Αλγόριθμοι Άμεσης Απόκρισης
description
Transcript of Αλγόριθμοι Άμεσης Απόκρισης
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2009
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2
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1 5
2 ; 92.1 . . . . . 102.2 . . . . . . . . . . . . . . . . . . . . 112.3 . . . . . . . . . . . . . . . . . . 13
3 153.1 . . . . . . . . . . . . . . . . . . . 163.2 . . . . . . 163.3 (h, k) . . . . . . . . . . . . . . 183.4 . . . 193.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Yao 254.1 . . . . . . . . . . . . . . . . . . . . . . . 254.2 . . . . . . . . . . . . . . . . . . . . . 264.3 . . . . . . . . . . . . 28
5 315.1 MARK . . . . . . . . . . . . . . . . . . . . . . . 315.2 . . . . . . . . . . 33
6 376.1 . . . . . . . . . . . 386.2 SLOWFIT . . . . . . 396.3 . . . . . . . . . 436.4 . . . . . . . . . . . . . . . . . . . 446.5 . . . . . . . . . . . . . . . 48
3
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4
7 537.1 . . . . . . . . . . . . . . . . . . . 537.2 : . . . . . . . . 557.3 ROBIN-HOOD . . . . . . . . . . . . . . . . . . 57
8 61
9 699.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 699.2 . . . . . . . . . . . . . . . . . . . . 719.3 . . . . . . . . . . . . . . . . 76
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1
- (online algorithms). . - . .
, . , - . - (competitive analy-sis), , () . (competitive ratio). ALG . , C
OPT() C ALG() . OPT() ALG() ()
5
-
6 1.
ALG , .
ALG . , ALG() -. - ALG (oblivious adversaries) C
OPT() C IE[ALG()] . , - ALG = 0. , C :
C = max
OPT()IE[ALG()]
.
, () - ALG C
IE[ALG()] C OPT() . OPT() ALG() () ALG , .
C = max
IE[ALG()]OPT()
0. .
-
7 - (, ). , , - .
; (ski-rental), (paging) , (load balancing), (bin packing), (call admission).
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8 1.
-
2
;
. . . , . . , . , . . .
. , . , . ;
9
-
10 2. ;
2.1
- . . , . , .
1 - B . 1 , - . , = B, . .
1. ; = B 2 1B .. - = B 2 1B . t > 0, ALGB(t) t OPT(t) t . B = maxtB(t) B(t) = ALGB(t)OPT(t) . B(t) 2 1/B, t > 0.
t OPT(t) = min{t, B}. t- ALGB(t) = t t < B (t t ) ALGB(t) =2B 1 t B (B 1 B 1 B). , B(t) = tmin{t,B} = 1 t < B B(t) =
2B1min{t,B} =
2B1B = 2 1B t B. , B(t) 2 1B ,
B 2 1B ., = B
. - 2 1/B. - () = ALG ()OPT() 21/B.
-
2.2. 11
: < B,
() = 1 +Bmin{, B} =
1 +B
> 1 +B 1
> 1 +B 1B
= 2 1/B
> B,
() = 1 +Bmin{, B} =
1 +BB
= 1 + 1B
> 2 1/B.
() > 2 1/B, , = maxt (t) () > 2 1/B.
2.2
. , , . , - , . 2 1/B B .
: ; , .
-. - , -, ( ).
= dB/e B . (= 1+
p5
2 ).
-
12 2. ;
2. - ; 5+
p5
4 1.81.. t > 0, ALG(t) t. B
ALG(t) =
8
-
2.3. 13
2.3
, . 1.81 - . , .
. , B 1 B. , dB/e 1 dB/e. , (= 1+
p5
2 ).
3. 5+
p5
4 1.81 ;
. . = dB/e. t > 0, ALG1(t) t ALG2(t) t . , ALG(t) t.
ALG1(t) =
t t < B2B 1 t B
ALG2(t) =
t t < +B 1 t
. , t > 0 ALG(t) IE[ALG(t)] = 12(ALG1(t) + ALG2(t)) . t OPT(t) = min{t, B}. , (t) = IE[ALG(t)]OPT(t) =
ALG1(t)+ALG2(t)2OPT(t) ,
-
14 2. ;
= maxt (t). t. t < , ALG1(t) = ALG2(t) = t OPT(t) = t. , (t) = 1. t < B, OPT(t) = t, ALG1(t) = t ALG2(t) = +B1. ,
(t) =ALG1(t) + ALG2(t)
2 OPT(t) =t+ +B 1
2t ... 5 +
p5
4.
t B, OPT(t) = B, ALG1(t) = 2B 1 ALG2(t) = + B 1.,
(t) =ALG1(t) + ALG2(t)
2 OPT(t) = + 3B 2
2B ... 5 +
p5
4.
