Online Load Balancing on Related Machines

Transcription

1 Onine Load Baancing on Reated Machines ABSTRACT Sungjin Im University of Caifornia at Merced Merced, CA, USA Debmaya Panigrahi Duke University Durham, NC, USA We give a constant-competitive agorithm for the probem of assigning jobs onine to machines with non-uniform speed caed reated machines) so as to optimize any q -norm of the machine oads. Previousy, such a resut was ony known for the specia case of the makespan, or. norm. We aso extend these resuts to obtain tight bounds for the probem of vector scheduing on reated machines, where no resuts were previousy known even for the makespan norm. To obtain our resuts, we empoy a convex reaxation of the q -norm objective and use a continuous greedy agorithm to sove this convex program onine. To round the fractiona soution, we then use a nove restructuring of the instance that we ca machine smoothing. This is a generic too that reduces a probem on reated machines to a set of probem instances on identica machines, and we hope it wi be usefu in other settings with non-uniform machine speeds as we. CCS CONCEPTS Theory of computation Onine agorithms; KEYWORDS onine agorithms, scheduing, oad baancing ACM Reference Format: Sungjin Im, Nathanie Ke, Debmaya Panigrahi, and Maryam Shadoo Onine Load Baancing on Reated Machines. In Proceedings of 50th Annua ACM SIGACT Symposium on the Theory of Computing STOC 8). ACM, New York, NY, USA, 4 pages. INTRODUCTION In this paper, we consider the onine oad baancing probem of assigning jobs to machines with non-uniform speeds caed reated machines). The onine oad baancing probem has a ong history for optimizing the makespan norm maximum machine oad or Permission to make digita or hard copies of a or part of this work for persona or cassroom use is granted without fee provided that copies are not made or distributed for profit or commercia advantage and that copies bear this notice and the fu citation on the first page. Copyrights for components of this work owned by others than the authors) must be honored. Abstracting with credit is permitted. To copy otherwise, or repubish, to post on servers or to redistribute to ists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. 208 Copyright hed by the owner/authors). Pubication rights icensed to the Association for Computing Machinery. ACM ISBN /8/06... $ Nathanie Ke Duke University Durham, NC, USA ke@cs.duke.edu Maryam Shadoo University of Caifornia at Merced Merced, CA, USA mshadoo@ucmerced.edu norm of machine oads) on machines of uniform speed identica machines) [,, 2, 20, 2, 23, 26, 30]. This probem has aso been studied for the more genera setting of unreated machines, where the processing time of a job on a machine is arbitrary, and a tight competitive ratio of Oogm) is known for the probem where m is the number of machines) [3, 0, 4]. In many situations, however, other q -norms of machine oads are more reevant than the makespan norm: e.g., the 2-norm is suitabe for disk storage [6, 8], whereas q between 2 and 3 is used for modeing energy consumption [2, 33, 37]. This has ed to the design of oad baancing agorithms for optimizing arbitrary q -norms or simpy q-norms) of machine oads, and tight competitive ratios of O) for identica machines [4] and Oq) for unreated machines [5, 4, 5] are known. However, for reated machines, the ony previous resut was a constant-competitive agorithm for the makespan norm, using the so-caed sowest fit agorithm [3, 3]. No resuts were known for any other q-norm prior to our work. Our first resut is to show the first constant-competitive agorithms for optimizing arbitrary q-norms on reated machines. Recent iterature has further expanded the scope of the job scheduing probem to vector jobs that have mutipe dimensions, the resuting probem being caed vector scheduing [8, 7, 29, 32]. This probem arises in scheduing on data centers where jobs with mutipe resource requirements have to be aocated to machine custers to make efficient use of imited resources such as CPU, memory, network bandwidth, and storage [9, 22, 28, 29, 3, 34]. Recenty, Im et a. [27] showed that for vector scheduing with the makespan norm, competitive ratios of Oog d/og og d) and Oog d + ogm) are tight for identica and unreated machines respectivey, where d is the number of dimensions and m is the number of machines. They aso extended these resuts to arbitrary q -norms. In many data center appications, the situation is between these two extremes of identica and unreated machines, and resembes the reated machines scenario. In other words, machines have non-uniform speeds and the oad created by a vector job on any dimension of a machine is inversey proportiona to the machine speed. But, no resuts were known prior to our work for vector scheduing on reated machines, either for the makespan norm or for arbitrary q norms. There are two natura variants of vector scheduing depending on whether machine speed is identica across dimensions we ca this homogeneous speeds) or different across dimensions we ca this heterogeneous speeds). We show a sharp contrast between these two settings: we give agorithms minimizing makespan and q norms in the homogeneous setting that match those for identica machines, 30

2 Sungjin Im, Nathanie Ke, Debmaya Panigrahi, and Maryam Shadoo and show ower bounds in the heterogeneous setting that match those for unreated machines. The ower bounds for vector scheduing on reated machines with heterogeneous speeds are deferred to the fu version of the paper because of space constraints. Preiminaries and Resuts: First, we set up some standard notation. In onine scheduing, a set of n jobs arrive onine and each job must be irrevocaby assigned to one of m machines immediatey on arriva. Each job j has a non-negative size p j. In vector scheduing, p j is a vector of d dimensions, p j = p j ),p j 2),...,p j d). Each machine i has a non-negative speed s i that is given offine. In vector scheduing, s i is a vector s i ), s i 2),..., s i d), where s i ) = s i 2) =... = s i d) denoted s i ) in the homogeneous setting. When job j is assigned to machine i, it produces a oad of p j /s i. In vector scheduing, the oad is p j k)/s i k) = p ij k) in dimension k. The oad produced by a set of jobs is the sum of their individua oads. The oad vector is denoted Λ = Λ, Λ 2,..., Λ m, where Λ i is the tota oad on machine i. For vector scheduing, every dimension k has its own oad vector, denoted Λk) = Λ k), Λ 2 k),..., Λ m k), where Λ i k) is the tota oad on machine i in dimension k. In vector scheduing, the makespan objective is given by: d max Λk) = d m max max Λ i k). k= k= i= For the probem of minimizing makespan in vector scheduing, we show the foowing resut. Theorem. For onine vector scheduing on reated machines for minimizing makespan: ) Section 3) For homogeneous speeds, we give a deterministic Oog d/og og d)-competitive agorithm. This is asymptoticay tight since it matches a known ower bound for identica machines [27]. 