M2R Parallel Systems Training exercises Exercises

n
•W1(n)
T1(n)
•D(n)
T∞(n)
•A= [a0, . . . , an−1]B= [b0, . . . , bm−1]
aibjai6=bj0≤i<n
0≤j < m
a0< a1< . . . < an−1b0< b1< . . . < bm−1
•X= [x0, . . . , xn+m−1]x0< x1< . . . < xn+m−1
A B
A n B m Cn
n+m=
(n+m)!
n!.m!X A B
n!'√2πnn
en
n=m

6=6=
<
6=
6=
W1(n, m)≤n+m−1
D
n≥m > 0 (B, A, X)m= 0
A A1= [a0, . . . , an/2−1]A2= [an/2, . . . , an−1]
α=an/2B B1B2B1= [b0, . . . , bj−1]
B α B2= [bj, . . . , bm−1]B α
•b0> α B1B2=B
•bm−1< α B1=B B2
•j bj−1< α < bj
A1B1X[0, . . . , n/2 + j−1]
A2B2X[n/2 + j, . . . , n +m−1]
A
B
j B
O(log2m)
D(m, n) = D(n, m)n<m
D(n, m)≤D(n/2, m) + O(log m)n≥m
D(n, 0) = O(1)
D(n, m) = O(log2(n+m))
W(n, m) =
n+m+o(n+m)p

a−1=b−1=−∞ an=bm= +∞
i∈ {0, . . . , n −1}A k ∈ {0, . . . , m}B
bk−1< aibk> ai
xi+k=ai
k aiO(1) m
O(1)
A B X
W1(n, m) = O(n+m)
i= 0, . . . , b√ncαi=ai√nj= 0, . . . , b√mcβj=bj√m
α−1=β−1 = −∞ αb√nc+1 =βb√mc+1 = +∞
i= 0, . . . , b√ncµi∈ {0, . . . , b√mc+ 1}βµi−1< αi< βµi
j= 0, . . . , b√mcνj∈ {0, . . . , b√nc+ 1}ανj−1< βj< ανj
µiνjO(1)
O(n+m)
O(log log n)
O(nlog log n)
D(n, m) = O(log log n)
O(n+m)

D(M)(n)W(M)
1(n)
D(n)W1(n)
D(M)(n) = log2n W (M)
1(n) = O(n)
D(M)(n) = log log n W (M)
1(n) = O(n)