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❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ✐♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ▼✐❝❤❛❡❧ ▲❡s♥✐❝❦ ❘♦② ❩❤❛♦ ❆✉❣✉st ✶✺✱ ✷✵✶✽ ❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶ ✴ ✶✼

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts ❇❛❝❦❣r♦✉♥❞ ✶ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥ ❉✐✣❝✉❧t✐❡s ❈♦♠♣✉t✐♥❣ ❇✐✜❧tr❛t✐♦♥s ✷ ●r❛❞❡s ♦❢ ❆♣♣❡❛r❛♥❝❡ ❘❡s♦❧✈✐♥❣ ❘❡❧❛t✐♦♥s ❈♦♠♠❡♥ts ✸ ❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✷ ✴ ✶✼

❉❡✜♥✐t✐♦♥s ❉❡✜♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛rt✐❛❧ ♦r❞❡r ♦♥ N ✷ ✿ ( x ✶ , y ✶ ) ≤ ( x ✷ , y ✷ ) ✐✛ x ✶ ≥ x ✷ ❛♥❞ y ✶ ≤ y ✷ ✳ ❆ ❜✐✜❧tr❛t✐♦♥ B ✐s ❛ s❡t ♦❢ ✈❡❝t♦r s♣❛❝❡s { B P } P ∈ N ✷ s✉❝❤ t❤❛t B P ⊂ B Q ❢♦r ❛❧❧ P ≤ Q ✳ ●✐✈❡♥ ❛ s❡t ♦❢ ♣♦✐♥ts S ❛♥❞ ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧ ♥✉♠❜❡r t ∈ N ✱ t❤❡ t s❦❡❧❡t♦♥ ♦❢ S ✐s t❤❡ ❣r❛♣❤ ✇❤♦s❡ ✈❡rt✐❝❡s ❛r❡ t❤❡ ♣♦✐♥ts ✐♥ S ❛♥❞ ❛♥❞ ❡❞❣❡ ❡①✐sts ❜❡t✇❡❡♥ t✇♦ ✈❡rt✐❝❡s ✐❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♣♦✐♥ts ✐s ≤ t ✳ ■♥ t❤❡ ❞❡❣r❡❡✲❘✐♣s ❜✐✜❧tr❛t✐♦♥ ♦❢ ❛ s❡t ♦❢ ♣♦✐♥ts S ✱ ❛ ✵✲s✐♠♣❧❡① P ❡①✐sts ❛t ❛ ♣♦✐♥t ( d , t ) ✐❢ ✐♥ t❤❡ t s❦❡❧❡t♦♥ ♦❢ S ✱ ✇❡ ❤❛✈❡ ❞❡❣ ( P ) ≥ d ✳ ❆ s✐♠♣❧❡① σ ❡①✐sts ❛t ❛ ♣♦✐♥t ✐❢ ❛❧❧ ♦❢ ✐ts ♣♦✐♥ts ❛♥❞ ❛❧❧ ♦❢ ✐ts ❡❞❣❡s ❡①✐st✳ ❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✸ ✴ ✶✼

❊①❛♠♣❧❡ ✷ ✶ ✷ ❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✹ ✴ ✶✼

Pr♦♣♦s✐t✐♦♥ ❋♦r ❛♥② s✐♠♣❧❡① ✐♥ ❛ ❞❡❣r❡❡✲❘✐♣s ❜✐✜❧tr❛t✐♦♥ ♦❢ ❛ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts ✱ t❤❡r❡ ❡①✐sts ❛ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts s✉❝❤ t❤❛t ❡①✐sts ❛t ❛ ♣♦✐♥t ✐✛ s✉❝❤ t❤❛t ✳ ❚❤❡ ♣♦✐♥ts ✐♥ ❛ ♠✐♥✐♠❛❧ s✉❝❤ ❛r❡ ❝❛❧❧❡❞ t❤❡ ❣r❛❞❡s ♦❢ ❛♣♣❡❛r❛♥❝❡ ♦❢ ✳ ■♥❝♦♠♣❛r❛❜❧❡ ●r❛❞❡s ❲❤❛t ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts s✉❝❤ t❤❛t ♣♦✐♥t A ❛♣♣❡❛rs❄ ❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✺ ✴ ✶✼