2) Deferred to fu version) For heterogeneous speeds, we give a ower bound of Ωogd + ogm) on the competitive ratio. This is asymptoticay tight since it matches a known upper bound for unreated machines [8, 27, 32]. Now we state our resuts for optimizing arbitrary q -norms. First, we consider the scaar scheduing probem. The q -norm objective is given by: m ) Λ q = Λ i ) q /q i= We obtain the foowing resut. Theorem 2. For onine scaar) scheduing on reated machine for minimizing q -norms Section 4), we give a deterministic agorithm with a constant competitive ratio. This is asymptoticay tight because onine scheduing has a constant ower bound even for identica machines [, 2, 20, 23, 26]. Next, we consider optimizing q -norms in vector scheduing. Our objective is given by: d max Λk) q = d m ) max Λ i k)) q /q k= k= i= We obtain the foowing resut. Theorem 3. For onine vector scheduing on reated machines for minimizing q -norms: ) Section 5) For homogeneous speeds, we give a deterministic agorithm with a competitive ratio of Oog c d) for some constant c. This is tight up to the vaue of the constant c, by a known ower bound for identica machines [27]. 2) Deferred to fu version) For heterogeneous speeds, we give a ower bound of Ωogd + q) on the competitive ratio. This is asymptoticay tight since it matches a known upper bound for unreated machines [27]. Note that Theorem 2 foows as a coroary of Theorem 3. However, our vector scheduing agorithm uses our scaar scheduing agorithm as a subroutine; consequenty, the proof of Theorem 3 reies on an independent proof of Theorem 2. Therefore, we present our scaar scheduing resuts before presenting our vector scheduing resuts for arbitrary q-norms. We note that homogeneous reated machines do not admit agorithms that optimize a norms simutaneousy unike identica machines. Thus we can ony have agorithms that optimize each specific q-norm. Techniques: A key too in our agorithms is what we ca machine smoothing. Imagine grouping together machines with simiar speeds. Then, one can empoy a two-stage agorithm that assigns each job to a machine group, and then use an identica machines agorithm within each machine group. Unfortunatey, this simpe grouping does not make the assignment probem easier as the number of machines in each group might be competey arbitrary. It turns out that the assignment of jobs to groups is faciitated if we can ensure that the cumuative processing power in a group exponentiay increases as we move to sower groups. The cumuative processing power for the makespan objective is simpy the sum of speeds of machines in the group; for other q -norms, this definition is suitaby generaized.) So, now we have two objectives: group machines with simiar speeds, but aso ensure exponentiay increasing processing powers of the groups in decreasing speed order. We show that we can transform any given instance into an instance that satisfies these goas simutaneousy, which we ca a smoothed instance, whie ony sacrificing a constant factor in the competitive ratio of the agorithm. It turns out that the machine smoothing technique is essentiay sufficient for soving the makespan minimization probem in vector scheduing, since the assignment of jobs to machine groups in a smoothed instance can be done by simuating the sowest-fit strategy used for scaar scheduing. However, for other q -norms, even for scaar scheduing, we need to work harder in designing the agorithm to assign jobs to machine groups in a smoothed instance. In particuar, we use a two-step approach. First, we use a continuous greedy agorithm on a suitaby chosen fractiona reaxation of the norm to produce a competitive fractiona soution. Next, we use an onine rounding agorithm to produce an integer assignment from the fractiona soution. In the case of vector scheduing for arbitrary q -norms, an additiona compication is caused by the fact that the continuous greedy agorithm can produce unbaanced oads on different dimensions since it foows the gradient for a singe objective, thereby eading to a arge competitive ratio. To avoid this difficuty, we use the assignment produced by the continuous greedy agorithm ony as an advice on the approximate speed of the machine group that a fractiona job shoud be assigned to. We then use a different agorithm to make the actua assignment of 3

3 Onine Load Baancing on Reated Machines the fractiona job to a machine group simiar to the advice, but not necessariy to the exact same group. Reated Work: In the interest of space, we wi ony state a sma subset of reated resuts and refer the reader to more detaied surveys [6, 35, 36] for other resuts. The onine job scheduing probem was introduced by Graham [24], who showed that ist scheduing has a competitive ratio of 2 /m) for the makespan objective on identica machines. Currenty, the best known upper bound is.920 [,, 2, 30], whie the best owerbound is.880 [, 2, 20, 23, 26]. For the reated machines setting, the sowest-fit agorithm is 2- competitive [3] given the optima makespan a priori), but for unreated machines, the optima competitive ratio is Θogm) [3, 0]. This probem was generaized to arbitrary q-norms by [4] for identica machines and by [5, 4] for unreated machines. The ony previous resut for reated machines was the competitive ratio of 2 achieved by the sowest-fit agorithm for the makespan norm [3]. For vector scheduing for identica machines, a PTAS is known for any fixed d [7]. For unreated machines, the best known approximation factor is Oogd/og ogd) [25] and a constant ower bound is known [7]. In the onine setting, Oog d)-competitive agorithms were given for identica machines [7, 32]. A recent work [27] improved these resuts by giving tight bounds of Oogd/og ogd) for identica machines and Oogd + ogm) for unreated machines. They aso extended these resuts to arbitrary q-norms, giving tight og d bounds of O og og d ) q q ) and Oogd + q) for identica and unreated machines. 