ss r s s t sts - - PowerPoint PPT Presentation

▶

Jul 12, 2023 109 likes •540 views

ss r s s t sts ts ss r P r

SLIDE 1

❆♥❛❧②s✐s ♦❢ r❛♥❞♦♠ ✇❛❧❦s ✉s✐♥❣ t❛❜✉ ❧✐sts

❑✳ ❆❧t✐s❡♥✱ ❙✳ ❉❡✈✐s♠❡s✱ ❆✳ ●❡r❜❛✉❞ ❛♥❞ P✳ ▲❛❢♦✉r❝❛❞❡

▲❛❜♦r❛t♦✐r❡ ❱❡r✐♠❛❣ ❯♥✐✈❡rs✐té ❏♦s❡♣❤ ❋♦✉r✐❡r✱ ●r❡♥♦❜❧❡ ✶

✶ ✴ ✶✽

SLIDE 2

✶

❘❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st

✷

❚❡r♠✐♥❛t✐♦♥

✸

❖♣t✐♠❛❧ ✉♣❞❛t❡ r✉❧❡s ❢♦r ♠✲❢r❡❡ ❣r❛♣❤s

✷ ✴ ✶✽

SLIDE 3

✶

❘❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st

✷

❚❡r♠✐♥❛t✐♦♥

✸

❖♣t✐♠❛❧ ✉♣❞❛t❡ r✉❧❡s ❢♦r ♠✲❢r❡❡ ❣r❛♣❤s

✸ ✴ ✶✽

SLIDE 4

❙✐♠♣❧❡ r❛♥❞♦♠ ✇❛❧❦ r♦✉t✐♥❣

1 2 3 4 5 6 7 8 9 1/4 1/4 1/4 1/4 source destination

❆❧❧ ❣r❛♣❤s ❛r❡ ✜♥✐t❡✱ s✐♠♣❧❡✱ ❝♦♥♥❡❝t❡❞✱ ✇✐t❤ ❛t ❧❡❛st t✇♦ ✈❡rt✐❝❡s✳

✹ ✴ ✶✽

SLIDE 5

❙✐♠♣❧❡ r❛♥❞♦♠ ✇❛❧❦ r♦✉t✐♥❣

1 2 3 4 5 6 7 8 9 1/4 1/4 1/4 1/4 source destination

❆t ❡❛❝❤ st❡♣✱ t❤❡ ✇❛❧❦❡r ♠♦✈❡s t♦ ❛ ♥❡✐❣❤❜♦r ❝❤♦s❡♥ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✳ ❚❤❡ ✇❛❧❦❡r r❡♣r❡s❡♥ts ❛ ♣❛❝❦❡t ❢♦r✇❛r❞❡❞ ❢r♦♠ ♥♦❞❡ t♦ ♥♦❞❡✳

✹ ✴ ✶✽

SLIDE 6

❙✐♠♣❧❡ r❛♥❞♦♠ ✇❛❧❦ r♦✉t✐♥❣

1 2 3 4 5 6 7 8 9 1/4 1/4 1/4 1/4 source destination

❆❞✈❛♥t❛❣❡s✿ ♥♦ r♦✉t✐♥❣ t❛❜❧❡❀ ❧♦❝❛❧ ❝♦♠♣✉t❛t✐♦♥s❀ s❝❛❧❛❜❧❡❀ r♦❜✉st❀ ❛♥❞ ❧♦❛❞✲❜❛❧❛♥❝❡❞✳ ❉r❛✇❜❛❝❦ ✿ s❧♦✇✳

✹ ✴ ✶✽

SLIDE 7

▼❡❛♥ ❤✐tt✐♥❣ t✐♠❡s

❍✐tt✐♥❣ t✐♠❡✿ ♥✉♠❜❡r ♦❢ st❡♣s t♦ ❞❡❧✐✈❡r t❤❡ ♣❛❝❦❡t t♦ ✐ts ✜♥❛❧ ❞❡st✐♥❛t✐♦♥✳ ▼♦t✐✈❛t✐♦♥✿ r❡❞✉❝✐♥❣ t❤❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s✳

✺ ✴ ✶✽

SLIDE 8

▼❡❛♥ ❤✐tt✐♥❣ t✐♠❡s

▲❡t (❳♥)♥✵ ❜❡ ❛ r❛♥❞♦♠ ✇❛❧❦ ♦♥ ❛ ❣r❛♣❤ ●✳ ❋♦r ❡✈❡r② t✇♦ ✈❡rt✐❝❡s ① ❛♥❞ ② ♦❢ ●✿ ❚❤❡ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② ✐s t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❍②(●) = ✐♥❢{♥ ✵ : ❳♥ = ②} . ❚❤❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② st❛rt✐♥❣ ❢r♦♠ ① ✐s t❤❡ ♠❡❛♥ ✈❛❧✉❡ ❊①❍②(●)✳

✺ ✴ ✶✽

SLIDE 9

▼❡❛♥ ❤✐tt✐♥❣ t✐♠❡s

▲❡t (❳♥)♥✵ ❜❡ ❛ r❛♥❞♦♠ ✇❛❧❦ ♦♥ ❛ ❣r❛♣❤ ●✳ ❋♦r ❡✈❡r② t✇♦ ✈❡rt✐❝❡s ① ❛♥❞ ② ♦❢ ●✿ ❚❤❡ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② ✐s t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❍②(●) = ✐♥❢{♥ ✵ : ❳♥ = ②} . ❚❤❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② st❛rt✐♥❣ ❢r♦♠ ① ✐s t❤❡ ♠❡❛♥ ✈❛❧✉❡ ❊①❍②(●)✳

