Fast intersection of half spaces
نویسنده
چکیده
T h e p r o b l e m of i n t e r s e c t i n g N h a l f s p a c e s in K s p a c e is t r a n s f o r m e d t o 2*K p r o b l e m s o f c o n s t r u c t i n g the c o n v e x hull of N p o i n t s in K s p a c e a n d a s i m p l e i n t e r s e c t i o n p r o b l e m . T h i s e n a b l e s one to i n t e r s e c t t h e N K d i m e n s i o n a l hal f s p a c e s in 0 < K * H ( N , K ) ) t i m e , w h e r e H(N,K) is the time r e q u i r e d to c o n s t r u c t the c o n v e x hu l l o f N p o i n t s in K s p a c e . F o r t w o a n d t h r e e d imens ions t h e a l g o r i t h m takes CKNIogN) t ime in t h e w o r s t c a s e , b u t u n d e r f a i r l y r o b u s t c o n d i t i o n s t h e e x p e c t e d t ime is o n l y 0 ( N ) . It is a l s o s h o w n t h a t a n a l g o r i t h m f o r i n t e r s e c t i o n of half s p a c e s c a n b e u s e d to c o n s t r u c t t h e c o n v e x hu l l o f p o i n t s in K s p a c e . T h u s , the i n t e r s e c t i o n o f hal f s p a c e s a n d c o n v e x h u l l o f p o i n t s p r o b l e m s a r e e s s e n t i a l l y e q u i v a l e n t . T h i s r e s e a r c h w a s p a r t i a l l y s u p p o r t e d b y the O f f i c e of N a v a l R e s e a r c h u n d e r c o n t r a c t n u m b e r N 0 0 0 1 4 7 6 C 0 8 2 9 .
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