2 MACHINE SMOOTHING One of the main ideas that we use throughout our agorithms is that of machine smoothing. There are two properties that we wish to derive from machine smoothing: that machines in a singe group have the same speed and that a sower group has processing power at east as much the sum over a its faster groups. To ensure both properties simutaneousy, simpy grouping the given machines is not sufficient instead, we need to modify machine speeds in the given instance and create new machines. The goa of this section is to show that such modification is vaid, i.e., there is a mapping between assignments made on the origina instance and the smoothed instance that ony changes the objective by a constant factor. We use a parameter γ := q/q ) for defining smoothed instances for the q-norm. For the makespan norm, we define γ :=. Using this parameter, we define the processing power of a group, denoted SG) for group G) SG) := i G s γ i ; note that for the makespan norm, the processing power is simpy the sum of speeds of the machines in the group, i.e., SG) := i G s i. Definition. An instance is smoothed if the machines are partitioned into groups G 0,G,... such that: Property : A machines in a group have equa speed. Property 2: SG ) SG 0 ) + SG ) + + SG ) for a groups G, where SG) := i G s γ i. Property 3: For any, the vaue of s γ i for machines i G is at east twice that of machines i G +. For the makespan norm, this means that machines in G are at east twice as fast as those in G +. Machine Smoothing for Makespan Norm. First, we give an agorithm that converts an arbitrary set of machines M to a smoothed set of machines M for the makespan norm see Fig. ). By scaing and rounding a machine speeds to powers of 2, we assume that a machine speeds are 2 t for some integer t 0 with the fastest machine having a speed of. Ceary, this changes the makespan by at most a factor of 2. First, we order the machines in M in non-increasing order of speed, breaking ties arbitrariy. Let us overoad notation and ca this sorted order M as we. The first group G 0 comprises ony the first machine in M, i.e., a machine with speed. To define a group G for, we discard the prefix of M comprising machines in G 0 G... G and from the remaining machines in M, choose a minima prefix such that their tota processing power sum of speeds) is exacty 2. Note that if the tota processing power of the remaining machines is at east 2, this is aways possibe since machine speeds are non-positive) powers of 2. If the tota processing power of the remaining machines is ess than 2, we simpy discard these remaining machines ca these discarded machines G L+ where L is the index of the ast group of machines). Thus, for each group G with 0 L, we have SG ) = 2. Let s min G ) denote the owest speed among a machines in G. To create our smoothed set of machines M, for each we repace G with a new set of machines G whose speeds are a equa to s min G ). The number of machines in G is chosen such that the processing power is preserved, i.e., SG ) = SG ). Note that G may have more machines than G. Finay, we combine a groups in G 0,G,... that have the same speed to form a new set of groups G 0,G,..., which form a smoothed instance. The next emma caims that any assignment in the origina instance can be mapped to an assignment in the smoothed instance, and vice-versa, such that the makespan ony changes by a constant. Lemma 4. For any set M of machines with homogeneous speeds in the case of vector scheduing), et M be the set of smoothed machines defined by the smoothing procedure for the makespan norm. Then there exist two mappings д : M M and д : M M such that if we are given a schedue on machines in M resp., M ), then scheduing jobs assigned to machine i on machine дi) resp., д i)) produces a schedue with makespan at most a constant factor more than that of the origina schedue on M resp., M ). Proof. To define our mappings, we wi use both sets of groups G 0,G,... on machines M and G 0,G,... on machines M. Note that {G } and {G } are groups defined on the same set of machines M ; so it is sufficient to output an assignment on G 0,G,....) First, we define the mapping д) see the bottom of Fig. for an iustration). We partition the machines in group G + for 0 L into sets {H i } i G, where a machines in H i are mapped to a singe machine i G. Since no machine in G + has a speed greater than s i and SG + ) 2SG ), we can ensure that SH i ) 3s i, e.g., by using a greedy partitioning agorithm. Thus, this mapping increases the makespan of a schedue by at most a factor of 3. Additionay, we map the unique machine in G 0 to itsef, eading to an overa increase in the makespan by at most a factor of 4. Now, we define the mapping д ) from machines M to M see the top of Fig. for an iustration). We partition the machines in 32

4 Sungjin Im, Nathanie Ke, Debmaya Panigrahi, and Maryam Shadoo machines tota) Figure : Iustration of machine smoothing. Here our origina instance of machines M consists of 24 machines, machine of speed, 3 machines of speed /2, 6 machines of speed /4, and 4 machines of speed /8. These machines are then grouped into G 0, G, and G 2 as shown. Our smoothed instance of machines M consists of groups G 0 machine of speed ), G 8 machines of speed /4) and G 2 32 machines of speed /8). The eft and right iustrations demonstrate mappings д ) and д), respectivey. group G for 0 L into sets {H i } i G, where a machines in H i are mapped to a singe machine i G. Since every machine in G has at east the speed of machines in G, SG ) = SG ), and a speeds are powers of 2, we can ensure that SH i ) = s i, e.g., by again using a greedy partitioning agorithm. Thus, this mapping does not increase the makespan of a schedue. Machine Smoothing for q-norms. For an arbitrary q-norm, we need to modify our machine smoothing procedure because the definition of processing power of a group, and consequenty that of a smoothed instance, is different from that for makespan. Thus in this section, we give an agorithm for converting machines M to a set of smoothed machines M when optimizing for arbitrary q-norms where q 2 since the case of q = is trivia). Reca that in the definition of smoothed instance for q-norms, we define tota processing power of a group SG ) = i G s γ i, where γ = q/q ). Define Γ := 2 /γ. Again by scaing and rounding, we can assume that the fastest machine has speed exacty and that a machines in M have speeds that are non-positive) powers Γ, i.e.,, Γ, Γ 2,.... This changes the q-norm by at most a factor of γ which is at most 2 since q 2. We again we order the machines in M in non-increasing order of speed, breaking ties arbitrariy. Let us overoad notation and ca this sorted order M as we. The first group G 0 comprises ony the first machine in M, i.e., a machine with speed. To define a group G for, we discard the prefix of M comprising machines in G 0 G... G and from the remaining machines in M, choose a minima prefix such that their tota processing power sum of s γ i ) is exacty 2. Note that if the tota processing power of the remaining machines is at east 2, this is aways possibe since machine speeds are non-positive) powers of Γ and thus each s γ i is a non-positive power of 2). If the tota processing power