. (t) 5+
p5
4 ,, = maxt (t) 5+
p5
4 .
-
3
- - . , , - P = {p1, p2, ..., pN} N . , , P k k < N . . pi, pi . pi (hit) . , (page fault) - pi . , () pi. .
. , .
15
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16 3.
3.1
, 60. -. :
LRU (least recently used): - .
FIFO (first-in-first-out) : - .
LIFO (last-in-first-out) : .
LFU (least frequently used) : .
LFD (longest forward distance) : .
. , LFD . (demand paging), . , . 2. - .
3.2 -
, LFD .
-
3.2. 17
, ( ;), . (.., ) , . , .
4. LFD .
. - LFD . . 1. ALG . i, i = 1, 2, ..., ||, ALGi :
1. ALGi i1 ALG.
2. i , ALGi - .
ALGi() ALGi ALG() ALG.
n = || . . OPT, i = 1 OPT1, - OPT1 i = 2 OPT2, . OPTn LFD .
-
18 3.
. AL-G, ALGi. i , ALG AL-Gi X [ {v}, X [ {u}, , X k 1 u, v . u = v, ALG ALGi ALGi() = ALG(). v 6= u ( , i- ). v, ALGi ALG u ALG v. - , - k 1. , k (.., ALG v), ALGi ALG .
v ALG ALGi , ALGi ALG. , v, , u v. ALG, ALGi. , ALGi v ALG. , v, ALGi u .
3.3 (h, k) .
k h h k. (h, k)- , k h k. h < k, .
-
3.4. 19
( FIFO ) Belady: , -. - Belady, .
3. LRU Belady FIFO .
(h, k) - ; , - . , .
3.4 -
LRU FI-FO -. FWF (flush when full) . FWF :
FWF: , .
, FWF . , FWF - . , ,
-
20 3.
. , , FWF .
( - LRU FWF) - kkh+1 (h, k)- . , k ((k, k)-). , -, ( - FIFO) kkh+1 .
- k . : 0 . i 1, i i 1 k . , i + 1, , (k + 1)- i- . k-. k- .
. - k-. bit bit . , bit 1, , - . k- . k-, k-. - . FWF - , FWF .
, -
-
3.4. 21
(h, k)- -.
5. ALG - k OPT h k. ALG kkh+1 OPT.
. - k-. , k- i 1, ALG k - . k ( k ). - . , k-, ALG .
i 1 q i. i i+ 1 ( i+ 1 ). OPT h1 q k . , OPT k(h1) = kh+1 . i , OPT k0 h , k0 . OPT.
, ALG k, , OPT kh+1 . , :
ALG() kk h+ 1 OPT() + k
0,
ALG() OPT() ALG OPT, , k0 k
-
22 3.
ALG k-.
1. LRU .
. LRU . LRU x k. x k. x, x - . , LRU k . x , LRU - x k . x k- - k 1 . , k- - k+1 , k. , LRU -.
FWF LRU . 5, 1, , - .
1. LRU FWF kkh+1 (h, k)- .
4. FIFO .
ALG k , ALG k .
-
3.5. 23
5. LRU FIFO -. , FWF .
5. 6. ALG - k OPT h k. - ALG kkh+1 OPT.
, . 2. FIFO kkh+1 (h, k)- .
3.5
k - k.
kkh+1 (h, k)- . , - LRU, FIFO, FWF (h, k)- .
- LFD . 2. k + 1 , LFD() LFD ||k + k 1k .. k-. k- - k ,
l||k
mk-
. k k- . , k +
l||k
m 1
k + ||k +k1k 1 = ||k + k 1k .
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24 3.
, . 7. ALG - k k.
. k + 1 p1, ..., pk+1. ALG . , 2.
: r1 i 2, ri ALG r1, ..., ri1 ( k+1 , - ). ALG .
, ALG LFD k.
LIFO LFU ( k). LIFO. p1, p2, ..., pk+1 .
= p1, p2, ..., pk, pk+1, pk, pk+1, ...
, LIFO k + 1 .
LFU. ` . :
= p`1, p`2, ..., p
`k1, (pk, pk+1)
`1.
LFU (k 1) ` . , . ` , LFU k.
-
4
Yao
Yao. - . - .
4.1
. . - : , v , , v( -). .
. , . , .
, . -
25
-
26 4. YAO
. , - . , , 1. , - . - . .
4.2
, - , , -, . , , - .
6. ;
, - .
