SLIDE 1
❈♦♥s❡♥s✉s ❙tr✐♥❣s ✇✐t❤ ❙♠❛❧❧ ▼❛①✐♠✉♠ ❉✐st❛♥❝❡ ❛♥❞ ❙♠❛❧❧ ❉✐st❛♥❝❡ ❙✉♠
▲❛✉r❡♥t ❇✉❧t❡❛✉1✱ ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞2
1 ❯♥✐✈❡rs✐té P❛r✐s✲❊st✱ ❋r❛♥❝❡ 2 ❚r✐❡r ❯♥✐✈❡rs✐t②✱ ●❡r♠❛♥②
sss trs t - - PowerPoint PPT Presentation
sss trs t - - PowerPoint PPT Presentation
sss trs t st st rt t 1 rs
SLIDE 2
SLIDE 3
❈♦♥s❡♥s✉s ❙tr✐♥❣ Pr♦❜❧❡♠s
■♥♣✉t✿ ❆ s❡t ✭♠✉❧t✐✲s❡t✮ ♦❢ str✐♥❣s✳ ❖✉t♣✉t✿ ❆ str✐♥❣ t❤❛t ✐s ❛ ❣♦♦❞ ❝♦♥s❡♥s✉s ♦❢ t❤❡ ✐♥♣✉t str✐♥❣s✳ s1 c b c a b a a s2 c b c a b c b s3 a b c c c c a s4 c c c a b c a s5 c b c a a c a s6 c b c a a c a s7 a b b a b a a s8 b b c a a c a
SLIDE 4
❈♦♥s❡♥s✉s ❙tr✐♥❣ Pr♦❜❧❡♠s
■♥♣✉t✿ ❆ s❡t ✭♠✉❧t✐✲s❡t✮ ♦❢ str✐♥❣s✳ ❖✉t♣✉t✿ ❆ str✐♥❣ t❤❛t ✐s ❛ ❣♦♦❞ ❝♦♥s❡♥s✉s ♦❢ t❤❡ ✐♥♣✉t str✐♥❣s✳ s1 c b c a b a a s2 c b c a b c b s3 a b c c c c a s4 c c c a b c a s5 c b c a a c a s6 c b c a a c a s7 a b b a b a a s8 b b c a a c a s c b c a b c a
SLIDE 5
❈♦♥s❡♥s✉s ❙tr✐♥❣ Pr♦❜❧❡♠s
■♥♣✉t✿ ❆ s❡t ✭♠✉❧t✐✲s❡t✮ ♦❢ str✐♥❣s✳ ❖✉t♣✉t✿ ❆ str✐♥❣ t❤❛t ✐s ❛ ❣♦♦❞ ❝♦♥s❡♥s✉s ♦❢ t❤❡ ✐♥♣✉t str✐♥❣s✳ s1 c b c a b a a s2 c b c a b c b s3 a b c c c c a s4 c c c a b c a s5 c b c a a c a s6 c b c a a c a s7 a b b a b a a s8 b b c a a c a s c b c a b c a
SLIDE 6
❈♦♥s❡♥s✉s ❙tr✐♥❣ Pr♦❜❧❡♠s
■♥♣✉t✿ ❆ s❡t ✭♠✉❧t✐✲s❡t✮ ♦❢ str✐♥❣s✳ ❖✉t♣✉t✿ ❆ str✐♥❣ t❤❛t ✐s ❛ ❣♦♦❞ ❝♦♥s❡♥s✉s ♦❢ t❤❡ ✐♥♣✉t str✐♥❣s✳ s1 c b c a b a a s2 c b c a b c b s3 a b c c c c a s4 c c c a b c a s5 c b c a a c a s6 c b c a a c a s7 a b b a b a a s8 b b c a a c a s a b c a b c a
SLIDE 7
❇❛s✐❝ ◆♦t❛t✐♦♥s
❙t❛♥❞❛r❞ str✐♥❣ ♥♦t❛t✐♦♥s✿ Σ ✜♥✐t❡ ❛❧♣❤❛❜❡t Σ∗ ✇♦r❞s ♦✈❡r Σ Σn {w ∈ Σ∗ | |w| = n} Σ≤n n
i=0 Σi
dH(u, v) ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡
- s✉❜str✐♥❣ r❡❧❛t✐♦♥✱
❛ u v ⇔ v = xuy ❋♦r ♠✉❧t✐✲s❡t ❛♥❞ ✿ r❛❞✐✉s ♦❢ ✭✇✳ r✳ t✳ ✮ ❞✐st❛♥❝❡ s✉♠ ♦❢ ✭✇✳ r✳ t✳ ✮
SLIDE 8
❇❛s✐❝ ◆♦t❛t✐♦♥s
❙t❛♥❞❛r❞ str✐♥❣ ♥♦t❛t✐♦♥s✿ Σ ✜♥✐t❡ ❛❧♣❤❛❜❡t Σ∗ ✇♦r❞s ♦✈❡r Σ Σn {w ∈ Σ∗ | |w| = n} Σ≤n n
i=0 Σi
dH(u, v) ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡
- s✉❜str✐♥❣ r❡❧❛t✐♦♥✱
❛ u v ⇔ v = xuy ❋♦r ♠✉❧t✐✲s❡t S ⊆ Σℓ ❛♥❞ v ∈ Σℓ✿ rH(v, S) = max{dH(v, u) | u ∈ S} r❛❞✐✉s ♦❢ S ✭✇✳ r✳ t✳ v✮ sH(v, S) =
u∈S dH(v, u)
❞✐st❛♥❝❡ s✉♠ ♦❢ S ✭✇✳ r✳ t✳ v✮
SLIDE 9
❚❤❡ ❈❧♦s❡st ❙tr✐♥❣ Pr♦❜❧❡♠
(r, s)-❈❧♦s❡st ❙tr✐♥❣
■♥st❛♥❝❡✿ ▼✉❧t✐✲s❡t S = {si | 1 ≤ i ≤ k} ⊆ Σℓ✱ ℓ ∈ N✱ dr, ds ∈ N✳ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ s ∈ Σℓ ✇✐t❤ rH(s, S) ≤ dr ❛♥❞ sH(s, S) ≤ ds❄
SLIDE 10
❚❤❡ ❈❧♦s❡st ❙tr✐♥❣ Pr♦❜❧❡♠
(r, s)-❈❧♦s❡st ❙tr✐♥❣
■♥st❛♥❝❡✿ ▼✉❧t✐✲s❡t S = {si | 1 ≤ i ≤ k} ⊆ Σℓ✱ ℓ ∈ N✱ dr, ds ∈ N✳ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ s ∈ Σℓ ✇✐t❤ rH(s, S) ≤ dr ❛♥❞ sH(s, S) ≤ ds❄ s1 c b c a b a a s2 c b c a b c b s3 a b c c c c a s4 c c c a b c a s5 c b c a a c a s6 c b c a a c a s7 a b b a b a a s8 b b c a a c a k = 8 ℓ = 7 dr = 2 ds = 20 rH(s, S) = 2 sH(s, S) = 16
SLIDE 11