of the remaining machines is ess than 2, we simpy discard these remaining machines ca these discarded machines G L+ where L is the index of the ast group of machines). Thus, for each group G with 0 L, we have SG ) = 2. Let s min G ) denote the owest speed among a machines in G. To create our smoothed set of machines M, for each we repace G with a new set of machines G whose speeds are a equa to s min G ). The number of machines in G is chosen such that the processing power is preserved, i.e., SG ) = SG ). Note that G may have more machines than G. Finay, we combine a groups in G 0,G,... that have the same speed to form a new set of groups G 0,G,..., which form a smoothed instance. We now generaize Lemma 4 to arbitrary q-norms. Lemma 5. For any set M of machines with homogeneous speeds in the case of vector scheduing), et M be the set of smoothed machines defined by the smoothing procedure for the q-norm. Then there exist two mappings д : M M and д : M M such that if we are given a schedue on machines in M resp., M ), then scheduing jobs assigned to machine i on machine дi) resp., д i)) produces a schedue with q-norm at most a constant factor more than that of the origina schedue on M resp., M ). Proof. To define our mappings, we wi use both sets of groups G 0,G,... on machines M and G 0,G,... on machines M. Note that {G } and {G } are groups defined on the same set of machines M ; so it is sufficient to output an assignment on G 0,G,....) First, we define the mapping д). We partition the machines in group G + for 0 L into sets {H i } i G, where a machines in H i are mapped to a singe machine i G. Since no machine in G + has a speed greater than s i and SG + ) 2SG ), we can ensure that SH i ) 3s γ i, e.g., by using a greedy partitioning agorithm. We now want to show that the q q -norm of the oads on 33

5 Onine Load Baancing on Reated Machines machines in H i ony increases by a factor of at most 3 q when the oads are reocated to machine i. Let u t denote the tota voume machine t H i i.e, sum of p j on machine t). Then, we have, t H i ) q ut = SH i ) s t t H i SH i ) t H i q ) sγ i = SH i ) q s γ t SH i ) u t s γ t ) q s γ t SH i ) ut s γ t t H i u t s i q ) q 3 q ) t H i u q t, s i where the first inequaity foows from the convexity of x q ; the second inequaity foows since SH i )/s γ i 3. Finay, note that the first group G 0 processes jobs reocated not ony from G but aso from G 0. Hence the q q -norm increases by a factor of at most 6 q, meaning that the optima q-norm increases by at most a constant factor. Now, we define the mapping д ) from machines M to M. We partition the machines in group G for 0 L into sets {H i } i G, where a machines in H i are mapped to a singe machine i G. Since every machine in G has at east the speed of machines in G, SG ) = SG ), and a sγ i are powers of 2, we can ensure that SH i ) = s γ i, e.g., by again using a greedy partitioning agorithm. Observe that H i = s γ i /s )γ, where s denotes the uniform speed of machines in G. We again et u t be the voume of jobs assigned on machine t H i. Thus we we have ) q ) q u t t H s H i i u t s t H i H i ) ) q q s i = H i s ) t H i u q ) t t H = i u q t, s i s i where the first inequaity again foows from the convexity of x q. The ast equaity foows since H i = s γ i /s )γ. Thus, we have shown that the q q -norm does not increase when jobs are moved from M to M foowing the mapping д ). 3 VECTOR SCHEDULING: MINIMIZING MAKESPAN ) og d Using machine smoothing, we now give ano og og d -competitive agorithm for makespan minimization on homogeneous) reated machines the first part of Theorem ). By Lemma 4, we assume throughout that we have a smoothed instance, and ony describe the agorithm to assign jobs to machine groups. Reca that each group is a set of identica machines, so we use the foowing theorem from [27] to assign jobs to individua machines in a group. Theorem 6 [27]). For vector jobs schedued on m identica machines, there is a deterministic agorithm that yieds a schedue with makespan norm of Oogd/ogd ogd) maxp j k) + max p j k)/m. k, j k j This theorem impies that it is sufficient to find an assignment of jobs to machine groups that bounds the foowing function h against the optima makespan norm opt: p j k) h := max max + max k, j J s k j J p j k) SG ) where J denotes the set of jobs assigned to group G. Here, s is the speed of any machine in G, and SG ) is the sum of speeds of a machines in G. Agorithm. Since a machines in a group have the same speed, we use s to denote the speed of any machine in group G. We assume that we know the vaue of the optima makespan opt within a constant factor by using a standard guess-and-doube technique. p We say that a group G is permissibe for job j if max j k) k s opt, i.e., assigning the job aone does not vioate the optima bound. The agorithm assigns a job j to the group G with the argest index i.e., sowest speed) among a permissibe groups for j. Anaysis. Let J denote the jobs assigned to group G. Since group p G is permissibe for any job in J, it foows that max j k) k, j J s opt. Furthermore, an optima soution can schedue jobs in J ony on machines in G for, since G is the sowest permissibe group for jobs in J. Thus, for any dimension k, p j k) SG ) opt j J =0 ) = SG ) + SG ) opt 2 SG ) opt =0 where the ast inequaity uses Property 2 of smoothed instances. Thus, h O) opt as desired. 4 SCALAR SCHEDULING: MINIMIZING q-norms In this section, we describe our agorithm for optimizing arbitrary q- norms for the scaar scheduing probem. For convenience, we wi work with the q q -norm i.e., the q-norm raised to the qth power) and give an O) q -competitive agorithm, which is equivaent to an O)-competitive agorithm for the q-norm. As in the previous section, we wi assume throughout that we have a smoothed instance; by Lemma 5, this ony oses a constant factor in the competitive ratio. Reca that the speed of a machine in group G is denoted s, and p j := p j /s denotes the processing time of job j on any machine in G. Now, suppose a set of jobs J is assigned to group G. Then, two natura ower bounds on the q q -norm of any assignment of these jobs to the individua machines ) q j J in G are given by: G p j, which is the q G q -norm of the optima fractiona assignment of these jobs, and j J p q j, which is the q q -norm of the optima assignment if G =. The next theorem, which foows from [9] and [24], states that the sum of these ower bounds can be attained in an onine assignment on any set of identica machines. We provide a proof for sake of competeness. ), 34

