Recherche des deux points les plus rapprochés

Q n >2
Q
n
2= Θ(n2)
P Q X P
Y P
|P|63
61
∆P PGPD
∆|PG|=b|P|
2c |PD|=b|P| 2c+ 1
PGPDδGδD
PGPDδ= min(δG, δD)
δ
PGPDδ
δ∆
P
X←Q
Y←Q
(X, Y )
(X, Y )
X63
(X)
(XG, XD, YG, YD)←(X, Y )
δG←(XG, YG)
δD←(XD, YD)
δ←min(δG, δD)
Y0←Y δ ∆
δ0←(Y0)
min(δ, δ0)

X Y XGXDYGYDPGPD
(X, Y )
XG←X[1,b|X|/2c]
XD←X[b|X|/2c+ 1,|X|]
x←X[b|X|/2c].
YG← |XG|
YD← |XD|
tG←1
tD←1
16i6|Y|
Y[i].6x
Y[tG]←T[i]
tG←tG+ 1
Y[tD]←T[i]
tD←tD+ 1
(XG, XD, YG, YD)
XG, XD, YG, YD∆
O(n)n=|X|=|Y|
b1. . . brY0
bi6=bj∈Y0d(bi, bj)< δ i, j ∈[|1, r|]i < j 6i+ 7
d(bi, bj)< δ
”⇐”
”⇒”bi6=bj∈Y0d(bi, bj)<δ i<j
bibjR δ ×2δ∆
biR P
R
δ×δ
PGδ
δ/2×δ/2√2δ/2< δ
R R
bi
i<j6i+ 7 d(bi, bj)< δ
O(|Y0|)O(n)

(Y0)
δ0←δ
16i6|Y0| − 1)
i+ 1 6j6max(i+ 7,|Y0|)
d(Y0[i], Y 0[j]) < δ0
δ0←d(Y0[i], Y 0[j])
δ0
T(n)n
T(n) = 2T(n/2) + O(n)n > 3
O(1) n63
T(n) = O(nlog n)
O(nlog n)