❚❤❡ ❈❧♦s❡st ❙tr✐♥❣ Pr♦❜❧❡♠
(r, s)-❈❧♦s❡st ❙tr✐♥❣
■♥st❛♥❝❡✿ ▼✉❧t✐✲s❡t S = {si | 1 ≤ i ≤ k} ⊆ Σℓ✱ ℓ ∈ N✱ dr, ds ∈ N✳ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ s ∈ Σℓ ✇✐t❤ rH(s, S) ≤ dr ❛♥❞ sH(s, S) ≤ ds❄ s1 c b c a b a a s2 c b c a b c b s3 a b c c c c a s4 c c c a b c a s5 c b c a a c a s6 c b c a a c a s7 a b b a b a a s8 b b c a a c a s a b c a b c a k = 8 ℓ = 7 dr = 2 ds = 20 rH(s, S) = 2 sH(s, S) = 16
SLIDE 12
❚❤❡ ❈❧♦s❡st ❙tr✐♥❣ Pr♦❜❧❡♠
(r, s)-❈❧♦s❡st ❙tr✐♥❣
■♥st❛♥❝❡✿ ▼✉❧t✐✲s❡t S = {si | 1 ≤ i ≤ k} ⊆ Σℓ✱ ℓ ∈ N✱ dr, ds ∈ N✳ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ s ∈ Σℓ ✇✐t❤ rH(s, S) ≤ dr ❛♥❞ sH(s, S) ≤ ds❄ s1 c b c a b a a s2 c b c a b c b s3 a b c c c c a s4 c c c a b c a s5 c b c a a c a s6 c b c a a c a s7 a b b a b a a s8 b b c a a c a s a b c a b c a k = 8 ℓ = 7 dr = 2 ds = 20 rH(s, S) = 2 sH(s, S) = 16
SLIDE 13
❚❤❡ ✏❙✉❜str✐♥❣ ❱❛r✐❛♥t✑
(r, s)-❈❧♦s❡st ❙✉❜str✐♥❣
■♥st❛♥❝❡✿ ▼✉❧t✐✲s❡t S = {si | 1 ≤ i ≤ k} ⊆ Σ≤ℓ✱ ℓ ∈ N✱ dr, ds, m ∈ N✳ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ s ∈ Σm✱ ♠✉❧t✐✲s❡t S′ = {s′
i | s′ i si, 1 ≤ i ≤ k} ⊆ Σm ✇✐t❤
rH(s, S′) ≤ dr ❛♥❞ sH(s, S′) ≤ ds❄
SLIDE 14
❚❤❡ ✏❙✉❜str✐♥❣ ❱❛r✐❛♥t✑
(r, s)-❈❧♦s❡st ❙✉❜str✐♥❣
■♥st❛♥❝❡✿ ▼✉❧t✐✲s❡t S = {si | 1 ≤ i ≤ k} ⊆ Σ≤ℓ✱ ℓ ∈ N✱ dr, ds, m ∈ N✳ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ s ∈ Σm✱ ♠✉❧t✐✲s❡t S′ = {s′
i | s′ i si, 1 ≤ i ≤ k} ⊆ Σm ✇✐t❤
rH(s, S′) ≤ dr ❛♥❞ sH(s, S′) ≤ ds❄ s1 a a c b c a b a a s2 b c b c a b c b s3 a a b c c s4 c c c a b c a c s5 c c b c a a c a s6 a c b c a a s7 a a b b a b a a s8 b b b c a a c a c c b k = 8 ℓ = 11 m = 4 dr = 3 ds = 7 rH(s, S) = 2 sH(s, S) = 16
SLIDE 15
❚❤❡ ✏❙✉❜str✐♥❣ ❱❛r✐❛♥t✑
(r, s)-❈❧♦s❡st ❙✉❜str✐♥❣
■♥st❛♥❝❡✿ ▼✉❧t✐✲s❡t S = {si | 1 ≤ i ≤ k} ⊆ Σ≤ℓ✱ ℓ ∈ N✱ dr, ds, m ∈ N✳ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ s ∈ Σm✱ ♠✉❧t✐✲s❡t S′ = {s′
i | s′ i si, 1 ≤ i ≤ k} ⊆ Σm ✇✐t❤
rH(s, S′) ≤ dr ❛♥❞ sH(s, S′) ≤ ds❄ s1 a a c b c a b a a s2 b c b c a b c b s3 a a b c c s4 c c c a b c a c s5 c c b c a a c a s6 a c b c a a s7 a a b b a b a a s8 b b b c a a c a c c b s a b c a k = 8 ℓ = 11 m = 4 dr = 3 ds = 7 rH(s, S) = 2 sH(s, S) = 16
SLIDE 16
❚❤❡ ✏❙✉❜str✐♥❣ ❱❛r✐❛♥t✑
(r, s)-❈❧♦s❡st ❙✉❜str✐♥❣
■♥st❛♥❝❡✿ ▼✉❧t✐✲s❡t S = {si | 1 ≤ i ≤ k} ⊆ Σ≤ℓ✱ ℓ ∈ N✱ dr, ds, m ∈ N✳ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ s ∈ Σm✱ ♠✉❧t✐✲s❡t S′ = {s′
i | s′ i si, 1 ≤ i ≤ k} ⊆ Σm ✇✐t❤
rH(s, S′) ≤ dr ❛♥❞ sH(s, S′) ≤ ds❄ s1 a a c b c a b a a s2 b c b c a b c b s3 a a b c c s4 c c c a b c a c s5 c c b c a a c a s6 a c b c a a s7 a a b b a b a a s8 b b b c a a c a c c b s a b c a k = 8 ℓ = 11 m = 4 dr = 3 ds = 7 rH(s, S) = 2 sH(s, S) = 16
SLIDE 17
❚❤❡ ✏❙✉❜str✐♥❣ ❱❛r✐❛♥t✑
(r, s)-❈❧♦s❡st ❙✉❜str✐♥❣
■♥st❛♥❝❡✿ ▼✉❧t✐✲s❡t S = {si | 1 ≤ i ≤ k} ⊆ Σ≤ℓ✱ ℓ ∈ N✱ dr, ds, m ∈ N✳ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ s ∈ Σm✱ ♠✉❧t✐✲s❡t S′ = {s′