1 2 3 4 5

❊✶❍✵(❑✻) = ✺

✺ ✴ ✶✽

SLIDE 10

▼❡❛♥ ❤✐tt✐♥❣ t✐♠❡s

▲❡t (❳♥)♥✵ ❜❡ ❛ r❛♥❞♦♠ ✇❛❧❦ ♦♥ ❛ ❣r❛♣❤ ●✳ ❋♦r ❡✈❡r② t✇♦ ✈❡rt✐❝❡s ① ❛♥❞ ② ♦❢ ●✿ ❚❤❡ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② ✐s t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❍②(●) = ✐♥❢{♥ ✵ : ❳♥ = ②} . ❚❤❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② st❛rt✐♥❣ ❢r♦♠ ① ✐s t❤❡ ♠❡❛♥ ✈❛❧✉❡ ❊①❍②(●)✳

1 2 3 4 5

❊✶❍✵(❙✻) = ✷ × ✺

✺ ✴ ✶✽

SLIDE 11

▼❡❛♥ ❤✐tt✐♥❣ t✐♠❡s

▲❡t (❳♥)♥✵ ❜❡ ❛ r❛♥❞♦♠ ✇❛❧❦ ♦♥ ❛ ❣r❛♣❤ ●✳ ❋♦r ❡✈❡r② t✇♦ ✈❡rt✐❝❡s ① ❛♥❞ ② ♦❢ ●✿ ❚❤❡ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② ✐s t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❍②(●) = ✐♥❢{♥ ✵ : ❳♥ = ②} . ❚❤❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② st❛rt✐♥❣ ❢r♦♠ ① ✐s t❤❡ ♠❡❛♥ ✈❛❧✉❡ ❊①❍②(●)✳

1 2 3 4 5 6 7 8 9

❊✶❍✵(●) = ✸✾

✺ ✴ ✶✽

SLIDE 12

❘❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st

❚❤❡ ✇❛❧❦❡r ❤❛s ❛ ✜♥✐t❡ ♠❡♠♦r② ✐♥ ✇❤✐❝❤ ✐t st♦r❡s ♦❝❝✉rr❡♥❝❡s ♦❢ ✈❡rt✐❝❡s ❛❧r❡❛❞② ✈✐s✐t❡❞✱ ❝❛❧❧❡❞ t❛❜✉ ❧✐st✳ ❚❤❡ ♦❝❝✉rr❡♥❝❡s ❛r❡ s♦rt❡❞ ✐♥ ❝❤r♦♥♦❧♦❣✐❝❛❧ ♦r❞❡r✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ❜❧♦❝❦s ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ t❛❜✉ ❧✐st✳ ❚❤❡ t❛❜✉ ❧✐st ✐s ❢✉❧❧ ✐❢ ❛❧❧ ❜❧♦❝❦s ❛r❡ ❛❧❧♦❝❛t❡❞✳ ❚❤❡ ♣♦❧✐❝② t♦ ✐♥s❡rt ♦r r❡♠♦✈❡ ♦❝❝✉rr❡♥❝❡s ♦❢ ✈❡rt✐❝❡s ✐♥ t❤❡ t❛❜✉ ❧✐st ✐s ❝❛❧❧❡❞ t❤❡ ✉♣❞❛t❡ r✉❧❡✳

3 1

last

ccurence

inserted ✭❛✮ ❚❤❡ t❛❜✉ ❧✐st ❤❛s ❧❡♥❣t❤ ✸ ✇✐t❤ ✷ ❛❧❧♦❝❛t❡❞ ❜❧♦❝❦s✳

2 5 1 2 4

last

ccurence

inserted ✭❜✮ ❚❤❡ t❛❜✉ ❧✐st ❤❛s ❧❡♥❣t❤ ✺ ❛♥❞ ✐s ❢✉❧❧✳

✻ ✴ ✶✽

SLIDE 13

❘❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st

■❢ t❤❡ t❛❜✉ ❧✐st ✐s ♥♦t ❢✉❧❧✱ t❤❡♥ t❤r❡❡ ❝❛s❡s ♠❛② ♦❝❝✉r✿

x t1 x t3 x t1 x t3 ✭❝✮ ❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ✐♥s❡rt❡❞✳ x t1 x t3 x t1 t3 ✭❞✮ ❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ✐♥s❡rt❡❞ ❛♥❞ ❛♥ ♦❧❞❡r ♦❝✲ ❝✉rr❡♥❝❡ ♦❢ t❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ❞✐s❝❛r❞❡❞✳ x t1 x t3 t1 x t3 ✭❡✮ ❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ♥♦t ✐♥s❡rt❡❞✳