6 Sungjin Im, Nathanie Ke, Debmaya Panigrahi, and Maryam Shadoo Theorem 7 [9], [24]). For scaar jobs schedued on m identica machines, there is a deterministic [ agorithm ) that yieds a schedue with q q -norm of O) q j p j q m m + j p q ] j. Proof. Our agorithm wi be to simpy schedue jobs greediy, i.e., for each job j that arrives, we schedue j on machine that currenty has the smaest oad. Let Λ be the oad vector produced by the agorithm, and et Λ i denote the oad of ith machines. Let p i be the oad of the ast job that was assigned to machine i,and et Λ i be the oad on machine without this ast job i.e., Λ i = Λ i p i ). Observe that Λ q q = Λ ) q i + p i 2 maxλ i, p i ) ) q i 2 q i i Λ i ) q q + p ) i 2 q m ) j p q j + p q m j j, as desired. The ast inequaity foows since the agorithm assigns ) greediy, and therefore Λ i j p j /m; we aso ceary have that i p i q j p q j. This theorem impies that it is sufficient for us to find an assignment of jobs to machine groups that bounds the foowing function h against the optima q q -norm: h := G ) j J p q j + G p q j. j J Once this assignment is obtained, Theorem 7 is invoked separatey on each machine group to obtain the fina assignment of jobs to individua machines within each group. The rest of this section describes the agorithm to assign jobs to machine groups in a smoothed instance. This agorithm has two parts: an onine fractiona agorithm that assigns jobs fractionay to machine groups, and an onine rounding agorithm that converts this fractiona assignment to an integra assignment. 4. Fractiona Agorithm A Continuous Greedy Agorithm) We first define the fractiona counterpart of the objective h defined above. Let x j be the fraction of job j assigned to group G in a fractiona assignment. Then, hx) := G ) j J p j x q j + G p q j x j. j J For brevity of notation, we denote the fractiona oad of group G by Λ x) := j x G j p j and separate the two terms in hx) as the oad-dependent objective f x) := G Λ ) q and the job-dependent objective дx) := j p j ) q x j. We aso ca their sum hx) the tota objective. The goa of the fractiona agorithm is to obtain a fractiona soution x that is O) q -competitive for the tota objective hx). Agorithm. Our agorithm is inspired by a continuous greedy strategy on the objective hx). To define it precisey, we denote the two terms in the derivative dhx) dx j by: α j := d f x) = G dx q Λ ) q p j G j = q Λ ) q pj s β j := dдx) ) = p dx j ) q pj q = j s The agorithm assigns an infinitesima fraction of the current job j to the machine group G that has the minimum vaue of η j := maxα j, β j ). Conceptuay, this is simiar to assigning to the machine that minimizes α j + β j, which is the derivative of the objective hx), but using the max instead of the sum makes our anaysis simper.) In case of a tie between machine groups, the foowing rue is used: If there is a machine group G with α j β j among the tied groups, then this machine group is used for the assignment. Note that by Property 3 of smoothed instances, no two machine groups can have an identica vaue of β j. It foows that there can ony be at most one machine group among the ones tied on η j that has η j = β j, i.e., α j β j. Hence, this rue is we-defined. If α j β j for a tied machine groups, then we divide the infinitesima job among the tied groups in proportion to G s γ, where γ = q/q ). These proportions are chosen to preserve the condition that the vaues of α j remain tied after the assignment see Caim 8 beow). Anaysis. Before giving any forma arguments, we first give some more intuition about the agorithm and a genera overview of our anaysis. Observe that if α j > β j for a machine groups G throughout the agorithm, then the optima strategy woud be to keep a the α j identica, since otherwise, moving an infinitesima fraction of j from a group with a higher vaue of α j to one with a ower vaue woud improve the objective. This property is ensured by the second tie-breaking rue, which is stated in Caim 8 and can be verified by a simpe cacuation that we show ater in the section. Caim 8. If job j is assigned in proportion to G s γ, where γ = q/q ), among machine groups G with identica vaues of α j, then the vaue of α j remains equa for these machine groups after the assignment. But, in genera, it might so happen that β j > α j for a suffix of machine groups reca that machine groups are ordered in decreasing speed, so the suffix represents sower machines). In this case, the agorithm might assign job j to faster groups even though this assignment makes the vaues of α j unequa for the different groups. But, in both these two types of assignments, the vaue of α j of a sower group never exceeds that of a faster group. Again we state this property in the next emma and formay prove it in ater in the section. Lemma 9. For any two groups G and G with s > s, and for any job j, it aways hods that α j α j. This impies that for any two groups G and G with s > s and any job j, α and β have exacty opposite reative orders throughout the agorithm: α j α j by Lemma 9) and β j < β j by Property 3 of smoothed instances). To get some more intuition about the agorithm, imagine extrapoating the discrete vaues of 35

7 Onine Load Baancing on Reated Machines x = vaues of apha = vaues of beta x x Group 0 Group x Group 2 decreasing speed x... Figure 2: Monotonicity of α j and β j vaues with machine speeds. The agorithm compares the circed vaues and chooses the machine group with the smaest of these vaues in this case, it assigns to machine group 2). The dotted ines show an interpoation between the discrete speeds of machine groups that aows us to think of α j and β j as continuous functions of machine speed. This interpoation is ony for intuitive purpose, and is not formay used in the agorithm. α j and β j between the machine groups. Then, α is a monotone non-decreasing and β a monotone decreasing function of machine speed. The agorithm woud ideay ike to assign the fractiona job to the point where these two curves cross, since it represents the minimum vaue of maxα, β) across a speeds. But, since this crossing point may not represent an actua machine group, the agorithm compares the two machine groups that stradde this crossing point and assigns the fractiona job to the group that minimizes maxα, β) among these two machine groups. This is iustrated in Fig. 2. Using the above interpretation that α and β are monotone functions of machine speed, we now sketch the rest of our anaysis before giving it formay. We fix an optima soution opt, and denote the fractiona agorithm s soution by ago; et the corresponding fractiona assignments be x opt and x ago. Now, for any assignment of a fractiona job by ago, the same fractiona job is assigned by opt to either the same machine group, or a sower one, or a faster one. It is easy to bound the objective due to assignments where ago and opt use the same machine group, so we