i | s′ i si, 1 ≤ i ≤ k} ⊆ Σm ✇✐t❤
rH(s, S′) ≤ dr ❛♥❞ sH(s, S′) ≤ ds❄ s1 a a c b c a b a a s2 b c b c a b c b s3 a a b c c s4 c c c a b c a c s5 c c b c a a c a s6 a c b c a a s7 a a b b a b a a s8 b b b c a a c a c c b s a b c a k = 8 ℓ = 11 m = 4 dr = 3 ds = 7 rH(s, S′) = 1 sH(s, S′) = 7
SLIDE 18
❚❤❡ ✏❖✉t❧✐❡r ❱❛r✐❛♥t✑
(r, s)-❈❧♦s❡st ❙tr✐♥❣ ✇✐t❤ ❖✉t❧✐❡rs
■♥st❛♥❝❡✿ ▼✉❧t✐✲s❡t S = {si | 1 ≤ i ≤ k} ⊆ Σℓ✱ ℓ ∈ N✱ dr, ds, t ∈ N✳ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ s ∈ Σℓ✱ S′ ⊆ S ✇✐t❤ |S′| = k − t ✇✐t❤ rH(s, S′) ≤ dr ❛♥❞ sH(s, S′) ≤ ds❄
SLIDE 19
❚❤❡ ✏❖✉t❧✐❡r ❱❛r✐❛♥t✑
(r, s)-❈❧♦s❡st ❙tr✐♥❣ ✇✐t❤ ❖✉t❧✐❡rs
■♥st❛♥❝❡✿ ▼✉❧t✐✲s❡t S = {si | 1 ≤ i ≤ k} ⊆ Σℓ✱ ℓ ∈ N✱ dr, ds, t ∈ N✳ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ s ∈ Σℓ✱ S′ ⊆ S ✇✐t❤ |S′| = k − t ✇✐t❤ rH(s, S′) ≤ dr ❛♥❞ sH(s, S′) ≤ ds❄ s1 c b c a b a a s2 c b c a b c b s3 a b c c c c a s4 c c c a b c a s5 c b c a a c a s6 c b c a a c a s7 a b b a b a a s8 b b c a a c a k = 8 ℓ = 7 t = 2 dr = 2 ds = 8 rH(s, S) = 2 sH(s, S) = 16
SLIDE 20
❚❤❡ ✏❖✉t❧✐❡r ❱❛r✐❛♥t✑
(r, s)-❈❧♦s❡st ❙tr✐♥❣ ✇✐t❤ ❖✉t❧✐❡rs
■♥st❛♥❝❡✿ ▼✉❧t✐✲s❡t S = {si | 1 ≤ i ≤ k} ⊆ Σℓ✱ ℓ ∈ N✱ dr, ds, t ∈ N✳ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ s ∈ Σℓ✱ S′ ⊆ S ✇✐t❤ |S′| = k − t ✇✐t❤ rH(s, S′) ≤ dr ❛♥❞ sH(s, S′) ≤ ds❄ s1 c b c a b a a s2 c b c a b c b s3 a b c c c c a s4 c c c a b c a s5 c b c a a c a s6 c b c a a c a s7 a b b a b a a s8 b b c a a c a k = 8 ℓ = 7 t = 2 dr = 2 ds = 8 rH(s, S) = 2 sH(s, S) = 16
SLIDE 21
❚❤❡ ✏❖✉t❧✐❡r ❱❛r✐❛♥t✑
(r, s)-❈❧♦s❡st ❙tr✐♥❣ ✇✐t❤ ❖✉t❧✐❡rs
■♥st❛♥❝❡✿ ▼✉❧t✐✲s❡t S = {si | 1 ≤ i ≤ k} ⊆ Σℓ✱ ℓ ∈ N✱ dr, ds, t ∈ N✳ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ s ∈ Σℓ✱ S′ ⊆ S ✇✐t❤ |S′| = k − t ✇✐t❤ rH(s, S′) ≤ dr ❛♥❞ sH(s, S′) ≤ ds❄ s1 c b c a b a a s2 c b c a b c b s3 a b c c c c a s4 c c c a b c a s5 c b c a a c a s6 c b c a a c a s7 a b b a b a a s8 b b c a a c a s c b c a b c a k = 8 ℓ = 7 t = 2 dr = 2 ds = 8 rH(s, S) = 2 sH(s, S) = 16
SLIDE 22
❚❤❡ ✏❖✉t❧✐❡r ❱❛r✐❛♥t✑
(r, s)-❈❧♦s❡st ❙tr✐♥❣ ✇✐t❤ ❖✉t❧✐❡rs
■♥st❛♥❝❡✿ ▼✉❧t✐✲s❡t S = {si | 1 ≤ i ≤ k} ⊆ Σℓ✱ ℓ ∈ N✱ dr, ds, t ∈ N✳ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ s ∈ Σℓ✱ S′ ⊆ S ✇✐t❤ |S′| = k − t ✇✐t❤ rH(s, S′) ≤ dr ❛♥❞ sH(s, S′) ≤ ds❄ s1 c b c a b a a s2 c b c a b c b s3 a b c c c c a s4 c c c a b c a s5 c b c a a c a s6 c b c a a c a s7 a b b a b a a s8 b b c a a c a s c b c a b c a k = 8 ℓ = 7 t = 2 dr = 2 ds = 8 rH(s, S) = 2 sH(s, S) = 7
SLIDE 23
❚❤❡ ✏r✲ ❛♥❞ s✲❱❛r✐❛♥ts✑
(r)-❈❧♦s❡st ❙tr✐♥❣✿ (r, s)-❈❧♦s❡st ❙tr✐♥❣ ✇✐t❤♦✉t ds ❜♦✉♥❞ (s)-❈❧♦s❡st ❙tr✐♥❣✿ (r, s)-❈❧♦s❡st ❙tr✐♥❣ ✇✐t❤♦✉t dr ❜♦✉♥❞ ▲✐❦❡✇✐s❡ ❢♦r s✉❜str✐♥❣✲ ❛♥❞ ♦✉t❧✐❡r✲✈❛r✐❛♥ts✳ ❚❤❡ r ✲ ❛♥❞ s ✲✈❛r✐❛♥ts ❛r❡ ✐♥t❡♥s✐✈❡❧② ✐♥✈❡st✐❣❛t❡❞✿ ♦✉r t❡r♠✐♥♦❧♦❣② ❝♦♠♠♦♥ ✐♥ ❧✐t❡r❛t✉r❡ r ❈❧♦s❡st ❙tr✐♥❣ ❈❧♦s❡st ❙tr✐♥❣ r ❈❧♦s❡st ❙✉❜str✐♥❣ ❈❧♦s❡st ❙✉❜str✐♥❣ s ❈❧♦s❡st ❙✉❜str✐♥❣ ❈♦♥s❡♥s✉s P❛tt❡r♥s