✻ ✴ ✶✽

SLIDE 14

❘❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st

■❢ t❤❡ t❛❜✉ ❧✐st ✐s ❢✉❧❧✱ t❤❡♥ t❤r❡❡ ❝❛s❡s ♠❛② ♦❝❝✉r✿

x t1 x t3 t4 t5 x t1 x t3 t5 ✭❢✮ ❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ✐♥s❡rt❡❞ ❛♥❞ ❛♥ ♦❝❝✉r✲ r❡♥❝❡ ♦❢ ❛♥♦t❤❡r ✈❡rt❡① ✐s ❞✐s❝❛r❞❡❞✳ x t1 x t3 t4 t5 x t1 t3 t4 t5 ✭❣✮ ❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ✐♥s❡rt❡❞ ❛♥❞ ❛♥ ♦❧❞❡r ♦❝✲ ❝✉rr❡♥❝❡ ♦❢ t❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ❞✐s❝❛r❞❡❞✳ x t1 x t3 t4 t5 t1 x t3 t4 t5 ✭❤✮ ❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ♥♦t ✐♥s❡rt❡❞✳

✻ ✴ ✶✽

SLIDE 15

❘❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st

❆t ❡❛❝❤ st❡♣✱ t❤❡ ✇❛❧❦❡r ♠♦✈❡s t♦ ❛ ♥❡✐❣❤❜♦r ❝❤♦s❡♥ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠ ❛♠♦♥❣ t❤♦s❡ ✇✐t❤♦✉t ♦❝❝✉rr❡♥❝❡ ✐♥ t❤❡ t❛❜✉ ❧✐st✳ ■❢ ❛❧❧ ♥❡✐❣❤❜♦rs ❤❛✈❡ ❛♥ ♦❝❝✉rr❡♥❝❡ ✐♥ t❤❡ t❛❜✉ ❧✐st✱ t❤❡ ✇❛❧❦❡r ♠♦✈❡s t♦ ❛ ♥❡✐❣❤❜♦r ❝❤♦s❡♥ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✳

1 2 3 4 5 6 7 8 9 1/2 1/2

2 5 2 7

1 2 3 4 5 6 7 8 9 1/4 1/4 1/4 1/4

2 8 6 7 6

✻ ✴ ✶✽

SLIDE 16

❋✐rst ■♥ ❋✐rst ❖✉t

❚❤❡ ❋■❋❖♠ ✉♣❞❛t❡ r✉❧❡s✱ ♠ ✵✳

x t1 t2

almost surely

x t1 t2 x t1 t2 t3 t4 t5

almost surely

x t1 t2 t3 t4

❍❡r❡✱ ♠ = ✺✳ ❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ✐♥s❡rt❡❞ ❛❧♠♦st s✉r❡❧②✳ ■❢ t❤❡ t❛❜✉ ❧✐st ✐s ❢✉❧❧✱ t❤❡♥ t❤❡ ♦❝❝✉r❡♥❝❡ ♦❢ t❤❡ ✈❡rt❡① st♦r❡❞ ✐♥ t❤❡ ❧❛st ❜❧♦❝❦ ✐s ❞✐s❝❛r❞❡❞✱ ❛❧♠♦st s✉r❡❧②✳

✼ ✴ ✶✽

SLIDE 17

❋✐rst ■♥ ❋✐rst ❖✉t

❚❤❡ ❋■❋❖✹✵ ✉♣❞❛t❡ r✉❧❡✳

✼ ✴ ✶✽

SLIDE 18

❋✐rst ■♥ ❋✐rst ❖✉t

❈♦♥tr❛r✐❧② t♦ t❤❡ s♥❛❦❡✱ t❤❡ ✇❛❧❦❡r ♥❡✈❡r ❞✐❡s✳

✼ ✴ ✶✽

SLIDE 19

❆ ♥♦♥ ❞❡t❡r♠✐♥✐st✐❝ ✉♣❞❛t❡ r✉❧❡

x t1 t2

almost surely

x t1 t2 x t1 t2 t3 t4 t5

probability 1/5

x t1 t2 t4 t5

❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ✐♥s❡rt❡❞ ❛❧♠♦st s✉r❡❧②✳ ■❢ t❤❡ t❛❜✉ ❧✐st ✐s ❢✉❧❧✱ t❤❡♥ ❛♥ ♦❝❝✉r❡♥❝❡ ✐♥ t❤❡ t❛❜✉ ❧✐st✱ ❝❤♦s❡♥ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✱ ✐s ❞✐s❝❛r❞❡❞✳ ❍❡r❡✱ t✸ ✐s ❞✐s❝❛r❞❡❞✳

✽ ✴ ✶✽

SLIDE 20

❍✐tt✐♥❣ t✐♠❡ ✐♥ ❧♦❧❧✐❧♦♣ ❣r❛♣❤

❋♦r t❤❡ s✐♠♣❧❡ r❛♥❞♦♠ ✇❛❧❦✱ ●✳ ❇r✐❣❤t✇❡❧❧ ❛♥❞ P✳ ❲✐♥❦❧❡r s❤♦✇❡❞ ✐♥ ✶✾✾✵✿ ♠❛①

:|V|=♥ ♠❛①

①,②∈V ❊①❍②(●) = ❊✶❍✵(▲♦❧❧✐❧♦♣♥) = ✹♥✸

✷✼ (✶ + ♦(✶)) . ❋♦r t❤❡ r❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st ❛♥❞ ✉♣❞❛t❡ r✉❧❡ ❋■❋❖✶✱ ❊✶❍✵(▲♦❧❧✐❧♦♣♥) = ✹♥✷ ✾ (✶ + ♦(✶)) .

