20110306 csseminar cg_illustrations_vyatkina
TRANSCRIPT
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÂû÷èñëèòåëüíàÿ ãåîìåòðèÿ:èëëþñòðàöèèÊèðà ÂÿòêèíàÑàíêò-Ïåòåðáóðãñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò6 ìàðòà 2011ã.
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏëàí äîêëàäà1 Âûïóêëûå îáîëî÷êè2 Ìèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòü3 Òðåóãîëüíèê ìèíèìàëüíîé ïëîùàäè
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÎïðåäåëåíèåÏóñòü S ⊂ R
2 íåêîòîðîå ïîäìíîæåñòâî ïëîñêîñòè
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÎïðåäåëåíèåÏóñòü S ⊂ R
2 íåêîòîðîå ïîäìíîæåñòâî ïëîñêîñòèÂûïóêëàÿ îáîëî÷êà CH(S) ìíîæåñòâà S íàèìåíüøååâûïóêëîå ìíîæåñòâî, ñîäåðæàùåå S
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÎïðåäåëåíèåÏóñòü S ⊂ R
2 íåêîòîðîå ïîäìíîæåñòâî ïëîñêîñòèÂûïóêëàÿ îáîëî÷êà CH(S) ìíîæåñòâà S íàèìåíüøååâûïóêëîå ìíîæåñòâî, ñîäåðæàùåå SÈëè: CH(S) ïåðåñå÷åíèå âñåõ âûïóêëûõ ìíîæåñòâ,ñîäåðæàùèõ S
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÄëÿ ìíîæåñòâà òî÷åêÏóñòü P ìíîæåñòâî èç n òî÷åê íà ïëîñêîñòè
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÄëÿ ìíîæåñòâà òî÷åêÏóñòü P ìíîæåñòâî èç n òî÷åê íà ïëîñêîñòèCH(P) âûïóêëûé ìíîãîóãîëüíèê ñ âåðøèíàìè âíåêîòîðûõ òî÷êàõ èç S
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÑëîæíîñòü ïîñòðîåíèÿÌîäåëü âû÷èñëåíèé: âåùåñòâåííîçíà÷íàÿ ÀÌðàâíîäîñòóïíàÿ àäðåñíàÿ ìàøèíà
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÑëîæíîñòü ïîñòðîåíèÿÌîäåëü âû÷èñëåíèé: âåùåñòâåííîçíà÷íàÿ ÀÌðàâíîäîñòóïíàÿ àäðåñíàÿ ìàøèíàÝëåìåíòàðíûå îïåðàöèè:àðèìåòè÷åñêèå îïåðàöèèîïåðàöèè ñðàâíåíèÿêîñâåííàÿ àäðåñàöèÿ ïàìÿòèÊèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÑëîæíîñòü ïîñòðîåíèÿÌîäåëü âû÷èñëåíèé: âåùåñòâåííîçíà÷íàÿ ÀÌðàâíîäîñòóïíàÿ àäðåñíàÿ ìàøèíàÝëåìåíòàðíûå îïåðàöèè:àðèìåòè÷åñêèå îïåðàöèèîïåðàöèè ñðàâíåíèÿêîñâåííàÿ àäðåñàöèÿ ïàìÿòèàíàëèòè÷åñêèå óíêöèèÊèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÑëîæíîñòü ïîñòðîåíèÿÌîäåëü âû÷èñëåíèé: âåùåñòâåííîçíà÷íàÿ ÀÌðàâíîäîñòóïíàÿ àäðåñíàÿ ìàøèíàÝëåìåíòàðíûå îïåðàöèè:àðèìåòè÷åñêèå îïåðàöèèîïåðàöèè ñðàâíåíèÿêîñâåííàÿ àäðåñàöèÿ ïàìÿòèàíàëèòè÷åñêèå óíêöèèÍèæíÿÿ îöåíêà ñëîæíîñòè ïîñòðîåíèÿ âûïóêëîé îáîëî÷êè:Ω(n log n)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìà
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp1
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp1
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp1
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp1
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp3
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp4
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp4
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp5
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp4
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp6p7
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp6p7
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp6p7
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp7
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp6p7
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp8
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp6p7
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp9
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp7
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp7
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp7
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp7
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp7
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàp7
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàÂñïîìîãàòåëüíàÿ ñòðóêòóðà äàííûõ: ñòåê