❍❛r❞♥❡ss
❆❧❧ t❤❡s❡ ♣r♦❜❧❡♠s ❛r❡ ✲❤❛r❞ ✭❡①❝❡♣t s ❈❧♦s❡st ❙tr✐♥❣✱ ✇❤✐❝❤ ✐s tr✐✈✐❛❧✮✳
SLIDE 24
❚❤❡ ✏r✲ ❛♥❞ s✲❱❛r✐❛♥ts✑
(r)-❈❧♦s❡st ❙tr✐♥❣✿ (r, s)-❈❧♦s❡st ❙tr✐♥❣ ✇✐t❤♦✉t ds ❜♦✉♥❞ (s)-❈❧♦s❡st ❙tr✐♥❣✿ (r, s)-❈❧♦s❡st ❙tr✐♥❣ ✇✐t❤♦✉t dr ❜♦✉♥❞ ▲✐❦❡✇✐s❡ ❢♦r s✉❜str✐♥❣✲ ❛♥❞ ♦✉t❧✐❡r✲✈❛r✐❛♥ts✳ ❚❤❡ (r)✲ ❛♥❞ (s)✲✈❛r✐❛♥ts ❛r❡ ✐♥t❡♥s✐✈❡❧② ✐♥✈❡st✐❣❛t❡❞✿ ♦✉r t❡r♠✐♥♦❧♦❣② ❝♦♠♠♦♥ ✐♥ ❧✐t❡r❛t✉r❡ (r)-❈❧♦s❡st ❙tr✐♥❣ ❈❧♦s❡st ❙tr✐♥❣ (r)-❈❧♦s❡st ❙✉❜str✐♥❣ ❈❧♦s❡st ❙✉❜str✐♥❣ (s)-❈❧♦s❡st ❙✉❜str✐♥❣ ❈♦♥s❡♥s✉s P❛tt❡r♥s
❍❛r❞♥❡ss
❆❧❧ t❤❡s❡ ♣r♦❜❧❡♠s ❛r❡ ✲❤❛r❞ ✭❡①❝❡♣t s ❈❧♦s❡st ❙tr✐♥❣✱ ✇❤✐❝❤ ✐s tr✐✈✐❛❧✮✳
SLIDE 25
❚❤❡ ✏r✲ ❛♥❞ s✲❱❛r✐❛♥ts✑
(r)-❈❧♦s❡st ❙tr✐♥❣✿ (r, s)-❈❧♦s❡st ❙tr✐♥❣ ✇✐t❤♦✉t ds ❜♦✉♥❞ (s)-❈❧♦s❡st ❙tr✐♥❣✿ (r, s)-❈❧♦s❡st ❙tr✐♥❣ ✇✐t❤♦✉t dr ❜♦✉♥❞ ▲✐❦❡✇✐s❡ ❢♦r s✉❜str✐♥❣✲ ❛♥❞ ♦✉t❧✐❡r✲✈❛r✐❛♥ts✳ ❚❤❡ (r)✲ ❛♥❞ (s)✲✈❛r✐❛♥ts ❛r❡ ✐♥t❡♥s✐✈❡❧② ✐♥✈❡st✐❣❛t❡❞✿ ♦✉r t❡r♠✐♥♦❧♦❣② ❝♦♠♠♦♥ ✐♥ ❧✐t❡r❛t✉r❡ (r)-❈❧♦s❡st ❙tr✐♥❣ ❈❧♦s❡st ❙tr✐♥❣ (r)-❈❧♦s❡st ❙✉❜str✐♥❣ ❈❧♦s❡st ❙✉❜str✐♥❣ (s)-❈❧♦s❡st ❙✉❜str✐♥❣ ❈♦♥s❡♥s✉s P❛tt❡r♥s
❍❛r❞♥❡ss
❆❧❧ t❤❡s❡ ♣r♦❜❧❡♠s ❛r❡ NP✲❤❛r❞ ✭❡①❝❡♣t (s)-❈❧♦s❡st ❙tr✐♥❣✱ ✇❤✐❝❤ ✐s tr✐✈✐❛❧✮✳
SLIDE 26
P❛r❛♠❡t❡rs
k ♥✉♠❜❡r ♦❢ ✐♥♣✉t str✐♥❣s ℓ ❧❡♥❣t❤ ♦❢ ✐♥♣✉t str✐♥❣s dr r❛❞✐✉s ❜♦✉♥❞ ds ❞✐st❛♥❝❡ s✉♠ ❜♦✉♥❞ |Σ| ❛❧♣❤❛❜❡t s✐③❡ m s✉❜str✐♥❣ ❧❡♥❣t❤ ✭(r, s)-❈❧♦s❡st ❙✉❜str✐♥❣✮ t ♥✉♠❜❡r ♦❢ ♦✉t❧✐❡rs ✭(r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦✮ k − t ♥✉♠❜❡r ♦❢ ✐♥❧✐❡rs ✭(r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦✮ ◆♦t❛t✐♦♥✿ r s ❈❧♦s❡st ❙tr✐♥❣ ♠❡❛♥s r s ❈❧♦s❡st ❙tr✐♥❣ ♣❛r❛♠❡t❡r✐s❡❞ ❜② ✳ ✜①❡❞✲♣❛r❛♠❡t❡r tr❛❝t❛❜❧❡✿ ❛❧❣♦r✐t❤♠ ✇✐t❤ r✉♥♥✐♥❣ t✐♠❡ ❢♦r r❡❝✉rs✐✈❡ ❛♥❞ ♣♦❧②♥♦♠✐❛❧ ✭ ✐s ✐♥♣✉t ❛♥❞ t❤❡ ♣❛r❛♠❡t❡r✮✳ ✲❤❛r❞♥❡ss ♥♦t ✜①❡❞ ♣❛r❛♠❡t❡r tr❛❝t❛❜❧❡✳
SLIDE 27
P❛r❛♠❡t❡rs
k ♥✉♠❜❡r ♦❢ ✐♥♣✉t str✐♥❣s ℓ ❧❡♥❣t❤ ♦❢ ✐♥♣✉t str✐♥❣s dr r❛❞✐✉s ❜♦✉♥❞ ds ❞✐st❛♥❝❡ s✉♠ ❜♦✉♥❞ |Σ| ❛❧♣❤❛❜❡t s✐③❡ m s✉❜str✐♥❣ ❧❡♥❣t❤ ✭(r, s)-❈❧♦s❡st ❙✉❜str✐♥❣✮ t ♥✉♠❜❡r ♦❢ ♦✉t❧✐❡rs ✭(r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦✮ k − t ♥✉♠❜❡r ♦❢ ✐♥❧✐❡rs ✭(r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦✮ ◆♦t❛t✐♦♥✿ (r, s)-❈❧♦s❡st ❙tr✐♥❣(p1, p2, . . . ) ♠❡❛♥s (r, s)-❈❧♦s❡st ❙tr✐♥❣ ♣❛r❛♠❡t❡r✐s❡❞ ❜② p1, p2, . . . ✳ ✜①❡❞✲♣❛r❛♠❡t❡r tr❛❝t❛❜❧❡✿ ❛❧❣♦r✐t❤♠ ✇✐t❤ r✉♥♥✐♥❣ t✐♠❡ ❢♦r r❡❝✉rs✐✈❡ ❛♥❞ ♣♦❧②♥♦♠✐❛❧ ✭ ✐s ✐♥♣✉t ❛♥❞ t❤❡ ♣❛r❛♠❡t❡r✮✳ ✲❤❛r❞♥❡ss ♥♦t ✜①❡❞ ♣❛r❛♠❡t❡r tr❛❝t❛❜❧❡✳
SLIDE 28
P❛r❛♠❡t❡rs
k ♥✉♠❜❡r ♦❢ ✐♥♣✉t str✐♥❣s ℓ ❧❡♥❣t❤ ♦❢ ✐♥♣✉t str✐♥❣s dr r❛❞✐✉s ❜♦✉♥❞ ds ❞✐st❛♥❝❡ s✉♠ ❜♦✉♥❞ |Σ| ❛❧♣❤❛❜❡t s✐③❡ m s✉❜str✐♥❣ ❧❡♥❣t❤ ✭(r, s)-❈❧♦s❡st ❙✉❜str✐♥❣✮ t ♥✉♠❜❡r ♦❢ ♦✉t❧✐❡rs ✭(r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦✮ k − t ♥✉♠❜❡r ♦❢ ✐♥❧✐❡rs ✭(r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦✮ ◆♦t❛t✐♦♥✿ (r, s)-❈❧♦s❡st ❙tr✐♥❣(p1, p2, . . . ) ♠❡❛♥s (r, s)-❈❧♦s❡st ❙tr✐♥❣ ♣❛r❛♠❡t❡r✐s❡❞ ❜② p1, p2, . . . ✳ ✜①❡❞✲♣❛r❛♠❡t❡r tr❛❝t❛❜❧❡✿ ∃ ❛❧❣♦r✐t❤♠ ✇✐t❤ r✉♥♥✐♥❣ t✐♠❡ f(k) × p(|x|) ❢♦r r❡❝✉rs✐✈❡ f ❛♥❞ ♣♦❧②♥♦♠✐❛❧ p ✭x ✐s ✐♥♣✉t ❛♥❞ k t❤❡ ♣❛r❛♠❡t❡r✮✳ W[1]✲❤❛r❞♥❡ss ⇒ ♥♦t ✜①❡❞ ♣❛r❛♠❡t❡r tr❛❝t❛❜❧❡✳