1 2 3 4 5 6 7 8 9 10 11 12 13 14 clique with 2n/3 vertices path with n/3 vertices

✾ ✴ ✶✽

SLIDE 21

✶

❘❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st

✷

❚❡r♠✐♥❛t✐♦♥

✸

❖♣t✐♠❛❧ ✉♣❞❛t❡ r✉❧❡s ❢♦r ♠✲❢r❡❡ ❣r❛♣❤s

✶✵ ✴ ✶✽

SLIDE 22

❚r❛♣

1 2 3 4

X0 X1 X2 X3 X4 X5 X6 X7

X0

1

X1

2 1

X2

3 2 1

X3

4 3 1

X4

2 4 1

X5

3 2 1

X6

4 3 1

X7

2 4 1

■♥ s♦♠❡ ❝❛s❡s✱ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❛ ✈❡rt❡① ✐s ♥❡✈❡r r❡❛❝❤❡❞✳

✶✶ ✴ ✶✽

SLIDE 23

❋✐♥✐t❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s

❆♥ ✉♣❞❛t❡ r✉❧❡ ✐s ❝❛❧❧❡❞ tr✐✈✐❛❧ ✐❢ t❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ♥❡✈❡r ✐♥s❡rt❡❞ ✐♥ t❤❡ t❛❜✉ ❧✐st ✇❤❡♥ t❤❡ t❛❜✉ ❧✐st ❝♦♥t❛✐♥s ♥♦ ❡❧❡♠❡♥t✳ ❚❤❡ ✉♥✐q✉❡ ✉♣❞❛t❡ r✉❧❡ ✇✐t❤ ③❡r♦ ❧❡♥❣t❤ ✐s tr✐✈✐❛❧✳ ❈♦♥s✐❞❡r ❛ tr✐✈✐❛❧ ✉♣❞❛t❡ r✉❧❡✳ ■❢ t❤❡ t❛❜✉ ❧✐st ✐s ✐♥✐t✐❛❧❧② ❡♠♣t②✱ t❤❡♥ ✐t r❡♠❛✐♥s ❡♠♣t② ❢♦r❡✈❡r ❛♥❞ t❤❡ ✇❛❧❦❡r ♣❡r❢♦r♠s ❛ s✐♠♣❧❡ r❛♥❞♦♠ ✇❛❧❦✳ ❈♦♥s❡q✉❡♥t❧②✱ ❛❧❧ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s ❛r❡ ✜♥✐t❡ ❢♦r ❛❧❧ ❣r❛♣❤s✳

✶✷ ✴ ✶✽

SLIDE 24

❋✐♥✐t❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s

❚❤❡♦r❡♠

❈♦♥s✐❞❡r ❛ ♥♦♥ tr✐✈✐❛❧ ✉♣❞❛t❡ r✉❧❡✳ ❆❧❧ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s ❛r❡ ✜♥✐t❡ ❢♦r ❛❧❧ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s✱ ✐❢ ❛♥❞ ♦♥❧② ✐❢✱ ✇❤❡♥ t❤❡ ❝✉rr❡♥t ✈❡rt❡① ❤❛s ♥♦ ♦❝❝✉rr❡♥❝❡ ✐♥ t❤❡ t❛❜✉ ❧✐st✿ ■❢ t❤❡ t❛❜✉ ❧✐st ✐s ♥♦t ❢✉❧❧✱ t❤❡♥ t❤❡ ♣r♦❜❛❜✐❧✐t② t♦ ✐♥s❡rt t❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ♣♦s✐t✐✈❡✳ ■❢ t❤❡ t❛❜✉ ❧✐st ✐s ❢✉❧❧✱ t❤❡♥ t❤❡ ♣r♦❜❛❜✐❧✐t② t♦ r❡♠♦✈❡ t❤❡ ❧❛st ❡❧❡♠❡♥t ✐s ♣♦s✐t✐✈❡✳

x t1 t2

probability > 0

x t1 t2 x t1 t2 t3 t4 t5

probability > 0

x t1 t2 t3 t4

✶✷ ✴ ✶✽

SLIDE 25

❋✐♥✐t❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✳

■❢ ♦♥❡ ♦❢ t❤❡ t✇♦ ❝♦♥❞✐t✐♦♥s ✐s ♥♦t ❢✉❧❧✜❧❧❡❞✱ t❤❡♥ ✇❡ ❡①❤✐❜✐t ❛ ❣r❛♣❤ s✉❝❤ t❤❛t✱ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t②✱ t❤❡ ✇❛❧❦❡r ♥❡✈❡r r❡❛❝❤❡s ❛ ❣✐✈❡♥ ✈❡rt❡①✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥✱ t❤❡ ✈❡rt❡① ✶ ✐s ♥❡✈❡r r❡♠♦✈❡❞ ❢r♦♠ t❤❡ t❛❜✉ ❧✐st✳ ❚❤✉s✱ t❤❡ ✈❡rt❡① ✵ ✐s ♥❡✈❡r r❡❛❝❤❡❞✳

1 5 4 3 2 6

X0

✶✷ ✴ ✶✽

SLIDE 26

❋✐♥✐t❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✳

■❢ ♦♥❡ ♦❢ t❤❡ t✇♦ ❝♦♥❞✐t✐♦♥s ✐s ♥♦t ❢✉❧❧✜❧❧❡❞✱ t❤❡♥ ✇❡ ❡①❤✐❜✐t ❛ ❣r❛♣❤ s✉❝❤ t❤❛t✱ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t②✱ t❤❡ ✇❛❧❦❡r ♥❡✈❡r r❡❛❝❤❡s ❛ ❣✐✈❡♥ ✈❡rt❡①✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥✱ t❤❡ ✈❡rt❡① ✶ ✐s ♥❡✈❡r r❡♠♦✈❡❞ ❢r♦♠ t❤❡ t❛❜✉ ❧✐st✳ ❚❤✉s✱ t❤❡ ✈❡rt❡① ✵ ✐s ♥❡✈❡r r❡❛❝❤❡❞✳