■♥❝♦♠♣❛r❛❜❧❡ ●r❛❞❡s ❲❤❛t ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♣♦✐♥ts s✉❝❤ t❤❛t ♣♦✐♥t A ❛♣♣❡❛rs❄ Pr♦♣♦s✐t✐♦♥ ❋♦r ❛♥② s✐♠♣❧❡① σ ✐♥ ❛ ❞❡❣r❡❡✲❘✐♣s ❜✐✜❧tr❛t✐♦♥ ♦❢ ❛ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts S ✱ t❤❡r❡ ❡①✐sts ❛ ✜♥✐t❡ s❡t ♦❢ ♣♦✐♥ts Σ s✉❝❤ t❤❛t σ ❡①✐sts ❛t ❛ ♣♦✐♥t Q ✐✛ ∃ P ∈ Σ s✉❝❤ t❤❛t P ≤ Q ✳ ❚❤❡ ♣♦✐♥ts ✐♥ ❛ ♠✐♥✐♠❛❧ s✉❝❤ Σ ❛r❡ ❝❛❧❧❡❞ t❤❡ ❣r❛❞❡s ♦❢ ❛♣♣❡❛r❛♥❝❡ ♦❢ σ ✳ ❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✺ ✴ ✶✼

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts ❇❛❝❦❣r♦✉♥❞ ✶ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥ ❉✐✣❝✉❧t✐❡s ❈♦♠♣✉t✐♥❣ ❇✐✜❧tr❛t✐♦♥s ✷ ●r❛❞❡s ♦❢ ❆♣♣❡❛r❛♥❝❡ ❘❡s♦❧✈✐♥❣ ❘❡❧❛t✐♦♥s ❈♦♠♠❡♥ts ✸ ❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✻ ✴ ✶✼

❙t❡♣ ✶✿ ❉❡❛❧ ✇✐t❤ ✵✲❙✐♠♣❧✐❝❡s ❢✉♥❝t✐♦♥ ●❡♥❡r❛t❡❱❡rt❡①▼✉❧t✐❣r❛❞❡s✭ d , k ✮ d k ← { d ( k , j ) : j � = k } Grades k ← { ( ✵ , ✵ ) } sort ( d k ) ❢♦r i ← ✵ ; i < size ( d k ); i + + ❞♦ ✇❤✐❧❡ i + ✶ < size ( d k ) ❛♥❞ d k [ i + ✶ ] = d k [ i ] ❞♦ i + + ❡♥❞ ✇❤✐❧❡ Grades k ← Grades k ∪ { ( i + ✶ , d k ( i )) } ❡♥❞ ❢♦r ❡♥❞ ❢✉♥❝t✐♦♥ ❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✼ ✴ ✶✼

❙✇❡❡♣✐♥❣ ❧✐♥❡ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✳ ❋♦r ❡❛❝❤ ✱ ♠❛✐♥t❛✐♥ ♠❛① ♠✐♥ ♠✐♥ ✳ ❚❤✐s ♠❛①✐♠✉♠ ❝❤❛♥❣❡s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✇❡ ❤❛✈❡ ❡♥❝♦✉♥t❡r❡❞ ❛ ❣r❛❞❡ ♦❢ ❛♣♣❡❛r❛♥❝❡ ❢♦r ✳ ❙t❡♣ ✷✿ ■♥❞✉❝t t♦ n ✲❙✐♠♣❧✐❝❡s Pr♦♣♦s✐t✐♦♥ ▲❡t σ ❜❡ ❛ n ✲s✐♠♣❧❡① ✇✐t❤ n ≥ ✶✱ τ ❜❡ ❛ ❢❛❝❡ ♦❢ σ ❛♥❞ P = σ \ τ ✳ ▲❡t D = ♠❛① Q ∈ τ d ( P , Q ) ✳ ▲❡t S σ , S τ , S P ❜❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ✇❤❡r❡ σ, τ, P ❡①✐st r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♥ S σ = S τ ∩ S P ∩ { ( x , y ) : y ≥ D } ✳ ❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✽ ✴ ✶✼