SLIDE 29
❙t❛t❡ ♦❢ t❤❡ ❆rt
Pr❡✈✐♦✉s ❧✐t❡r❛t✉r❡✿ (r)-❈❧♦s❡st ❙tr✐♥❣ (s)-❈❧♦s❡st ❙tr✐♥❣ (r)-❈❧♦s❡st ❙✉❜str✐♥❣ (s)-❈❧♦s❡st ❙✉❜str✐♥❣ ✭(s)-❈❧♦s❡st ❙✉❜str✐♥❣(ℓ, m)✮ r s ❈❧♦s❡st ❙tr✐♥❣ r s ❈❧♦s❡st ❙✉❜str✐♥❣ r ❈❧♦s❡st ❙tr✐♥❣✲✇♦ s ❈❧♦s❡st ❙tr✐♥❣✲✇♦ r s ❈❧♦s❡st ❙tr✐♥❣✲✇♦ ❖✉r ❈♦♥tr✐❜✉t✐♦♥✿ r s ❈❧♦s❡st ❙tr✐♥❣ r s ❈❧♦s❡st ❙✉❜str✐♥❣ s ❈❧♦s❡st ❙✉❜str✐♥❣ r ❈❧♦s❡st ❙tr✐♥❣✲✇♦ s ❈❧♦s❡st ❙tr✐♥❣✲✇♦ r s ❈❧♦s❡st ❙tr✐♥❣✲✇♦
SLIDE 30
❙t❛t❡ ♦❢ t❤❡ ❆rt
Pr❡✈✐♦✉s ❧✐t❡r❛t✉r❡✿ (r)-❈❧♦s❡st ❙tr✐♥❣ (s)-❈❧♦s❡st ❙tr✐♥❣ (r)-❈❧♦s❡st ❙✉❜str✐♥❣ (s)-❈❧♦s❡st ❙✉❜str✐♥❣ ✭(s)-❈❧♦s❡st ❙✉❜str✐♥❣(ℓ, m)✮ (r, s)-❈❧♦s❡st ❙tr✐♥❣ (r, s)-❈❧♦s❡st ❙✉❜str✐♥❣ r ❈❧♦s❡st ❙tr✐♥❣✲✇♦ s ❈❧♦s❡st ❙tr✐♥❣✲✇♦ r s ❈❧♦s❡st ❙tr✐♥❣✲✇♦ ❖✉r ❈♦♥tr✐❜✉t✐♦♥✿ r s ❈❧♦s❡st ❙tr✐♥❣ r s ❈❧♦s❡st ❙✉❜str✐♥❣ s ❈❧♦s❡st ❙✉❜str✐♥❣ r ❈❧♦s❡st ❙tr✐♥❣✲✇♦ s ❈❧♦s❡st ❙tr✐♥❣✲✇♦ r s ❈❧♦s❡st ❙tr✐♥❣✲✇♦
SLIDE 31
❙t❛t❡ ♦❢ t❤❡ ❆rt
Pr❡✈✐♦✉s ❧✐t❡r❛t✉r❡✿ (r)-❈❧♦s❡st ❙tr✐♥❣ (s)-❈❧♦s❡st ❙tr✐♥❣ (r)-❈❧♦s❡st ❙✉❜str✐♥❣ (s)-❈❧♦s❡st ❙✉❜str✐♥❣ ✭(s)-❈❧♦s❡st ❙✉❜str✐♥❣(ℓ, m)✮ (r, s)-❈❧♦s❡st ❙tr✐♥❣ (r, s)-❈❧♦s❡st ❙✉❜str✐♥❣ (r)-❈❧♦s❡st ❙tr✐♥❣✲✇♦ (s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦ (r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦ ❖✉r ❈♦♥tr✐❜✉t✐♦♥✿ r s ❈❧♦s❡st ❙tr✐♥❣ r s ❈❧♦s❡st ❙✉❜str✐♥❣ s ❈❧♦s❡st ❙✉❜str✐♥❣ r ❈❧♦s❡st ❙tr✐♥❣✲✇♦ s ❈❧♦s❡st ❙tr✐♥❣✲✇♦ r s ❈❧♦s❡st ❙tr✐♥❣✲✇♦
SLIDE 32
❙t❛t❡ ♦❢ t❤❡ ❆rt
Pr❡✈✐♦✉s ❧✐t❡r❛t✉r❡✿ (r)-❈❧♦s❡st ❙tr✐♥❣ (s)-❈❧♦s❡st ❙tr✐♥❣ (r)-❈❧♦s❡st ❙✉❜str✐♥❣ (s)-❈❧♦s❡st ❙✉❜str✐♥❣ ✭(s)-❈❧♦s❡st ❙✉❜str✐♥❣(ℓ, m)✮ (r, s)-❈❧♦s❡st ❙tr✐♥❣ (r, s)-❈❧♦s❡st ❙✉❜str✐♥❣ (r)-❈❧♦s❡st ❙tr✐♥❣✲✇♦ (s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦ (r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦ ❖✉r ❈♦♥tr✐❜✉t✐♦♥✿ (r, s)-❈❧♦s❡st ❙tr✐♥❣ (r, s)-❈❧♦s❡st ❙✉❜str✐♥❣ (s)-❈❧♦s❡st ❙✉❜str✐♥❣(ℓ, m) r ❈❧♦s❡st ❙tr✐♥❣✲✇♦ s ❈❧♦s❡st ❙tr✐♥❣✲✇♦ r s ❈❧♦s❡st ❙tr✐♥❣✲✇♦
SLIDE 33
❙t❛t❡ ♦❢ t❤❡ ❆rt
Pr❡✈✐♦✉s ❧✐t❡r❛t✉r❡✿ (r)-❈❧♦s❡st ❙tr✐♥❣ (s)-❈❧♦s❡st ❙tr✐♥❣ (r)-❈❧♦s❡st ❙✉❜str✐♥❣ (s)-❈❧♦s❡st ❙✉❜str✐♥❣ ✭(s)-❈❧♦s❡st ❙✉❜str✐♥❣(ℓ, m)✮ (r, s)-❈❧♦s❡st ❙tr✐♥❣ (r, s)-❈❧♦s❡st ❙✉❜str✐♥❣ (r)-❈❧♦s❡st ❙tr✐♥❣✲✇♦ (s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦ (r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦ ❖✉r ❈♦♥tr✐❜✉t✐♦♥✿ (r, s)-❈❧♦s❡st ❙tr✐♥❣ (r, s)-❈❧♦s❡st ❙✉❜str✐♥❣ (s)-❈❧♦s❡st ❙✉❜str✐♥❣(ℓ, m) (r)-❈❧♦s❡st ❙tr✐♥❣✲✇♦ (s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦ (r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦
SLIDE 34
❘❡s✉❧ts ❢♦r (r, s)-❈❧♦s❡st ❙tr✐♥❣
k dr ds |Σ| ℓ ❘❡s✉❧t ♣ ✕ ✕ ✕ ✕ FPT ✕ ♣ ✕ ✕ ✕ FPT ✕ ✕ ♣ ✕ ✕ FPT ✕ ✕ ✕ ✷ ✕ NP✲❤❛r❞ ✕ ✕ ✕ ✕ ♣ FPT
SLIDE 35
❘❡s✉❧ts ❢♦r (r, s)-❈❧♦s❡st ❙tr✐♥❣
k dr ds |Σ| ℓ ❘❡s✉❧t ♣ ✕ ✕ ✕ ✕ FPT ✕ ♣ ✕ ✕ ✕ FPT ✕ ✕ ♣ ✕ ✕ FPT ✕ ✕ ✕ ✷ ✕ NP✲❤❛r❞ ✕ ✕ ✕ ✕ ♣ FPT
