News from the online traveling repairman

16 Pages·2003·0.24 MB·English

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Theoretical Computer Science 295 (2003) 279–294
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News from the online traveling repairman
Sven O. Krumkea;∗;1, Willem E. de Paepeb, Diana Poensgena,
Leen Stougiec;d;2
aKonrad-Zuse-Zentrum fu(cid:15)r Informationstechnik Berlin, Department Optimization, Takustr. 7,
0-14195 Berlin-Dahlem, Germany
bDepartment of Technology Management, Technical University of Eindhoven, P.O. Box 513,
5600MB Eindhoven, The Netherlands
cDepartment of Mathematics, Technical University of Eindhoven, P.O. Box 513, 5600MB Eindhoven,
The Netherlands
dCentre for Mathematics and Computer Science (CWI), P.O. Box 94079, NL-1090 GB Amsterdam,
The Netherlands
Received 26 September 2001; received in revised form 13 December 2001; accepted 27 December 2001
Abstract
In the traveling repairman problem (TRP), a tour must be found through every one of a set of
points (cities) in some metric space such that the weighted sum of completion times of the cities
is minimized. Given a tour, the completion time of a city is the time traveled on the tour before
thecityisreached.Intheonlinetravelingrepairmanproblem(OLTRP)requestsforvisitstocities
arrive online while the repairman is traveling. We analyze the performance of algorithms for the
online problem using competitive analysis, where the cost of an online algorithm is compared
to that of an optimal o7ine algorithm.
We show how to use techniques from online-scheduling to obtain a deterministic algorithm
√
with a competitive ratio of (1+ 2)2¡5:8285 for the OLTRP in general metric spaces. We also
present a randomized algorithm which achieves a competitive ratio of 4=ln3¡3:6410 against
an oblivious adversary. Our results extend to the “dial-a-ride” generalization L-OLDARP of the
OLTRP, where objects have to be picked up and delivered by a server. This improves upon
the previously best competitive ratio of 9 for the OLTRP on the real line and, moreover, the
results are valid for any metric space. For the case of the L-OLDARP our algorithms are the Arst
competitive algorithms.
We also derive the Arst lower bounds for the competitive ratio of randomized algorithms for
the OLTRP and the L-OLDARP against an oblivious adversary. Our lower bounds are (ln16+1)=
∗Corresponding author.
E-mail addresses: [email protected] (S.O. Krumke), [email protected] (W.E. de Paepe), poensgen@
zib.de (D. Poensgen), [email protected] (L. Stougie).
1Research supported by the German Science Foundation (DFG, grant Gr 883=10).
2Supported by the TMR Network DONET of the European Community ERB TMRX-CT98-0202.
0304-3975/03/$-see front matter (cid:2)c 2002 Elsevier Science B.V. All rights reserved.
PII: S0304-3975(02)00409-7
280 S.O. Krumke et al./Theoretical Computer Science 295 (2003) 279–294
(ln16−1)¿2:1282fortheL-OLDARPontheline,(4e−5)=(2e−3)¿2:41041fortheL-OLDARP
on general metric spaces, 2 for the OLTRP on the line, and 7 for the OLTRP on general metric
3
spaces.
(cid:3)c 2002 Elsevier Science B.V. All rights reserved.
Keywords: Online Algorithms; Competitive analysis; Traveling repairman; Latency; Dial-a-ride problem
1. Introduction
In the traveling repairman problem (TRP) [1] a server must visit a set of m points
p ;:::;p in a metric space. The server starts in a designated point 0 of the metric
1 m
space, called the origin, and travels at most at unit speed. Given a tour through the m
points, the completion time C of point p is deAned as the time traveled by the server
j j
on the tour until it reaches p (j=1;:::;m). Each point p has a weight w, and the
j j j
object(cid:1)ive of the TRP is to And the tour that minimizes the total weighted completion
time m wC. This objective is also referred to as the latency.
j=1 j j
In this paper we consider an online version of the TRP called the online traveling
repairman problem (OLTRP). Requests for visits to points are released over time while
the repairman (the server) is traveling. In the online setting the completion time of a
request r at point p with release time t is the Arst time at which the repairman visits
j j j
p after the release time t . The online model allows the server to wait. However,
j j
waiting yields an increase in the completion times of the points still to be served.