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàÂñïîìîãàòåëüíàÿ ñòðóêòóðà äàííûõ: ñòåêÊîððåêòíîñòü àëãîðèòìà:
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàÂñïîìîãàòåëüíàÿ ñòðóêòóðà äàííûõ: ñòåêÊîððåêòíîñòü àëãîðèòìà:Ïóñòü Pi = p1, . . . , pi
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 39: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/39.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàÂñïîìîãàòåëüíàÿ ñòðóêòóðà äàííûõ: ñòåêÊîððåêòíîñòü àëãîðèòìà:Ïóñòü Pi = p1, . . . , piÏî çàâåðøåíèè i-é èòåðàöèè ñòåê ñîäåðæèò âåðøèíûCH(Pi )
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 40: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/40.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàÂñïîìîãàòåëüíàÿ ñòðóêòóðà äàííûõ: ñòåêÊîððåêòíîñòü àëãîðèòìà:Ïóñòü Pi = p1, . . . , piÏî çàâåðøåíèè i-é èòåðàöèè ñòåê ñîäåðæèò âåðøèíûCH(Pi )Âðåìåííàÿ ñëîæíîñòü: O(n log n)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 41: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/41.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàÂñïîìîãàòåëüíàÿ ñòðóêòóðà äàííûõ: ñòåêÊîððåêòíîñòü àëãîðèòìà:Ïóñòü Pi = p1, . . . , piÏî çàâåðøåíèè i-é èòåðàöèè ñòåê ñîäåðæèò âåðøèíûCH(Pi )Âðåìåííàÿ ñëîæíîñòü: O(n log n)Íàõîæäåíèå p1: O(n)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 42: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/42.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàÂñïîìîãàòåëüíàÿ ñòðóêòóðà äàííûõ: ñòåêÊîððåêòíîñòü àëãîðèòìà:Ïóñòü Pi = p1, . . . , piÏî çàâåðøåíèè i-é èòåðàöèè ñòåê ñîäåðæèò âåðøèíûCH(Pi )Âðåìåííàÿ ñëîæíîñòü: O(n log n)Íàõîæäåíèå p1: O(n)Ñîðòèðîâêà âåðøèí: O(n log n)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 43: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/43.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàÂñïîìîãàòåëüíàÿ ñòðóêòóðà äàííûõ: ñòåêÊîððåêòíîñòü àëãîðèòìà:Ïóñòü Pi = p1, . . . , piÏî çàâåðøåíèè i-é èòåðàöèè ñòåê ñîäåðæèò âåðøèíûCH(Pi )Âðåìåííàÿ ñëîæíîñòü: O(n log n)Íàõîæäåíèå p1: O(n)Ñîðòèðîâêà âåðøèí: O(n log n)Îáõîä: O(n)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàÂñïîìîãàòåëüíàÿ ñòðóêòóðà äàííûõ: ñòåêÊîððåêòíîñòü àëãîðèòìà:Ïóñòü Pi = p1, . . . , piÏî çàâåðøåíèè i-é èòåðàöèè ñòåê ñîäåðæèò âåðøèíûCH(Pi )Âðåìåííàÿ ñëîæíîñòü: O(n log n)Íàõîæäåíèå p1: O(n)Ñîðòèðîâêà âåðøèí: O(n log n)Îáõîä: O(n)Âðåìÿ ïðîïîðöèîíàëüíî îáùåìó ÷èñëó îïåðàöèé ñî ñòåêîì
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàÂñïîìîãàòåëüíàÿ ñòðóêòóðà äàííûõ: ñòåêÊîððåêòíîñòü àëãîðèòìà:Ïóñòü Pi = p1, . . . , piÏî çàâåðøåíèè i-é èòåðàöèè ñòåê ñîäåðæèò âåðøèíûCH(Pi )Âðåìåííàÿ ñëîæíîñòü: O(n log n)Íàõîæäåíèå p1: O(n)Ñîðòèðîâêà âåðøèí: O(n log n)Îáõîä: O(n)Âðåìÿ ïðîïîðöèîíàëüíî îáùåìó ÷èñëó îïåðàöèé ñî ñòåêîìÊàæäàÿ èç âåðøèí p2, . . . , pn ïîìåùàåòñÿ â ñòåê ðîâíîîäèí ðàç
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàÂñïîìîãàòåëüíàÿ ñòðóêòóðà äàííûõ: ñòåêÊîððåêòíîñòü àëãîðèòìà:Ïóñòü Pi = p1, . . . , piÏî çàâåðøåíèè i-é èòåðàöèè ñòåê ñîäåðæèò âåðøèíûCH(Pi )Âðåìåííàÿ ñëîæíîñòü: O(n log n)Íàõîæäåíèå p1: O(n)Ñîðòèðîâêà âåðøèí: O(n log n)Îáõîä: O(n)Âðåìÿ ïðîïîðöèîíàëüíî îáùåìó ÷èñëó îïåðàöèé ñî ñòåêîìÊàæäàÿ èç âåðøèí p2, . . . , pn ïîìåùàåòñÿ â ñòåê ðîâíîîäèí ðàçÑëåäîâàòåëüíî, îáùåå ÷èñëî îïåðàöèé óäàëåíèÿ íåïðåâîñõîäèò n − 1Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä îáõîäà ðýõåìàÂñïîìîãàòåëüíàÿ ñòðóêòóðà äàííûõ: ñòåêÊîððåêòíîñòü àëãîðèòìà:Ïóñòü Pi = p1, . . . , piÏî çàâåðøåíèè i-é èòåðàöèè ñòåê ñîäåðæèò âåðøèíûCH(Pi )Âðåìåííàÿ ñëîæíîñòü: O(n log n)Íàõîæäåíèå p1: O(n)Ñîðòèðîâêà âåðøèí: O(n log n)Îáõîä: O(n)Âðåìÿ ïðîïîðöèîíàëüíî îáùåìó ÷èñëó îïåðàöèé ñî ñòåêîìÊàæäàÿ èç âåðøèí p2, . . . , pn ïîìåùàåòñÿ â ñòåê ðîâíîîäèí ðàçÑëåäîâàòåëüíî, îáùåå ÷èñëî îïåðàöèé óäàëåíèÿ íåïðåâîñõîäèò n − 1Çàòðàòû ïàìÿòè: O(n)Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä Äæàðâèñà