SLIDE 36
❇r❛♥❝❤✐♥❣ ❆❧❣♦r✐t❤♠
❋♣t✲❜r❛♥❝❤✐♥❣ ❛❧❣♦r✐t❤♠ ❢♦r (r)-❈❧♦s❡st ❙tr✐♥❣(dr)✿1 dr = 2 s1 c b c a b a a s2 c b c a b c b s3 a b c c c c a s4 c c c a b c a s5 c b c a a c a s6 c b c a a c a s7 a b b a b a a s8 b b c a a c a s a b c a b c a
1●r❛♠♠✱ ◆✐❡❞❡r♠❡✐❡r✱ ❘♦ss♠❛♥✐t❤✱ ❆❧❣♦r✐t❤♠✐❝❛✱ ✷✵✵✸✳
SLIDE 37
❇r❛♥❝❤✐♥❣ ❆❧❣♦r✐t❤♠
❋♣t✲❜r❛♥❝❤✐♥❣ ❛❧❣♦r✐t❤♠ ❢♦r (r)-❈❧♦s❡st ❙tr✐♥❣(dr)✿1 dr = 2 s1 c b c a b a a s2 c b c a b c b s3 a b c c c c a s4 c c c a b c a s5 c b c a a c a s6 c b c a a c a s7 a b b a b a a s8 b b c a a c a s a b c a b c a s′ = c b c a b a a
1●r❛♠♠✱ ◆✐❡❞❡r♠❡✐❡r✱ ❘♦ss♠❛♥✐t❤✱ ❆❧❣♦r✐t❤♠✐❝❛✱ ✷✵✵✸✳
SLIDE 38
❇r❛♥❝❤✐♥❣ ❆❧❣♦r✐t❤♠
❋♣t✲❜r❛♥❝❤✐♥❣ ❛❧❣♦r✐t❤♠ ❢♦r (r)-❈❧♦s❡st ❙tr✐♥❣(dr)✿1 dr = 2 s1 c b c a b a a s2 c b c a b c b s3 a b c c c c a s4 c c c a b c a s5 c b c a a c a s6 c b c a a c a s7 a b b a b a a s8 b b c a a c a s a b c a b c a s′ = c b c a b a a
1●r❛♠♠✱ ◆✐❡❞❡r♠❡✐❡r✱ ❘♦ss♠❛♥✐t❤✱ ❆❧❣♦r✐t❤♠✐❝❛✱ ✷✵✵✸✳
SLIDE 39
❊①t❡♥❞❡❞ ❇r❛♥❝❤✐♥❣ ❆❧❣♦r✐t❤♠
- ♦❛❧✿ ❊①t❡♥❞ ❜r❛♥❝❤✐♥❣ ❛❧❣♦r✐t❤♠ t♦ (r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦(dr, t)✳
❈❤♦✐❝❡ ♦❢ ♦✉t❧✐❡rs ❝❛♥ ❜❡ ❛❞❞❡❞ t♦ t❤❡ ❜r❛♥❝❤✐♥❣ ✭ ✐s ❛ ♣❛r❛♠❡t❡r✮✳
- ❡♥❡r❛❧ ❜r❛♥❝❤✐♥❣ s✐♠✐❧❛r✿
❜r❛♥❝❤ ❜② ♠✐s♠❛t❝❤❡s ❜❡t✇❡❡♥ ❝❛♥❞✐❞❛t❡ ❛♥❞ ✐♥♣✉t str✐♥❣✳ ▼❛✐♥ ♣r♦❜❧❡♠✿ ❲❤❛t ✐s ❛ ❣♦♦❞ ✐♥✐t✐❛❧ ❝❛♥❞✐❞❛t❡ str✐♥❣❄ s♦♠❡ ✐♥♣✉t str✐♥❣ ❤♦✇ ❞♦ ✇❡ s❛t✐s❢② ❜♦✉♥❞❄ s♦♠❡ str✐♥❣ ✇✐t❤ ❧♦✇ ❞✐st❛♥❝❡ s✉♠ ❤♦✇ t♦ ❜♦✉♥❞ ❞❡♣t❤ ♦❢ ❜r❛♥❝❤✐♥❣❄ ❇♦✉❝❤❡r ❛♥❞ ▼❛✱ ❇▼❈ ❇✐♦✐♥❢♦r♠❛t✐❝s✱ ✷✵✶✶✳
SLIDE 40
❊①t❡♥❞❡❞ ❇r❛♥❝❤✐♥❣ ❆❧❣♦r✐t❤♠
- ♦❛❧✿ ❊①t❡♥❞ ❜r❛♥❝❤✐♥❣ ❛❧❣♦r✐t❤♠ t♦ (r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦(dr, t)✳
❈❤♦✐❝❡ ♦❢ ♦✉t❧✐❡rs ❝❛♥ ❜❡ ❛❞❞❡❞ t♦ t❤❡ ❜r❛♥❝❤✐♥❣ ✭t ✐s ❛ ♣❛r❛♠❡t❡r✮✳2
- ❡♥❡r❛❧ ❜r❛♥❝❤✐♥❣ s✐♠✐❧❛r✿
❜r❛♥❝❤ ❜② dr + 1 ♠✐s♠❛t❝❤❡s ❜❡t✇❡❡♥ ❝❛♥❞✐❞❛t❡ ❛♥❞ ✐♥♣✉t str✐♥❣✳ ▼❛✐♥ ♣r♦❜❧❡♠✿ ❲❤❛t ✐s ❛ ❣♦♦❞ ✐♥✐t✐❛❧ ❝❛♥❞✐❞❛t❡ str✐♥❣❄ s♦♠❡ ✐♥♣✉t str✐♥❣ ❤♦✇ ❞♦ ✇❡ s❛t✐s❢② ❜♦✉♥❞❄ s♦♠❡ str✐♥❣ ✇✐t❤ ❧♦✇ ❞✐st❛♥❝❡ s✉♠ ❤♦✇ t♦ ❜♦✉♥❞ ❞❡♣t❤ ♦❢ ❜r❛♥❝❤✐♥❣❄
2❇♦✉❝❤❡r ❛♥❞ ▼❛✱ ❇▼❈ ❇✐♦✐♥❢♦r♠❛t✐❝s✱ ✷✵✶✶✳
SLIDE 41
❊①t❡♥❞❡❞ ❇r❛♥❝❤✐♥❣ ❆❧❣♦r✐t❤♠
- ♦❛❧✿ ❊①t❡♥❞ ❜r❛♥❝❤✐♥❣ ❛❧❣♦r✐t❤♠ t♦ (r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦(dr, t)✳
❈❤♦✐❝❡ ♦❢ ♦✉t❧✐❡rs ❝❛♥ ❜❡ ❛❞❞❡❞ t♦ t❤❡ ❜r❛♥❝❤✐♥❣ ✭t ✐s ❛ ♣❛r❛♠❡t❡r✮✳2
- ❡♥❡r❛❧ ❜r❛♥❝❤✐♥❣ s✐♠✐❧❛r✿
❜r❛♥❝❤ ❜② dr + 1 ♠✐s♠❛t❝❤❡s ❜❡t✇❡❡♥ ❝❛♥❞✐❞❛t❡ ❛♥❞ ✐♥♣✉t str✐♥❣✳ ▼❛✐♥ ♣r♦❜❧❡♠✿ ❲❤❛t ✐s ❛ ❣♦♦❞ ✐♥✐t✐❛❧ ❝❛♥❞✐❞❛t❡ str✐♥❣❄ s♦♠❡ ✐♥♣✉t str✐♥❣ ❤♦✇ ❞♦ ✇❡ s❛t✐s❢② ds ❜♦✉♥❞❄ s♦♠❡ str✐♥❣ ✇✐t❤ ❧♦✇ ❞✐st❛♥❝❡ s✉♠ ❤♦✇ t♦ ❜♦✉♥❞ ❞❡♣t❤ ♦❢ ❜r❛♥❝❤✐♥❣❄
2❇♦✉❝❤❡r ❛♥❞ ▼❛✱ ❇▼❈ ❇✐♦✐♥❢♦r♠❛t✐❝s✱ ✷✵✶✶✳
SLIDE 42
❊①t❡♥❞❡❞ ❇r❛♥❝❤✐♥❣ ❆❧❣♦r✐t❤♠
▼❛❥♦r✐t② str✐♥❣ sm✿ ♣✐❝❦ ❛ ♠♦st ❢r❡q✉❡♥t s②♠❜♦❧ ✐♥ ❡❛❝❤ ❝♦❧✉♠♥✳ ❊①❛♠♣❧❡✿ s1 d b a b b b b s2 d a a b c c d s3 d a a b c c d s4 a a c c c c d s5 a a c b c c d s6 a c a b d b d sm d a a b c c d
▲❡♠♠❛
✳ Pr♦❜❧❡♠✿ ❖✉t❧✐❡rs✦ ❉✐s♣✉t❡❞ ❝♦❧✉♠♥s✳
▲❡♠♠❛