Decisions are revocable as long as they have not been executed.
In the dial-a-ride generalization L-OLDARP (for “latency online dial-a-ride problem”)
each request speciAes a ride from one point in the metric space, its source, to another
point, its destination. The server can serve only one ride at a time, and preemption of
rides is not allowed: once a ride is started it has to be Anished without interruption.
An online algorithm does not know about the existence of a request before its release
time. It must base its decisions solely on past information. A common way to evaluate
the quality of online algorithms is competitive analysis [5]: An algorithm is called
c-competitive if its cost on any input sequence is at most c times the cost of an
optimal o7ine algorithm.
1.1. Related work
Feuerstein and Stougie [7] presented a 9-competitive algorithm for the OLTRP on
the real line and a 15-competitive algorithm for the L-OLDARP on the real line for the
special case that the server has inAnite capacity. In the same paper lower bounds of
√
1+ 2 and 3 on the competitive ratio of any deterministic algorithm for the OLTRP
and the L-OLDARP, respectively, are proved.
The o7ine traveling repairman problem TRP is known to be NP-hard [1]. In the
special case when the metric space is the real line, the TRP can be solved in polynomial
time [1]. However, up to now, the complexity of the TRP on weighted trees remains
unsettled. Approximation algorithms for the TRP have been studied in [2,4,8].
S.O. Krumke et al./Theoretical Computer Science 295 (2003) 279–294 281
Table 1
Deterministic upper and lower bounds for the OLTRP and the L-OLDARP
Deterministic Previous best Deterministic
UB UB LB
√ √
OLTRP (1+ 2)2¡5:8285 9, see [7] 1+ 2, see [7]
(Corollary 8) (real line)
√
L-OLDARP (1+ 2)2¡5:8285 15, see [7] 3, see [7]
(Theorem 7) (real line, server capac. ∞)
Table 2
Randomized upper and lower bounds for the OLTRP and the L-OLDARP
Randomized Randomized
UB LB
OLTRP 4 ¡3:6410 (Corollary 12) general:
ln3
7 (Theorem 16)
3
real line:
2 (Theorem 17)
L-OLDARP 4 ¡3:6410 (Theorem 11) general:
ln3
4e−5¿2:4104 (Theorem 14)
2e−3
real line:
ln16+1¿2:1282 (Theorem 15)
ln16−1
1.2. Contribution and paper outline
Our main results are competitive algorithms and randomized lower bounds for the
OLTRP and the L-OLDARP. Our algorithms improve the competitive ratios given in [7],
and, moreover, the results are valid for any metric space and not just the real line.
For the case of the L-OLDARP our algorithms are the Arst competitive algorithms. The
randomized lower bounds are the Arst ones for the OLTRP and the L-OLDARP.
Our algorithms are adaptations of the GREEDY-INTERVAL algorithm for online
scheduling presented in [9,10] and of the randomized version given in [6]. Our lower
bound results are obtained by applying Yao’s principle in conjunction with a technique
of Seiden [12]. An overview of the results is given in Tables 1 and 2.
In Section 2 we deAne the problems formally. Section 3 contains the deterministic
algorithm INTERVAL and the proof of its competitive ratio. In Section 4 we present
(cid:15)
the randomized algorithm RANDINTERVAL which achieves a better competitive ratio.
(cid:15)
In Section 5 we prove lower bounds on the competitive ratio of randomized algorithms
against an oblivious adversary.
2. Preliminaries
An instance of the online dial-a-ride problem OLDARP consists of a metric space
M=(X;d) with a distinguished origin o∈X and a sequence (cid:20)=r ;:::;r of requests.
1 m
282 S.O. Krumke et al./Theoretical Computer Science 295 (2003) 279–294
We assume that M has the property that for all pairs of points (x;y)∈M there is a
continuous path p:[0;1]→X in X with p(0)=x and p(1)=y of length d(x;y) (see
[3] for a thorough discussion of this model). Examples of metric spaces that satisfy the
above condition are the Euclidean space Rp and a metric space induced by a connected
undirected edge-weighted graph.