1 5 4 3 2 6

4 3 2 1 X4

✶✷ ✴ ✶✽

SLIDE 27

❋✐♥✐t❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✳

■❢ ♦♥❡ ♦❢ t❤❡ t✇♦ ❝♦♥❞✐t✐♦♥s ✐s ♥♦t ❢✉❧❧✜❧❧❡❞✱ t❤❡♥ ✇❡ ❡①❤✐❜✐t ❛ ❣r❛♣❤ s✉❝❤ t❤❛t✱ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t②✱ t❤❡ ✇❛❧❦❡r ♥❡✈❡r r❡❛❝❤❡s ❛ ❣✐✈❡♥ ✈❡rt❡①✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥✱ t❤❡ ✈❡rt❡① ✶ ✐s ♥❡✈❡r r❡♠♦✈❡❞ ❢r♦♠ t❤❡ t❛❜✉ ❧✐st✳ ❚❤✉s✱ t❤❡ ✈❡rt❡① ✵ ✐s ♥❡✈❡r r❡❛❝❤❡❞✳

1 5 4 3 2 6

5 3 2 1 X5

✶✷ ✴ ✶✽

SLIDE 28

❋✐♥✐t❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✳

■❢ ♦♥❡ ♦❢ t❤❡ t✇♦ ❝♦♥❞✐t✐♦♥s ✐s ♥♦t ❢✉❧❧✜❧❧❡❞✱ t❤❡♥ ✇❡ ❡①❤✐❜✐t ❛ ❣r❛♣❤ s✉❝❤ t❤❛t✱ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t②✱ t❤❡ ✇❛❧❦❡r ♥❡✈❡r r❡❛❝❤❡s ❛ ❣✐✈❡♥ ✈❡rt❡①✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥✱ t❤❡ ✈❡rt❡① ✶ ✐s ♥❡✈❡r r❡♠♦✈❡❞ ❢r♦♠ t❤❡ t❛❜✉ ❧✐st✳ ❚❤✉s✱ t❤❡ ✈❡rt❡① ✵ ✐s ♥❡✈❡r r❡❛❝❤❡❞✳

1 5 4 3 2 6

6 5 2 1 X6

✶✷ ✴ ✶✽

SLIDE 29

❋✐♥✐t❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✳

■❢ ♦♥❡ ♦❢ t❤❡ t✇♦ ❝♦♥❞✐t✐♦♥s ✐s ♥♦t ❢✉❧❧✜❧❧❡❞✱ t❤❡♥ ✇❡ ❡①❤✐❜✐t ❛ ❣r❛♣❤ s✉❝❤ t❤❛t✱ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t②✱ t❤❡ ✇❛❧❦❡r ♥❡✈❡r r❡❛❝❤❡s ❛ ❣✐✈❡♥ ✈❡rt❡①✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥✱ t❤❡ ✈❡rt❡① ✶ ✐s ♥❡✈❡r r❡♠♦✈❡❞ ❢r♦♠ t❤❡ t❛❜✉ ❧✐st✳ ❚❤✉s✱ t❤❡ ✈❡rt❡① ✵ ✐s ♥❡✈❡r r❡❛❝❤❡❞✳ ❚❤❡ r❡❝✐♣r♦❝❛❧ ❛ss❡rt✐♦♥ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ t❤❡♦r② ♦❢ ▼❛r❦♦✈ ❝❤❛✐♥s ♦♥ ✜♥✐t❡ s♣❛❝❡✳

1 5 4 3 2 6

6 5 2 1 X6

✶✷ ✴ ✶✽

SLIDE 30

✶

❘❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st

✷

❚❡r♠✐♥❛t✐♦♥

✸

❖♣t✐♠❛❧ ✉♣❞❛t❡ r✉❧❡s ❢♦r ♠✲❢r❡❡ ❣r❛♣❤s

✶✸ ✴ ✶✽

SLIDE 31

❖♣t✐♠❛❧ ✉♣❞❛t❡ r✉❧❡s ❢♦r ♠✲❢r❡❡ ❣r❛♣❤s

❆ ❣r❛♣❤ ✐s ♠✲❢r❡❡ ✐❢ ❛ t❛❜✉ r❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ ❋■❋❖♠ ✉♣❞❛t❡ r✉❧❡ ❝❛♥♥♦t ✈✐s✐t ❛ ✈❡rt❡① ✐♥ ✐ts ❝✉rr❡♥t t❛❜✉ ❧✐st✱ ✉♥❧❡ss t❤❡ ❝✉rr❡♥t ✈❡rt❡① ❤❛s ❞❡❣r❡❡ ♦♥❡✳ ❆❧❧ ❣r❛♣❤s ❛r❡ ✶✲❢r❡❡✳ ❆ ❣r❛♣❤ ✐s ✷✲❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② tr✐❛♥❣❧❡ ✇✐t❤ ❛ ❞❡❣r❡❡ t✇♦ ✈❡rt❡①✳ ❊✈❡r② ♠ ✶ ✲r❡❣✉❧❛r ❣r❛♣❤ ✐s ♠✲❢r❡❡✳ ❆ ❣r❛♣❤ ✇✐t❤ ❣✐rt❤ ❣r❡❛t❡r t❤❛♥ ♦r ❡q✉❛❧ t♦ ♠ ✷ ✐s ♠✲❢r❡❡✳ ■❢ ✶ ❦ ♠✱ t❤❡♥ ❛❧❧ ♠✲❢r❡❡ ❣r❛♣❤s ❛r❡ ❦✲❢r❡❡✳