■❢ t❤❡ ✐♥st❛♥❝❡ ❤❛s ❛ s♦❧✉t✐♦♥✱ t❤❡♥ ★❞✐s♣✳ ❝♦❧✉♠♥s ✳ ❆❧❣♦r✐t❤♠✿ st❛rt ✇✐t❤ ❜r❛♥❝❤ ♦✈❡r ♠✐s♠❛t❝❤❡s ✭♦r ❞❡❝❧❛r❡ ♦✉t❧✐❡r✮ ❞❡♣t❤ ❜♦✉♥❞✿ ✳
SLIDE 43
❊①t❡♥❞❡❞ ❇r❛♥❝❤✐♥❣ ❆❧❣♦r✐t❤♠
▼❛❥♦r✐t② str✐♥❣ sm✿ ♣✐❝❦ ❛ ♠♦st ❢r❡q✉❡♥t s②♠❜♦❧ ✐♥ ❡❛❝❤ ❝♦❧✉♠♥✳ ❊①❛♠♣❧❡✿ s1 d b a b b b b s2 d a a b c c d s3 d a a b c c d s4 a a c c c c d s5 a a c b c c d s6 a c a b d b d sm d a a b c c d
▲❡♠♠❛
rH(s, S) ≤ dr ⇒ dH(sm, s) ≤ 2dr✳ Pr♦❜❧❡♠✿ ❖✉t❧✐❡rs✦ ❉✐s♣✉t❡❞ ❝♦❧✉♠♥s✳
▲❡♠♠❛
■❢ t❤❡ ✐♥st❛♥❝❡ ❤❛s ❛ s♦❧✉t✐♦♥✱ t❤❡♥ ★❞✐s♣✳ ❝♦❧✉♠♥s ✳ ❆❧❣♦r✐t❤♠✿ st❛rt ✇✐t❤ ❜r❛♥❝❤ ♦✈❡r ♠✐s♠❛t❝❤❡s ✭♦r ❞❡❝❧❛r❡ ♦✉t❧✐❡r✮ ❞❡♣t❤ ❜♦✉♥❞✿ ✳
SLIDE 44
❊①t❡♥❞❡❞ ❇r❛♥❝❤✐♥❣ ❆❧❣♦r✐t❤♠
▼❛❥♦r✐t② str✐♥❣ sm✿ ♣✐❝❦ ❛ ♠♦st ❢r❡q✉❡♥t s②♠❜♦❧ ✐♥ ❡❛❝❤ ❝♦❧✉♠♥✳ ❊①❛♠♣❧❡✿ s1 d b a b b b b s2 d a a b c c d s3 d a a b c c d s4 a a c c c c d s5 a a c b c c d s6 a c a b d b d sm d a a b c c d
▲❡♠♠❛
rH(s, S) ≤ dr ⇒ dH(sm, s) ≤ 2dr✳ Pr♦❜❧❡♠✿ ❖✉t❧✐❡rs✦ ❉✐s♣✉t❡❞ ❝♦❧✉♠♥s✳
▲❡♠♠❛
■❢ t❤❡ ✐♥st❛♥❝❡ ❤❛s ❛ s♦❧✉t✐♦♥✱ t❤❡♥ ★❞✐s♣✳ ❝♦❧✉♠♥s ✳ ❆❧❣♦r✐t❤♠✿ st❛rt ✇✐t❤ ❜r❛♥❝❤ ♦✈❡r ♠✐s♠❛t❝❤❡s ✭♦r ❞❡❝❧❛r❡ ♦✉t❧✐❡r✮ ❞❡♣t❤ ❜♦✉♥❞✿ ✳
SLIDE 45
❊①t❡♥❞❡❞ ❇r❛♥❝❤✐♥❣ ❆❧❣♦r✐t❤♠
▼❛❥♦r✐t② str✐♥❣ sm✿ ♣✐❝❦ ❛ ♠♦st ❢r❡q✉❡♥t s②♠❜♦❧ ✐♥ ❡❛❝❤ ❝♦❧✉♠♥✳ ❊①❛♠♣❧❡✿ t = 2 s1 d b a b b b b s2 d a a b c c d s3 d a a b c c d s4 a a c c c c d s5 a a c b c c d s6 a c a b d b d sm d a a b c c d
▲❡♠♠❛
rH(s, S) ≤ dr ⇒ dH(sm, s) ≤ 2dr✳ Pr♦❜❧❡♠✿ ❖✉t❧✐❡rs✦ ❉✐s♣✉t❡❞ ❝♦❧✉♠♥s✳
▲❡♠♠❛
■❢ t❤❡ ✐♥st❛♥❝❡ ❤❛s ❛ s♦❧✉t✐♦♥✱ t❤❡♥ ★❞✐s♣✳ ❝♦❧✉♠♥s ✳ ❆❧❣♦r✐t❤♠✿ st❛rt ✇✐t❤ ❜r❛♥❝❤ ♦✈❡r ♠✐s♠❛t❝❤❡s ✭♦r ❞❡❝❧❛r❡ ♦✉t❧✐❡r✮ ❞❡♣t❤ ❜♦✉♥❞✿ ✳
SLIDE 46
❊①t❡♥❞❡❞ ❇r❛♥❝❤✐♥❣ ❆❧❣♦r✐t❤♠
❘❡✜♥❡❞ ♠❛❥♦r✐t② str✐♥❣ s⋄
m✿
∃ s②♠❜✳ ✇✐t❤ ❛t ❧❡❛st ❛s ♠❛♥② ♦❝❝✳ ❛s ♠❛❥♦r✐t② s②♠❜♦❧ ♠✐♥✉s t ⇒ ✉s❡ ⋄✳ ❊①❛♠♣❧❡✿ t = 2 s1 d b a b b b b s2 d a a b c c d s3 d a a b c c d s4 a a c c c c d s5 a a c b c c d s6 a c a b d b d s⋄
m
⋄ a ⋄ b c ⋄ d
▲❡♠♠❛
rH(s, S) ≤ dr ⇒ dH(sm, s) ≤ 2dr✳ Pr♦❜❧❡♠✿ ❖✉t❧✐❡rs✦ ❉✐s♣✉t❡❞ ❝♦❧✉♠♥s✳
▲❡♠♠❛
■❢ t❤❡ ✐♥st❛♥❝❡ ❤❛s ❛ s♦❧✉t✐♦♥✱ t❤❡♥ ★❞✐s♣✳ ❝♦❧✉♠♥s ✳ ❆❧❣♦r✐t❤♠✿ st❛rt ✇✐t❤ ❜r❛♥❝❤ ♦✈❡r ♠✐s♠❛t❝❤❡s ✭♦r ❞❡❝❧❛r❡ ♦✉t❧✐❡r✮ ❞❡♣t❤ ❜♦✉♥❞✿ ✳
SLIDE 47
❊①t❡♥❞❡❞ ❇r❛♥❝❤✐♥❣ ❆❧❣♦r✐t❤♠
❘❡✜♥❡❞ ♠❛❥♦r✐t② str✐♥❣ s⋄
m✿
∃ s②♠❜✳ ✇✐t❤ ❛t ❧❡❛st ❛s ♠❛♥② ♦❝❝✳ ❛s ♠❛❥♦r✐t② s②♠❜♦❧ ♠✐♥✉s t ⇒ ✉s❡ ⋄✳ ❊①❛♠♣❧❡✿ t = 2 s1 d b a b b b b s2 d a a b c c d s3 d a a b c c d s4 a a c c c c d s5 a a c b c c d s6 a c a b d b d s⋄
m
⋄ a ⋄ b c ⋄ d
▲❡♠♠❛
rH(s, S) ≤ dr ⇒ dH(sm, s) ≤ 2dr✳ Pr♦❜❧❡♠✿ ❖✉t❧✐❡rs✦ ❉✐s♣✉t❡❞ ❝♦❧✉♠♥s✳
▲❡♠♠❛
■❢ t❤❡ ✐♥st❛♥❝❡ ❤❛s ❛ s♦❧✉t✐♦♥✱ t❤❡♥ ★❞✐s♣✳ ❝♦❧✉♠♥s ✳ ❆❧❣♦r✐t❤♠✿ st❛rt ✇✐t❤ ❜r❛♥❝❤ ♦✈❡r ♠✐s♠❛t❝❤❡s ✭♦r ❞❡❝❧❛r❡ ♦✉t❧✐❡r✮ ❞❡♣t❤ ❜♦✉♥❞✿ ✳
SLIDE 48
❊①t❡♥❞❡❞ ❇r❛♥❝❤✐♥❣ ❆❧❣♦r✐t❤♠
❘❡✜♥❡❞ ♠❛❥♦r✐t② str✐♥❣ s⋄
m✿
∃ s②♠❜✳ ✇✐t❤ ❛t ❧❡❛st ❛s ♠❛♥② ♦❝❝✳ ❛s ♠❛❥♦r✐t② s②♠❜♦❧ ♠✐♥✉s t ⇒ ✉s❡ ⋄✳ ❊①❛♠♣❧❡✿ t = 2 s1 d b a b b b b s2 d a a b c c d s3 d a a b c c d s4 a a c c c c d s5 a a c b c c d s6 a c a b d b d s⋄
m
⋄ a ⋄ b c ⋄ d
▲❡♠♠❛
rH(s, S) ≤ dr ⇒ dH(sm, s) ≤ 2dr✳ Pr♦❜❧❡♠✿ ❖✉t❧✐❡rs✦ ❉✐s♣✉t❡❞ ❝♦❧✉♠♥s✳
