2009

2

1 2 ; 2.1 . . . . . 2.2 . . . . . . . . . . . . . . . . . . . . 2.3 . . . . . . . . . . . . . . . . . . 3 3.1 . . . . . . . . . . . . . . . . 3.2 . . . 3.3 (h, k ) . . . . . . . . . . . 3.4 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 9 10 11 13 15 16 16 18 19 23

4 Yao 25 4.1 . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 . . . . . . . . . . . . . . . . . . . . . 26 4.3 . . . . . . . . . . . . 28 5 31 5.1 MARK . . . . . . . . . . . . . . . . . . . . . . . 31 5.2 . . . . . . . . . . 33 6 6.1 . . . . . 6.2 SLOWFIT 6.3 . . . 6.4 . . . . . . . . . . . . . 6.5 . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 38 39 43 44 48

4

7 53 7.1 . . . . . . . . . . . . . . . . . . . 53 7.2 : . . . . . . . . 55 7.3 ROBIN-HOOD . . . . . . . . . . . . . . . . . . 57 8 61

9 69 9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 9.2 . . . . . . . . . . . . . . . . . . . . 71 9.3 . . . . . . . . . . . . . . . . 76

1 (online algorithms). . . . , . , . (competitive analysis), , () . (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

IE[ALG( )]

OPT ( )

. , () 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).

8

1.

2 ;

. . . , . . , . , . . . . , . , . ; 9

10

2. ;

2.1

. . , . , . 1 B . 1 , . , = B , . . 1. ; = B 1 2 B . . 1 = B 2 B . t > 0, ALGB (t) t OPT(t) t . ALGB (t) B = maxt B (t) B (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 t 2B 1 B ). , B (t) = min{ = 1 t < B B (t) = min = t,B } {t,B }1 = 2 B t B . , B (t) 2 1 B 2 B . 2B 1 B 1 , B

, = B . 2 1/B . ALG ( ) ( ) = OPT( ) 2 1/B .

2.2. : < B ,

11

( ) =

1+B 1+B B1 B1 = >1+ >1+ = 2 1/B min{ , B } B 1+B 1+B 1 = =1+ > 2 1/B. min{ , B } B B

> B ,

( ) =

( ) > 2 1/B , , = maxt (t) ( ) > 2 1/B .

2.2

. , , . , , . 2 1/B B . : ; , . . , , ( ). = dB/e B p 1+ 5 . (= 2 ).

12

2. ;

2. ; p 5+ 5 4 1.81. . t > 0, ALG(t) t. B

1 t + 1 , . 1 B B 1 . t OPT(t) = (t) 2 min{t, B }. , (t) = ALG , OPT(t) = maxt (t). t. t < , ALG(t) = 2t OPT(t) = 2t ( t < < B ). , (t) = 1. t < B ,

8 t i1 i 3. IE[k+1 ].

IE[k+1 ] = =

k+1 X i=2 k +1 X i=2

(IE[i ] IE[i1 ]) IE[i i1 ].

. IE[i i1 ] i 1 , i- . (i 1) , i 1 i+2 kk . , i 1 , k k i + 2 (i 1) . ,

j 1 1 X ki+2 ki+2 IE[i i1 ] = j 1 k k j =1 = j i- j 1 j . P1 j 1 = 1/x2 . j =1 j (1 x) ,

k . ki+2

IE[k+1 ] =

k+1 X i=2

X1 k =k = kHk ki+2 i i=1

k

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. 1 2 N N . . 2 1/N . , N (N 1) , 1, N . 1 N 1. , , 2N 1. , N

6.2. SLOWFIT

39

N 1 , N . , 2 1/N . , . r w ` r. , r, ` ( r ). , r N `. , w N ` + w OPT( ) ` + N . , OPT( ) w . , GR ( ) : GR ( ) = ` + w

w 1 = `+ + 1 w N N 1 OPT( ) + 1 OPT( ) N 1 = 2 OPT( ) N

.

6.2

SLOWFIT

, . , SLOWFIT . SLOWFIT , . -

40

6.

SLOWFIT, . SLOWFIT : , . . . i, j 2,

j = arg min{`i1 (k ) + ri (k ) 2}k

(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 ) . SLOWFIT 0 ( SLOWFIT ) 1. , i 1, i = 2i1 , SLOWFIT i t SLOWFIT i1 -

42

6.