✶✹ ✴ ✶✽

SLIDE 32

❖♣t✐♠❛❧ ✉♣❞❛t❡ r✉❧❡s ❢♦r ♠✲❢r❡❡ ❣r❛♣❤s

❆ ❣r❛♣❤ ✐s ♠✲❢r❡❡ ✐❢ ❛ t❛❜✉ r❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ ❋■❋❖♠ ✉♣❞❛t❡ r✉❧❡ ❝❛♥♥♦t ✈✐s✐t ❛ ✈❡rt❡① ✐♥ ✐ts ❝✉rr❡♥t t❛❜✉ ❧✐st✱ ✉♥❧❡ss t❤❡ ❝✉rr❡♥t ✈❡rt❡① ❤❛s ❞❡❣r❡❡ ♦♥❡✳ ❆❧❧ ❣r❛♣❤s ❛r❡ ✶✲❢r❡❡✳ ❆ ❣r❛♣❤ ✐s ✷✲❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② tr✐❛♥❣❧❡ ✇✐t❤ ❛ ❞❡❣r❡❡ t✇♦ ✈❡rt❡①✳ ❊✈❡r② (♠ + ✶)✲r❡❣✉❧❛r ❣r❛♣❤ ✐s ♠✲❢r❡❡✳ ❆ ❣r❛♣❤ ✇✐t❤ ❣✐rt❤ ❣r❡❛t❡r t❤❛♥ ♦r ❡q✉❛❧ t♦ ♠ + ✷ ✐s ♠✲❢r❡❡✳ ■❢ ✶ ❦ < ♠✱ t❤❡♥ ❛❧❧ ♠✲❢r❡❡ ❣r❛♣❤s ❛r❡ ❦✲❢r❡❡✳

✶✹ ✴ ✶✽

SLIDE 33

❖♣t✐♠❛❧ ✉♣❞❛t❡ r✉❧❡s ❢♦r ♠✲❢r❡❡ ❣r❛♣❤s

❆ ❣r❛♣❤ ✐s ♠✲❢r❡❡ ✐❢ ❛ t❛❜✉ r❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ ❋■❋❖♠ ✉♣❞❛t❡ r✉❧❡ ❝❛♥♥♦t ✈✐s✐t ❛ ✈❡rt❡① ✐♥ ✐ts ❝✉rr❡♥t t❛❜✉ ❧✐st✱ ✉♥❧❡ss t❤❡ ❝✉rr❡♥t ✈❡rt❡① ❤❛s ❞❡❣r❡❡ ♦♥❡✳ ❆❧❧ ❣r❛♣❤s ❛r❡ ✶✲❢r❡❡✳ ❆ ❣r❛♣❤ ✐s ✷✲❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② tr✐❛♥❣❧❡ ✇✐t❤ ❛ ❞❡❣r❡❡ t✇♦ ✈❡rt❡①✳ ❊✈❡r② (♠ + ✶)✲r❡❣✉❧❛r ❣r❛♣❤ ✐s ♠✲❢r❡❡✳ ❆ ❣r❛♣❤ ✇✐t❤ ❣✐rt❤ ❣r❡❛t❡r t❤❛♥ ♦r ❡q✉❛❧ t♦ ♠ + ✷ ✐s ♠✲❢r❡❡✳ ■❢ ✶ ❦ < ♠✱ t❤❡♥ ❛❧❧ ♠✲❢r❡❡ ❣r❛♣❤s ❛r❡ ❦✲❢r❡❡✳

1 2 3 4 5

❚❤❡ ❝❧✐q✉❡ ✇✐t❤ ✻ ❡❧❡♠❡♥ts ✐s ✺✲r❡❣✉❧❛r✱ ❤❡♥❝❡ ✹✲❢r❡❡✳

✶✹ ✴ ✶✽

SLIDE 34

❖♣t✐♠❛❧ ✉♣❞❛t❡ r✉❧❡s ❢♦r ♠✲❢r❡❡ ❣r❛♣❤s

❆ ❣r❛♣❤ ✐s ♠✲❢r❡❡ ✐❢ ❛ t❛❜✉ r❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ ❋■❋❖♠ ✉♣❞❛t❡ r✉❧❡ ❝❛♥♥♦t ✈✐s✐t ❛ ✈❡rt❡① ✐♥ ✐ts ❝✉rr❡♥t t❛❜✉ ❧✐st✱ ✉♥❧❡ss t❤❡ ❝✉rr❡♥t ✈❡rt❡① ❤❛s ❞❡❣r❡❡ ♦♥❡✳ ❆❧❧ ❣r❛♣❤s ❛r❡ ✶✲❢r❡❡✳ ❆ ❣r❛♣❤ ✐s ✷✲❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② tr✐❛♥❣❧❡ ✇✐t❤ ❛ ❞❡❣r❡❡ t✇♦ ✈❡rt❡①✳ ❊✈❡r② (♠ + ✶)✲r❡❣✉❧❛r ❣r❛♣❤ ✐s ♠✲❢r❡❡✳ ❆ ❣r❛♣❤ ✇✐t❤ ❣✐rt❤ ❣r❡❛t❡r t❤❛♥ ♦r ❡q✉❛❧ t♦ ♠ + ✷ ✐s ♠✲❢r❡❡✳ ■❢ ✶ ❦ < ♠✱ t❤❡♥ ❛❧❧ ♠✲❢r❡❡ ❣r❛♣❤s ❛r❡ ❦✲❢r❡❡✳

