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❈♦♥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 ▼❋❈❙ ✷✵✶✽

❈♦♥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✳

❈♦♥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✳ s 1 c b c a b a a s 2 c b c a b c b s 3 a b c c c c a s 4 c c c a b c a s 5 c b c a a c a s 6 c b c a a c a s 7 a b b a b a a s 8 b b c a a c a

❈♦♥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✳ s 1 c b c a b a a s 2 c b c a b c b s 3 a b c c c c a s 4 c c c a b c a s 5 c b c a a c a s 6 c b c a a c a s 7 a b b a b a a s 8 b b c a a c a s c b c a b c a

❈♦♥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✳ s 1 c b c a b a a s 2 c b c a b c b s 3 a b c c c c a s 4 c c c a b c a s 5 c b c a a c a s 6 c b c a a c a s 7 a b b a b a a s 8 b b c a a c a s c b c a b c a

❈♦♥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✳ s 1 c b c a b a a s 2 c b c a b c b s 3 a b c c c c a s 4 c c c a b c a s 5 c b c a a c a s 6 c b c a a c a s 7 a b b a b a a s 8 b b c a a c a s a b c a b c a

❋♦r ♠✉❧t✐✲s❡t ❛♥❞ ✿ r❛❞✐✉s ♦❢ ✭✇✳ r✳ t✳ ✮ ❞✐st❛♥❝❡ s✉♠ ♦❢ ✭✇✳ r✳ t✳ ✮ ❇❛s✐❝ ◆♦t❛t✐♦♥s ❙t❛♥❞❛r❞ str✐♥❣ ♥♦t❛t✐♦♥s✿ Σ ✜♥✐t❡ ❛❧♣❤❛❜❡t Σ ∗ ✇♦r❞s ♦✈❡r Σ { w ∈ Σ ∗ | | w | = n } Σ n � n Σ ≤ n i =0 Σ i d H ( u, v ) ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡ � s✉❜str✐♥❣ r❡❧❛t✐♦♥✱ ❛ u � v ⇔ v = xuy

❇❛s✐❝ ◆♦t❛t✐♦♥s ❙t❛♥❞❛r❞ str✐♥❣ ♥♦t❛t✐♦♥s✿ Σ ✜♥✐t❡ ❛❧♣❤❛❜❡t Σ ∗ ✇♦r❞s ♦✈❡r Σ { w ∈ Σ ∗ | | w | = n } Σ n � n Σ ≤ n i =0 Σ i d H ( u, v ) ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡ � s✉❜str✐♥❣ r❡❧❛t✐♦♥✱ ❛ u � v ⇔ v = xuy ❋♦r ♠✉❧t✐✲s❡t S ⊆ Σ ℓ ❛♥❞ v ∈ Σ ℓ ✿ r H ( v, S ) = max { d H ( v, u ) | u ∈ S } r❛❞✐✉s ♦❢ S ✭✇✳ r✳ t✳ v ✮ s H ( v, S ) = � u ∈ S d H ( v, u ) ❞✐st❛♥❝❡ s✉♠ ♦❢ S ✭✇✳ r✳ t✳ v ✮

❚❤❡ ❈❧♦s❡st ❙tr✐♥❣ Pr♦❜❧❡♠ ( r , s )- ❈❧♦s❡st ❙tr✐♥❣ ▼✉❧t✐✲s❡t S = { s i | 1 ≤ i ≤ k } ⊆ Σ ℓ ✱ ℓ ∈ N ✱ ■♥st❛♥❝❡ ✿ d r , d s ∈ N ✳ ■s t❤❡r❡ ❛♥ s ∈ Σ ℓ ✇✐t❤ ◗✉❡st✐♦♥ ✿ r H ( s, S ) ≤ d r ❛♥❞ s H ( s, S ) ≤ d s ❄

❚❤❡ ❈❧♦s❡st ❙tr✐♥❣ Pr♦❜❧❡♠ ( r , s )- ❈❧♦s❡st ❙tr✐♥❣ ▼✉❧t✐✲s❡t S = { s i | 1 ≤ i ≤ k } ⊆ Σ ℓ ✱ ℓ ∈ N ✱ ■♥st❛♥❝❡ ✿ d r , d s ∈ N ✳ ■s t❤❡r❡ ❛♥ s ∈ Σ ℓ ✇✐t❤ ◗✉❡st✐♦♥ ✿ r H ( s, S ) ≤ d r ❛♥❞ s H ( s, S ) ≤ d s ❄ s 1 c b c a b a a s 2 c b c a b c b k = 8 s 3 a b c c c c a ℓ = 7 s 4 c c c a b c a d r = 2 s 5 c b c a a c a d s = 20 s 6 c b c a a c a r H ( s, S ) = 2 s 7 a b b a b a a s H ( s, S ) = 16 s 8 b b c a a c a

