,u,,}, which can create any schedule, is said to be a sc h e dule d se t In this paper, the such set contains all on-time jobs of the schedule... THE FAST ALGORITHM FOR FOUNDING NONPREEM
Trang 1Te chi Tin hoc ,'<1Dieu klJie'n hoc, T.l7, S.l (2001),21-3
TRIN NHAT TIEN
Abstract In [2[we presented an O ( n2 1 0gn ) algorithm to determine a shedule with maximal number of on-time jobs in minimal processing tme for problem 1 Tj 12::U j in the case that release dates and due dates are satisfied I, :S1 :S ", :SIn, where Ij : =[T j , dj] ; (i.e., Tj <Tk '*d j ~ dk)
In this paper, we would exten the above algorithm to determine a schedule of the sme problem but with any number of on-time jobs in minimal processing time The time for this problem is O ( n 3. logn)
Tom t~t Trong [2[chung t6i cli trrnh bay th udt toan O( n 2. og n) cle'xac dinh thai gian bie'u v o'i so luo'ng IO'n n hfit cric c6 g v iecclung han va t hoi gian xrl'ly it nhit cho vin cle1 Tj 12:: U j, trong d I j := [Tj, dj]'
ma Tj <Tk ' *d j ~ dk.
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cu a cling van cle nh u'ng so luo'ng cong v iec clung h an la tuy y nhung thoi gian xd: 11 la it nhat
1 S OME BA S I C C O NC EPT S
Some conceptions in the paper are presented in [2] Now we would remind some concepts and
notations related to " job ", "r e aliz t ition " an " chedule ". The following data can be specified for each
jo u :
- Tu is a release d te, on which u becomes avaiable for processing;
- d" is a due date, by which u sh uld ide lly be completed;
- t I Lis a proces ing time (or length) of tz ,
We assume that the ab ve data are nonnegative integers and are regarded as parameters of job
u For convenience we will also use a concept "pre-fob" u; it is a pair (I, tu), where I" = [Tu , du ] is
it s active ar a. A pre-jo u such that i; <::::d., - T" is said to be a fob.
R" : =[b , ,,c,, ] (bu isa starting time , C" isa completion time)
is said to be a realizatio of jo u on machine A jo u is said to be completed o t me (or a on-time fob) ifc" <:::: d,,; otherwise a jo u issaid to be late.
Let I = iri, di ] and I = [ r ), d ) ] be active areas ofcorresponding jobs i and i,respectively Then
the area Ii is said to be a ead of area Ij (or area I j is behind area I J and denoted by I :SI j ifan only ifr i <:: : r j and d, ~ d j
Similarly I, -< I j ifand only if I , :S I j an I, 1- Ij [ B,G] := [MinT j ,Maxd j ] is said to be an actiue area of the system; where B is a r leas date and Gisa due date of it
Let U=IU l, U 2, " , u.,} be a subset ofjo s onthe system Sup ose that S :={RUl' R U2' "., Ru s is
aset of realizatio s ofcorresponding jo s U l, U 2, ' " u, such that R u, nRu) = 0,ViI e f i f = 1,2, "., s
Then S (or U : =1R",) is said to be a schedule o the set U ofthe system (or a schedule of the system)
U : = l R", is also said to be a processing area of the schedule S.
A realizatio Ru of job u in sched le S iswritten b R,,(S) or u(S) and sometime only by {u}.