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä Äæàðâèñàp1
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä Äæàðâèñàp1
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä Äæàðâèñàp1
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä Äæàðâèñàp1
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä ÄæàðâèñàÊîððåêòíîñòü:
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä ÄæàðâèñàÊîððåêòíîñòü:Íà êàæäîì øàãå ñòðîèòñÿ î÷åðåäíîå ðåáðî CH(P)
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌåòîä ÄæàðâèñàÊîððåêòíîñòü:Íà êàæäîì øàãå ñòðîèòñÿ î÷åðåäíîå ðåáðî CH(P)Âðåìåííàÿ ñëîæíîñòü: O(nh), ãäå h ÷èñëî âåðøèí CH(P)Àëãîðèòì, ÷óâñòâèòåëüíûé ê âûõîäóÇàòðàòû ïàìÿòè: O(n)
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÕðîíîëîãèÿ1972: ìåòîä îáõîäà ðýõåìà, O(n log n)1973: ìåòîä Äæàðâèñà, O(nh)1986: àëãîðèòì Êèðêïàòðèêà è Çåéäåëÿ, O(n log h)Ñëîæíûé àëãîðèòì
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÕðîíîëîãèÿ1972: ìåòîä îáõîäà ðýõåìà, O(n log n)1973: ìåòîä Äæàðâèñà, O(nh)1986: àëãîðèòì Êèðêïàòðèêà è Çåéäåëÿ, O(n log h)Ñëîæíûé àëãîðèòì1996: àëãîðèòì ×åíà, O(n log h)Ïðîñòîé àëãîðèòì
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀëãîðèòì ×åíàÇàèêñèðóåì m ≤ n
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m⌉àçîáüåì P íà r ïîäìíîæåñòâ P1, . . . , Pr
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m⌉àçîáüåì P íà r ïîäìíîæåñòâ P1, . . . , Pr
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀëãîðèòì ×åíàp1
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Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀëãîðèòì ×åíàp1
p2
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 84: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/84.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåäâàðèòåëüíûé àíàëèç ñëîæíîñòèÏðåäîáðàáîòêà:
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 85: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/85.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåäâàðèòåëüíûé àíàëèç ñëîæíîñòèÏðåäîáðàáîòêà:Ïîñòðîåíèå CH(Pi ): O(m logm) âðåìåíè
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 86: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/86.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåäâàðèòåëüíûé àíàëèç ñëîæíîñòèÏðåäîáðàáîòêà:Ïîñòðîåíèå CH(Pi ): O(m logm) âðåìåíèÏîñòðîåíèå âñåõ âûïóêëûõ îáîëî÷åê:O(rm logm) = O(n logm) âðåìåíè
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 87: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/87.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåäâàðèòåëüíûé àíàëèç ñëîæíîñòèÏðåäîáðàáîòêà:Ïîñòðîåíèå CH(Pi ): O(m logm) âðåìåíèÏîñòðîåíèå âñåõ âûïóêëûõ îáîëî÷åê:O(rm logm) = O(n logm) âðåìåíèÎñíîâíîé öèêë:
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 88: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/88.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåäâàðèòåëüíûé àíàëèç ñëîæíîñòèÏðåäîáðàáîòêà:Ïîñòðîåíèå CH(Pi ): O(m logm) âðåìåíèÏîñòðîåíèå âñåõ âûïóêëûõ îáîëî÷åê:O(rm logm) = O(n logm) âðåìåíèÎñíîâíîé öèêë:Ïîñòðîåíèå îïîðíîé ïðÿìîé ê CH(Pi ): O(logm) âðåìåíè
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 89: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/89.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåäâàðèòåëüíûé àíàëèç ñëîæíîñòèÏðåäîáðàáîòêà:Ïîñòðîåíèå CH(Pi ): O(m logm) âðåìåíèÏîñòðîåíèå âñåõ âûïóêëûõ îáîëî÷åê:O(rm logm) = O(n logm) âðåìåíèÎñíîâíîé öèêë:Ïîñòðîåíèå îïîðíîé ïðÿìîé ê CH(Pi ): O(logm) âðåìåíèÍàõîæäåíèå î÷åðåäíîé âåðøèíû CH(P): O(r logm)âðåìåíè