, . SLOWFIT i . 12. SLOWFIT 8 . . SLOWFIT k , k = 2k r1 (N ). k k . h. h = 0, 20 2OPT( ), SLOWFIT 0 . h > 0, r h. h 1 h1 r h1 = 2h1 r1 (N ), OPT( ) OPT(h1 r ) > 2h1 r1 (N ). SLOWFIT : SLOWFIT ( ) =h X k=0 h X k=0

SLOWFIT k (k )

h X k=0

2k

=

2 2k r1 (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. blog N c N . . k = blog N c. N 0 = 2k k + 1 . , N 0 . k , . i i = 1, ..., k 1, (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 = blog N c + 1.

6.4

O (log N ) . Lp , p 1 . Lp . `j j , j = 1, ..., N . , Lp :

N !1/p X p |`|p = `j .j =1

, :

max{`j } = lim |`|p .j p!1

Lp :

6.4.

45

p-GREEDY: , p- . . 15. p 1, p-GREEDY 1 21/p Lp . 1

. n N . ` j j Lp . yij 2 {0, 1} i j (yij = 1) (yij = 0). P , i yij ri (j ) = `j ri (j ) i j . , ij j i 0j = 0 . i, p- . , p- . N X j =1

p ij

p i1,j

(x + a)p xp x = i1,j , a = yij ri (j ) 0, p 1. x [0, +1] ,N X j =1 p p ij i1,j

N X (i1,j + yij ri (j ))p p i1,j . j =1

, . 4. p 1, t 0, ai 0, i = 1, ..., k . ,k X i=1

N X j =1

(nj + yij ri (j ))p p nj .

(6.2)

((t + ai )p tp )

t+

k X i=1

ai

!p

tp .

46

6.

. ai = 0 i = 1, ..., k . Pk Pk ai > 0 = i=1 ai i = ai / . , i=1 i = 1. z p [0, 1),

(t + ai )p =

(1 i )t + i t +

(1 i )tp + i t +

k X i=1

k X i=1

ai

ai

!p

!!p(6.3)

i = 1, ..., k . (6.3), k X i=1

((t + ai )p tp ) tp =

k Xi=1

(1 i ) k ai !p tp

!

+

t+

t+

k X i=1

k X i=1

ai

!p

k X i=1

i

p- :N X j =1

|n |p p

=

p nj

=

n X N X p = ij p i1,j . i=1 j =1

N X n X p ij p i1,j j =1 i=1

(6.2), ,

6.4. 4,

47

|n |p p =

n X N X i=1 j =1 N X n X j =1 i=1 N X j =1

(nj + yij ri (j ))p p nj (nj + yij ri (j ))p p njn X i=1

!

Pnp

(nj +

yij ri (j ))p p nj .

i=1 yij ri (j ) = `j Minkowski ( Lp )

at , bt 0,

P k

t=1

(at + bt )

1/p

P

k t=1

ap t

1/pN X j =1

+

P

k p t=1 bt

1/p

p 1

|n |p p

0 1 1 1p !p N !p N N X X X p p @ A nj + `j p nj =

N X j =1

(nj +

p ` j)

p nj

j =1

|n |p + |` |p

p

j =1

j =1

|n |p p

p p |` |p p c = |n |p / |` |p , 2c (c + 1) 1 c 21/p . 1

ez 1 + z , p 1 p-GREEDY 21/p ln 1.4427p. 2 1 . 5. Ap c Lp . , Ap cN 1/p N .

48

6.

. ` . ` Lp c p p cN 1/p maxj ` j . , |` |p N maxj `j |` |p N 1/p maxj ` j . , maxj `j |`|p . |`|p c|` |p , maxj `j cN 1/p maxj ` j. p-GREEDY p ln Lp , 2 .

3. p-GREEDY p = ln N e log N 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 P j (e) e2Pj aLj 1 (e)+ aLj1 (e) ,

Lj (e) =

ij Pi :e2Pi

X

bj (e),

j (e) = Lj (e) , L j (e) = bj (e) .