In the paper we assume that Ru(S) C Iu, therefore the schedule S isregarded as a set ofdisju ctve realzatio s ofon-time jobs
We note some following parameters of the schedule S :
- ~ : =s is a number of realizations or a number of jo s;
- t ,,.: =2 : =1t-; is a processing time (or a length)
- b s := Min {bu,} is a starting time;
:=Max {c",} isa completion time ;
Trang 2T RI N NHAT T I EN
- [b s, c s] is an activ e a r a ofschedule S
Let u : = (I u, ta ) be a jo , [ X , Y] be a time area We define apre-job v: = (I v , t v ) o [ X , Y]such
as I " = L ; n [ X , Yj , t ; = i ; and we write v = u i [ X , Y j
For a set of jo s U = { Ul,U2, ,U } , we denote a set of pre-jobs on [ X , Y ] by U i [ X , Y ] = { U l i [ X , Y j , U2 T [ X , Y j , ,u T [ X , Y ] }
We say that a schedule S isin th e a r ea [ X , Yj if its active area [ b s, cs ] 5;;[X , Y ]
Note that we define a s che dul e only on the set of J o s, not on a set of pre-jobs A set of jobs
U = { Ul' U2, ,u,,}, which can create any schedule, is said to be a sc h e dule d se t In this paper, the
such set contains all on-time jobs of the schedule Sometime for s hedule S having scheduled set
{U l , U2, u ,}, we also write S = {U l, U2, ,u }
We denote pro lem [T]by following:
[T]: 1[ r , [ L U J ,where U J = 0 ifCJ < d J, UJ= 1otherwise
This problem me ns that the system has n J o s with different release dates r J, they are available
processin on one machine, we have to construct a nonpreemptive schedule with a minimal number
oflate jobs (i.e., amaximal number ofon-time jo s) Weknow that the problem isstron ly NP-hard,
auth rs H Kis ,T Ibaraki and H Mine (1979) provided an O(n2 ) algorithm for problem [T]in the
case that release dates an due date are similarly ordered ( i.e r J < r » = d J ~ d k ). We like to
express this case b following:
[Kt]: 1[ 1 1 :5 1 2 :5 : 5I n [ M a x L U J ,where UJ= 1 ifCJ ~ dJ , UJ= 0 otherwise
The problem is to build a nonpreemptive schedule with maximal number of on-time jobs
Now we would pay attention to following special cases:
-[Tl]: 1 ] 11 : 5 1 : 5 : 5In[ M a x L U J and MinLU J J
The problem is to construct a nonpreemptive shedule with a maximal number ofo -time jo s and
furthermore in minimal processing time In ]2]wepresented an O(n210gn) algorithm for the pro lem
[T2]: 1 [ 11 : 5 1 : 5 : 5In [ L U J= sand MinLUJ J
The problem is to construct a nonpreemptive schedule with a fixed number of on-time jobs (i.e., s )
and furthermore in minimal processing time In this paper, we extend the O(n21 gn) algorithm in [2]to solve the problem [T2]
We would remind some following concepts and notations pre ented in [2]
Let R; = r bi , C i] and R J = [ bJ, c l] be realizations of corresponding jobs i and J,respectively We
say that R , is ah e ad of R J (or R J is behind R; and write R, :5 R J if and only ifthey satisfy one from
two following condito s: 1)i == J and b i ~ b i; 2) it: - J and I,: 5 I i' Similarly we write R , -< R J
Let P = { U l, U 2, ,ur n} an Q = { V l, V2 , ,vm } be s hedules with the same number ofjo s We
say that P is a h ea d o f Q (or Q is beh i nd P) and wrie P : 5 Q ifand o ly if R u : 5 Rv , \ji = 1,2, ,m
Similarly we write P -< Q
A schedule S = {Ul' U 2 , ,ur n } is said to be R- s chedule in [ X , Y]ifit is in the area and realiz
-tons [ bu." c u.] have following forms:
CUm = M in{ d um , Y } ; b Um = C U m -tu m;
cu = Min {d Uil b U '+ l } bu = CU - i «, \j i = m - 1,m - 2, ,2, 1
Let P = { U l , U 2,"" Up} and Q = { Vl,V2 , " V ' l} be R-schedules in [ X , Y ] We say that Pi s
R - bette r th a Q and denote by P > - r Q +- one of the following conditions satisfied:
(rd p > q (i.e., P has the number of jobs more than Q )
(r 2 ) P = q and tI ' < t Q (i.e P has the processing time less than Q ) ;
(r 3 ) P = q and t r = tQ and b I ' >bQ (i.e., P has the starting time later than Q ) ;
h )p = q and t t - = tQ an bI ' = b Q and Q : 5 P (i.e P isbehind Q); With i=1,2,3,4, if P > - Q in
the sens (r i ) , we write P > - r. Q
Trang 3THE FAST ALGORITHM FOR FOUNDING NONPREEMPTIVE SCHEDULE 23
We say that schedule S is R - best if and only if it is R-schedule having:
(r o l a maximal n mber of jobs completed on time;
( r02) a minimal processing time ts from schedules satisfying above condition;
(r03) a latest starting time bs from schedules satisfying above condition;
and it is
In the case that the R-best schedule has only 1 job (.e., 1 realization), we call it R-best
realization
Let P = {Ul,U2, ,Up} be R-schedule in [ Xp,Yp] and Q = {Vl,V2, "" V q } be R-schedule in
[ XQ,YQ ] , where Yp:::; XQ, i ; « I v1.
