rs s rt r - PowerPoint PPT Presentation

■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s ◆ór❛ ❙③❛❦á❝s ✭✇✐t❤ ❘♦❜❡rt ❏❛❥❝❛②✱ ❚❛t✐❛♥❛ ❏❛❥❝❛②♦✈❛ ❛♥❞ ▼ár✐❛ ❙③❡♥❞r❡✐✮ ❯♥✐✈❡rs✐t② ♦❢ ❙③❡❣❡❞✱ ❇♦❧②❛✐ ■♥st✐t✉t❡ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❙❡♠✐❣r♦✉♣s ✷✵✶✽✱ ▲✐s❜♦♥ ◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

❞✐r❡❝t❡❞✿ ✱ ✇❤❡r❡ ♠❛♣s ❛r❡ ❣✐✈❡♥ ❞✐r❡❝t❡❞✱ ❡❞❣❡✲❝♦❧♦r❡❞✿ ✶ ●r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ❜✐❥❡❝t✐♦♥ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ P❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ♣❛rt✐❛❧ ♦♥❡✲t♦✲♦♥❡ ♠❛♣ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ P❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❣r❛♣❤s ❉❡✜♥✐t✐♦♥ ●r❛♣❤✿ � V � ◮ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡✿ Γ = ( V , E ) ✱ ✇❤❡r❡ E ⊆ ✷ ◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

❞✐r❡❝t❡❞✱ ❡❞❣❡✲❝♦❧♦r❡❞✿ ✶ ●r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ❜✐❥❡❝t✐♦♥ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ P❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ♣❛rt✐❛❧ ♦♥❡✲t♦✲♦♥❡ ♠❛♣ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ P❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❣r❛♣❤s ❉❡✜♥✐t✐♦♥ ●r❛♣❤✿ � V � ◮ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡✿ Γ = ( V , E ) ✱ ✇❤❡r❡ E ⊆ ✷ ◮ ❞✐r❡❝t❡❞✿ Γ = ( V , E ) ✱ ✇❤❡r❡ ♠❛♣s α, ω : E → V ❛r❡ ❣✐✈❡♥ ◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

●r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ❜✐❥❡❝t✐♦♥ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ P❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ♣❛rt✐❛❧ ♦♥❡✲t♦✲♦♥❡ ♠❛♣ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ P❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❣r❛♣❤s ❉❡✜♥✐t✐♦♥ ●r❛♣❤✿ � V � ◮ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡✿ Γ = ( V , E ) ✱ ✇❤❡r❡ E ⊆ ✷ ◮ ❞✐r❡❝t❡❞✿ Γ = ( V , E ) ✱ ✇❤❡r❡ ♠❛♣s α, ω : E → V ❛r❡ ❣✐✈❡♥ ◮ ❞✐r❡❝t❡❞✱ ❡❞❣❡✲❝♦❧♦r❡❞✿ Γ = ( V ; E ✶ , · · · , E n ) ◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

P❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ♣❛rt✐❛❧ ♦♥❡✲t♦✲♦♥❡ ♠❛♣ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ P❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❣r❛♣❤s ❉❡✜♥✐t✐♦♥ ●r❛♣❤✿ � V � ◮ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡✿ Γ = ( V , E ) ✱ ✇❤❡r❡ E ⊆ ✷ ◮ ❞✐r❡❝t❡❞✿ Γ = ( V , E ) ✱ ✇❤❡r❡ ♠❛♣s α, ω : E → V ❛r❡ ❣✐✈❡♥ ◮ ❞✐r❡❝t❡❞✱ ❡❞❣❡✲❝♦❧♦r❡❞✿ Γ = ( V ; E ✶ , · · · , E n ) ●r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ❜✐❥❡❝t✐♦♥ ϕ : V → V ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ ◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

P❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❣r❛♣❤s ❉❡✜♥✐t✐♦♥ ●r❛♣❤✿ � V � ◮ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡✿ Γ = ( V , E ) ✱ ✇❤❡r❡ E ⊆ ✷ ◮ ❞✐r❡❝t❡❞✿ Γ = ( V , E ) ✱ ✇❤❡r❡ ♠❛♣s α, ω : E → V ❛r❡ ❣✐✈❡♥ ◮ ❞✐r❡❝t❡❞✱ ❡❞❣❡✲❝♦❧♦r❡❞✿ Γ = ( V ; E ✶ , · · · , E n ) ●r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ❜✐❥❡❝t✐♦♥ ϕ : V → V ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ P❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ♣❛rt✐❛❧ ♦♥❡✲t♦✲♦♥❡ ♠❛♣ ψ : V → V ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ ◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

t❤❡ ❣r❛♣❤ r❡❝♦♥str✉❝t✐♦♥ ❝♦♥❥❡❝t✉r❡✿ ▲❡t ❜❡ ❛♥ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡ ❣r❛♣❤ ♦♥ ♣♦✐♥ts✳ ❉❡❝❦ ✿ ♠✉❧t✐s❡t ♦❢ s♣❛♥♥❡❞ s✉❜❣r❛♣❤s ♦❢ ♦♥ ✶ ♣♦✐♥ts ❈♦♥❥❡❝t✉r❡✿ ❉❡❝❦ ❞❡t❡r♠✐♥❡s ✳ ❚❤❡ r❡❛s♦♥ t❤✐s ✐s ❤❛r❞✿ s♦♠❡ ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ s✉❜❣r❛♣❤s ♦♥ ✶ ♣♦✐♥ts ❞♦♥✬t ❡①t❡♥❞ t♦ ❛✉t♦♠♦r♣❤✐s♠s✳ ❲❤② ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s❄ ◮ t♦ ❤❛✈❡ ❛♥ ❛❧❣❡❜r❛✐❝ t♦♦❧ t♦ ✉♥❞❡rst❛♥❞ ❣r❛♣❤s ✇✐t❤ tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s ◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

❚❤❡ r❡❛s♦♥ t❤✐s ✐s ❤❛r❞✿ s♦♠❡ ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ s✉❜❣r❛♣❤s ♦♥ ✶ ♣♦✐♥ts ❞♦♥✬t ❡①t❡♥❞ t♦ ❛✉t♦♠♦r♣❤✐s♠s✳ ❲❤② ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s❄ ◮ t♦ ❤❛✈❡ ❛♥ ❛❧❣❡❜r❛✐❝ t♦♦❧ t♦ ✉♥❞❡rst❛♥❞ ❣r❛♣❤s ✇✐t❤ tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s ◮ t❤❡ ❣r❛♣❤ r❡❝♦♥str✉❝t✐♦♥ ❝♦♥❥❡❝t✉r❡✿ ▲❡t Γ ❜❡ ❛♥ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡ ❣r❛♣❤ ♦♥ n ♣♦✐♥ts✳ ❉❡❝❦ (Γ) ✿ ♠✉❧t✐s❡t ♦❢ s♣❛♥♥❡❞ s✉❜❣r❛♣❤s ♦❢ Γ ♦♥ n − ✶ ♣♦✐♥ts ❈♦♥❥❡❝t✉r❡✿ ❉❡❝❦ (Γ) ❞❡t❡r♠✐♥❡s Γ ✳ ◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

❲❤② ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s❄ ◮ t♦ ❤❛✈❡ ❛♥ ❛❧❣❡❜r❛✐❝ t♦♦❧ t♦ ✉♥❞❡rst❛♥❞ ❣r❛♣❤s ✇✐t❤ tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s ◮ t❤❡ ❣r❛♣❤ r❡❝♦♥str✉❝t✐♦♥ ❝♦♥❥❡❝t✉r❡✿ ▲❡t Γ ❜❡ ❛♥ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡ ❣r❛♣❤ ♦♥ n ♣♦✐♥ts✳ ❉❡❝❦ (Γ) ✿ ♠✉❧t✐s❡t ♦❢ s♣❛♥♥❡❞ s✉❜❣r❛♣❤s ♦❢ Γ ♦♥ n − ✶ ♣♦✐♥ts ❈♦♥❥❡❝t✉r❡✿ ❉❡❝❦ (Γ) ❞❡t❡r♠✐♥❡s Γ ✳ ❚❤❡ r❡❛s♦♥ t❤✐s ✐s ❤❛r❞✿ s♦♠❡ ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ s✉❜❣r❛♣❤s ♦♥ n − ✶ ♣♦✐♥ts ❞♦♥✬t ❡①t❡♥❞ t♦ ❛✉t♦♠♦r♣❤✐s♠s✳ ◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s