❚❤❡ ❈❧♦s❡st ❙tr✐♥❣ Pr♦❜❧❡♠ ( r , s )- ❈❧♦s❡st ❙tr✐♥❣ ▼✉❧t✐✲s❡t S = { s i | 1 ≤ i ≤ k } ⊆ Σ ℓ ✱ ℓ ∈ N ✱ ■♥st❛♥❝❡ ✿ d r , d s ∈ N ✳ ■s t❤❡r❡ ❛♥ s ∈ Σ ℓ ✇✐t❤ ◗✉❡st✐♦♥ ✿ r H ( s, S ) ≤ d r ❛♥❞ s H ( s, S ) ≤ d s ❄ s 1 c b c a b a a s 2 c b c a b c b k = 8 s 3 a b c c c c a ℓ = 7 s 4 c c c a b c a d r = 2 s 5 c b c a a c a d s = 20 s 6 c b c a a c a s 7 r H ( s, S ) = 2 a b b a b a a s H ( s, S ) = 16 s 8 b b c a a c a s a b c a b c a

❚❤❡ ❈❧♦s❡st ❙tr✐♥❣ Pr♦❜❧❡♠ ( r , s )- ❈❧♦s❡st ❙tr✐♥❣ ▼✉❧t✐✲s❡t S = { s i | 1 ≤ i ≤ k } ⊆ Σ ℓ ✱ ℓ ∈ N ✱ ■♥st❛♥❝❡ ✿ d r , d s ∈ N ✳ ■s t❤❡r❡ ❛♥ s ∈ Σ ℓ ✇✐t❤ ◗✉❡st✐♦♥ ✿ r H ( s, S ) ≤ d r ❛♥❞ s H ( s, S ) ≤ d s ❄ s 1 c b c a b a a s 2 c b c a b c b k = 8 s 3 a b c c c c a ℓ = 7 s 4 c c c a b c a d r = 2 s 5 c b c a a c a d s = 20 s 6 c b c a a c a s 7 r H ( s, S ) = 2 a b b a b a a s H ( s, S ) = 16 s 8 b b c a a c a s a b c a b c a

❚❤❡ ✏❙✉❜str✐♥❣ ❱❛r✐❛♥t✑ ( r , s )- ❈❧♦s❡st ❙✉❜str✐♥❣ ▼✉❧t✐✲s❡t S = { s i | 1 ≤ i ≤ k } ⊆ Σ ≤ ℓ ✱ ℓ ∈ N ✱ ■♥st❛♥❝❡ ✿ d r , d s , m ∈ N ✳ ■s t❤❡r❡ ❛♥ s ∈ Σ m ✱ ◗✉❡st✐♦♥ ✿ ♠✉❧t✐✲s❡t S ′ = { s ′ i � s i , 1 ≤ i ≤ k } ⊆ Σ m ✇✐t❤ i | s ′ r H ( s, S ′ ) ≤ d r ❛♥❞ s H ( s, S ′ ) ≤ d s ❄

❚❤❡ ✏❙✉❜str✐♥❣ ❱❛r✐❛♥t✑ ( r , s )- ❈❧♦s❡st ❙✉❜str✐♥❣ ▼✉❧t✐✲s❡t S = { s i | 1 ≤ i ≤ k } ⊆ Σ ≤ ℓ ✱ ℓ ∈ N ✱ ■♥st❛♥❝❡ ✿ d r , d s , m ∈ N ✳ ■s t❤❡r❡ ❛♥ s ∈ Σ m ✱ ◗✉❡st✐♦♥ ✿ ♠✉❧t✐✲s❡t S ′ = { s ′ i � s i , 1 ≤ i ≤ k } ⊆ Σ m ✇✐t❤ i | s ′ r H ( s, S ′ ) ≤ d r ❛♥❞ s H ( s, S ′ ) ≤ d s ❄ s 1 a a c b c a b a a k = 8 s 2 b c b c a b c b ℓ = 11 s 3 a a b c c m = 4 s 4 c c c a b c a c d r = 3 s 5 c c b c a a c a d s = 7 s 6 a c b c a a s 7 r H ( s, S ) = 2 a a b b a b a a s H ( s, S ) = 16 s 8 b b b c a a c a c c b