, 1/2 ( ) 1. , 2.
-, - ( - ). - 1 ( ). 1/2.
-
4.2. 27
1/2 1+1/2 /2. , 2.
2, .
, - . , - , - . -, :
, p 1 p.
, , - .
p , . p-Random - p. ( p) . , , p-Random .
p-Random - p . , .
, p-Random p ( ) 1. , 1/p.
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28 4. YAO
-, - . 1 . p - 1 p. p 1 + (1 p) = p( 1). , - p(1) .
p p . , - p-Random ( ) p . , p = 21 (- ) - p-Random 1/p = 2 1/.
4.3
- . , - . , Yao .
. P . ALG . , ALG() ALG OPT() P . CP(ALG) - ALG P - C
IE[OPT()] C IE[ALG()]
-
4.3. 29
- . , - . ,
IE[ALG()] C IE[OPT()]
ALG() OPT() ALG , .
Yao - - P. - . - . = 0 .
Yao. , - . , q , - 1 q . q - . , . , q + (1 q).
, . - .
-
30 4. YAO
-. 1 ( ). q + (1 q).
- , - . , (1 q) q+(1q)(1q) . , - .
- - q, q. -, - , q = 11/ - 2 1/. Yao, 2 1/ .
-
5
- MARK - . -, Yao. , , - .
5.1 MARK
MARK - k. 2Hk , Hk k- (harmonic number) Hk =
Pki=1
1i . ln k < Hk < 1+ln k
MARK .
MARK: , . p , p . , p , ,
31
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32 5.
. - p , .
8. MARK 2Hk .
. - k-. k-, k-. i- k- - . k- k-.
i- k-. mi . k-, . , mi , mi . , k mi () .
j- ( k-) i- k- kmi(j1)k(j1) .
j- , . mikj+1 . , i- k-
mi +kmiXj=1
mik j + 1 = mi +mi(Hk Hmi) = mi(Hk Hmi + 1) miHk.
. k- - mi, i- (i 1)- k-,
-
5.2. 33
k +mi . , k- i ( ), i- (i 1)- k- mi. k- - m1. , 12
Pimi.
HkP
imi, . 7. MARK k Hk N = k + 1.
8. MARK Hk ( k = 2 N = 4).
5.2 -
- k. Yao. 9. ALG - k. Hk.
. I k + 1 . 1 I. i > 1, i- i k I {i1} .
k-. , , LFD - - () Hk .
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34 5.
i 2, k . , LFD i, i+ 1. , k 1 .
A. k + 1 , i, A - q. i ( ). , k I{i}, q 1/k. , Hk, kHk.
, - . - , , ( - ) . - I k+1 , - . ., k-, k + 1 - , k-. .
3. k + 1
-
5.2. 35
kHk.
. i i . , 1 = 0, 2 = 1, i > i1 i 3. IE[k+1].
IE[k+1] =k+1Xi=2
(IE[i] IE[i1])
=k+1Xi=2
IE[i i1].
. IE[i i1] i1 , i- . (i 1)- , i 1 ki+2k . , i 1 , k k i+ 2 (i 1) . ,
IE[i i1] =1Xj=1
j
1 k i+ 2
k
j1 k i+ 2k
=k
k i+ 2 .
j i- j 1 j.
P1j=1 j(1 x)j1 = 1/x2. -
,
IE[k+1] =k+1Xi=2
k
k i+ 2 = kkXi=1
1
i= kHk
-
36 5.
.
3 , k- kHk. A 1/k, - Hk . , .
-
6
- . . , - . i hri(1), ri(2), ..., ri(N)i. ri(j) i j . (makespan) .
- (unrelated machines). - . (related machines), i wi j sj. i j ri(j) = wi/sj. - (, sj = 1 j) - (identical machines). - (restricted assignments) . . i j ri(j) = wi/sj j i, ri(j) =1.
37
-
38 6.
. :
GREEDY: , .
, GREEDY - . , SLOWFIT . - .
6.1
, , - , . , :
GREEDY : -, . , - .
.
10. - 2 1N N .. 2 1/N . , N(N 1) , 1, N . - 1 N 1. , , 2N1. , N
-
6.2. SLOWFIT 39
N 1 , N . -, 2 1/N .
, . r w ` r. , r, ` ( r ). , r N`. , N` + w OPT() ` + wN ., OPT() w. , GR() :
GR() = `+ w
= `+w
N+
1 1
N
w
OPT() +1 1
N
OPT()
=
2 1
N
OPT()
.
6.2 SLOWFIT
, . , - SLOWFIT . - SLOWFIT, . -
-
40 6.