Each request r =(t ;a ;b ;w) speciAes a ride, deAned by two points: a ∈X, the
j j j j j j
ride’s source, and b ∈X, its destination, and a weight w¿0. Each request r is
j j j
released at a nonnegative time t ¿0, its release time. For t¿0 we denote by (cid:20) ((cid:20) )
j 6t =t
the set of requests in (cid:20) released no later than time t (exactly at time t). A server is
located at the origin o∈X at time 0 and can move at most at unit speed. We consider
the case in which the server can serve only one ride at a time, and in which preemption
of rides is not allowed.
An online algorithm learns of the existence of request r only at its release time
j
t . In particular, it has neither information about the release time of the last request
j
nor about the total number of requests. Hence, at any moment in time t, an online
algorithm can base its decisions only on the requests in (cid:20) . An o7ine algorithm has
6t
complete knowledge about the sequence (cid:20) already at time 0.
Given a sequence (cid:20) of requests, a feasible route for (cid:20) is a sequence of moves of
the server such that the following conditions are satisAed: (a) The server starts in the
origin o, (b) each ride requested in (cid:20) is served, and (c) the repairman does not start
to serve any ride (in its source) before its release time. Let CS denote the completion
j
time of request r on a feasible route S. The length of a route S, denoted by
j
l(S), is deAned as the diPerence between the time when S is completed and its start
time.
De(cid:22)nition1 (L-OLDARP, OLTRP). Given a request se(cid:1)quence (cid:20)=r1;:::;rm the problem
L-OLDARP is to And a feasible route S minimizing mj=1wjCjS. The online traveling
repairman problem (OLTRP) is the special case of L-OLDARP in which for each request
r source and destination coincide, i.e., a =b .
j j j
We denote by ALG((cid:20)) the objective function value of the solution produced by an
algorithm ALG on input (cid:20). We use OPT to denote an optimal o7ine algorithm.
De(cid:22)nition2 (Competitive deterministic algorithm). A deterministic online algorithm
ALG for the L-OLDARP is c-competitive, if there exists a constant c such that for
any request sequence (cid:20): ALG((cid:20))6c·OPT((cid:20)).
A randomized online algorithm is a probability distribution over a set of deterministic
online algorithms. The objective value produced by a randomized algorithm is therefore
a random variable. In this paper we analyze the performance of randomized online
algorithms against an oblivious adversary. An oblivious adversary knows the online
algorithm and the distributions it uses, but does not see the realizations of the random
choices made by the online algorithm and therefore has to generate a request sequence
in advance. We refer to [5] for details on the various adversary models.
S.O. Krumke et al./Theoretical Computer Science 295 (2003) 279–294 283
De(cid:22)nition3 (Competitive randomized algorithm). A randomized online algorithm
RALG is c-competitive against an oblivious adversary if for any request sequence (cid:20): E
[RALG((cid:20))]6c·OPT((cid:20)).
3. Adeterministicalgorithm
Our deterministic strategy is an adaption of the GREEDY-INTERVAL algorithm pre-
sented in [9,10] for online scheduling. The proof of performance borrows concepts of
the proofs in [9,10].
Algorithm INTERVAL
(cid:1)
Phase 0: In this phase the algorithm is initialized.
Set L to be the earliest time when a request could be completed by OPT. We can
assume that L¿0, since L=0 means that there are requests released at time 0 with
source and destination o. These requests are served at no cost. For i=0;1;2;:::; deAne
√
B :=(cid:15)i−1L, where (cid:15)∈[1+ 2;3] is Axed.
i
Phase i, for i=1;2;::: : At time B compute a transportation schedule S for the set
i i
of yet unserved requests released up to time B with the following properties:
i
(i) Schedule S starts at the endpoint x of schedule S (we set x :=o).
i i−1 i−1 0
(ii) Schedule S ends at a point x with an empty server such that d(o;x)6B.
i i i i
(iii) The length of schedule S, denoted by l(S), satisAes
i i
(cid:2)
B if i=1
1
l(S)6
i B +B if i¿2:
i i−1
(iv) The transportation schedule S maximizes the sum of the weights of requests
i
served among all schedules satisfying (i)–(iii).