We define a operation, which is called R - connection and denoted by P EElT Q, to connect P to Q The result of the operation is schedule S, having following realizations:
[b1 l ( S ) c1 l ( S ) = [ b1l, ( Q ) , c V, (Q) ] V z= q, q - 1, ,1;
[ bur ( S ) , cu" ( S ) , where cUI'( S ) = Min {dup, bQ}; bu, ( S ) = cUI'(S) - tu,.;
[ bu, ( S ) ,cu, ( S ) , where cu, ( S ) = Min{du"bu'+l(S)}; bu, ( S ) = cu, ( S ) - t«
A schedule S = {U l U2, , urn} is said to be L - schedule in[X, Y ] if it is in the area and realizations
[ b u, , c u , ] have following forms:
b U1 = Max{X,rUl}; CUl = b Ul +tUl;
b u, = Max{cU'_ T u,} c u, = bu , +tu" Vi = 2,3, m
Let P = {Ul, U2, ,up} an Q = { vi. V2 , ""v'l} be L-schedules in [X, Y]. We say that P is
L-better than Q and denote by P :> - 1 Q + - o e of the following conditions satisfied:
( 1 1 ) p> q (i.e., P has the number ofjo s more than Q);
(1 2 ) p = q and Cp < cQ (i.e., P has the completion time earler than Q ) ;
(13 ) p = q and Cp = cQ and C u, ( P) :::;c1 l (Q), Vi = 1,2, ,p - 1;
(14 ) P = q and C u, ( P ) = c1l,( Q ), Vi = 1,2, ,p and P:::S Q (i.e P is ahead of Q ) ;
For i= 1,2,3,4, if P:>-I Q in the sense ( Ii ) , we wrie P :> - 1 , Q
We say that schedule S is L- b es t if and only if it is L-schedule having:
(10 1 ) a maximal number of jobs completed on time;
( 10 2) a earlest completion time C c,' from schedules satisfying above condition;
(10 3 ) a earlest completion time of realizations from schedules satisfying above conditio ;
and it is
(104) ahead of all schedules satisfying above condition
Let P = {Ul,U2, Up} be L-schedule in [Xp,Yp] and Q = {Vl,V2 , ,Vq} be L-sched le in
[ XQ,YQ] , where Yp:::; X q, i.; « Iv1'
We define a operation, which is called L - connection and denoted by P EElI Q, to connect Q to P
The result of the operation is schedule S, having following realizations:
[ bu, (S ) cu , ( S ) = [bu, ( P ) [ cu , ( P ) ' V i = 1,2, ,p;
[b 1 ( S ) , CVl (S ) , where b Vl (S ) = M ax{cp, T V,}; C 1l1 (S) = b Vl (S) +tv, ;
[ bv,(S), cv , (S)] , where b11,(S) = Max {cv, _1(S) , Tv , } Cv, (S) = b v, (S) +tv" V i= 2, 3, ,q
We say that schedule S is s-optimal if and only if it is R-schedule having:
( o just s jobs completed on time;
(02) a minimal processing time ts from schedules satisfying above condition;
(03) a latest starting time b ') from schedules satisfying above condition;
and it is
(04) behind all schedules satisfying above condition
Conclusion. According to the above conceptions, the s-optimal schedule is just R-best schedule having s on-time jobs Therefore solving problem [T2] is just determining the s optimal schedule