SLOWFIT, .
SLOWFIT: , . . . i, - j 2,
j = argmink
{`i1(k) + ri(k) 2} (6.1)
`i1(j) j i 1 SLOWFIT ri(j) i j. , .
11. . - OPT() , SLOWFIT SLOWFIT() 2.. SLOWFIT , SLOWFIT() 2.
. OPT() SLOWFIT t. - t. , t OPT(). - OPT(), t 1 j j t 1 .
f `t1(f) OPT(). f 6= N `t1(N) + rt(N)
-
6.2. SLOWFIT 41
2OPT() 2 rt(N) t. , t N .
= {j : j > N} f ( f `t1(j) > OPT() j 2 ). , t 1 , t1 . g j , i0 f . , rg(i0) OPT() f i0, rg(f) rg(i0) OPT()., f , `t1(f) OPT() `g(f) OPT(). , g, `g1(f) + rg(f) 2OPT() 2. f j, g . .
SLOWFIT, - SLOWFIT .
SLOWFIT: . -, SLOWFIT . 0, 0 r1(N) . - SLOWFIT0 ( SLOWFIT) - 1. , i 1, i = 2i1, SLOWFITi t SLOWFITi1 -
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42 6.
, . SLOWFITi .
12. SLOWFIT 8 .
. SLOWFIT k, k = 2kr1(N). k k . - h. h = 0, 20 2OPT(), SLOWFIT0 . h > 0, r h. h 1 h1r h1 = 2h1r1(N),
OPT() OPT(h1r) > 2h1r1(N).
SLOWFIT :
SLOWFIT() =hX
k=0
SLOWFITk(k) hX
k=0
2k
=hX
k=0
2 2kr1(N) = 2(2h+1 1)r1(N).
, SLOWFIT() < 8 OPT().
, :
13. . ALG , OPT() ) ALG() c (. c = 2 SLOWFIT ). ALG , ALG() 4c OPT(). , ALG 4c-.
-
6.3. 43
6.3 -
-. . , .
14. blogNc - N .
. k = blogNc. N 0 = 2k k+1 . , N 0 . k , . i i = 1, ..., k1, (a, b), 1 a b . i - . , 1 .
, i k 2ki ,, N 0/2 , N 0/4 ,..., k. -, i i k, i. 1 0 2 1. , j < k, j+1. , k + 1 k + 1.
-
44 6.
( ) . , i i i + 1. , . k , , k . , , , .
, k + 1 =blogNc+ 1.
6.4
O(logN) . Lp , p 1 . Lp.
`j j, j = 1, ..., N . , Lp :
|`|p =
NXj=1
`pj
!1/p.
, :
maxj
{`j} = limp!1
|`|p.
Lp :
-
6.4. 45
p-GREEDY: , p- .
. 15. p 1, p-GREEDY 1
21/p1 Lp.
. n N . `j j Lp . yij 2 {0, 1} i j (yij = 1) (yij = 0).,
Pi yijri(j) = `
j ri(j) i
j. , ij j i 0j = 0 .
i, - p- . , p- .
NXj=1
pij pi1,j
NXj=1
(i1,j + yijri(j))p pi1,j
.
(x + a)p xp x = i1,j, a = yijri(j) 0, p 1. x [0,+1] ,
NXj=1
pij pi1,j
NXj=1
(nj + yijri(j))
p pnj. (6.2)
, . 4. p 1, t 0, ai 0, i = 1, ..., k. ,
kXi=1
((t+ ai)p tp)
t+
kXi=1
ai
!p tp.
-
46 6.
. ai = 0 i = 1, ..., k . - ai > 0 =
Pki=1 ai i = ai/. ,
Pki=1 i = 1.
zp [0,1),
(t+ ai)p =
(1 i)t+ i
t+
kXi=1
ai
!!p
(1 i)tp + it+
kXi=1
ai
!p(6.3)
i = 1, ..., k. (6.3),
kXi=1
((t+ ai)p tp) tp
kXi=1
(1 i) k!
+
t+
kXi=1
ai
!p kXi=1
i
=
t+
kXi=1
ai
!p tp
p- :
|n|pp =NXj=1
pnj
=NXj=1
nXi=1
pij pi1,j
=
nXi=1
NXj=1
pij pi1,j
.
(6.2), ,
-
6.4. 47
4,
|n|pp nXi=1
NXj=1
(nj + yijri(j))
p pnj
=NXj=1
nXi=1
(nj + yijri(j))
p pnj
NXj=1
(nj +
nXi=1
yijri(j))p pnj
!.