If i=1, then follow S starting at time B . If i¿2, follow S starting at time (cid:30)B until
1 1 i i
(cid:30)B , where (cid:30):=((cid:15)+1)=((cid:15)((cid:15)−1)).
i+1
√
To justify the correctness of the algorithm Arst notice that (cid:30)¿1 for any (cid:15)¿1+ 2.
Moreover, it holds that the transportation schedule S computed at time B can actually
i i
be Anished before time (cid:30)B , the time when transportation schedule S , computed at
i+1 i+1
time B , needs to be started: S is Anished latest at time B +B =2B 6(cid:15)+1B =(cid:30)B
i+1 1 1 1 1 (cid:15)−1 1 2
since (cid:15)63. For i¿2, schedule S is also Anished in time: By condition (iii), l(S)6B+
i i i
B =(1+1=(cid:15))B. Hence, schedule S is Anished latest at time (cid:30)B +(1+1)B =((cid:15)+
i−1 i i i (cid:15) i
1)=((cid:15)−1)B =(cid:30)B .
i i+1
Lemma4. Let R be the set of requests served by schedule S computed at time B,
i i i
i=1;2;:::; and let R∗ be the set of requests in the optimal o?ine solution which are
i
284 S.O. Krumke et al./Theoretical Computer Science 295 (2003) 279–294
completed in the time interval (B ;B]. Then
i−1 i
(cid:1)k (cid:1)k
w(R)¿ w(R∗) for k =1;2;::: :
i i
i=1 i=1
Proof. We Arst argue that for any k¿1 we can obtain from the optimal o7ine solution
S∗ a schedule S which starts in the origin, has length at most B , ends with an empty
k
server at a point with distance at most B from the origin, and which serves all requests
(cid:3) k
in k R∗.
i=1 i
Consider the optimal o7ine transportation schedule S∗. Start at the origin and follow
S∗ for the Arst B time units with the modiAcation that, if a request is picked up in
k
S∗ before time B but not delivered before time B , omit this action. Observe that
k k
this implies that the server is empty at the end of this schedule. We thereby obtain a
(cid:3)
schedule S of length at most B which serves all requests in k R∗. Since the server
k i=1 i
moves at unit speed, it follows that S ends at a point with distance at most B from
k
the origin.
We now consider phase k and show that by the end of phase k, at least requests of
(cid:1)
weight k w(R∗) have been scheduled by INTERVAL . If k=1, the transportation
i=1 i (cid:15)
schedule S obtained as outlined above satisAes already all conditions (i)–(iii) required
by INTERVAL . If k¿2, then condition (i) might be violated, since S starts in the
(cid:15)
origin. However, we can obtain a new schedule S(cid:3) from S starting at the endpoint
x of the schedule from the previous phase, moving the empty server from x
k−1 k−1
to the origin and then following S. Since d(x ;o)6B , the new schedule S(cid:3) has
k−1 k−1
length at most B +l(S)6B +B which means that it satisAes all the properties
k−1 k−1 k
(i)–(iii) required by INTERVAL .
(cid:15) (cid:3)
Recall that schedule S and thus also S(cid:3) serves all requests in k R∗. Possibly,
(cid:3) i=1 i
some of the requests from k R∗ have already been served by INTERVAL in previous
i=1 i (cid:15)
phases. As omitting requests can never increase the length of a transportation schedule,
in phase k, INTERVAL can schedule at least all requests from
(cid:15)
(cid:4) (cid:5)(cid:6)(cid:4) (cid:5)
(cid:3)k k(cid:3)−1
R∗ R :
i i
i=1 i=1
Consequently, the weight of all requests served in schedules S ;:::;S of INTERVAL
(cid:3) (cid:1) 1 k (cid:15)
is at least w( k R∗)= k w(R∗) as claimed.
i=1 k i=1 k
The previous lemma gives us the following bound on the number of phases that
INTERVAL uses to process a given input sequence (cid:20).
