Distributed Algo Notes

7/29/2019 Distributed Algo Notes

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G = (V, E)

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pp

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Mp

p|M| |M|M

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n nn

i

i id

i id

n

n 1m i id

mll l

l l

n

n

&


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=

i = id

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nn

2n

nO(n2)

n

ni=1 i = O(n

2)

O(n2)O(n2)

nO(n2)

O(n) O(1)

O(n)

&


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l

l

O(1)O(n)

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d 1m 0l id

l min{i, l}m m + 1

m d 1m = d

M j =

M

m m + 1=

l min{i, l}M

j = k

m = d

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15/37

vs

v s

v = s vv v

v = sv v

w w vv = s v

i O(diD) di i Di di

n DO(n)

i O(diD)O(di) di i

s

ss

sv v

v

v

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n m O(m + n)

s

s s

s n 1

n 1

m

O(m) m

&


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v sn

sC v

A v

v v = sv

vCv

CC

vu A

u AM u u A

M

u C

v = sw A

w vv

v ww

wv v = s n1

C

&


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O(n)s n

1

O(n2)m

n O(n2 + m) m O(n2)O(n2 + n2) = O(2n2) = O(n2)

G n mG O(n2)

GT G

T w =

eT w(e)

O(m)

O(m) (v, u) G vu O(n)

O(n) n mO(m)

T

TO(logn)

O(mlogn)

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G = (V, E)|V

|=

n |E| = m

n 1Ci

Cie =

(u, v) v Ci u / Ci

id

v e v

e

CC

&


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O(m)O(logn)

O(mlogn)

s M

s

v = s M u vM u

&


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O(1)O((dmax 1)D)

dmax

(x, k)

(x, k)

(x, k)

(x, k)

O(n)

O(m)

O((dmax 1)D)

&


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w(e)e

e

xx

y

Dx[y] x

y Cx[y]

x Dx[y]Cx[y]

(x, y) Dx[y] = Dy[x] = w(x, y) Cx[y] =Cy[x] = (x, y)

Dx +

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xn 1

w xy

Dw[y] + w(x, w) < Dx[y]x y w

Dx[y] = Dw[y] + w(x, y)

Cx[y] = (x, w)

xn x

O(dxn) dx x n

D D

n 1D

D

xx

x M yx M y Cx[y]

ii

&


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nx O(dxn)

dx x O(nm)

O(Dnm)

O(n2m)

n 1 O(n)O(1)

x

x

xx y Cx[y]

O(mlogn)

O(nlogn+m)

O(n) O(1)m

mO(m2)

&


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g

s s

xs y x y

x s

y x s xy y y

s gy x s

s g

y x s x

y y xs

sT s

g

x T

g xy T y s g

y xT

&


26/37

T

Tx

T xx

gx x T

s gT

g

gh g

hg h x x

T

z x z xg z

O(m) m

O(n)n

x O(|S| |G |dx)S G dx

x

zg

&


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z T gz

gz

z zT

x T xg T x

T

x

T zz

T

Tg

x xz

y ty u z y

y tu T

y Tu y T

xT

y T yu T

g

Tg O (| T |)

Tx O(| G | dx) G

dx x

&


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n

NN

z

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NN

interconnectionnetwork

ij i

jj

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n

x

M NM/N

PiPj Pk Pj

f(N 1)log(N 1)

N 1 f(M/N)log(M/N)

NPi

Pi1 Pi+1P1 PN

N m m m = Nj k P(j, k) 0 j m1

0 k m 1 P(j, k)P(j + 1, k) P(j 1, k) P(j, k + 1) P(j, k 1)

d0 d1 N = 2d1

i i + 1i 1 d 1

0

P0, P1, . . . , P N1

Pi Pj j =

2i2i+1N

forfor

0iN/21N/2iN1

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N = 2q q 1 NP0, P1, . . . , P N1

q qPj Pi j

i

n x1, x2, . . . , xn

N N N

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nn t(n)

n N

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n nn2

n!logn! nlogn

nlogn

n nn2 n3

nx x < 2.5

nlogn

f(n) g(n)

g(n) f(n) (f(n))c n0 g(n) cf(n)

n n0g(n) f(n) O(f(n))

c n0 g(n) cf(n)n n0

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(n2)O(n2.5) (nlogn) O(nlogn)

n nn2

P(n 1, n 1) P(0, 0)P(n

1, n

1)

2(n 1)(n)

Speedup =worstcaserunningtimeoffastestknownsequentialalgorithmf orproblem

worstcaserunningtimeofparallelalgorithm

nn p(n)

=

nn c(n)

c(n) = p(n) t(n)

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(T(n))n (T(n)/N)

N

Efficiency =worstcaserunningtimeoffastestknownsequentialalgorithmf orproblem

costofparallelalgorithmefficiency 1

n n

N P1, P2, . . . , P ND N

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P1D

A(1)i = 0 (logN 1)

j = 2i + 1 2i+1

PjA(j 2i)

A(j)

A ND

A(i)

DO(logN)

N

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Distributed Algo Notes