We call the schedule constructed by authors Kise, Ibaraki and Mine (197 ) K-schedule We cal their algorithm K - a l gorithm. We assume that this schedule has just mjobs, it is the maximal number
Trang 4TRINH NHAT TIEN
K = { Xl , X 2, , X " , } be K-schedule, [ b ; { K) , c ; {K) ] : = [ bx , (K ) , c x , (K) ] be the realizatio x ; {K) We
U, = {J obs u II I x, : SI" -< I x , + J f o r t = 1 , 2 , , m - 1 ;
i.e., we can put in order njobs from U to m+1following subsets:
[ J,o {I UO' U 2 ' U " } ;
UI { XI , U2 I, ,U"II}, W Irere XI UI I;
U2 { X 2, U2· ·· , U2 - ,n? } werhe X2 UI2 ;
i = Xi) Ui , , Ui J were Xi == Ui ;
Ur n , = { X r n J U ;fl , u; ~m }, whe r Xr n == u; n ;
s et s of Job s U , and other notion s be such as (1) We have followi n r es u lt:
For k : = 1,2, m - 1, if U k c ontain s W i EW s u h that
CXk (K) Sc W l (W ) and s - U + 1) <m - k
Ui , doe s n ' conta in an y job WEW.
(2) (3) ( )
th en
U k does n ' c ontam a nex t Job Wi + l EW;
Lemma 2 The a ss umption s are the s ame a s in the L emma 1 W e have fo l lowin g r esu l t:
b W l ( W ) <C X k (K) a nd J - 2 < k : then
U k - { x doe s n't contain a preceding Job w J - 1 EW ;
( 5 ) (6)
Urn doe s n't contain a Job w W s uc h th at C W (W) <C x , (K) (7) Lemma 3 The ass umptions are th e s ame a s i n th e L e m ma 1 W e ha ve f o llowi ng re s ult: Th e r e i s not any int e e r k (0S k S m) s u h that U i ; {xd contain s 2 n eigh bou r ing Jo b s wi, Wi + l E W
Lemma 4 The a ss umptions ar e th e same a s i n th e L e mma 1 We have fo l owing re s ult :
Wi EU; UUi + 1 U UU i+ m- U{ xi + m- ,+ d , 'it = 1,2, ,s - 1 ( 8 )
and w , E U UU + l U UUr n
Trang 54 s'-OPTIMAL SCHEDULE
From result of Lemma 4 we define concept "s * - op ti ma!" schedule related to the s - optimal sched
Lemma 5 The a s sumpt io ns a re the same as in the D ef i iiition. 1 W e have [o l lo unn q result:
U , l I B, Cj then bX ,(K ) - Sb s
Definition 2 The assumpto s are the same as in the Definition 1 For d = 1,2, , m, with
q (1-S q -S m- d+1), we defne following concepts:
w , is q * - optimal schedule o U, 7 i I B, Cj (10)
W , ~is q - optimal schedule on 0 , 7 l I bw' - ' +1,CJ,
d
(11)
IJ , 'f : ={V , l, V}, ,V , I} is said to be a i n fu l set o f q - op ti mo l schedules o the set U , 7if and only
v , is q " - optimal schedule o (U ,7 - {x,d) l I B , C j
V ; IS q "-optimal schedule on ( U,; - { x,d ) l ' Ib '-' +1,C j ,
d
(12)
(1 3 )
d
set has R-or de r if t s ' <t S ' + 1 ; b s ' < b ,, ,+ 1 ; s' ~ Si + 1 ,Vi = 1,2 , , p - 1
o set U,7
Lemma 6 or 1-S -S s- 1J e t d + 1' - d + 1' d + 2' , d + m - •- 1 ) e a s s e m 0 q -opt im a l s ch ed ule s o n th e s et U ,7+ 1'
Trang 6TRINH N H AT T I EN
Every schedule from the s stem A ; ~has to contain any s h edule from the system A:~ ~ as its
"ending part" with (q - 1) Jobs.