(cid:15)
Corollary5. Suppose that the optimum o?ine schedule is completed in the interval
(B ;B ] for some p¿1. Then the number of phases of the Algorithm INTERVAL
p−1 p (cid:15)
is at most p. Schedule S computed at time B by INTERVAL is completed no later
p p (cid:15)
than time (cid:30)B .
p+1
Proof. By Lemma 4 the weight of all requests scheduled in the Arst p phases equals
the total weight of all requests. Hence all requests must be scheduled within the Arst
S.O. Krumke et al./Theoretical Computer Science 295 (2003) 279–294 285
p phases. Since, by construction of INTERVAL , schedule S computed in phase p
(cid:15) p
completes by time (cid:30)B , the claim follows.
p+1
To prove competitiveness of INTERVAL we need an elementary lemma which can
(cid:15)
be proven by induction.
Lemma6. Let a;b ∈R+ for i=1;:::;p, for which
(cid:1) (cid:1)i i 0
(i) p a = p b;
(cid:1)i=1 i (cid:1)i=1 i
(ii) p(cid:1) a¿ p(cid:1) b for all 16p(cid:3)6p.
i=1 i i=1 i
(cid:1) (cid:1)
Then p "a6 p "b for any nondecreasing sequence 06" 6···6" .
i=1 i i i=1 i i 1 p
Theorem7.√AlgorithmINTERV√AL(cid:15) is(cid:15)((cid:15)+1)=((cid:15)−1)-competitiveforthe√L-OLDARPfor
any(cid:15)∈[1+ 2;3].For(cid:15)=1+ 2,thisyieldsacompetitiveratioof(1+ 2)2¡5:8285.
Proof. Let (cid:20)=r ;:::;r be any sequence of requests. By deAnition of INTERVAL ,
1 m (cid:15)
each request served in schedule S completes no later than time (cid:30)B =(cid:15) + 1=
i i+1
(cid:15)((cid:15)−1)B . Summing over all phases 1;:::;p yields
i+1
(cid:15)+1 (cid:1)p (cid:15)+1 (cid:1)p
INTERVAL ((cid:20))6 B w(R)=(cid:15)· B w(R): (1)
(cid:15) (cid:15)((cid:15)−1) i+1 i (cid:15)−1 i−1 i
i=1 i=1
(cid:1) (cid:1)
From Lemma 4 we know that k w(R)¿ k w(R∗) for k=1;2;:::; and from
(cid:1) i=1 i (cid:1)i=1 i
Corollary 5 we know that p w(R) = p w(R∗). Therefore, application of
i=1 i i=1 i
Lemma 6 to the sequences a :=w(R) and b :=w(R∗) with the weighing sequence
i i i i
" :=B ; i=1;:::;p, gives
i i−1
(cid:15)+1 (cid:1)p (cid:15)+1 (cid:1)p
(cid:15)· B w(R)6(cid:15)· B w(R∗): (2)
(cid:15)−1 i−1 i (cid:15)−1 i−1 i
i=1 i=1
Denote by C∗ the completion time of request r in the optimal o7ine solution OPT((cid:20)).
j j
For each request r denote by (B ;B ] the interval that contains C∗. Then
j #j #j+1 j
(cid:15)+1 (cid:1)p (cid:15)+1 (cid:1)m (cid:15)+1 (cid:1)m
(cid:15)· B w(R∗)=(cid:15)· B w 6(cid:15)· w C∗: (3)
(cid:15)−1 i−1 i (cid:15)−1 #j j (cid:15)−1 j j
i=1 j=1 j=1
Eqs. (1)–(3) together yield
(cid:15)+1
INTERVAL ((cid:20))6(cid:15)· ·OPT((cid:20)):
(cid:15) (cid:15)−1
√
The value (cid:15)=1+ 2 minimizes the function f((cid:15)):=(cid:15)((cid:15)+1)=((cid:15)−1) in the interval
√ √
[1+ 2;3], yielding (1+ 2)2¡5:8285 as competitive ratio.
√ √
Corollary8. For (cid:15)=1+ 2, algorithm INTERVAL is (1+ 2)2-competitive for the
(cid:15)
OLTRP.