1 2 3 4 5 6 7 8 9 X0 X1 X2

8 5

❚❤❛t ❣r❛♣❤ ✐s ♥♦t ✷✲❢r❡❡ ❜❡❝❛✉s❡ ❳✸ ❜❡❧♦♥❣s t♦ {✺, ✽} ❛❧♠♦st s✉r❡❧②✳

✶✹ ✴ ✶✽

SLIDE 35

❖♣t✐♠❛❧ ✉♣❞❛t❡ r✉❧❡s ❢♦r ♠✲❢r❡❡ ❣r❛♣❤s

❆ ❣r❛♣❤ ✐s ♠✲❢r❡❡ ✐❢ ❛ t❛❜✉ r❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ ❋■❋❖♠ ✉♣❞❛t❡ r✉❧❡ ❝❛♥♥♦t ✈✐s✐t ❛ ✈❡rt❡① ✐♥ ✐ts ❝✉rr❡♥t t❛❜✉ ❧✐st✱ ✉♥❧❡ss t❤❡ ❝✉rr❡♥t ✈❡rt❡① ❤❛s ❞❡❣r❡❡ ♦♥❡✳ ❆❧❧ ❣r❛♣❤s ❛r❡ ✶✲❢r❡❡✳ ❆ ❣r❛♣❤ ✐s ✷✲❢r❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② tr✐❛♥❣❧❡ ✇✐t❤ ❛ ❞❡❣r❡❡ t✇♦ ✈❡rt❡①✳ ❊✈❡r② (♠ + ✶)✲r❡❣✉❧❛r ❣r❛♣❤ ✐s ♠✲❢r❡❡✳ ❆ ❣r❛♣❤ ✇✐t❤ ❣✐rt❤ ❣r❡❛t❡r t❤❛♥ ♦r ❡q✉❛❧ t♦ ♠ + ✷ ✐s ♠✲❢r❡❡✳ ■❢ ✶ ❦ < ♠✱ t❤❡♥ ❛❧❧ ♠✲❢r❡❡ ❣r❛♣❤s ❛r❡ ❦✲❢r❡❡✳

1 2 3 4 5 6 7 8 9 X0 X1 X2 X3

8 5

❚❤❛t ❣r❛♣❤ ✐s ✷✲❢r❡❡ ❜❡❝❛✉s❡ ✐t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② tr✐❛♥❣❧❡ ✇✐t❤ ❛ ❞❡❣r❡❡ t✇♦ ✈❡rt❡①✳

✶✹ ✴ ✶✽

SLIDE 36

❚❤❡♦r❡♠

❋♦r ❡✈❡r② ♥♦♥♥❡❣❛t✐✈❡ ✐♥t❡❣❡r ♠ ❛♥❞ ❢♦r ❡✈❡r② ♠✲❢r❡❡ ❣r❛♣❤✱ ❋■❋❖♠ ②✐❡❧❞s ♠✐♥✐♠❛❧ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s ❛♠♦♥❣ ❛❧❧ ✉♣❞❛t❡ r✉❧❡s ✇✐t❤ ❧❡♥❣t❤ ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♠✳

✶✺ ✴ ✶✽

SLIDE 37

❚❤❡♦r❡♠

❋♦r ❡✈❡r② ♥♦♥♥❡❣❛t✐✈❡ ✐♥t❡❣❡r ♠ ❛♥❞ ❢♦r ❡✈❡r② ♠✲❢r❡❡ ❣r❛♣❤✱ ❋■❋❖♠ ②✐❡❧❞s ♠✐♥✐♠❛❧ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s ❛♠♦♥❣ ❛❧❧ ✉♣❞❛t❡ r✉❧❡s ✇✐t❤ ❧❡♥❣t❤ ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♠✳

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✳

▲❡t ♠ ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❆ ✇❛❧❦❡r (❳♥)♥✵ ❜❛❝❦tr❛❝❦s ❛t st❡♣ ♥ ✐❢ ❳♥ ❜❡❧♦♥❣s t♦ {❳♠❛①{♥−✷,✵}, . . . , ❳♠❛①{♥−♠−✶,✵}} ✇❤✐❧❡ t❤❡r❡ ❡①✐sts ❛ ♥❡✐❣❤❜♦r ♦❢ ❳♥−✶ ✇❤✐❝❤ ❞♦❡s ♥♦t ❤❛✈❡ ❛♥② ♦❝❝✉rr❡♥❝❡ ✐♥ t❤❡ t❛❜✉ ❧✐st✳ ❖♥ ♦♥❡ ❤❛♥❞✱ ✇❡ s❤♦✇ t❤❛t ❛♥② t❛❜✉ r❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ ❋■❋❖♠ ✉♣❞❛t❡ r✉❧❡ ❞♦❡s ♥♦t ❜❛❝❦tr❛❝❦ ♦♥ ❛♥② ♠✲❢r❡❡ ❣r❛♣❤✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ st❛rt✐♥❣ ❢r♦♠ ❛♥② t❛❜✉ r❛♥❞♦♠ ✇❛❧❦✱ ✇❡ r❡♠♦✈❡ ❛❧❧ ❜❛❝❦tr❛❝❦s t♦ ❣❡t ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss t❤❛t ❢♦❧❧♦✇s t❤❡ s❛♠❡ ❧❛✇ t❤❛♥ ❛ t❛❜✉ r❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ ❋■❋❖♠ ✉♣❞❛t❡ r✉❧❡✳