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 90: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/90.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåäâàðèòåëüíûé àíàëèç ñëîæíîñòèÏðåäîáðàáîòêà:Ïîñòðîåíèå CH(Pi ): O(m logm) âðåìåíèÏîñòðîåíèå âñåõ âûïóêëûõ îáîëî÷åê:O(rm logm) = O(n logm) âðåìåíèÎñíîâíîé öèêë:Ïîñòðîåíèå îïîðíîé ïðÿìîé ê CH(Pi ): O(logm) âðåìåíèÍàõîæäåíèå î÷åðåäíîé âåðøèíû CH(P): O(r logm)âðåìåíèÍàõîæäåíèå âñåõ âåðøèí CH(P):O(hr logm) = O(h · n
m· logm) âðåìåíè
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 91: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/91.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåäâàðèòåëüíûé àíàëèç ñëîæíîñòèÏðåäîáðàáîòêà:Ïîñòðîåíèå CH(Pi ): O(m logm) âðåìåíèÏîñòðîåíèå âñåõ âûïóêëûõ îáîëî÷åê:O(rm logm) = O(n logm) âðåìåíèÎñíîâíîé öèêë:Ïîñòðîåíèå îïîðíîé ïðÿìîé ê CH(Pi ): O(logm) âðåìåíèÍàõîæäåíèå î÷åðåäíîé âåðøèíû CH(P): O(r logm)âðåìåíèÍàõîæäåíèå âñåõ âåðøèí CH(P):O(hr logm) = O(h · n
m· logm) âðåìåíèÎáùåå âðåìÿ ðàáîòû àëãîðèòìà: O ((
n + h · nm
)
logm)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 92: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/92.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåäâàðèòåëüíûé àíàëèç ñëîæíîñòèÏðåäîáðàáîòêà:Ïîñòðîåíèå CH(Pi ): O(m logm) âðåìåíèÏîñòðîåíèå âñåõ âûïóêëûõ îáîëî÷åê:O(rm logm) = O(n logm) âðåìåíèÎñíîâíîé öèêë:Ïîñòðîåíèå îïîðíîé ïðÿìîé ê CH(Pi ): O(logm) âðåìåíèÍàõîæäåíèå î÷åðåäíîé âåðøèíû CH(P): O(r logm)âðåìåíèÍàõîæäåíèå âñåõ âåðøèí CH(P):O(hr logm) = O(h · n
m· logm) âðåìåíèÎáùåå âðåìÿ ðàáîòû àëãîðèòìà: O ((
n + h · nm
)
logm)Ïðè m = h: O(n log h)Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäè×àñòè÷íîå ïîñòðîåíèå âûïóêëîé îáîëî÷êèPartialHull(P,m)1 r ← ⌈ nm⌉2 P → P1, . . . ,Pr3 for i ← 1 to r4 do Ïîñòðîèòü CH(Pi )5 p1 ← òî÷êà èç P ñ íàèìåíüøåé îðäèíàòîé(â ñëó÷àå íåîäíîçíà÷íîñòè ñàìàÿ ëåâàÿ èç òàêîâûõ)6 for k ← 1 to m7 do Íàéòè âåðøèíó pk+1 âûïóêëîé îáîëî÷êè CH(P)8 if pk+1 = p19 then return ((p1, . . . , pk))10 return m ñëèøêîì ìàëîÊèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäè×àñòè÷íîå ïîñòðîåíèå âûïóêëîé îáîëî÷êèPartialHull(P,m)1 r ← ⌈ nm⌉2 P → P1, . . . ,Pr3 for i ← 1 to r O(r ·m logm) = O(n logm)4 do Ïîñòðîèòü CH(Pi )5 p1 ← òî÷êà èç P ñ íàèìåíüøåé îðäèíàòîé(â ñëó÷àå íåîäíîçíà÷íîñòè ñàìàÿ ëåâàÿ èç òàêîâûõ)6 for k ← 1 to m O(m · r logm) = O(n logm)7 do Íàéòè âåðøèíó pk+1 âûïóêëîé îáîëî÷êè CH(P)8 if pk+1 = p19 then return ((p1, . . . , pk))10 return m ñëèøêîì ìàëîÊèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäè×àñòè÷íîå ïîñòðîåíèå âûïóêëîé îáîëî÷êèPartialHull(P,m)1 r ← ⌈ nm⌉2 P → P1, . . . ,Pr3 for i ← 1 to r4 do Ïîñòðîèòü CH(Pi )5 p1 ← òî÷êà èç P ñ íàèìåíüøåé îðäèíàòîé(â ñëó÷àå íåîäíîçíà÷íîñòè ñàìàÿ ëåâàÿ èç òàêîâûõ)6 for k ← 1 to m7 do Íàéòè âåðøèíó pk+1 âûïóêëîé îáîëî÷êè CH(P)8 if pk+1 = p19 then return ((p1, . . . , pk))10 return m ñëèøêîì ìàëîÂðåìÿ ðàáîòû: O(n logm)Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÈòîãîâûé àëãîðèòìConvexHull(P)1 for t ← 1, 2, 3, . . .2 do m← min22
t
, n3 L← PartialHull(P,m)4 if L 6= m ñëèøêîì ìàëî5 then return L
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÈòîãîâûé àëãîðèòìConvexHull(P)1 for t ← 1, 2, 3, . . .2 do m← min22
t
, n3 L← PartialHull(P,m)4 if L 6= m ñëèøêîì ìàëî5 then return LÀëãîðèòì çàâåðøèò ðàáîòó ïðè t = ⌈log log h⌉
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèÂðåìÿ âûïîëíåíèÿ èòåðàöèè t: O(n log 22
t
) = O(n · 2t)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèÂðåìÿ âûïîëíåíèÿ èòåðàöèè t: O(n log 22
t
) = O(n · 2t)Îáùåå âðåìÿ ðàáîòû àëãîðèòìà: ⌈log log h⌉∑
t=1
O(n · 2t)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèÂðåìÿ âûïîëíåíèÿ èòåðàöèè t: O(n log 22
t
) = O(n · 2t)Îáùåå âðåìÿ ðàáîòû àëãîðèòìà: ⌈log log h⌉∑