❚❤❡ ✏❙✉❜str✐♥❣ ❱❛r✐❛♥t✑ ( r , s )- ❈❧♦s❡st ❙✉❜str✐♥❣ ▼✉❧t✐✲s❡t S = { s i | 1 ≤ i ≤ k } ⊆ Σ ≤ ℓ ✱ ℓ ∈ N ✱ ■♥st❛♥❝❡ ✿ d r , d s , m ∈ N ✳ ■s t❤❡r❡ ❛♥ s ∈ Σ m ✱ ◗✉❡st✐♦♥ ✿ ♠✉❧t✐✲s❡t S ′ = { s ′ i � s i , 1 ≤ i ≤ k } ⊆ Σ m ✇✐t❤ i | s ′ r H ( s, S ′ ) ≤ d r ❛♥❞ s H ( s, S ′ ) ≤ d s ❄ s 1 a a c b c a b a a s 2 b c b c a b c b k = 8 ℓ = 11 s 3 a a b c c m = 4 s 4 c c c a b c a c d r = 3 s 5 c c b c a a c a d s = 7 s 6 a c b c a a s 7 a a b b a b a a r H ( s, S ) = 2 s 8 b b b c a a c a c c b s H ( s, S ) = 16 s a b c a

❚❤❡ ✏❙✉❜str✐♥❣ ❱❛r✐❛♥t✑ ( r , s )- ❈❧♦s❡st ❙✉❜str✐♥❣ ▼✉❧t✐✲s❡t S = { s i | 1 ≤ i ≤ k } ⊆ Σ ≤ ℓ ✱ ℓ ∈ N ✱ ■♥st❛♥❝❡ ✿ d r , d s , m ∈ N ✳ ■s t❤❡r❡ ❛♥ s ∈ Σ m ✱ ◗✉❡st✐♦♥ ✿ ♠✉❧t✐✲s❡t S ′ = { s ′ i � s i , 1 ≤ i ≤ k } ⊆ Σ m ✇✐t❤ i | s ′ r H ( s, S ′ ) ≤ d r ❛♥❞ s H ( s, S ′ ) ≤ d s ❄ s 1 a a c b c a b a a s 2 b c b c a b c b k = 8 ℓ = 11 s 3 a a b c c m = 4 s 4 c c c a b c a c d r = 3 s 5 c c b c a a c a d s = 7 s 6 a c b c a a s 7 a a b b a b a a r H ( s, S ) = 2 s 8 b b b c a a c a c c b s H ( s, S ) = 16 s a b c a

❚❤❡ ✏❙✉❜str✐♥❣ ❱❛r✐❛♥t✑ ( r , s )- ❈❧♦s❡st ❙✉❜str✐♥❣ ▼✉❧t✐✲s❡t S = { s i | 1 ≤ i ≤ k } ⊆ Σ ≤ ℓ ✱ ℓ ∈ N ✱ ■♥st❛♥❝❡ ✿ d r , d s , m ∈ N ✳ ■s t❤❡r❡ ❛♥ s ∈ Σ m ✱ ◗✉❡st✐♦♥ ✿ ♠✉❧t✐✲s❡t S ′ = { s ′ i � s i , 1 ≤ i ≤ k } ⊆ Σ m ✇✐t❤ i | s ′ r H ( s, S ′ ) ≤ d r ❛♥❞ s H ( s, S ′ ) ≤ d s ❄ s 1 a a c b c a b a a s 2 b c b c a b c b k = 8 ℓ = 11 s 3 a a b c c m = 4 s 4 c c c a b c a c d r = 3 s 5 c c b c a a c a d s = 7 s 6 a c b c a a s 7 a a b b a b a a r H ( s, S ′ ) = 1 s 8 s H ( s, S ′ ) = 7 b b b c a a c a c c b s a b c a

❚❤❡ ✏❖✉t❧✐❡r ❱❛r✐❛♥t✑ ( r , s )- ❈❧♦s❡st ❙tr✐♥❣ ✇✐t❤ ❖✉t❧✐❡rs ▼✉❧t✐✲s❡t S = { s i | 1 ≤ i ≤ k } ⊆ Σ ℓ ✱ ℓ ∈ N ✱ ■♥st❛♥❝❡ ✿ d r , d s , t ∈ N ✳ ■s t❤❡r❡ ❛♥ s ∈ Σ ℓ ✱ ◗✉❡st✐♦♥ ✿ S ′ ⊆ S ✇✐t❤ | S ′ | = k − t ✇✐t❤ r H ( s, S ′ ) ≤ d r ❛♥❞ s H ( s, S ′ ) ≤ d s ❄