Pn
i=1 yijri(j) = `j
Minkowski ( Lp)
Pkt=1 (at + bt)
p1/p Pkt=1 apt1/p + Pkt=1 bpt1/p p 1
at, bt 0,
|n|pp NXj=1
(nj + `j)p
NXj=1
pnj
0@ NX
j=1
pnj
! 1p
+
NXj=1
`pj
! 1p
1Ap NXj=1
pnj
=|n|p + |`|p
p |n|pp
|`|pp c = |n|p / |`|p, 2cp (c+ 1)p c 1
21/p1 .
ez 1+ z, p-GREEDY 1
21/p1 pln 2 1.4427p.
.
5. Ap c Lp ., Ap cN1/p N .
-
48 6.
. ` . ` Lp c cN1/pmaxj `j . , |`|pp N maxj `pj |`|p N1/pmaxj `j . , maxj `j |`|p. - |`|p c|`|p , maxj `j cN1/pmaxj `j .
p-GREEDY - pln 2 Lp, - .
3. p-GREEDY p = lnN e logN N .
6.5
, . , : , (weighted). ( ), .
- . G = (V,E) rj = (sj, tj, bj) bj sj tj. , bj bj : E ! R+. rj Pj sj tj e Pj bj(e).
-. , . , , (
-
6.5. 49
). , , .
.
ROUTEEXP > 0 = (1 + ). rj = (sj, tj, bj), rj Pj
Pe2Pj a
Lj1(e)+j(e) aLj1(e),
Lj(e) =Xij
Pi:e2Pi
bj(e),
Lj(e) =Lj(e)
,
j(e) =bj(e)
.
, rj cj(e) = aLj1(e)+j(e) aLj1(e) = aLj1(e)
aj(e) 1.
16. G = (V,E) m . OPT() , ROUTEEXP() = O(logm) .
. a = (1 + ),
ax 1 x, x 2 [0, 1] (6.4)
P1, . . . , Pn ( P 1 , . . . , P n ) - ROUTEEXP ( ) -
-
50 6.
r1, . . . , rn .Xe2Pj
aLj1(e)aj(e) 1 X
e2P jaLj1(e)
aj(e) 1
Pj
Xe2P j
aLn(e)aj(e) 1
Lk(e) k
Xe2P j
aLn(e)j(e),
(6.4)
nX
j=1
Xe2Pj
aLj1(e)aj(e) 1 X
j
Xe2P j
aLn(e)j(e).
Xe
Xj:e2Pj
aLj1(e)aj(e) 1 X
e
aLn(e)X
j:e2P jj(e)
Xe
(aLn(e) 1) Xe
aLn(e)X
j:e2P jj(e)
1 - OPT(r1, . . . , rn) X
e
aLn(e) m Xe
aLn(e).
-
6.5. 51
, < 1, Xe
aLn(e) m1
Ln(e) loga
m
1
Ln(e) loga
m
1
ROUTEEXP() = O(logm) (6.5)
ROUTEEXP ROUTE-EXP 13.
-
52 6.
-
7
- . . j- - sj fj, j- dj = fj sj. fj dj - j. , . ALG() ALG , .
, GREEDY 2 1N , - . , 2 1N . , SLOWFIT .
7.1
, , , T , . , T
53
-
54 7.
, .
- .
17. ( ), T ( ), ROUTE-KNOWNT , O(logNT )-.
. - [(k1)T, kT ] k. ( ), : Gk 2TN + 3T
uki,h 0 i < 2T 1 h N , tki+1 0 i < 2T 1, ski 0 i < T , .
i , h . Gk 5NT 2N :
2NT N eki,h = (uki,h, uki+1,h), - ( ) .
2NT N (uki+1,h, tki+1) 0 i < 2T 1 1 h N , ( ).
NT (ski , uki,h) 0 i < T 1 h N , ( ).
rj = hrj(1), . . . , rj(N)i, dj (k 1)T + l, (0 l < T ), r0j skl , tkl+dj ,
-
7.2. : 55
bj(eki,h) = rj(h), l i < l + dj, bj(e) = 0 e. ROUTEEXP, , r0j.
r0j k, Gk. , rj h. , h (.. t = (k 1)T + i) ek1i+T,h, eki,h. Gk O(logNT ) , ROUTE-KNOWNT O(logNT )-.
7.2 :
. , .
18. - . (
pN).
. Li(t) i t ( - t, ). t, . , L1(t) L2(t) Lq(t)(t) Li(t) = 0 i > q(t). - , .
1. , N - . , OPT() 1.
2. , N Li(t) Li+1(t) + 1, 1 i q 1.
-
56 7.
3. r i L(k)i > 0, ALG r i. k , .