✶✺ ✴ ✶✽

SLIDE 38

❚❤❡♦r❡♠

❋♦r ❡✈❡r② ♥♦♥♥❡❣❛t✐✈❡ ✐♥t❡❣❡r ♠ ❛♥❞ ❢♦r ❡✈❡r② ♠✲❢r❡❡ ❣r❛♣❤✱ ❋■❋❖♠ ②✐❡❧❞s ♠✐♥✐♠❛❧ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s ❛♠♦♥❣ ❛❧❧ ✉♣❞❛t❡ r✉❧❡s ✇✐t❤ ❧❡♥❣t❤ ❧❡ss t❤❛♥ ♦r ❡q✉❛❧ t♦ ♠✳

❙❦❡t❝❤ ♦❢ ♣r♦♦❢✳

1 2 3 4 5 6 7 8 9 X0 X1 X2 = X5 X3 X4 X6

✭❛✮ ❚❤❡ ✇❛❧❦❡r ❜❛❝❦tr❛❝❦s ❛t st❡♣ ✺✳

1 2 3 4 5 6 7 8 9

✭❜✮ ❲❡ r❡♠♦✈❡ t❤❡ ❜❛❝❦✲ tr❛❝❦s✳

✶✺ ✴ ✶✽

SLIDE 39

❙✐♥❝❡ ❛❧❧ ❣r❛♣❤s ❛r❡ ✶✲❢r❡❡✿

❈♦r♦❧❧❛r②

❋♦r ❡✈❡r② ✉♣❞❛t❡ r✉❧❡ ❘ ♦❢ ❧❡♥❣t❤ ✵ ♦r ✶✱ t❤❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s ❛ss♦❝✐❛t❡❞ t♦ ❘ ❛r❡ ❣r❡❛t❡r t❤❛♥ ♦r ❡q✉❛❧ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s ❛ss♦❝✐❛t❡❞ t♦ ❋■❋❖✶✳

❨♦✉r ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s ❛r❡ ❛❧✇❛②s ❧♦✇❡r ✐❢ s♦♠❡♦♥❡ ❢♦❧❧♦✇s ②♦✉✳ ■t ❞♦❡s ♥♦t ❤♦❧❞ ♦♥ ❛❧❧ ❣r❛♣❤s ❢♦r t✇♦ ♣❡♦♣❧❡ ♦r ♠♦r❡✳

✶✻ ✴ ✶✽

SLIDE 40

❇❡tt❡r s♦♠❡t✐♠❡s t♦ ❢♦r❣❡t

1 2 3

1

1 2 3

2 1

1 2 3

3 2 1

✶✼ ✴ ✶✽

SLIDE 41

❈♦♠♣❛r✐s♦♥s ❜❡t✇❡❡♥ ❋■❋❖♠ ✉♣❞❛t❡ r✉❧❡s✱ ♠ ✵

❑♥✿ ❝❧✐q✉❡ ✇✐t❤ ♥ ✈❡rt✐❝❡s✳ ▲♥✿ ♣❛t❤ ✇✐t❤ ♥ ✈❡rt✐❝❡s✳ ❋♥✿ ✢♦✇❡r ❣r❛♣❤ ✇✐t❤ ♥ ♣❡t❛❧s✳

1 2 3 4 5 6 7 8 9

✭❛✮ ❋✹

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

✭❜✮ ❋✼

✶✽ ✴ ✶✽

SLIDE 42

❈♦♠♣❛r✐s♦♥s ❜❡t✇❡❡♥ ❋■❋❖♠ ✉♣❞❛t❡ r✉❧❡s✱ ♠ ✵

❑♥✿ ❝❧✐q✉❡ ✇✐t❤ ♥ ✈❡rt✐❝❡s✳ ▲♥✿ ♣❛t❤ ✇✐t❤ ♥ ✈❡rt✐❝❡s✳ ❋♥✿ ✢♦✇❡r ❣r❛♣❤ ✇✐t❤ ♥ ♣❡t❛❧s✳ ✵ ✶ ✷ ✸ ✹ ♠ ✺ ✵ × ❑✸ ❑✸ ❑✸ ❑✸ ❑✸ ✶ × × ❑✸ ❑✸ ❑✸ ❑✸ ✷ ❋✼ ❋✹ × ❑✹ ❑✹ ❑✹ ✸ ▲✹ ▲✹ ▲✹ × ❑✺ ❑✺ ✹ ▲✹ ▲✹ ▲✹ ▲✺ × ❑✻ ❦ ✺ ▲✹ ▲✹ ▲✹ ▲✺ ▲✻      ▲♠+✷ ✐❢ ♠ < ❦ , × ✐❢ ♠ = ❦ , ❑❦+✷ ✐❢ ♠ > ❦ .

✶✽ ✴ ✶✽