Corollary The s" -optimal schedule has to contain any schedule from t h e s stem A ;- I as its "endin g part " with (s - 1) Jobs.
5.1 Main idea of algorithm
By the ab ve results, our algorithm will co strue ted by following steps:
-First determine K-s hedule K = {XI, X2, , xm} byK-algorithm wih time O ( n ) or by Lawler's
algorithm with the time O ( n ogn).
- Lemma 4 and Lemma 5 determine the positio of the s-optirn al schedule W in comparise with
K-schedule Here ifW = {WI,W2,""W.,} then
ui, E U,UUi+1 U UUi+ rn - , U{ xi+m- +d, Vi = 1,2, , s - 1; (14)
w E U UU., + I U UUrn an b Sbw.
For d : ~ s, s - 1,s - 2, ,2,1, p t q :=s - d+I
To create W, we construct the system A l of all s hedules, which could become W, these such
schedules equally have property (14) By Lemma 6, this system will created rec urssively by 3 following
algorithms:
1 / Algorithm SBASE will create the basic system A! , i.e the system of I ' - optima l schedules on
the set U ;; one from these schedules wil'become "an ending part" { w , } of optimal s hedule W.
2/ Procedure SSTEP willfom the well-known system A:~~~of ( q - 1) *-op t m schedules on the set V~+I determine a system A ;~ofq' - opt imal schedules on the set U,7;one from these schedules will
become "an ending p art " { WIl' Wd+ l , , Wd+(.- J ofthe optimal schedule W
3/ Algorithm USE-SSTEP will from the basic system A! apply ( s- l) times the procedure SSTEP, '11 L I ,,2,,3 ".-1 n I " • - { T ' 1 " 1 " }
we WI obt ain successivery systems / 1 " _ 1' / 1 _ '"' ' / 1 , I ' wnere / 1 1' - JI, 2'"' ' m- ( , - I ) ,
1 t = ( Wi ',Vj') Suppose W : = {WI , W2, . WI'} then Wi isjust the desirable s' - op t imal schedule
Let a set of jo s U = {xl, x 2, , x }, according to the definito of R- best s hedule, we can create a procedure to find R-best schedule with 1job (i.e., R-b es t realization) { } on U and write:
{x} : = = R - JO B ( { l x2, , xk }) ;
In the case the set isrestricted by the time area IX,Y], wewrite:
{ } : = RB - JOB ({ xl, x2, ,xk} r IX, V ]
Processing time ofthis procedure isO ( k ) Weneed note that, may be {xl, x 2, . xk} r I X,Y]is not a set ofjobs, therefore there isnot such { }
Input: - U = {xl X 2, , xk} is the set ofjobs such a I x 1 ::SI x ::S ::SI xk ;
- b is a starting time of the area time;
- S is R-schedule on the set of jobs {yl, y2, , yh} such as I x - < Iy1 :SIy2 :S :SI y
Output: - Z = { ZI' Z 2, , Zp} is a set of R-schedules, every schedule Z, is created by R- c onn ecti on
ofR -best realization o U to S;
- K = { I, k2' ,kp} isa set index corres ponding to Z ; p = ~Z
Method: The algorithm applies the procedure RB - J O B to determine a R-best realizatio o U, if
there is the such realization then connects it to S.