286 S.O. Krumke et al./Theoretical Computer Science 295 (2003) 279–294
4. Animprovedrandomizedalgorithm
In this section we use techniques developed in [6] to devise a randomized algorithm
RANDINTERVAL . At the beginning, RANDINTERVAL chooses a random number
(cid:15) (cid:15)
$∈(0;1] according to the uniform distribution. From this moment on, the algorithm
is completely deterministic, working in the same way as the deterministic algorithm
INTERVAL presented in the previous .section For i¿0 deAne B(cid:3):=(cid:15)i−1+$L, where
(cid:15) i
again L is the earliest time that a request could be completed by OPT. As stated before
in the case of INTERVAL we can assume that L¿0.
(cid:15)
The diPerence between RANDINTERVAL and INTERVAL is that all phases are
(cid:15) (cid:15)
deAned using B(cid:3):=(cid:15)i−1+$L instead of B :=(cid:15)i−1L, i¿1. To justify the accuracy of
i i
RANDINTERVAL , note that in the correctness proof for the deterministic version, we
(cid:15)
only made use of the fact that B =(cid:15)B for i¿0. This also holds for the B(cid:3). Hence,
i+1√ i i
anychoiceoftheparameter(cid:15)∈[1+ 2;3]yieldsacorrectversionofRANDINTERVAL .
(cid:15)
We will show later that the optimal choice for RANDINTERVAL is (cid:15)=3.
(cid:15)
The proof of Lemma 4 also holds also with B replaced by B(cid:3) for each i¿0. We
i i
thus obtain the following lemma.
Lemma9. Let R be the set of requests scheduled in phase i¿1 of Algorithm RAND
i
INTERVAL and denote by R∗ the set of requests that are completed by OPT in the
(cid:15) i
time interval (B(cid:3) ;B(cid:3)]. Then
i−1 i
(cid:1)k (cid:1)k
w(R)¿ w(R∗) for k =1;2;::: :
i i
i=1 i=1
We can now use the proof of Theorem 7 with Lemma 4 replaced by Lemma 9.
This enables us to conclude that for a sequence (cid:20)=r ;:::;r of requests the expected
1 m
objective function value of RANDINTERVAL satisAes
(cid:15)
(cid:7) (cid:8)
(cid:15)+1 (cid:1)m
E[RANDINTERVAL [((cid:20))]6E (cid:15)· B(cid:3) w
(cid:15) (cid:15)−1 #j j
j=1
(cid:15)+1 (cid:1)m
= (cid:15)· w E[B(cid:3) ]; (4)
(cid:15)−1 j #j
j=1
where (B(cid:3) ;B(cid:3) ] is the interval containing the completion time C∗ of request r in
#j #j+1 j j
the optimal solution OPT((cid:20)).
To prove a bound on the performance of RANDINTERVAL we compute E[B(cid:3) ].
(cid:15) #j
Notice that B(cid:3) is the largest value (cid:15)k+$L; k∈Z, which is strictly smaller than C∗.
#j j
Lemma10. Let z¿L and $∈(0;1] be a random variable uniformly distributed
on (0;1]. DeAne B by B:= max{(cid:15)k+$L:(cid:15)k+$L¡z and k∈Z}. Then, E[B]((cid:15) − 1)=
((cid:15) ln (cid:15))·z.
S.O. Krumke et al./Theoretical Computer Science 295 (2003) 279–294 287
Proof. Suppose that (cid:15)kL6z¡(cid:15)k+1L for some k¿0. Observe that
(cid:2)
(cid:15)k−1+$L if $¿log z ;
B= (cid:15) (cid:15)kL
(cid:15)k+$L otherwise:
Hence
(cid:9) (cid:9)
log (z=(cid:15)kL) 1
E[B]= (cid:15) (cid:15)k+$Ld$+ (cid:15)k−1+$Ld$
0 log (z=(cid:15)kL)
(cid:15)
(cid:10) (cid:11) (cid:10) (cid:11)
1 log(cid:15)(z=(cid:15)kL) 1 1 ((cid:15)−1)
=(cid:15)kL (cid:15)$ +(cid:15)k−1L (cid:15)$ = ·z:
ln (cid:15) ln (cid:15) ((cid:15) ln (cid:15))
0 log (z=(cid:15)kL)
(cid:15)
This completes the proof.