- . . 1.
L = hL1(t), . . . , Lq(t)(t)i L0 = hL1(t0), . . . , Lq(t0)(t0), L < L0, Lj(t) = Lj(t0) j < i Li(t) < Li(t0) 1 i q(t0)
: . N , -
. ( .)
. k k
L(k) = hL(k)1 , . . . , L(k)q(k)i, k. L(k) (L(k)i L(k)i+1 + 1), = k., k k+1 L(k) < L(k+1).
Pqki=1 L
(k)i = N
, .
, k + 1
L(k)i = L(k)i+1 > 0
i 1, . :
1. i i + 1 . ( ALG , 3)
-
7.3. ROBIN-HOOD 57
2. , {i, i + 1}. - ALG i, i+ 1.
3. i + 1 ALG.
4. , .
k+1. L(k+1)i = L(k)i +1 L(k) < L(k+1).
. , l = L1(t) = ALG()
N =qX
i=1
Li(t) q1Xi=0
(l i) = ql q(q 1)2
q l, l (q 1) Lq(t) > 0. l q = l l(l+1)2 N , l p2N(1 + o(1)). ,
ALG() = (pN).
7.3 ROBIN-HOOD
. , . - (N2/3), O(
pN)) .
rj sj lj - Mj . Li(sj) i rj (- t sj ).
-
58 7.
j j r1, r2, . . . , rj. j - B(sj) OPT , B(s0) = 0 B(sj) = max{B(sj1), lj, 1N (lj +
Pi Li(sj))},
. i sj, Li(sj)
pN B(sj).
B , B(sj1) < B(sj), . , (, ) (, ).
ROBIN-HOOD r1 . rj, sj , i 2 Mj sj Li(sj) >>>>>:16 2" i = 1, . . . , n.13 + " i = n+ 1, . . . , 2n.
12 + " i = 2n+ 1, . . . , 3n.
, OPT() = n. , , FIRST-FIT ( BEST-FIT) , " n6 n ,
n2 n, n
n . , 5n3 .
FF() 1710OPT() + 2. :
w(r) =
8>>>>>>>>>>>>>>>:
65r 0 r 16 .95r 110 16 r 13 .65r +
110
13 r 12 .
1 12 < r.
= r1, r2, . . . , rn, W () =Pn
i=1w(ri), B, W (B) =
Ptj=1w(rij), {rij |j = 1, . . . , t}
B. , .
6. B W (B) 1710 , W () 1710OPT().
. 12 ,
w(ri)
ri 3
2 12 rij ri
tXj=1
rij 13 . . r1 13 > r2, r2 > 13 ,
w(r1)+w(r2) =6
5r1+
9
5r2 =
6
5(r1+r2)+
3
5r2 >
6
5(1)+ 3
5 > 1
3. m = 3. .
13 12 m. .
9. B r1 rm.
Pmi=1w(ri) = 1 > 0, (i) m = 1 r1 12 , (ii)Pm
i=1 ri 1 59.. m = 1, r1 > 12 > 0.
m 2, , r1 r2 > .
Pmi=1 ri = 1 . eB
-
66 8.
r3, . . . , rm 1, 2, 1 + 2 = r1 + r2 + i ri. 8 eB,
mXi=3
w(ri) + w(1) + w(2) 1 (8.1)
w 95 12 ,
w(1) + w(2) w(r1) + w(r2) + 95
(8.1) mXi=1
w(ri) +9
5
mXi=3
w(ri) + w(1) + w(2) 1,
1 =mXi=1
w(ri) 1 95.
5
9
.
W (Bi) = 1 i i > 0 i < 12 Bi. Bi ri1, ri2, . . .
l > 1. 9, 1 < i l:
i 1Xj
ri1j i1 +5
9i1.
Bi1 (i) 9, i 12 .
l1Xi=1
i 95
lXi=2
(i i1) = 95(l 1) 9
5 12< 1.
l 1, Pl
i=1 i 2. , l = 1 .
-
67
l 9, m (FF() = m + l). 8 1. ,
W () =X
w(ri) m+lX
i=1
W (Bi) = m+lX
i=1
(1 i) = m+ l lX
i=1
i
FF() 2. FF() W () + 2. 6
FF() 1710
OPT() + 2.
.
-
68 8.
-
9
, . , . , . , -, . , , . , , , - . .
9.1
( , - ) ( ) . - , , -.
i- ri = (si, ti, bi, pi) - si ti,
69
-
70 9.
bi, pi ., - (throughput) :
.
.
. , , , .
, - , .