Algorithm:
Trang 72 := 2+1;
Algorithm:
Begin
For 2 := 2 to r d
begin
end;
21"" PI' 11 21"" P 2" ' " 11 21"" Pr - 1 , 2, ··· , P ,
End
Trang 82 bz ' <bz' < < bz' <bZ' + l \i t = 1, 2 , r :
3. Z i1 z :2 z :1 ', Zi1+ l , \v tJ = 1 2, , ,r;
Proposition 5 L et R m pr oce dur e JOB - SCHE D ULES be the s et of d-optimal sc hedule s, then Z t S
the s et of ( d +I)- optimal schedule s on c or r s pon d ing se t s of Job s and eve ry s u c h s ch e dule c onta i n s
c o r respo nding Si E Ra s it s "e nd i ng par t JJ.
Proposiion 6
1 T he numb e of s h edule s m t h e se t Z isp:::; k +r , w h e e k = U , r = U R.
2 Th e p r o c ss i ng t me of pr oce dure JO B - SC H E D ULES is r.O(k.logk)
4/ Procedure unifies 2sets of schedules, having R-order: UN fO N( P, Q,X, Y; T )
Input: - P = {PI , P 2 , , PI ' } isa set of R-schedules, where PI >- P 2 >- >- PI ';
- Q = {Ql, Q 2, , Q } is a set of R-schedules, where Ql >-r Q2 >- >- Q ,
- [ X , Y1 isthe time area
Output:
Method: The procedure is similar as unifyin 2 ordered sets of integers
5.3 Main algor ithms
Let K = { X I , X 2, ,X m } be K-schedule, let sets ofjobs U , and other notions be such as (1) There are 3 following main algorithms:
a Au i ary procedure BASE: BASE(U , b ; 1)
Input: U: = {X O, XI,x 2, ,xk} ; b isa integer
Output: J : = (W , V) is the pair of 2 sets of 1 * - opti m a l schedule o U,
Method: The algorithm applies procedure RB - J OB to determine 1 - optimal schedule on U.
Algor-ithm:
Begin
if there is { xkl}: = RB - JOB({x O ,XI , x 2 , ,xk } i [ b,e l
then WI := {X k l} an p:= 1 else put W := 0 and p: =0;
Repeat
i=i+l
if there is {x k, } := RB - J OB( { xk ,- d l , x ,-d 2 , , x } r [ b w' _ l +1,e l
then Wi := { k, } and p := i
Until p <t ; ( i.e., there is not {xk , } )
i: =1;
ifthere is {yi Ll}: = RB - JOB({ XI, x2 , x 3, ,x k} r [ B, e l
then VI := {yiLl } and q :=1 else put V :=0 and q :=0 ;
Repeat
if there is {x h, } : =RB - J OB ( { X h, - l + l, xiL , - d2 , , x } r [ b V , _ l +1,e l
then V i : ={x i L,} an q : =i;
Until q <i; (i.e., there is not { " , }) ;
End;
Trang 9Input: Ut, b ; (K) , for i=.s.s-L, ,m are the same as (1)
Output: A ~: = {~1, 1/ +1, 1, } J is the system of 1* -opt i m a l schedules o set U :
Method: The algorithm applies the procedure BAS E ( m - s+1) times
Algorithm: For ~ : = s To m Do BA SE(U ; *,b ; (K) ; J/) ;
Proposition 7
1.A ~ i s Ju s t the s ys tem of 1*-op t m al s chedule s on U :
2 ~ W/ and ~ V/ :s; n , for i= s, s - 1, , m, whe r e 1/ = (W /, V/).
3 1/ r s determined a fte r t h e ti m e O((n : 2 )
