k tr r ss str - - PowerPoint PPT Presentation

❆ ✭✶ ✰ ε✮✲❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r 1✲❝❡♥t❡r ✐♥ Rd

❙❛♥❣✲❙✉❜ ❑✐♠ ❇❛r❜❛r❛ ❙❝❤✇❛r③✇❛❧❞ ❊✉r♦❈● ✷✵✷✵✱ ❲ür③❜✉r❣✱ ▼❛r❝❤ ✶✻✲✶✽✱ ✷✵✷✵

❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✱ ●❡r♠❛♥② s❝❤✇❛r③✇❛❧❞❅✉♥✐✲❜♦♥♥✳❞❡

❚❤❡ k✲❝❡♥t❡r ♣r♦❜❧❡♠

• ❈❧❛ss✐❝ ❝❧✉st❡r✐♥❣ ♣r♦❜❧❡♠
• ❙❡t ♦❢ ♣♦✐♥t P ✐♥ Rd
• ❋✐♥❞ k ❜❛❧❧s ♦❢ ♠✐♥✐♠✉♠ r❛❞✐✉s r∗

❝♦✈❡r✐♥❣ ❛❧❧ ♣♦✐♥ts ♦❢ P ⇔ ❋✐♥❞ s❡t ♦❢ ❝❡♥t❡rs C ♠✐♥✐♠✐③✐♥❣ max

p∈P min c∈C d(p, c)

❣❡♥❡r❛❧ k k = 1 ❜♦✉♥❞❡❞ d ◆P✲❤❛r❞ O(n) ❆P❳✲❤❛r❞ ❣❡♥❡r❛❧ d ◆P✲❤❛r❞ ♥♦ ❛❧❣♦r✐t❤♠ ❆P❳✲❤❛r❞ ♣♦❧② ✐♥ ❞

✶

❆♣♣r♦①✐♠❛t✐♥❣ t❤❡ 1✲❝❡♥t❡r ♣r♦❜❧❡♠

c∗

• ❙❡t ♦❢ ♣♦✐♥ts P ✐♥ Rd
• ❋✐♥❞ ❛ ❜❛❧❧ ♦❢ ♠✐♥✐♠✉♠ r❛❞✐✉s r∗

❝♦✈❡r✐♥❣ ❛❧❧ ♣♦✐♥ts ♦❢ P ⇔ ❋✐♥❞ ❝❡♥t❡r c∗ ♠✐♥✐♠✐③✐♥❣ max

p∈P d(p, 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 + 1 ε4.5 log 1 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 3✱ . . .✱ 1 i+1 ✐♥ st❡♣ i

• ❆r✐t❤♠❡t✐❝ ♠❡❛♥ ♦❢ ❛❧❧ ♣♦✐♥ts s♦ ❢❛r
• ❘❡♣❡❛t ⌈ 1

ε2 ⌉ st❡♣s

• ✭✶✰ε✮ ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ t✐♠❡ O( nd

ε2 ) ✸

❙tr✉❝t✉r❛❧ Pr♦♣❡rt② ▲❡♠♠❛ ✭❋✐rst st❡♣✮

l B B′ ≥ |l| m m′ r′ c r

⊲ ●✐✈❡♥ ❛ ❜❛❧❧ B = B(c, r) ⊲ P❧❛❝❡ ❛ ❧✐♥❡ s❡❣♠❡♥t l ✐♥ B ❛s ❢❛r ❛✇❛② ❢r♦♠ c ❛s ♣♦ss✐❜❧❡ ⊲ ❚❤✐s ❞❡✜♥❡s ❛ ❜❛❧❧ B′ = B(c, r′) t❛♥❣❡♥t✐❛❧ t♦ l ✐♥ ✐ts ♠✐❞♣♦✐♥t m ⊲

r′ r ♦♥❧② ❞❡♣❡♥❞s ♦♥ |l| r

⊲ ❆♥② ❧✐♥❡ s❡❣♠❡♥t ❢✉❧❧② ✐♥ B ✇✐t❤ ❧❡♥❣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❡♣✮

B B′ ≥ |l| m′ r′ c r B′′ r′′ m l

• ●✐✈❡♥ t✇♦ ❜❛❧❧s B = B(c, r) ❛♥❞

B′ = (c, r′) ❛r♦✉♥❞ t❤❡ s❛♠❡ ❝❡♥t❡r

• P❧❛❝❡ ❛ ❧✐♥❡ s❡❣♠❡♥t l ✐♥ ✐t

❛s ❢❛r ❛✇❛② ❢r♦♠ c ❛s ♣♦ss✐❜❧❡ ✇✐t❤ ♦♥❡ ❡♥❞♣♦✐♥t ✐♥ B′

• ❚❤✐s ❞❡✜♥❡s B′′ = B(c, r′′)

t❛♥❣❡♥t✐❛❧ t♦ l ❛t s♦♠❡ ♣♦✐♥t m

• 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 ❣r❛❞✐❡♥t ❞❡s❝❡♥t O( nd

ε )

O(nd k

k ε )

❛❧❣♦r✐t❤♠ ✉s✐♥❣ ⌊ 2

ε⌋ ♣♦✐♥ts

❣✉❡ss✐♥❣ ♣♦✐♥ts ❝♦r❡s❡t O( nd

ε + 1 ε4.5 log 1 ε)

O(nd k

k ε )

❛❧❣♦r✐t❤♠ ✉s✐♥❣ O( 1

ε) ♣♦✐♥ts

❣✉❡ss✐♥❣ ♣♦✐♥ts

❚❤❛♥❦ ②♦✉✦

✻