❙t❡♣ ✷✿ ■♥❞✉❝t t♦ n ✲❙✐♠♣❧✐❝❡s Pr♦♣♦s✐t✐♦♥ ▲❡t σ ❜❡ ❛ n ✲s✐♠♣❧❡① ✇✐t❤ n ≥ ✶✱ τ ❜❡ ❛ ❢❛❝❡ ♦❢ σ ❛♥❞ P = σ \ τ ✳ ▲❡t D = ♠❛① Q ∈ τ d ( P , Q ) ✳ ▲❡t S σ , S τ , S P ❜❡ t❤❡ s❡t ♦❢ ♣♦✐♥ts ✇❤❡r❡ σ, τ, P ❡①✐st r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♥ S σ = S τ ∩ S P ∩ { ( x , y ) : y ≥ D } ✳ ❙✇❡❡♣✐♥❣ ❧✐♥❡ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✳ ❋♦r ❡❛❝❤ x ∈ N ✱ ♠❛✐♥t❛✐♥ ♠❛① ( D , ♠✐♥ ( x , y ) ∈ S P y , ♠✐♥ ( x , y ) ∈ S τ y ) ✳ ❚❤✐s ♠❛①✐♠✉♠ ❝❤❛♥❣❡s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✇❡ ❤❛✈❡ ❡♥❝♦✉♥t❡r❡❞ ❛ ❣r❛❞❡ ♦❢ ❛♣♣❡❛r❛♥❝❡ ❢♦r σ ✳ ❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✽ ✴ ✶✼

Ps❡✉❞♦✲❝♦❞❡ ❢✉♥❝t✐♦♥ ❈♦♠❜✐♥❡▼✉❧t✐❣r❛❞❡s✭ S P , S τ , D ✮ G ← {} i ✶ ← ✵ , i ✷ ← ✵ d = ♠❛① ( D , S P [ i ✶ ] . y , S τ [ i ✷ ] . y ) ✇❤✐❧❡ i ✶ < | S P | ❛♥❞ i ✷ < | S τ | ❞♦ minX ← ♠✐♥ ( S P [ i ✶ ] . x , S τ [ i ✷ ] . x ) ✐❢ S P [ i ✶ ] . x == minX t❤❡♥ i ✶ + + ❡♥❞ ✐❢ ✐❢ S τ [ i ✷ ] . x == minX t❤❡♥ i ✷ + + ❡♥❞ ✐❢ d ′ = ♠❛① ( D , S P [ i ✶ ] . y , S τ [ i ✷ ] . y ) ✐❢ d ′ > d t❤❡♥ G ← G ∪ { ( minX , d ) } d ← d ′ ❡♥❞ ✐❢ ❡♥❞ ✇❤✐❧❡ ❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✾ ✴ ✶✼ ❡♥❞ ❢✉♥❝t✐♦♥

▼❛t❤❡♠❛t✐❝❛❧ ❇❛❝❦❣r♦✉♥❞ ●✐✈❡♥ ❛ ❜✐✜❧tr❛t✐♦♥ B ✱ ✇❡ ❣❡t ❛ ❝❤❛✐♥ ❝♦♠♣❧❡① ♦❢ ✷✲❉ ❜✐♣❡rs✐st❡♥❝❡ ♠♦❞✉❧❡s d j + ✶ d j d j − ✶ → · · · d ✶ C j + ✶ → C j → C j − ✶ → C ✵ → ✵ ❛♥❞ H j ( B ) ∼ = ❦❡r d j / ✐♠ d j + ✶ . ❘■❱❊❚ r❡q✉✐r❡s t❤❡ C i t♦ ❜❡ ❢r❡❡ ♦r ♦♥❡✲❝r✐t✐❝❛❧✳ ❘♦② ❩❤❛♦ ❈♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❉❡❣r❡❡✲❘✐♣s ❇✐✜❧tr❛t✐♦♥s ❆✉❣✉st ✶✺✱ ✷✵✶✽ ✶✵ ✴ ✶✼