, rj (e) j (e) j cj (e) = aLj1 (e)+ aLj1 (e) = aLj1 (e) a 1 . 16. G = (V, E ) m . OPT( ) , ROUTEEXP ( ) = O (log m) . . a = (1 + ),

ax 1 x,

x 2 [0, 1]

(6.4)

P1 , . . . , Pn ( P1 , . . . , Pn ) ROUTEEXP ( ) -

50

6.

r1 , . . . , rn .

e2Pj

X

X j (e) j (e) aLj1 (e) a 1 aLj1 (e) a 1 e2Pj

Pj

e2Pj

k (e) k L X aLn (e) j (e),

X

j (e) aLn (e) a 1

(6.4) n X X j =1 e2Pj

e2Pj

XX j (e) aLj1 (e) a 1 aLn (e) j (e).j e2Pj

X Xe

j :e2Pj

X X j (e) aLj1 (e) a 1 aLn (e) j (e)e j :e2Pj

Xe

(aLn (e) 1)

Xe

aLn (e)

j :e2Pj

X

j (e)

1 OPT(r1 , . . . , rn )

Xe

aLn (e) m

Xe

aLn (e) .

6.5. , < 1,

51

Xe

m 1 m n (e) loga L 1 m Ln (e) loga 1 aLn (e)

ROUTEEXP ( ) = O (log m)

(6.5)

ROUTEEXP ROUTEEXP 13.

52

6.

7 . . j - sj fj , j - dj = fj sj . fj dj j . , . ALG( ) ALG , . , GREEDY 1 2 N , . , 1 2 N . , SLOWFIT .

7.1

, , , T , . , T 53

54

7.

, . . 17. ( ), T ( ), ROUTE-KNOWN T , O (log N T )-. . [(k 1)T, kT ] k . ( ), : Gk 2T N + 3T uk i,h 0 i < 2T 1 h N , tk i+1 0 i < 2T 1, sk i 0 i < T , . i , h . Gk 5N T 2N :k k 2N T N ek i,h = (ui,h , ui+1,h ), ( ) . k 2N T N (uk i+1,h , ti+1 ) 0 i < 2T 1 1 h N , ( ). k N T (sk i , ui,h ) 0 i < T 1 h N , ( ).

rj = hrj (1), . . . , rj (N )i, dj (k 1)T + l, (0 l < T ), 0 k sk rj l , tl+dj ,

7.2. :

55

bj (ek i,h ) = rj (h), l i < l + dj , bj (e) = 0 e.

ROUTEEXP, 0 , rj . 0 rj k , Gk . , rj h. , h (.. t = (k 1)T + i) 1 k ek i+T,h , ei,h . Gk O (log N T ) , ROUTEKNOWNT O(log N T )-.

7.2

:

. , . 18. . p ( N ). . 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 (k) Li > 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) = hL1 , . . . , Lq(k) i, k . (k ) (k) L(k) (Li Li+1 + 1), = k . , k P k (k) k+1 L(k) < L(k+1) . q = N i=1 Li , . , k + 1

(k )

(k )

Li

(k)

= Li+1 > 0

(k)

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. Li = Li +1 L(k) < L(k+1) . . , l = L1 (t) = ALG( ) (k+1) (k )

N=

q X i=1

Li (t)

q 1 X i=0

(l i) = ql

q (q 1) 2

q l, l (q 1) Lq (t) > 0. l l(l+1) q = l 2 N , l

p

2N (1 + o(1)). ,

ALG( ) = ( N ).

p

7.3

ROBIN-HOOD

. , . (N 2/3 ), p O ( N )) . rj sj lj Mj . Li (sj ) i rj ( t sj ).

58

7.

j j r1 , r2 , . . . , rj . j B (sj ) OP T , B (s0 ) = 0 P 1 B (sj ) = max{B (sj 1 ), lj , N (lj + i Li (sj ))}, . p i sj , Li (sj ) N B (sj ). B , B (sj 1 ) < B (sj ), . , (, ) (, ). ROBIN-HOOD r1 . rj , sj , p i 2 Mj sj Li (sj ) < N B (sj ). , rj . ( , .) 19. ROBIN-HOOD (2 N + 1). . rj P i Li (sj ) N B (sj ). p N , . ROBINHOOD rj m.

p

Lm (sj ) + lj

p

N (B (sj ) + OPT(j )) + OPT(j )

p m sj , Lm (sj ) + lj N B (sj ) + OPT(j ). m st(j ) S m [st(j ) , sj ). k 2 S Mk ( k ) st(j ) ( rk m ). st(j ) ( ) p N ,

h=|

k2S

[

Mk |

p

N.

