❆ ✭✶ ✰ ε ✮✲❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r 1 ✲❝❡♥t❡r ✐♥ R d ❙❛♥❣✲❙✉❜ ❑✐♠ ❇❛r❜❛r❛ ❙❝❤✇❛r③✇❛❧❞ ❊✉r♦❈● ✷✵✷✵✱ ❲ür③❜✉r❣✱ ▼❛r❝❤ ✶✻✲✶✽✱ ✷✵✷✵ ❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✱ ●❡r♠❛♥② s❝❤✇❛r③✇❛❧❞❅✉♥✐✲❜♦♥♥✳❞❡
❚❤❡ k ✲❝❡♥t❡r ♣r♦❜❧❡♠ • ❈❧❛ss✐❝ ❝❧✉st❡r✐♥❣ ♣r♦❜❧❡♠ • ❙❡t ♦❢ ♣♦✐♥t P ✐♥ R d • ❋✐♥❞ k ❜❛❧❧s ♦❢ ♠✐♥✐♠✉♠ r❛❞✐✉s r ∗ ❝♦✈❡r✐♥❣ ❛❧❧ ♣♦✐♥ts ♦❢ P ⇔ ❋✐♥❞ s❡t ♦❢ ❝❡♥t❡rs C ♠✐♥✐♠✐③✐♥❣ max p ∈ P min c ∈ C d ( p, c ) k = 1 ❣❡♥❡r❛❧ k ❜♦✉♥❞❡❞ d ◆P✲❤❛r❞ O ( n ) ❆P❳✲❤❛r❞ ◆P✲❤❛r❞ ❣❡♥❡r❛❧ d ♥♦ ❛❧❣♦r✐t❤♠ ❆P❳✲❤❛r❞ ♣♦❧② ✐♥ ❞ ✶
❆♣♣r♦①✐♠❛t✐♥❣ t❤❡ 1 ✲❝❡♥t❡r ♣r♦❜❧❡♠ • ❙❡t ♦❢ ♣♦✐♥ts P ✐♥ R d • ❋✐♥❞ ❛ ❜❛❧❧ ♦❢ ♠✐♥✐♠✉♠ r❛❞✐✉s r ∗ ❝♦✈❡r✐♥❣ ❛❧❧ ♣♦✐♥ts ♦❢ P ⇔ ❋✐♥❞ ❝❡♥t❡r c ∗ ♠✐♥✐♠✐③✐♥❣ max p ∈ P d ( p, c ∗ ) c ∗ • p ✇✐t❤ d ( p, c ∗ ) < εr ∗ ✐s ❛ (1 + ε ) ✲❛♣♣r♦①✐♠❛t✐♦♥ • ❆♥② ♣♦✐♥t p ∈ P ✐s ❛ ✷✲❛♣♣r♦①✐♠❛t✐♦♥ • ■❞❡❛✿ ❯s❡ ❢❛rt❤❡st ♣♦✐♥t t♦ ✐♠♣r♦✈❡ • ❇✞ ❛❞♦✐✉ ❡t✳❛❧✳✱ ❑✉♠❛r ❡t✳❛❧✳✭✷✵✵✼✮✿ ❇✉✐❧❞ ❝♦r❡✲s❡t ♦❢ s✐③❡ O ( 1 ε ) ✐♥ t✐♠❡ O ( nd ε 4 . 5 log 1 1 ε + ε ) • ❙✐♠♣❧❡r ❣r❛❞✐❡♥t ❞❡s❝❡♥t❄ ✷
◆❛✐✈❡ ●r❛❞✐❡♥t✲❉❡s❝❡♥t • ❇❡❣✐♥ ✇✐t❤ ❛r❜✐tr❛r② ♣♦✐♥t p ∈ P • ❋✐♥❞ ❢❛rt❤❡st ♣♦✐♥t • ▼♦✈❡ ❝❡♥t❡r str❛✐❣❤t t♦✇❛r❞s ✐t 1 2 ✱ 1 1 3 ✱ . . . ✱ i +1 ✐♥ st❡♣ i • • ❆r✐t❤♠❡t✐❝ ♠❡❛♥ ♦❢ ❛❧❧ ♣♦✐♥ts s♦ ❢❛r • ❘❡♣❡❛t ⌈ 1 ε 2 ⌉ st❡♣s • ✭✶✰ ε ✮ ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ t✐♠❡ O ( nd ε 2 ) ✸