From Lemma 10 we can conclude that E[B(cid:3) ] = ((cid:15)−1)=((cid:15) ln (cid:15))·C∗. Using this
#j j
result in inequality (4), we obtain
(cid:15)+1 (cid:15)−1 (cid:1)m (cid:15)+1
E[RANDINTERVAL ((cid:20))]6(cid:15)· · w C∗ = ·OPT((cid:20)):
(cid:15) (cid:15)−1 (cid:15) ln (cid:15) j j ln (cid:15)
j=1
√
Minimizing the function g((cid:15)):=((cid:15)+1)=(ln (cid:15)) over the interval [1+ 2;3], we conclude
that the best choice is (cid:15)=3.
Theorem11. Algorithm RANDINTERVAL(cid:15) is ((cid:15) + 1)√=(ln (cid:15))-competitive for the
L-OLDARP against an oblivious adversary, where (cid:15)∈[1+ 2;3]. Choosing (cid:15)=3 yields
a competitive ratio of 4=ln 3¡3:6410 for RANDINTERVAL against an oblivious
(cid:15)
adversary.
Corollary12. For (cid:15)=3, algorithm RANDINTERVAL is (4=ln 3)-competitive for the
(cid:15)
OLTRP against an oblivious adversary.
5. Lowerbounds
In this section we show lower bounds for the competitive ratio of any randomized
algorithm against an oblivious adversary for the problem L-OLDARP. The basic method
for deriving such a lower bound is Yao’s principle (see also [5,11,12]). Let X be
a probability distribution over input sequences ’={(cid:20) : x ∈ X}. We denote the ex-
x
pected cost of the deterministic algorithm ALG according to the distribution X on ’
by E [ALG[((cid:20) )]. Yao’s principle can now be stated as follows.
X x
Theorem13 (Yao’s principle). Let {ALG : y ∈ Y} denote the set of deterministic
y
online algorithms for an online minimization problem. If X is a distribution over
input sequences {(cid:20) : x∈X} such that
x
inf E [ALG ((cid:20) )]¿cQE [OPT((cid:20) )]
X y x X x
y∈Y
288 S.O. Krumke et al./Theoretical Computer Science 295 (2003) 279–294
Fig. 1. The star with three rays used in the lower bound construction.
for some real number cQ¿1, then cQ is a lower bound on the competitive ratio of any
randomized algorithm against an oblivious adversary.
We use a method explained in [12] to compute a suitable distribution once our ground
set of request sequences has been Axed.
5.1. A General lower bound for the L-OLDARP
We provide a general lower bound where the metric space is a star, which consists
of three rays of length 2 each. The center of the star is the origin, denoted by o. The
server capacity equals one.
Let the rays of the star be named A;B and C. Let x be the point on A with distance
A
x to o, 0¡x62. Points on B and C will be denoted in the same manner. Fig. 1 contains
an illustration of the star-shaped metric space.
Let k∈N, k¿2 be arbitrary. At time 0 there is one request from o to 1 with
A
weight 1, denoted by r =(0;o;1 ;1). With probability ) there are no further requests.
1 A
With probability 1 − ) there are k requests at time 2x, where x∈(0;1] is chosen
(cid:12)
according to a density function p. The density p satisAes )+ 1p(x)dx=1 and will
0
be determined suitably in the sequel. Each of the k requests released at time 2x has
weight 1. With probability 1(1−)) these requests will be given from 2x to 2x and
2 B B
with probability 1(1−)) from 2x to 2x . This yields a probability distribution X
2 C C
over the set ’={(cid:20) : x∈[0;1]; R ∈ {B;C}} of request sequences where (cid:20) =r
x;R 0;R 1
for R∈{B;C}, and
(cid:20) =r ;(2x;2x ;2x ;1);:::;(2x;2x ;2x ;1) for 0¡x 61 and R∈{B;C}:
x;R 1 (cid:13) R R (cid:14)(cid:15) R R (cid:16)
k times
Fig. 2 illustrates the construction.
We Arst calculate the expected cost E [OPT((cid:20) )] of an optimal o7ine algorithm
X x;R
with respect to the distribution X on ’. With probability ) there is only request r to
1
be served, and in this case the o7ine cost is 1. Now consider the situation where there
are k additional requests at position 2x or 2x . Clearly, the optimal o7ine cost is
B C