ony focus on the ast two categories, which we ca red and bue assignments respectivey. For red assignments, opt assigns to a sower machine group than ago; thus, its vaue of β is greater than the vaue of β = α for ago. Now, note that the job-dependent objective дx opt ) is simpy given by the sum over β of a assignments of opt, since β is independent of the current oads of machine groups. This aows us to bound the increase in the tota objective hx ago ) due to red assignments against the vaue of дx opt ) see Lemma 0). x Simiary, for bue assignments, opt assigns to a faster machine group than ago; thus, by Lemma 9, its vaue of α is greater than the vaue of α = β for ago. One woud then hope to use a simiar argument as above, that the vaue of f x opt ) is the sum over α of a assignments of opt. Unfortunatey, this is not true since the vaues of α are dependent on the current oads of ago, which may be different from the oads in opt. Instead, we need to use a more goba argument for bue assignments. Note that if a assignments were bue, then using Caim 8, we can bound the vaue of f x ago ) gobay against the vaue of f x opt ). There are two main chaenges in generaizing this argument: first, it is possibe that β > α for ago in a bue assignment, which makes bounding f x ago ) insufficient, and second, red and bue assignments may be intereaved thereby invaidating the goba argument that assumed a assignments to be bue. Overcoming these technica hurdes for bue assignments constitutes the buk of our anaysis in particuar see Lemma for the fina bound of the increase in hx ago ) due to bue assignments). This competes the overview of our anaysis. To make the above arguments forma, we begin with giving proofs of Caim 8. Proof of Caim 8. Reca that α j := q Λ ) q pj s ) Therefore its derivative with respect to an assignment x j is: dα j = qq ) Λ dx ) q 2 p2 j j s 2 Substituting for Λ using ) we have: dα j dx j = qq ) s α j p j q ) q 2 q p 2 j G. s 2 2) G To keep α j vaues equa whie dividing x j infinitesimay among the groups, we shoud assign mass inversey proportiona to dα j dx j to each group G. However, since a G aready have equa α j upon the assignment, a terms in dα j dx except s j and G are common across these groups. Thus, each group shoud receive mass in proportion to s 2 q 2)/q ) G = s γ G. Next, we give a forma proof of Lemma 9, which states that: for any two groups G and G with s > s, and for any job j, it aways hods that α j α j. Proof of Lemma 9. First, note that the emma hods for a jobs if it does for any singe job since α j /p j is a function independent of job j. We now prove the emma by showing that it inductivey hods for the current job j at any time. For the sake of contradiction, suppose the property is vioated for the first time by the current fractiona assignment, which impies that α j = α j before the assignment and α j < α j after the assignment. Now, note that β j > β j by Property 3 of smoothed instances. Therefore, the agorithm can make an assignment on G ony if G and G are tied with η j = α j = α j = η j. But in this case, by the second tie-breaking rue, the agorithm assigns job j to groups G and G in proportion to G s γ and G sγ, where γ = q/q ). This assignment preserves α j = α j by Caim 8, which is a contradiction. 36

8 Sungjin Im, Nathanie Ke, Debmaya Panigrahi, and Maryam Shadoo Red and Bue Assignments. We now give the detaied anaysis that bounds hx ago ) against hx opt ). We distinguish between two types of assignments, red and bue assignments that we precisey define beow. In the rest of the proof, we overoad notation to denote a fractiona job assigned in a singe step of the fractiona agorithm by j.) For a fractiona job j, et optj) resp., agoj)) be the machine group on which it is assigned by opt resp., ago). We ca the assignment of a fractiona job j a red assignment if opt assigns j on a sower machine group, i.e., if s optj) < s agoj) ; we ca it a bue assignment if opt assigns j on a faster machine group, i.e., s optj) > s agoj). If optj) = agoj) = G, we ca it a red assignment if β j α j when the assignment was made; ese, we ca it a bue assignment. We wi anayze the tota increase in the objective hx ago ) caused by red and bue assignments separatey. Note that there was a specia case in the agorithm when machine groups were tied, where we assigned a fractiona job to mutipe machine groups. However, in this case, by Property 2 of smoothed instances, at east haf of the job is assigned to the sowest tied machine group. Since η j = α j for a tied groups in this case, the increase in hx) overa is at most a constant factor times the increase of hx) on the sowest machine group. Therefore, in this anaysis, we wi ony consider the sowest machine group in this scenario. Thus, agoj) is we-defined in a cases. Anaysis for Red Assignments. We first bound the contribution from red assignments. Lemma 0. The tota increase in hx ago ) due to red assignments of ago is at most twice the job-dependent objective дx opt ) of opt. Proof. Consider a fractiona job j that undergoes a red assignment in the agorithm. We have two cases. First, suppose s optj) < s agoj). Given that we ony consider the assignment on the sowest group in case of a tie, we can concude that: maxα optj)j, β optj)j ) = η optj)j > η agoj)j = maxα agoj)j, β agoj)j ) α agoj)j α optj)j by Lemma 9). Therefore, β optj)j > α agoj)j. But, since β optj)j > β agoj)j by Property 3 of smoothed instances, we have: α agoj)j + β agoj)j < 2β optj)j. Next, suppose optj) = agoj). In this case, α agoj)j + β agoj)j 2 maxα agoj)j, β agoj)j ) = 2β agoj)j = 2β optj)j, where the second to ast equaity foows from the definition of red assignments. To compete the proof of the emma, we note that the sum of β optj)j across a fractiona jobs is precisey equa to дx opt ). Anaysis for Bue Assignments. We are eft to bound the tota increase in hx ago ) due to bue assignments. For bue assignments, opt assigns the fractiona jobs to faster machine groups than ago. Our goa is to show the foowing emma. Lemma. The tota increase in objective due to bue assignments in ago is at most O) q times the oad-dependent objective of opt. Using an argument simiar to above emma, we can bound the increase in hx ago ) due to bue assignments against the vaue of α optj)j. But, unike the fact that the sum of β optj)j across a jobs gives the vaue of дx opt ), adding the vaues of α optj)j across a jobs does not yied the vaue of f x opt ). This is because α optj)j are defined based on ago s current oads, which may be different from opt s oads, whereas β optj)j is oad-independent. To understand the intuition behind our anaysis of bue assignments, et us imagine an ideaized scenario where α j > β j throughout the agorithm. In