❙tr✉❝t✉r❛❧ Pr♦♣❡rt② ▲❡♠♠❛ ✭❋✐rst st❡♣✮ ⊲ ●✐✈❡♥ ❛ ❜❛❧❧ B = B ( c, r ) B ⊲ P❧❛❝❡ ❛ ❧✐♥❡ s❡❣♠❡♥t l ✐♥ B l ❛s ❢❛r ❛✇❛② ❢r♦♠ c ❛s ♣♦ss✐❜❧❡ m ⊲ ❚❤✐s ❞❡✜♥❡s ❛ ❜❛❧❧ B ′ = B ( c, r ′ ) r ′ m ′ t❛♥❣❡♥t✐❛❧ t♦ l ✐♥ ✐ts ♠✐❞♣♦✐♥t m ≥ | l | r ′ r ♦♥❧② ❞❡♣❡♥❞s ♦♥ | l | ⊲ c B ′ r ⊲ ❆♥② ❧✐♥❡ s❡❣♠❡♥t ❢✉❧❧② ✐♥ B ✇✐t❤ r ❧❡♥❣t❤ ≥ | l | ❤❛s ✐ts ♠✐❞♣♦✐♥t ✐♥ B ′ ⊲ ❈❛♥ ❛ss✉♠❡ ✜rst t✇♦ ♣♦✐♥ts ❤❛✈❡ ❞✐st❛♥❝❡ > (1 + ε ) r ∗ ⊲ ▼✐❞♣♦✐♥t ❧✐❡s ✐♥ ❜❛❧❧ B ′ ✇✐t❤ r ′ r ❞❡✜♥❡❞ ❜② | l | r = 1+ ε 1 ✹
❙tr✉❝t✉r❛❧ Pr♦♣❡rt② ▲❡♠♠❛✭❊✈❡r② ❢✉rt❤❡r st❡♣✮ • ●✐✈❡♥ t✇♦ ❜❛❧❧s B = B ( c, r ) ❛♥❞ B B ′ = ( c, r ′ ) ❛r♦✉♥❞ t❤❡ s❛♠❡ ❝❡♥t❡r l • P❧❛❝❡ ❛ ❧✐♥❡ s❡❣♠❡♥t l ✐♥ ✐t m ❛s ❢❛r ❛✇❛② ❢r♦♠ c ❛s ♣♦ss✐❜❧❡ ✇✐t❤ ♦♥❡ ❡♥❞♣♦✐♥t ✐♥ B ′ m ′ r ′′ B ′′ • ❚❤✐s ❞❡✜♥❡s B ′′ = B ( c, r ′′ ) c B ′ ≥ | l | t❛♥❣❡♥t✐❛❧ t♦ l ❛t s♦♠❡ ♣♦✐♥t m r r ′ r ′′ r ❞❡♣❡♥❞s ♦♥ | l | r ❛♥❞ r ′ • r • ❆♥② ❧✐♥❡ s❡❣♠❡♥t ❢✉❧❧② ✐♥ B ✇✐t❤ ❧❡♥❣t❤ ≥ | l | ❛♥❞ ♦♥❡ ❡♥❞♣♦✐♥t ✐♥ B ′ ❤❛s t❤❡ ♣♦✐♥t m ′ ❛t t❤❡ s❛♠❡ r❡❧❛t✐✈❡ ♣♦s✐t✐♦♥ ❛s m ✐♥s✐❞❡ B ′′ ◆❡❡❞ t♦ ❦♥♦✇ r ∗ ❄ ◆♦✱ t❤❡ r❡❧❛t✐✈❡ ♣♦s✐t✐♦♥ ♦❢ ♠ ❛❧s♦ ❞❡♣❡♥❞s ♦♥❧② ♦♥ | l | r ❛♥❞ r ′ r ✺
❘❡s✉❧t 1 ✲❝❡♥t❡r k ✲❝❡♥t❡r k O ( nd ε ) ❣r❛❞✐❡♥t ❞❡s❝❡♥t ε ) O ( nd k ✉s✐♥❣ ⌊ 2 ❛❧❣♦r✐t❤♠ ε ⌋ ♣♦✐♥ts ❣✉❡ss✐♥❣ ♣♦✐♥ts k O ( nd ε 4 . 5 log 1 1 ε ) ❝♦r❡s❡t ε + ε ) O ( nd k ✉s✐♥❣ O ( 1 ❛❧❣♦r✐t❤♠ ε ) ♣♦✐♥ts ❣✉❡ss✐♥❣ ♣♦✐♥ts ❚❤❛♥❦ ②♦✉✦ ✻
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