- , - (.. ) :
1. (bidirectional mode). - hu, vi - hu, vi hv, ui, c, c .
2. (undirected mode). - hu, vi - , - . , c, c .
, - , , .
-
9.2. 71
9.2
, - . . , , -.
ALG . - u(e) e Lj(e) - e ( ) j ALG. Aj , j , , Pk rk, k 2 Aj.
Lj(e) =1
u(e)Xk2Aje2Pk
bk
Ln(e) 1 e = r1, . . . , rn. :
cj(e) = u(e)[Lj(e) 1], (9.1)
D - . , , , ( ), pi = D bi, ri =(si, ti, bi, pi). , . , 0 " 1 ri, i = 1, 2, . . . n
bi mine
u(e)
" logD + 1 + "= b(e)
,bi = O
u(e)
logD
-
72 9.
" 1 1logD .
AAP" D (.. ) , 0 " 1. bi b(") pi = D bi ri. = 21+ 1"D. j- rj = (sj, tj, bj, pj) P
Xe2P
bju(e)
cj1(e) pj.
rj , . Lj(e) cj(e) . " .
21. , - AAP .
. , Ln(e) 1 e. = r1, . . . , rn, A rj ( Pj) AAP. .
k 2 A. AAP, - rk
Xe2Pk
bku(e)
ck1(e) pk = D bk.
Xe2P
ck1(e)u(e)
D.
-
9.2. 73
e 2 Pk
D ck1(e)u(e) = Lk1(e) 1
= Lk(e)bku(e) 1
Lk(e) 1(" logD+1+") 1
bku(e) 1(" logD+1+") .
= Lk(e)1
(" log ) 1
" log = " logD + 1 + ".
x = 2x log . . .
= Lk(e)
21" 1.
Lk(e) 2 1"D + 2 1" 21+ 1"D = , Lk(e) 1. 22. , - AAP
21+1" log + 1 = O(2
1" + 2
1" logD).
. r1, r2, . . . rn . A rj ( Pj) AAP, A AAP, . C , C =
Pe cn(e).
:
1.C (21+ 1" log )
Xj2A
pj,
.
-
74 9.
2. Xj2A
pj C,
- .
1 2, ,
AAP() =Xj2A
pj OPT() AAP() +Xj2A
pj.
|A|, - AAP. , . , . , rk . X
e
ck1(e) 21+ 1" log X
j2A{k}pj, (9.2)
Xe
ck(e) 21+ 1" log Xj2A
pj. (9.3)
Xe
ck(e)Xe
ck1(e) =Xe2Pk
(ck(e) ck1(e)) 21+ 1"pk log . (9.4)
e 2 Pk,[ck(e) ck1(e)] = u(e)[Lk1(e)+
bku(e) Lk1(e)]
= u(e)Lk1(e)[bku(e) 1]
u(e)Lk1(e)hlog bku(e)
i21"
= 21" log
hck1(e)u(e) + 1
i bk
= 21" log
hbku(e)ck1(e) + bk
i.
-
9.2. 75
:
x 1 = 2x log 1, log bku(e)
1",
2y 1 y2 1" y 2 [0, 1"]. ,
Xe2Pk
[ck(e) ck1(e)] 2 1" log Xe2Pk
bku(e)
ck1(e) + bk
= 21" log
Xe2Pk
bku(e)
ck1(e) +Xe2Pk
bk
!
2 1" log pk +
Xe2Pk
bk
!
AAP
2 1" log (pk +D bk)
= 21" log 2pk.
1. , P j , -
rj. -, AAP, - ,
pj < minsjtj-
P
Xe2P
bju(e)
cj1(e) Xe2P j
bju(e)
cj1(e)
-
76 9.
,Xj2A
pj 0 .
bi mineu(e)
" logDT + 1 +
1
"
pi = Dbi(fiai) ri. = 21+ 1"DT . j- rj = (sj, tj, bj, pj, aj, fj) P sj tj X
ajt
-
80 9.
-
[1] D. Achlioptas, M. Chrobak, and J. Noga. Competitive analysis of ran-domized paging algorithms. Proceedings of the 4th Annual EuropeanSymposium on Algorithms (ESA 96), LNCS 1136, Springer, pp. 419-430, 1996.
[2] S. Albers. Lecture notes on competitive online algorithms. BRICS Lec-ture Series LS-96-2, AArhus University, Denmark, 1996.
[3] J. Aspnes, Y. Azar, A. Fiat, S. Plotkin, and O. Waarts. On-line routingof virtual circuits with applications to load balancing and machinescheduling. Journal of the ACM, 44(3), pp. 486-504, 1997.