this case, by the second tie-breaking rue, ago maintains equa vaues of α j across a machine groups for a jobs j. Note that this is an optima fractiona assignment for the oaddependent objective f x); therefore, f x ago ) f x opt ). The same argument works even if α j is not equa for a groups, provided a jobs are bue, by repacing uniformity of α j by the monotonicity property from Lemma 9. However, there are two main difficuties with generaizing this argument further: ) First, for a bue assignment of job j to machine group agoj), it may be the case that β agoj)j > α agoj)j. In this case, we need to bound β agoj)j and not α agoj)j in order to bound the increase in the tota objective α agoj)j + β agoj)j. 2) Second, we need to account for the fact that not a assignments are bue, and a genera instance can intereave red and bue assignments. To address the first issue, we specificay consider the bue assignments with β agoj)j > α agoj)j ; et us ca them specia assignments. For a such specia assignments, we modify ago to ago by additionay assigning the fractiona job to the machine group denoted agoj) + ) that is immediatey faster than agoj). The idea behind this additiona dummy assignment is that α agoj) + j η agoj)j irrespective of whether η agoj)j = β agoj)j or η agoj)j = α agoj)j. Therefore, for the specia assignments, we can bound the increase in the tota objective of ago by α agoj) + j due to the dummy assignments that we added. Correspondingy, we modify opt to opt by adding a second copy of each such fractiona job to optj). Note that for specia bue assignments, we have the strict inequaity s optj) > s agoj) ; ese, we woud ca it a red assignment. Hence, these additiona dummy assignments are aso bue assignments. We wi estabish that these modifications do not significanty change the objectives of the two soutions ago and opt Lemmas 2 and 3). To hande the second issue, we modify opt further to opt by adding the oad due to red assignments in ago to each machine group in opt. This aows us to view the red assignments as bue assignments for the purposes of this anaysis, since opt now has a copy of every red job on the same machine group as ago. Again, we estabish that this transformation does not significanty change the objective of opt Lemma 4). Once we have modified ago to ago and opt to opt respectivey, we are abe to show a strong property of the oad profies of ago and opt, namey that for any prefix of machine groups starting with the fastest group, the tota oad of opt is at east as much as that of ago. Informay, this means that opt is even more biased than ago in terms of assigning jobs to faster machines. But, note that Lemma 9 asserts that ago is aready biased toward faster machines than the optima aocation for the oad-dependent 37

9 Onine Load Baancing on Reated Machines objective f x), and it turns out, so is ago. Combining these facts, we bound the increase in the tota objective of ago against the oad-dependent objective of opt Lemma 6). We now give a forma proof for the increase in the tota objective of ago due to bue assignments, based on the outine presented above. Reca that there are three parts to this anaysis: a) modifying ago to ago and opt to opt respectivey do not significanty change their objectives, b) modifying opt to opt does not significanty change its objective, and c) the increase in the tota objective of ago due to bue assignments can be bounded against the oaddependent objective of opt. First, we show that in handing the first issue, modifying ago to ago and opt to opt respectivey do not significanty change the objectives of the respective soutions. The first emma is immediate. Lemma 2. The oad-dependent objective f x opt ) in opt is at most 2 q times the corresponding objective f x opt ) in opt. Lemma 3. The tota objective hx ago ) due to bue assignments in ago is at most twice the oad-dependent objective f x ago ) due to bue assignments in ago. Proof. We consider two cases. First, suppose α agoj)j β agoj)j. This is not a specia bue assignment. In this case, α agoj)j + β agoj)j 2α agoj)j. Since ago has at east as much oad on every machine group as ago, it foows that the tota increase of objective in ago due to assignments in this case is at most twice the oad-dependent objective of ago. Next, suppose α agoj)j < β agoj)j in a bue assignment. This is a specia bue assignment, and we have s optj) > s agoj), as noted earier. In this case, β agoj) + j < β agoj)j, but η agoj) + j η agoj)j. Therefore, α agoj) + j β agoj)j and α agoj) + j α agoj)j. Therefore, we have α agoj)j + β agoj)j 2α agoj) + j. But, for every specia assignment to machine group agoj) in ago, there is a corresponding assignment to group agoj) + in ago. Therefore, the tota increase of objective in ago due to specia assignments is at most twice the oad-dependent objective of ago. Now, we show that in handing the second issue, modifying opt to opt does not significanty change the objective. Lemma 4. The oad-dependent objective f x opt ) in opt is at most 2 q times the oad-dependent objective f x opt ) in opt pus 2 q+ times the job-dependent objective дx opt ) in opt. Proof. We cassify machine groups into two groups. The first type of group is one where the oad in opt is at east its oad from red assignments in ago. The oad in opt for such groups is at most twice the oad in opt. Therefore for these machine groups, the oaddependent objective in opt is at most 2 q times oad-dependent objective in opt. The second type of machine group is one where the red oad in ago is more than the oad in opt. The oad in opt for such machine groups is at most twice the red oad in ago. Therefore by Lemma 0, the oad-dependent objective in opt is at most 2 2 q times the job-dependent objective дx opt ) in opt. We wi now be abe to appy our high eve approach and show that the oad-dependent objective of ago is bounded by that of opt. We first show the foowing emma on oad profies, which formaizes our earier intuition. Lemma 5. Consider two oad profies ψ and ξ over the machine groups with the foowing properties: ) First condition) For any prefix G of machine groups in decreasing order of speeds, the tota job voumes satisfy: G G ψ G s G G ξ G s. 2) Second condition) There exists a µ such that for any two machine groups G and G, we have: ξ q s G µ ξ q s G. Then, the oad-dependent objective of oad profie ψ is at east µ q q times the oad-dependent objective of oad profie ξ. Proof. First, we transform the oad profie ξ to χ so as to change the vaue of µ to in the second condition. For any group G, we set χ so that it satisfies χ q s q ξ G :s s s = min G G. Since χ ξ for any machine group G, the first condition hods for ψ and χ as we. Furthermore, by definition of χ, it