t=1
O(n · 2t)
⌈log log h⌉∑
t=1
n · 2t < n · 2⌈log log h⌉+1 < 4n · 2log log h = 4n log h
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèÂðåìÿ âûïîëíåíèÿ èòåðàöèè t: O(n log 22
t
) = O(n · 2t)Îáùåå âðåìÿ ðàáîòû àëãîðèòìà: ⌈log log h⌉∑
t=1
O(n · 2t)
⌈log log h⌉∑
t=1
n · 2t < n · 2⌈log log h⌉+1 < 4n · 2log log h = 4n log hÂðåìåííàÿ ñëîæíîñòü àëãîðèòìà ×åíà: O(n log h)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèÂðåìÿ âûïîëíåíèÿ èòåðàöèè t: O(n log 22
t
) = O(n · 2t)Îáùåå âðåìÿ ðàáîòû àëãîðèòìà: ⌈log log h⌉∑
t=1
O(n · 2t)
⌈log log h⌉∑
t=1
n · 2t < n · 2⌈log log h⌉+1 < 4n · 2log log h = 4n log hÂðåìåííàÿ ñëîæíîñòü àëãîðèòìà ×åíà: O(n log h)Çàòðàòû ïàìÿòè: O(n)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏëàí äîêëàäà1 Âûïóêëûå îáîëî÷êè2 Ìèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòü3 Òðåóãîëüíèê ìèíèìàëüíîé ïëîùàäè
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÄàíî ìíîæåñòâî P èç n òî÷åê íà ïëîñêîñòè
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÄàíî ìíîæåñòâî P èç n òî÷åê íà ïëîñêîñòèÏîñòðîèòü îêðóæíîñòü ìèíèìàëüíîãî ðàäèóñà,ñîäåðæàùóþ âíóòðè âñå òî÷êè èç P
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÑâîéñòâàÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòü äëÿ Pîïðåäåëåíà îäíîçíà÷íî
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÑâîéñòâàÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòü äëÿ Pîïðåäåëåíà îäíîçíà÷íîïðîõîäèòëèáî ðîâíî ÷åðåç äâå òî÷êè èç P, êîòîðûå îïðåäåëÿþò ååäèàìåòð,Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÑâîéñòâàÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòü äëÿ Pîïðåäåëåíà îäíîçíà÷íîïðîõîäèòëèáî ðîâíî ÷åðåç äâå òî÷êè èç P, êîòîðûå îïðåäåëÿþò ååäèàìåòð,ëèáî ïî êðàéíåé ìåðå ÷åðåç òðè òî÷êè èç P
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÎáîçíà÷åíèÿÏóñòü 1 ≤ i ≤ n
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÎáîçíà÷åíèÿÏóñòü 1 ≤ i ≤ n
Pi = p1, . . . , pi
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÎáîçíà÷åíèÿÏóñòü 1 ≤ i ≤ n
Pi = p1, . . . , pi
Ci ìèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòü äëÿ Pi
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèåíåðàöèÿ ñëó÷àéíîé ïåðåñòàíîâêèRandomPermutation(A)1 for k ← n downto 22 do rndIndex ← Random(k)3 A[k]↔ A[rndIndex ]
A ìíîæåñòâî ýëåìåíòîâ, ïðåäñòàâëåííîå ïðè ïîìîùèìàññèâàRandom(k) ãåíåðèðóåò ñëó÷àéíîå öåëîå èç äèàïàçîíà[1, k] çà âðåìÿ O(1)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèåíåðàöèÿ ñëó÷àéíîé ïåðåñòàíîâêèRandomPermutation(A)1 for k ← n downto 22 do rndIndex ← Random(k)3 A[k]↔ A[rndIndex ]
A ìíîæåñòâî ýëåìåíòîâ, ïðåäñòàâëåííîå ïðè ïîìîùèìàññèâàRandom(k) ãåíåðèðóåò ñëó÷àéíîå öåëîå èç äèàïàçîíà[1, k] çà âðåìÿ O(1)
∀a ∈ A, âåðîÿòíîñòü íàõîæäåíèÿ a íà ìåñòå j â ñëó÷àéíîéïåðåñòàíîâêå ñîñòàâëÿåò 1/n, ãäå 1 ≤ j ≤ nÊèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèËåììà 1Åñëè pi âíóòðè Ci−1, òî Ci = Ci−1Åñëè pi âíå Ci−1, òî Ci ïðîõîäèò ÷åðåç pi
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèËåììà 1Åñëè pi âíóòðè Ci−1, òî Ci = Ci−1Åñëè pi âíå Ci−1, òî Ci ïðîõîäèò ÷åðåç pi
Ci-1=Ci
pi
Ci+1
pi+1
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏîñòðîåíèåMinCir le(P)1 Ñãåíåðèðîâàòü ñëó÷àéíóþ ïåðåñòàíîâêó òî÷åê èç P2 C2 ← îêðóæíîñòü ñ äèàìåòðîì p1p23 for i ← 3 to n4 do if pi âíóòðè Ci−15 then Ci ← Ci−16 else Ci ←MinCir le-1(Pi−1, pi)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèËåììà 2q òî÷êà íà ïëîñêîñòè, òàêàÿ, ÷òî ñóùåñòâóåò îïèñàííàÿîêðóæíîñòü äëÿ P , ïðîõîäÿùàÿ ÷åðåç q
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèËåììà 2q òî÷êà íà ïëîñêîñòè, òàêàÿ, ÷òî ñóùåñòâóåò îïèñàííàÿîêðóæíîñòü äëÿ P , ïðîõîäÿùàÿ ÷åðåç q
Cqi ìèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòü äëÿ Pi ,ïðîõîäÿùàÿ ÷åðåç q