4 Processing ttme of the algo r ithm is 2 : ~ '1 -· ' + 1 )O(( n ) 2 ).
By the method mentioned in the Proposition 2, 1/ is determined after the time O(n7.logn7),
therefore the algorithm SBASE needs only the time 2:; :'1 - ·,+1)O(n~.lognn
2/ Procedure SSTEP: S S T EP(A ;~ ~ ~, A ; ~ ) ; d = s - 1,s - 2, ,2,1
Input Jl 1 d + ' 1 - I' _ {1'I - l1'1 d + l ' - 1 d +2 ' , ' 1'/-1 / + " ' -(1 1 - 1 ) }'I S t eh system 0f()*'q - 1 - optima l sc ehdlu es on t ehe sseet
U '~ + I'
Output:
A ; ~: = {1, ; ' , l ;' + l' . l, + m - l/ ) is the system of q' - optimal schedules o the set U , l'
Method: The algorithm applies procedure JOB - SG H E D U L ES to connect jobs ofUd U U,/ + l U U
Ud + m - " U{ Xd + m - d to schedules of A;~ ~ ~,after that by procedure U NION to unify the created
sets of schedules
Algorithm:
Begin
For i: =d To ( d +m - s ) Do
begin
JOB - SG H EDU LES(U i , b ,(K) , Wi'~/; [ , e );
JOB - SG H ED U L ES ( U i - { ; }, B , W ; ';/; 9, g )
J O B - SGB ED U L ES ( { Xi + l } , B, V; ' ; } {, h ) ;
U N I O N([ , } , bi lK) , G; W : ,) ; U N ION(9 ,}{, B, G;v ; ') ;
end;
End;
Proposition 8 For d: =s - 1,s - 2, ,2,1 and q: =s - d+ 1:
2 , For i = d,d + l, " d+ m : ~ W : , and ~ V : ' : S; n i, w h e ren i= n i + n i + l + , + n m,
, T h f hi ' h , ,,,d + m -' l O( l ) *
3, e pr ocessi t q ti m e 0 te a q rit m t S l J i=d ni' o ri, , ni +1
3/ Algorithm USF,-SSTEP:
In ut: A is the system of 1 - op tt ' mal schedules on U : ,
Output: A ; l' A ; 2: " A ; l, Al are the desirable systems of schedules
Method: The algorithm applies procedure SSTEP (s-l) times with input A ~,
Algorithm: For d : = - =s - 1 DownTo 1 Do SSTE P( A:~ ~ , A : ~ )
Theorem 1 I n the output of algorithm USE - SS T EP s uppo s e
A " 1 ,- , {T J , ' l' 12'" " , 'm -(,,- I ) },Jl T ,' - - (W 1, 1 V) a d W 1 - - {WI ,W 2 , "W"} ,
then Wi is Ju s t th e d es
Trang 10ir-TRINH N AT TIEN
Proof By Proposi on 7, A! isjust the system of I*-optimalschedules on U ;. By propositio 8, A ; ~is the system ofq* - o t m a l schedules on the set V~,for d = s - 1,s - 2, 2,1 Algorithm USE-SSTEP
following systems ofschedules: A~ _ I' A ;- 2, , A; - l, Al , suppose Al := {1 i', '2" , ,1 ~' _ (" _ ) } ' 1 1 '=
(WI, V rl and WI = {W 1,W2, , WI'}, then by definitions 2, 3, WI isjust the desirable s" - optimal
schedule
L:I : ( ~ " - ,O( n i logni) n 7+1' for d = s - 1,s - 2, ,2,1
L ( L O(ni 10gni).n; + I = L (L O(nd + i lognd +; ).n~ + i + l
m
: s ; (m - s+1)'(L O(nk lognk) n + I ·
k = 1
(15)
L nk.(lognk) n ~ + 1 < ( o gn) '(L nk.n Z + l ,
L nk.n~ + 1 = L(n~ - n~ + l ·n~ + 1 = L n~.n~ + 1 - L(n~ + 1)2
<L n~.n~ - L(n~ + r l2 = L((n~)2 - (n~ + 1)2)
( * ) 2 ( * ) 2 ( * ) 2 ( , ) 2 ( * ) 2 ( * ) 2 ( * ) 2
The abo e calculatio s implies the proof
I).O(n210gnJ , where m is the maximal number of on-time Jobs, s is the fixed numbe r of on-time fo b s ( s :s;m) (i.e., O(n310gn)).
above corollary the time for this case isO(n 2 10gn).
REFERENCES
Com-puter Science and Cybernetics 15 (1) (1999) 66-76
Re eived July 1 4, 2 000
Vietnam National Univer ity,