ❛❧❣♦r✐t❤♠✿ ✶✳ T ← MST ✷✳ ❆❞❞ ♠✐♥✐♠✉♠ ♣❡r❢❡❝t ♠❛t❝❤✐♥❣ ♦❢ ♦❞❞ ❞❡❣r❡❡ ✈❡rt✐❝❡s ✐♥ T ✸✳ E tour ← ❊✉❧❡r t♦✉r ✹✳ ❚♦✉r ← ✈❡rt✐❝❡s ✐♥ ♦r❞❡r ♦❢ ❛♣♣❡❛r❛♥❝❡ ✐♥ E tour
❆♣♣r♦①✐♠❛t✐♦♥ ❆❧❣♦r✐t❤♠s ❆ ✸ / ✷ ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r ▼❡tr✐❝ ❚❙P✿ ❈❤r✐st♦✜❞❡s c ( M ) ≤ ✶ ✷❖P❚ : c ( M ) ≤ ✶ ✷ c ( ❚♦✉r ) ≤ c ( ❚♦✉r ) ≤ c ( ❖P❚ )
❆♣♣r♦①✐♠❛t✐♦♥ ❆❧❣♦r✐t❤♠s ▲♦✇❡r ❜♦✉♥❞s ❚❤❡♦r❡♠ ❋♦r α < ✶✽✺ ✶✽✹ ✱ t❤❡r❡ ✐s ♥♦ α ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r ▼❡tr✐❝ ❚❙P✳ ❚❤❡♦r❡♠ ❚❤❡r❡ ✐s ♥♦ ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r ❣❡♥❡r❛❧ ❚❙P✳
▲♦❝❛❧ ❙❡❛r❝❤
▼❡t❛❤❡✉r✐st✐❝s ❚r②✐♥❣ ❞✐✛❡r❡♥t ❝❤❛♥❣❡s ✴ ❛❝❝❡♣t✐♥❣ ✇♦rs❡♥✐♥❣ ❝❤❛♥❣❡s✿
❈♦♥t❡♥ts ❆♣♣r♦① ❆❧❣♦r✐t❤♠s ▲♦❝❛❧ ❙❡❛r❝❤ ✰ ▼❡t❛❤❡✉r✐st✐❝s � ❙❡t ❈♦✈❡r � � ❙❛t✐s✜❛❜✐❧✐t② � � ❚r❛✈❡❧✐♥❣ ❙❛❧❡s♠❛♥ � � ❙❝❤❡❞✉❧✐♥❣ � ❑♥❛♣s❛❝❦ � ❇✐♥ ♣❛❝❦✐♥❣
❉▼✽✻✺ ✖ ❍❡✉r✐st✐❝s ❛♥❞ ❆♣♣r♦①✐♠❛t✐♦♥ ❆❧❣♦r✐t❤♠s ❚❡❛❝❤❡rs✿ ▼❛r❝♦ ❈❤✐❛r❛♥❞✐♥✐ ✫ ▲❡♥❡ ❋❛✈r❤♦❧❞t ❈r❡❞✐ts✿ ✶✵ ❊❈❚❙ Pr❡r❡q✉✐s✐t❡s✿ � Pr♦❣r❛♠♠✐♥❣ ✭❉▼✺✵✷✱ ❉▼✺✵✸✱ ❉▼✺✺✵✮ � ❆❧❣♦r✐t❤♠s ❛♥❞ ❉❛t❛str✉❝t✉r❡s ✭❉▼✺✵✼✮ � ❈♦♠♣❧❡①✐t② ❛♥❞ ❈♦♠♣✉t❛❜✐❧✐t② ✭❉▼✺✵✽✱ ❉▼✺✺✸✮ � ▲✐♥❡❛r ❛♥❞ ■♥t❡❣❡r Pr♦❣r❛♠♠✐♥❣ ✭❉▼✺✺✾✱ ❉▼✺✹✺✱ ❉▼✺✺✹✱ ❉▼✽✼✶✮ ▲❛♥❣✉❛❣❡✿ ❊♥❣❧✐s❤ ❛♥❞✴♦r ❉❛♥✐s❤ ❈❧❛ss❡s✿ ■♥tr♦✿ ✷ h × ✷✵❀ tr❛✐♥✐♥❣✿ ✷ h × ✶✺ ▼❛t❡r✐❛❧✿ ❙❧✐❞❡s ✰ t❡①t❜♦♦❦ ✰ ❛rt✐❝❧❡s ✰ st❛rt✐♥❣ ❝♦❞❡ ❆ss❡ss♠❡♥t✿ Pr❛❝t✐❝❛❧ ♣r♦❥❡❝ts ✰ ♦r❛❧ ❡①❛♠
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