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèËåììà 2q òî÷êà íà ïëîñêîñòè, òàêàÿ, ÷òî ñóùåñòâóåò îïèñàííàÿîêðóæíîñòü äëÿ P , ïðîõîäÿùàÿ ÷åðåç q
Cqi ìèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòü äëÿ Pi ,ïðîõîäÿùàÿ ÷åðåç qÅñëè pi âíóòðè C
qi−1, òî C
qi = C
qi−1Åñëè pi âíå C
qi−1, òî C
qi ïðîõîäèò ÷åðåç pi
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏîñòðîåíèå: îäíà èêñèðîâàííàÿ òî÷êàMinCir le-1(P,q)1 Ñãåíåðèðîâàòü ñëó÷àéíóþ ïåðåñòàíîâêó òî÷åê èç P2 C1 ← îêðóæíîñòü ñ äèàìåòðîì p1q3 for j ← 2 to n4 do if pj âíóòðè Cj−15 then Cj ← Cj−16 else Cj ←MinCir le-2(Pj−1, pj , q)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèËåììà 3q1, q2 òî÷êè íà ïëîñêîñòè, òàêèå, ÷òî ñóùåñòâóåòîïèñàííàÿ îêðóæíîñòü äëÿ P , ïðîõîäÿùàÿ ÷åðåç q1 è q2
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèËåììà 3q1, q2 òî÷êè íà ïëîñêîñòè, òàêèå, ÷òî ñóùåñòâóåòîïèñàííàÿ îêðóæíîñòü äëÿ P , ïðîõîäÿùàÿ ÷åðåç q1 è q2
Cq1,q2i ìèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòü äëÿ Pi ,ïðîõîäÿùàÿ ÷åðåç q1 è q2
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèËåììà 3q1, q2 òî÷êè íà ïëîñêîñòè, òàêèå, ÷òî ñóùåñòâóåòîïèñàííàÿ îêðóæíîñòü äëÿ P , ïðîõîäÿùàÿ ÷åðåç q1 è q2
Cq1,q2i ìèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòü äëÿ Pi ,ïðîõîäÿùàÿ ÷åðåç q1 è q2Åñëè pi âíóòðè C
q1,q2i−1 , òî C
q1,q2i = C
q1,q2i−1Åñëè pi âíå C
q1,q2i−1 , òî C
q1,q2i ïðîõîäèò ÷åðåç pi
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏîñòðîåíèå: äâå èêñèðîâàííûõ òî÷êèMinCir le-2(P,q1, q2)1 C0 ← îêðóæíîñòü ñ äèàìåòðîì q1q22 for k ← 1 to n3 do if pk âíóòðè Ck−14 then Ck ← Ck−15 else Ck ← îêðóæíîñòü, ïðîõîäÿùàÿ ÷åðåç q1, q2 è pk
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèMinCir le-2(P):
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèMinCir le-2(P):Âðåìÿ ðàáîòû: O(n)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèMinCir le-2(P):Âðåìÿ ðàáîòû: O(n)MinCir le-1(P):
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 128: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/128.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèMinCir le-2(P):Âðåìÿ ðàáîòû: O(n)MinCir le-1(P):Âåðîÿòíîñòü âûçîâà MinCir le-2 íà èòåðàöèè j íåïðåâîñõîäèò 2/j
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 129: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/129.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèMinCir le-2(P):Âðåìÿ ðàáîòû: O(n)MinCir le-1(P):Âåðîÿòíîñòü âûçîâà MinCir le-2 íà èòåðàöèè j íåïðåâîñõîäèò 2/jÎæèäàåìîå âðåìÿ ðàáîòû: O(n) +n∑
j=2
O(j) 2j= O(n)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 130: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/130.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèMinCir le-2(P):Âðåìÿ ðàáîòû: O(n)MinCir le-1(P):Âåðîÿòíîñòü âûçîâà MinCir le-2 íà èòåðàöèè j íåïðåâîñõîäèò 2/jÎæèäàåìîå âðåìÿ ðàáîòû: O(n) +n∑
j=2
O(j) 2j= O(n)MinCir le(P):
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 131: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/131.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèMinCir le-2(P):Âðåìÿ ðàáîòû: O(n)MinCir le-1(P):Âåðîÿòíîñòü âûçîâà MinCir le-2 íà èòåðàöèè j íåïðåâîñõîäèò 2/jÎæèäàåìîå âðåìÿ ðàáîòû: O(n) +n∑
j=2
O(j) 2j= O(n)MinCir le(P):Âåðîÿòíîñòü âûçîâà MinCir le-1 íà èòåðàöèè i íåïðåâîñõîäèò 3/i
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 132: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/132.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèMinCir le-2(P):Âðåìÿ ðàáîòû: O(n)MinCir le-1(P):Âåðîÿòíîñòü âûçîâà MinCir le-2 íà èòåðàöèè j íåïðåâîñõîäèò 2/jÎæèäàåìîå âðåìÿ ðàáîòû: O(n) +n∑
j=2