satisfies the second condition with µ =. Finay, note that by the second condition on ξ, χ q s q ξ G :s s s = min G G µ ξ q s G. 3) Now, we use an exchange argument to transform ψ without increasing its oad-dependent objective unti for every machine group G, we have ψ χ. In each step of the exchange, we identify the sowest machine group G where ψ < χ. By the first condition, there must be a machine group G with s > s such that ψ > χ and for every prefix G of machine groups in decreasing order of speeds containing G but not containing G, the foowing strict inequaity hods: ψ G s > χ G s. 4) G G G G Furthermore, using the second condition with now µ = ), we have that ψ q s G > χq s G χq s G > ψ q s G. 5) Now, we move an infinitesima job voume from group G to group G in ψ. Inequaity 5) impies that the oad-dependent objective of ψ decreases due to this move. Furthermore, both conditions of the emma continue to remain vaid by Eqns. 4) and 5). Such moves are repeatedy performed to obtain a oad profie ψ with at most the oad-dependent objective of ψ, but additionay satisfying ψ χ for a machine groups G. At this point, the emma hods for the transformed oad profie χ with µ =. To transate this back to the origina oad profie ξ, 38

10 Sungjin Im, Nathanie Ke, Debmaya Panigrahi, and Maryam Shadoo note that Eqn. 3) impies that χ µ /q ) ξ for every machine group G. We now appy Lemma 5 to ago and opt to get our desired bound. Lemma 6. The oad-dependent objective of ago is at most 2 q times the oad-dependent objective of opt. Proof. In Lemma 5, we set ψ to the oad profie of opt and ξ to the oad profie of ago. The first condition of Lemma 5 foows from the foowing observations: a) for bue assignments in ago, s optj) s agoj) ; b) for red assignments in ago, the same fractiona job j is assigned to agoj) in transforming opt to opt ; c) finay, for specia assignments added in transforming ago to ago, we have s optj) > s agoj), i.e., s optj) s agoj) +. We now check the second condition of Lemma 5. From Lemma 9, the condition hods with µ = for ago. In ago, the oad Λ + on a machine group increases by the tota oad due to specia s assignments on machine group G, i.e., by at most Λ s Λ +. But, by Lemma 9, Λ Λ +. Therefore, the oad on machine group G + increases by at most a factor of 2. It foows that the second condition of Lemma 5 hods with µ = /2 q. Now, the emma foows by appying Lemma 5. Combining Lemmas 2, 3, 4, and 6, we obtain the desired bound for bue assignments, i.e., prove Lemma. Lemmas and 0 impy that the agorithm is O) q -competitive on objective hx), as desired. 4.2 Rounding Agorithm In this section, we give an onine rounding procedure that converts a fractiona assignment to an integra assignment for machine groups with a oss of O) q in the tota objective hx). Agorithm. Reca that x j denotes the fraction of job j assigned to machine group G, where the machine groups are denoted G 0,G,... from the fastest to the sowest group. Let mj) denote the median of job j, which we define to be the machine group that satisfies mj) x j /2 and mj) x j /2. In our integra assignment, we assign job j to group G mj). Anaysis. Let x be the soution produced by the rounding agorithm. We wi show the foowing emma. Lemma 7. After rounding x to x, we have hx ) O) q hx). We consider the oad-dependent and job-dependent objectives f x ) and дx ) separatey. Lemma 8. f x ) 4 q f x). Proof. We perform two transformations on assignment x such that f x) can ony increase by a factor of at most 4 q, and then compare the resuting assignment with x. Our first transformation converts x to an assignmenty, as foows. For each job j, we set y j := 2x j for mj). We then aocate the remainder of the job to group G mj), i.e., y mj)j := mj) y j. For the rest of the groups mj) +, we set y j := 0. Observe that based on the definition of mj), this is aways possibe. It is straightforward to see that f y) 2 q f x) since for any given group, we are at most doubing its oad. Then, we construct an optima prefix-constrained schedue, y w.r.t. f. We say that a schedue is prefix-constrained if each job j s assignment is restricted to machine groups G 0,G,...G jm). Note that y is prefix-constrained. Therefore, f y ) f y). We construct y as foows. We consider machines groups in increasing order of their speeds. For a group G in consideration, we consider each job j with mj) =. We continuousy assign job j to the machine group with the minimum α := qλ ) q /s or equivaenty to α, j ). Note that α has no dependency on p j unike α, j. Since we ony consider the oad-dependent objective, it is an easy exercise to show that this continuous process yieds an optima assignment w.r.t. f. Formay, one can show that the monotonicity of α vaues, i.e., α 0 α..., hods throughout. The optimaity foows from this monotonicity property and the fact that two machine groups continue to have the same α vaues once they do. We observe that at east haf of each job j is assigned to G mj) in y. Due to the monotonicity property, j is assigned to machine group mj) east when α 0 = α = α 2 =... = α mj) before j starts getting assigned. In this case, job j is assigned to machine group G in proportion to S), which impies G mj) gets at east haf of the job j since SG mj) ) SG 0 ) + SG ) + SG 2 ) SG jm) ) Property 2). Finay, we obtain the fina assignment x from y by etting each job j to be fuy assigned to G mj). This increases each machine group s oad by a factor of at most 2, therefore we have f x ) 2 q f y ). Lemma 9. дx ) 2 дx). Proof. Consider a transformation of x to y where for each job j, we sety j := 2x j for mj)+. We then aocate the remainder of the job to group G mj), i.e., y mj)j := mj)+ y j. For the rest of the groups mj), we set y j := 0. Since mj) x j /2, for each job j and machine group G, we have y j 2x j. Thus, дy) 2 дx). On the other hand, since every job j is assigned to G mj) in x and to machine groups that are no faster than G mj) in y, it foows that дx ) дy). Combining Lemmas 8 and 9, we obtain Lemma 7. 5 VECTOR SCHEDULING: MINIMIZING q-norms As in the previous section on scaar scheduing, we wi assume throughout that we have a smoothed instance; by Lemma 5, this ony oses a constant factor in the competitive ratio. We aso assume that q c ogd for a arge enough constant c, since otherwise, we can use a scheduing agorithm for unreated machines in [27] to find an Oogd + q) = Oogd)-competitive assignment. As for the scaar case, we derive two natura ower bounds on the q q - norm of any assignment of these jobs to the individua machines ) q j J in G. The first bound is max k G p j k), which is the G q q -norm of the optima fractiona assignment of these jobs, and q, the second bound is max k j J p j k)) which is the q q -norm 39