7.3. ROBIN-HOOD

59

k k2S lh ,

P

Xk2S

lk h OPT(j )

p

N OPT(j ).

(7.1)

, m rj

Li (sj ) + lj Lm (st(j ) ) + < p

Xk2S

lk + lj

N Bt(j ) +

Xk 2S

lk + lj ,

m rt(j ) ,

p (2 N + 1)OPT(j ), 7.1 lk OPT(j ), k j .

60

7.

8 (bin packing) . , ( ), ( 1). r1 , r2 , . . . ri 1 . , . . , () , FIRST-FIT BEST-FIT. FIRST-FIT . BEST-FIT, , ri , 1 ri . 20. FIRST-FIT BEST-FIT 17 . , , 10 FF( )

17 OPT( ) + 1 10 17 OPT( ) 8. 10

, FF( )

BEST-FIT. 61

62

8.

. 5 . 3 17 10 . = r1 , . . . , r3n ,

ri =

8 > > > > < > > > > :

1 6 1 3 1 2

2" i = 1, . . . , n. +" +" i = n + 1, . . . , 2n. i = 2n + 1, . . . , 3n.

, OPT( ) = n. , , FIRST-FIT ( BEST-FIT ) , " n n , n n, n 6 2 n n . , 53 . FF( ) :17 OPT( ) 10

+ 2.

w(r) =

8 6 r > 5 > > > > > > > r < 9 5 > 6 > r+ > 5 > > > > > : 1

0 r 1 . 61 10 1 10

1 6 1 3 1 2

r 1 . 3 r 1 . 2 < r. Pn

= r1 , r2 , . . . , rn , W ( ) = i=1 w (ri ), Pt B , W (B ) = j =1 w (rij ), {rij |j = 1, . . . , t} B . , . 6. B W (B ) 17 , 10

W ( )

17 OPT( ). 10

. 1 , 2

w(ri ) 3 17 < . ri 2 10

63 , t X

3X 3 17 w(rij ) rij 1 < 2 j =1 2 10 j =1

t

. , ri > 1 2 rij ri t X

1 rij < , 2 j =1 7 . 10

, rij 1 , rij > 1 , 3 3 1 ri = j

t X j =1

w(rij ) r2 , r2 > 1 , 3 3

6 9 6 3 6 3 w(r1 ) + w(r2 ) = r1 + r2 = (r1 + r2 ) + r2 > (1 ) + > 1 5 5 5 5 5 53. m = 3. . 1 1 m. 3 2 .

9. B r1 rm . Pm 1 Pm i=1 w(ri ) = 15 > 0, (i) m = 1 r1 2 , (ii) i=1 ri 1 9 . . m = 1, r1 > 1 > 0. 2 m 2, , r1 r2 > . Pm e i=1 ri = 1 . B

66

8.

r3 , . . . , rm 1 , 2 , e , 1 + 2 = r1 + r2 + i ri . 8 B w m X i=3

w(ri ) + w(1 ) + w(2 ) 19 5

(8.1)

1 2

,

9 w(1 ) + w(2 ) w(r1 ) + w(r2 ) + 5 (8.1) m X i=1

X 9 w(ri ) + w(ri ) + w(1 ) + w(2 ) 1, 5 i=3 1 =m X i=1

m

9 w(ri ) 1 . 5

. W (Bi ) = 1 i i > 0 i < 1 2 i i Bi . Bi r1 , r2 ,... l > 1. 9, 1 < i l:

5 9

i 1

Xj

5 i1 rj i1 + i1 . 91 2

Bi1 (i) 9, i . l 1 X i=1

9X 9 9 1 i (i i1 ) = (l 1 ) < 1. 5 i=2 5 5 2 Pli=1

l

l 1, .

i 2. , l = 1

67 l 9, m (FF( ) = m + l). 8 1. ,

W ( ) =

FF( ) 2.