O(j) 2j= O(n)MinCir le(P):Âåðîÿòíîñòü âûçîâà MinCir le-1 íà èòåðàöèè i íåïðåâîñõîäèò 3/iÎæèäàåìîå âðåìÿ ðàáîòû: O(n) +
n∑
i=3
O(i) 3i= O(n)
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 133: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/133.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèMinCir le-2(P):Âðåìÿ ðàáîòû: O(n)MinCir le-1(P):Âåðîÿòíîñòü âûçîâà MinCir le-2 íà èòåðàöèè j íåïðåâîñõîäèò 2/jÎæèäàåìîå âðåìÿ ðàáîòû: O(n) +n∑
j=2
O(j) 2j= O(n)MinCir le(P):Âåðîÿòíîñòü âûçîâà MinCir le-1 íà èòåðàöèè i íåïðåâîñõîäèò 3/iÎæèäàåìîå âðåìÿ ðàáîòû: O(n) +
n∑
i=3
O(i) 3i= O(n)Îæèäàåìîå âðåìÿ ïîñòðîåíèÿ ìèíèìàëüíîé îïèñàííîéîêðóæíîñòè: O(n)Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 134: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/134.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÀíàëèç ñëîæíîñòèMinCir le-2(P):Âðåìÿ ðàáîòû: O(n)MinCir le-1(P):Âåðîÿòíîñòü âûçîâà MinCir le-2 íà èòåðàöèè j íåïðåâîñõîäèò 2/jÎæèäàåìîå âðåìÿ ðàáîòû: O(n) +n∑
j=2
O(j) 2j= O(n)MinCir le(P):Âåðîÿòíîñòü âûçîâà MinCir le-1 íà èòåðàöèè i íåïðåâîñõîäèò 3/iÎæèäàåìîå âðåìÿ ðàáîòû: O(n) +
n∑
i=3
O(i) 3i= O(n)Îæèäàåìîå âðåìÿ ïîñòðîåíèÿ ìèíèìàëüíîé îïèñàííîéîêðóæíîñòè: O(n)Çàòðàòû ïàìÿòè: O(n)Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 135: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/135.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏëàí äîêëàäà1 Âûïóêëûå îáîëî÷êè2 Ìèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòü3 Òðåóãîëüíèê ìèíèìàëüíîé ïëîùàäè
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 136: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/136.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÄàíî ìíîæåñòâî P èç n òî÷åê íà ïëîñêîñòè
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 137: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/137.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÄàíî ìíîæåñòâî P èç n òî÷åê íà ïëîñêîñòèÍàéòè òðåóãîëüíèê ìèíèìàëüíîé ïëîùàäè ñ âåðøèíàìè âòðåõ ðàçëè÷íûõ âåðøèíàõ èç P
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 138: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/138.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÍåëîêàëüíîñòü
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÍåëîêàëüíîñòü
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 140: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/140.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåîáðàçîâàíèå äâîéñòâåííîñòèp = (a, b) → ℓ: y = ax + b
ℓ: y = kx + d → p = (−k , d)
x
y
x
y
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 141: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/141.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåîáðàçîâàíèå äâîéñòâåííîñòèp = (a, b) → ℓ: y = ax + b
ℓ: y = kx + d → p = (−k , d)
x
y
x
y
p1
l1
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 142: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/142.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåîáðàçîâàíèå äâîéñòâåííîñòèp = (a, b) → ℓ: y = ax + b
ℓ: y = kx + d → p = (−k , d)
x
y
x
y
p1
p2
l1
l2
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 143: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/143.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåîáðàçîâàíèå äâîéñòâåííîñòèp = (a, b) → ℓ: y = ax + b
ℓ: y = kx + d → p = (−k , d)
x
y
x
y
p1
p2
p3
l1
l2
l3
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 144: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/144.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåîáðàçîâàíèå äâîéñòâåííîñòèp = (a, b) → ℓ: y = ax + b
ℓ: y = kx + d → p = (−k , d)
x
y
x
y
p1
p2
p3
p4
l1
l2
l4
l3
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 145: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/145.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåîáðàçîâàíèå äâîéñòâåííîñòèp = (a, b) → ℓ: y = ax + b