▲❡♠♠❛
■❢ t❤❡ ✐♥st❛♥❝❡ ❤❛s ❛ s♦❧✉t✐♦♥✱ t❤❡♥ ★❞✐s♣✳ ❝♦❧✉♠♥s ≤ 4dr✳ ❆❧❣♦r✐t❤♠✿ st❛rt ✇✐t❤ ❜r❛♥❝❤ ♦✈❡r ♠✐s♠❛t❝❤❡s ✭♦r ❞❡❝❧❛r❡ ♦✉t❧✐❡r✮ ❞❡♣t❤ ❜♦✉♥❞✿ ✳
SLIDE 49
❊①t❡♥❞❡❞ ❇r❛♥❝❤✐♥❣ ❆❧❣♦r✐t❤♠
❘❡✜♥❡❞ ♠❛❥♦r✐t② str✐♥❣ s⋄
m✿
∃ s②♠❜✳ ✇✐t❤ ❛t ❧❡❛st ❛s ♠❛♥② ♦❝❝✳ ❛s ♠❛❥♦r✐t② s②♠❜♦❧ ♠✐♥✉s t ⇒ ✉s❡ ⋄✳ ❊①❛♠♣❧❡✿ t = 2 s1 d b a b b b b s2 d a a b c c d s3 d a a b c c d s4 a a c c c c d s5 a a c b c c d s6 a c a b d b d s⋄
m
⋄ a ⋄ b c ⋄ d
▲❡♠♠❛
rH(s, S) ≤ dr ⇒ dH(sm, s) ≤ 2dr✳ Pr♦❜❧❡♠✿ ❖✉t❧✐❡rs✦ ❉✐s♣✉t❡❞ ❝♦❧✉♠♥s✳
▲❡♠♠❛
■❢ t❤❡ ✐♥st❛♥❝❡ ❤❛s ❛ s♦❧✉t✐♦♥✱ t❤❡♥ ★❞✐s♣✳ ❝♦❧✉♠♥s ≤ 4dr✳ ❆❧❣♦r✐t❤♠✿ st❛rt ✇✐t❤ s⋄
m
❜r❛♥❝❤ ♦✈❡r dr + 1 ♠✐s♠❛t❝❤❡s ✭♦r ❞❡❝❧❛r❡ ♦✉t❧✐❡r✮ ❞❡♣t❤ ❜♦✉♥❞✿ 6dr + t✳
SLIDE 50
❘❡s✉❧ts ❢♦r (r, s)-❈❧♦s❡st ❙✉❜str✐♥❣
ℓ k m dr ds |Σ| ❘❡s✉❧t ✕ ✕ ♣ ✕ ✕ ♣ FPT ♣ ♣ ✕ ✕ ✕ ✕ FPT ♣ ✕ ✕ ✕ ♣ ✕ FPT ♣ ✕ ✕ ✕ ✕ ♣ FPT ♣ ✕ ♣ ♣ ✕ ✕ W[1]✲❤❛r❞ ✕ ♣ ✕ ♣ ♣ ♣ W[1]✲❤❛r❞ ✕ ♣ ♣ ♣ ♣ ✕ W[1]✲❤❛r❞
SLIDE 51
❘❡s✉❧ts ❢♦r (r, s)-❈❧♦s❡st ❙✉❜str✐♥❣
ℓ k m dr ds |Σ| ❘❡s✉❧t ✕ ✕ ♣ ✕ ✕ ♣ FPT ♣ ♣ ✕ ✕ ✕ ✕ FPT ♣ ✕ ✕ ✕ ♣ ✕ FPT ♣ ✕ ✕ ✕ ✕ ♣ FPT ♣ ✕ ♣ ♣ ✕ ✕ W[1]✲❤❛r❞ ✕ ♣ ✕ ♣ ♣ ♣ W[1]✲❤❛r❞ ✕ ♣ ♣ ♣ ♣ ✕ W[1]✲❤❛r❞
SLIDE 52
(r, s)-❈❧♦s❡st ❙✉❜str✐♥❣(ℓ, m)
❚❤❡♦r❡♠
(s)-❈❧♦s❡st ❙✉❜str✐♥❣(ℓ, m) ✐s W[1]✲❤❛r❞✳ ❘❡❞✉❝t✐♦♥ ❢r♦♠ ♠✉❧t✐✲❝♦❧♦✉r❡❞ ❝❧✐q✉❡✿ ▲❡t ❜❡ ❛ ❣r❛♣❤ ✇✐t❤ ♣❛rt✐t✐♦♥ s✉❝❤ t❤❛t ❡✈❡r② ✱ ✱ ✐s ❛♥ ✐♥❞❡♣❡♥❞❡♥t s❡t✳
SLIDE 53
(r, s)-❈❧♦s❡st ❙✉❜str✐♥❣(ℓ, m)
❚❤❡♦r❡♠
(s)-❈❧♦s❡st ❙✉❜str✐♥❣(ℓ, m) ✐s W[1]✲❤❛r❞✳ ❘❡❞✉❝t✐♦♥ ❢r♦♠ ♠✉❧t✐✲❝♦❧♦✉r❡❞ ❝❧✐q✉❡✿ ▲❡t G = (V, E) ❜❡ ❛ ❣r❛♣❤ ✇✐t❤ ♣❛rt✐t✐♦♥ V = V1 ∪ . . . ∪ Vkc s✉❝❤ t❤❛t ❡✈❡r② Vi✱ 1 ≤ i ≤ kc✱ ✐s ❛♥ ✐♥❞❡♣❡♥❞❡♥t s❡t✳
SLIDE 54
(r, s)-❈❧♦s❡st ❙✉❜str✐♥❣(ℓ, m)
❚❤❡♦r❡♠
(s)-❈❧♦s❡st ❙✉❜str✐♥❣(ℓ, m) ✐s W[1]✲❤❛r❞✳ ❘❡❞✉❝t✐♦♥ ❢r♦♠ ♠✉❧t✐✲❝♦❧♦✉r❡❞ ❝❧✐q✉❡✿ ▲❡t G = (V, E) ❜❡ ❛ ❣r❛♣❤ ✇✐t❤ ♣❛rt✐t✐♦♥ V = V1 ∪ . . . ∪ Vkc s✉❝❤ t❤❛t ❡✈❡r② Vi✱ 1 ≤ i ≤ kc✱ ✐s ❛♥ ✐♥❞❡♣❡♥❞❡♥t s❡t✳
a b c d e f V1 : $ a c e V2 : $ b d f E1 : $ ⋄ a c ⋄ E2 : $ ⋄ a d ⋄ E3 : $ ⋄ a ⋄ e E4 : $ ⋄ b c ⋄ E5 : $ ⋄ b ⋄ e E6 : $ ⋄ ⋄ c f E7 : $ ⋄ ⋄ d e s : $ a d e
Repeat N = 36 times
SLIDE 55
❘❡s✉❧t ❢♦r t❤❡ ❖✉t❧✐❡r ❱❛r✐❛♥ts
❚❤❡♦r❡♠
(s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦(ds, ℓ, k − t) ✐s W[1]✲❤❛r❞✳ ❲❡ ❦♥♦✇ r ❈❧♦s❡st ❙tr✐♥❣✲✇♦ ✐s ✲❤❛r❞✱ ❜✉t ✳ ✳ ✳
❖♣❡♥ Pr♦❜❧❡♠
r ❈❧♦s❡st ❙tr✐♥❣✲✇♦ ✱ r ❈❧♦s❡st ❙tr✐♥❣✲✇♦ ✱ r ❈❧♦s❡st ❙tr✐♥❣✲✇♦ ✳
SLIDE 56
❘❡s✉❧t ❢♦r t❤❡ ❖✉t❧✐❡r ❱❛r✐❛♥ts
❚❤❡♦r❡♠
(s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦(ds, ℓ, k − t) ✐s W[1]✲❤❛r❞✳ ❲❡ ❦♥♦✇ (r)-❈❧♦s❡st ❙tr✐♥❣✲✇♦(t = 0, |Σ| = 2) ✐s NP✲❤❛r❞✱ ❜✉t ✳ ✳ ✳
❖♣❡♥ Pr♦❜❧❡♠
(r)-❈❧♦s❡st ❙tr✐♥❣✲✇♦(|Σ|, k − t)✱ (r)-❈❧♦s❡st ❙tr✐♥❣✲✇♦(|Σ|, dr)✱ (r)-❈❧♦s❡st ❙tr✐♥❣✲✇♦(|Σ|, dr, k − t)✳
SLIDE 57
❘❡s✉❧t ❢♦r t❤❡ ❖✉t❧✐❡r ❱❛r✐❛♥ts
(r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦(ℓ, |Σ|) ∈ FPT ✭tr✐✈✐❛❧✮✳
❚❤❡♦r❡♠
(r, s)-❈❧♦s❡st ❙tr✐♥❣✲✇♦(ℓ, |Σ|, dr, ds, (k − t)) ❞♦❡s ♥♦t ❛❞♠✐t ❛ ♣♦❧②♥♦♠✐❛❧ ❦❡r♥❡❧ ✉♥❧❡ss coNP ⊆ NP/Poly✳
SLIDE 58