X

w(ri ) m +

l X i=1

W (Bi ) = m +

l X i=1

(1 i ) = m + l

l X i=1

i

FF( ) W ( ) + 2. 6 FF( )

17 OPT( ) + 2. 10

.

68

8.

9 , . , . , . , , . , , . , , , - . .

9.1

( , ) ( ) . , , . i- ri = (si , ti , bi , pi ) si ti , 69

70

9.

bi , pi . , (throughput) : . . . , , , . , , . , - (.. ) : 1. ( bidirectional mode ). hu, v i hu, v i hv, ui, c, c . 2. ( undirected mode ). hu, v i , . , 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) =

X 1 bk u(e) k2Ae2Pkj

Ln (e) 1 e = r1 , . . . , rn . :

D . , , , ( ), pi = D bi , ri = (si , ti , bi , pi ). , . , 0 " 1

cj (e) = u(e)[Lj (e) 1],

(9.1)

ri ,

i = 1, 2, . . . n

bi mine

u(e) = b (e) " log D + 1 + " u(e) log D

,

bi = O

72

9.

" 1 1 log D .

AAP" D (.. ) , 0 " 1. 1 bi b (") pi = D bi ri . = 21+ " D. j - rj = (sj , tj , bj , pj ) P

X bj cj 1 (e) pj . u ( e ) e2P rj , . Lj (e) cj (e) . " . 21. , AAP . . , Ln (e) 1 e. = r1 , . . . , rn , A rj ( Pj ) AAP. . k 2 A. AAP, rk

X bk ck1 (e) pk = D bk . u ( e ) e2Pk

X ck1 (e)e2P

u(e)

D.

9.2. e 2 Pk

73

D

ck1 (e) u(e)

= Lk1 (e) 1 = Lk (e) u(e) 1 Lk (e) (" log D+1+") 1bk u(e)1 1 bk

1 . (" log D+1+")

= Lk (e) (" log ) 1 " log = " log D + 1 + ". x = 2x log . . .

=1 1

Lk (e) 2"1 1

1.

Lk (e) 2 " D + 2 " 21+ " D = , Lk (e) 1. 22. , AAP

21+ " log + 1 = O(2 " + 2 " log D).. r1 , r2 , . . . rn . A rj ( Pj ) AAP, A AAP, . C P , C = e cn (e). : 1.

1

1

1

C (21+ " log )

1

.

Xj 2A

pj ,

74 2.

9.

. 1 2, , AAP( ) =

X j 2A

pj C,

|A|, AAP. , . , . , rk .

Xj 2A

pj OPT( ) AAP( ) +

X j 2A

pj .

Xe

ck1 (e) 21+ " log

1

j 2A{k}

X

pj ,

(9.2)

Xe

ck (e) 21+ " log X

1

Xj 2A

pj .

(9.3)

e 2 Pk ,

Xe

ck (e)

Xe

ck1 (e) =

e2Pk

(ck (e) ck1 (e)) 21+ " pk log .

1

(9.4)

[ck (e) ck1 (e)] = u(e)[Lk1 (e)+ u(e) Lk1 (e) ] = u(e)Lk1 (e) [ u(e) 1] u(e)1 " bk

bk

Lk1 (e)

= 2 log = 2 log 1 "

h h

h

log

bk u(e)

ck1 (e) u(e)

+ 1 bk + bk . i

i

i

2"

1

bk c (e) u(e) k1

9.2. :

75

x 1 = 2x log 1,

log

bk 1 , u(e) "

2y 1 y 2 " X

1

y 2 [0, ]. ,

1 "

e2Pk

X bk [ck (e) ck1 (e)] 2 log ck1 (e) + bk u ( e ) e2P1 " k

X bk X = 2 log ck1 (e) + bk u ( e ) e2 P e2P1 " k k

!

2 log pk +

1 "

e2Pk

X

bk

!

AAP

2 " log (pk + D bk ) = 2 " log 2pk . 1. , Pj , rj . , AAP, , 1

1

pj 0 .

1 bi min u(e) " log DT + 1 + e "

pi = D bi (fi ai ) ri . = 21+ " DT . j - rj = (sj , tj , bj , pj , aj , fj ) P sj tj

1

rj , . Lj (e, t) cj (e, t) . , rj aj t < fj ,

X X bj cj 1 (e, t) pj . u(e) a t