ℓ: y = kx + d → p = (−k , d)
x
y
x
y
p1
p2
p3
p4
l13
p13
l1
l2
l4
l3
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåîáðàçîâàíèå äâîéñòâåííîñòèp = (a, b) → ℓ: y = ax + b
ℓ: y = kx + d → p = (−k , d)
x
y
x
y
p1
p5
p3l13
p13
l1l3
l5
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
![Page 147: 20110306 csseminar cg_illustrations_vyatkina](https://reader030.vdocuments.net/reader030/viewer/2022020307/55a342f21a28ab0f4b8b47dd/html5/thumbnails/147.jpg)
Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÏðåîáðàçîâàíèå äâîéñòâåííîñòèp = (a, b) → ℓ: y = ax + b
ℓ: y = kx + d → p = (−k , d)
x
y
x
y
p1
p5
p3l13
p13
l1l3
l5
h
h
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÄâîéñòâåííàÿ ïîñòàíîâêà çàäà÷è
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÄâîéñòâåííàÿ ïîñòàíîâêà çàäà÷èÏóñòü pipjpk òðåóãîëüíèê ìèíèìàëüíîé ïëîùàäè
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÄâîéñòâåííàÿ ïîñòàíîâêà çàäà÷èlkpij
Ïóñòü pipjpk òðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÍà äâîéñòâåííîé ïëîñêîñòè: pij âåðøèíà ãðàíè,èíöèäåíòíîé lk
Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÄâîéñòâåííàÿ ïîñòàíîâêà çàäà÷èlkpij
Ïóñòü pipjpk òðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÍà äâîéñòâåííîé ïëîñêîñòè: pij âåðøèíà ãðàíè,èíöèäåíòíîé lkÄëÿ êàæäîé ïðÿìîé íåîáõîäèìî ïðîñìîòðåòü O(n) âåðøèíÊèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÄâîéñòâåííàÿ ïîñòàíîâêà çàäà÷èlkpij
Ïóñòü pipjpk òðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÍà äâîéñòâåííîé ïëîñêîñòè: pij âåðøèíà ãðàíè,èíöèäåíòíîé lkÄëÿ êàæäîé ïðÿìîé íåîáõîäèìî ïðîñìîòðåòü O(n) âåðøèíÂðåìåííàÿ ñëîæíîñòü: O(n2)Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÄâîéñòâåííàÿ ïîñòàíîâêà çàäà÷èlkpij
Ïóñòü pipjpk òðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèÍà äâîéñòâåííîé ïëîñêîñòè: pij âåðøèíà ãðàíè,èíöèäåíòíîé lkÄëÿ êàæäîé ïðÿìîé íåîáõîäèìî ïðîñìîòðåòü O(n) âåðøèíÂðåìåííàÿ ñëîæíîñòü: O(n2)Çàòðàòû ïàìÿòè: O(n2)Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèËèòåðàòóðà: ìîíîãðàèè è êóðñ ëåêöèé1 M. de Berg, O. Cheong, M. van Kreveld, M. Overmars,Computational Geometry: Algorithms and Appli ations,Third Edition, Springer, 2008.2 J. O'Rourke, Computational Geometry in C, Se ond Edition,Cambridge University Press, 1998.3 J.-D. Boissonnat, M. Yvine , Geometrie algorithmique,Edis ien e international, Paris, 1995.4 Ô. Ïðåïàðàòà, Ì. Øåéìîñ, Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ:ââåäåíèå, ïåð. ñ àíãë., Ì., Ìèð, 1989.5 D. Mount, Computational Geometry: Le ture Notes, Fall 2002,http://www. s.umd.edu/~mount/754/Le ts/754le ts.pdfÊèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè
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Âûïóêëûå îáîëî÷êèÌèíèìàëüíàÿ îïèñàííàÿ îêðóæíîñòüÒðåóãîëüíèê ìèíèìàëüíîé ïëîùàäèËèòåðàòóðà: ñòàòüè1 R. L. Graham, An e ient algorithms for determining the onvex hull of a nite planar set, Info. Pro . Lett., 1:132133,1972.2 R. A. Jarvis, On the identi ation of the onvex hull of a niteset of points in the plane, Info. Pro . Lett., 2:1821, 1973.3 D. G. Kirkpatri k, R. Seidel, The ultimate planar onvex hullalgorithm, SIAM J. Comput., 15(1):287299, 1986.4 T. Chan, Optimal output-sensitive onvex hull algorithms intwo and three dimensions, Dis r. Comp. Geom., 16:361368,1996.5 E. Welzl, Smallest en losing disks (balls and ellipsoids), inH. Maurer (Ed.), New Results and New Trends in ComputerS ien e, LNCS, 555:359370, Springer, 1991.6 B. Chazelle, L. J. Guibas, D. T. Lee, The power of geometri duality, BIT, 25:7690, 1985.Êèðà Âÿòêèíà Âû÷èñëèòåëüíàÿ ãåîìåòðèÿ: èëëþñòðàöèè