[4] B. Awerbuch, Y. Azar, E. F. Grove, M.-Y. Kao, P. Krishnan, and J. S.Vitter. Load balancing in the Lp norm. Proceedings of the 36th AnnualSymposium on Foundations of Computer Science (FOCS 95), pp. 383-391, 1995.
[5] B. Awerbuch, Y. Azar, and S. Plotkin. Throughput-competitive on-linerouting. Proceedings of the 34th Annual Symposium on Foundations ofComputer Science (FOCS 93), pp. 32-40, 1993.
[6] B. Awerbuch, Y. Bartal, A. Fiat, and A. Rosen. Proceedings of the 5thAnnual ACM-SIAM Symposium on Discrete Algorithms (SODA 94), pp.312-320, 1994.
[7] Y. Azar. On-line load balancing. Online Algorithms, The state of theart. LNCS 1442, Springer, pp. 178-195, 1996.
[8] Y. Azar, A. Z. Broder, and A. Karlin. On-line load balancing. Theoreti-cal Computer Science, 130(1), pp. 73-84, 1994.
81
-
82
[9] Y. Azar, J. Naor, and R. Rom. The competitiveness of on-line assign-ments. Journal of Algorithms, 22(1), pp. 221-237, 1995.
[10] Y. Bartal. On-line Computation and Network Algorithms, LectureNotes. Hebrew University of Jerusalem, Israel, 1997.
[11] L. A. Belady. A study of replacement algorithms for virtual storagecomputers. IBM Systems Journal, 5, pp. 78-101, 1966.
[12] L. A. Belady, R. A. Nelson, and G. S. Shedler. An anomaly in space-time characteristics of certain programs running in a paging machine.Communications of the ACM, 12(6), pp. 349-353, 1969.
[13] A. Borodin and R. El-Yaniv. Online computation and competitive anal-ysis. Cambridge University Press, 1998.
[14] A. Borodin, S. Irani, P. Raghavan, and B Schieber. Competitive pagingwith locality of reference. Journal of Computer and System Sciences,50(2), pp. 244-258, 1995.
[15] I. Caragiannis. Better bounds for online load balancing on unrelatedmachines. Proceedings of the 19th Annual ACM-SIAM Symposium onDiscrete Algorithms (SODA 08), pp. 972-981, 2008.
[16] I. Caragiannis, C. Kaklamanis, and E. Papaioannou. Ecient on-linefrequency allocation and call control in cellular networks. Theory ofComputing Systems, 35(5), pp. 521-543, 2002.
[17] I. Caragiannis, A. V. Fishkin, C. Kaklamanis, and E. Papaioan-nou. Randomized on-line algorithms and lower bounds for computinglarge independent sets in disk graphs. Discrete Applied Mathematics,155(2), pp. 119-136, 2007.
[18] A. Fiat, R. M. Karp, M. Luby, L. A. McGeoch, D. D. Sleator, and N. E.Young. On competitive paging algorithms. Journal of Algorithms, 12,pp. 685-699, 1991.
[19] R. L. Graham. Bounds on certain multiprocessor anomalies. Bell Sys-tems Technical Journal, 45, pp. 1563-1581, 1996.
[20] M. X. Goemans. Advanced Algorithms, Lecture Notes, MIT Laboratoryof Computer Science, 1993.
-
83
[21] A. R. Karlin, C. Kenyon, and D. Randall. Dynamic TCP acknowledge-ment and other stories about e/(e 1). Proceedings on 33rd AnnualACM Symposium on Theory of Computing (STOC 01), pp. 502-509,2001.
[22] A. R. Karlin, M. S. Manasse, L. Rudolph, and D. D. Sleator. Competi-tive snoopy caching. Algorithmica, 3, pp. 77-119, 1988.
[23] R. J. Lipton and A. Tomkins. Online interval scheduling. Proceedingsof the 5th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA94), pp. 302-311, 1994.
[24] R. Motwani and P. Raghavan. Randomized Algorithms. Cambridge U-niversity Press, 1995.
[25] D. D. Sleator and R. E. Tarjan. Amortized eciency of list update andpaging rules. Communications of the ACM, 28(2), pp. 202-208, 1985.
[26] E. Torng. A unified analysis of paging and caching. Algorithmica, 20,pp. 175-200, 1998.
[27] J. Ullman. The performance of a memory allocation algorithm. Techni-cal Report 100, Dept. of Electrical Engineering, Princeton University,1971.
[28] A. C. Yao. Probabilistic computations: Towards a unified measure ofcomplexity. Proceedings of the 18th Annual Symposium on Foundationsof Computer Science (FOCS 77), 1977.