Online Vector Scheduling and Generalized Load Balancing

0.34 MB·English

by  Xiaojun Zhu

Online Vector Scheduling and Generalized Load
Balancing
Xiaojun Zhu∗†, Qun Li†, Weizhen Mao† and Guihai Chen∗
∗ State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, P. R. China
† Department of Computer Science, the College of William and Mary, Williamsburg, VA, USA
Email: {njuxjzhu,liqun,wm}@cs.wm.edu, [email protected]
Abstract—We give an approximation-preserving reduction of vectors, and the other providing O(lnd) approximation
2 from vector scheduling problem (VS) to generalized load bal- with running time polynomial in nd, where n is the number
1 ancing problem (GLB). The reduction bridges existing results of of vectors. The third algorithm is a randomized algorithm,
0 the two problems. Specifically, the hardness result for VS holds
whichassignseachvectortoauniformlyandrandomlychosen
2 alsoforGLBandanyalgorithmforGLBcanbeusedtosolveVS.
Based on this, we get two new results. First, GLB does not have partition.ItgivesO(lndm/lnlndm)approximationwithhigh
v
constant approximation algorithms that run in polynomial time probability,wheremisthenumberofpartitions(servers).For
o
unless P = NP. Second, there is an online algorithm (vectors fixed d, there exists a polynomial time approximation scheme
N
coming in an online fashion) that solves VS with approximation
(PTAS) [1]. A PTAS has also been proposed for a wide class
boundelog(md),whereeisthenaturalnumber,misthenumber
5 of cost functions (rather than max) [2].
of partitions and d is the dimension of vectors. The algorithm
2
is borrowed from GLB literature and is very simple in that
Generalized load balancing is recently introduced to model
each vector only needs to minimize the L norm of the
] ln(md) theeffectofwirelessinterference[3][4].Eachjobincurscosts
C resultingload.However,itisunclearwhetherthisalgorithmruns
in polynomial time, due to the irrational and non-integer nature to all machines, no matter which machine it is assigned to.
C
ofln(md).Weaddressthisissuebyroundingln(md)tothenext The exact cost incurred by a job to a specific machine is
s. integer.Weprovethattheresultingalgorithmrunsinpolynomial dependent on which machine the job is assigned to. The load
c time with approximation bound elog(md)+ lne(lmogd()e+)1 which is of a machine is the total cost incurred by all the jobs, instead
[ in O(ln(md)). This improves the O(ln2d) bound of the existing ofjustthejobsitserves.Thismodeliswellsuitedforwireless
polynomial time algorithm for VS.
1 transmission, since, in wireless network, a user may influence
v I. INTRODUCTION all APs in its transmission range due to the broadcast nature
9
of wireless signal. To solve the generalized load balancing
2 Scheduling with costs is a very well studied problem in
problem, the current solution is an online algorithm, adapted
7 combinatorial optimization. The traditional paradigm assumes
5 from the recent progress in online scheduling on traditional
single-cost scenario: each job incurs a single cost to the
. model[5].Thesolution,thoughprovidesgoodapproximation,
1 machine that it is assigned to. The load of a machine is the
is rather simple: each job selects the machine to minimize the
1 total cost incurred by the jobs it serves. The objective is to
2 minimize the makespan, the maximum machine load. Vector Lτ norm of the resulting loads at all machines where τ is a
1 parameter.
schedulingandgeneralizedloadbalancingextendthescenario
:
v in different directions. Toavoidconfusion,wekeepthetwotermsjobandmachine
i
X Vector scheduling assumes that each job incurs a vector unchanged for generalized load balancing, while refer to job
cost to the machine that it is assigned to. The load of a and machine in the vector scheduling model as vector and
r
a machine is defined as the maximum cost among all dimen- partition respectively.
sions. The objective is to minimize the makespan. Vector
We make two contributions. First, we present an approach
scheduling is a multi-dimensional generalization of the tra-
to encode any vector scheduling instance by an instance of
ditional paradigm. It finds application in multi-dimensional
generalized load balancing problem. This encoding method
resource scheduling in parallel query optimization [1], where
brings the recent progress in generalized load balancing into
it is not suitable to represent different requests (such as
the vector scheduling domain, and takes the hardness result
CPU, memory, network resource, etc.) as a single scalar.
of vector scheduling to generalized load balancing problem.
To solve vector scheduling, there are three approximation
Second, the existing solution to GLB needs to compute the
solutions [1]. Two of them are deterministic algorithms based
L norm (l is the number of machines), but it is unclear
on derandomization of a randomized algorithm, with one lnl
whether this norm can be computed in polynomial time. We
providing O(ln2d) approximation1, where d is the dimension
eliminate this uncertainty by rounding lnl to the next integer,
guaranteeing polynomial running time. In addition, we prove
The work was done when the first author was visiting the College of
WilliamandMary. that the approximation loss due to rounding is small.
1In this paper, e denotes the natural number, ln(·) denotes the natural
logarithm,andlog(·)denotesthelogarithmbase2. We conclude this section by the following two definitions.
A. Vector Scheduling A. Creating GLB Instances
We are given positive integers n,d,m. There are a set Comparing VS to GLB, we can find that they mainly differ
(cid:80)
V of n rational and d-dimensional vectors p ,p ,...,p in the subscripts of max and . Our construction is inspired
1 2 n
from [0,∞)d. Denote vector p = (p ,...,p ). We need by this observation.
i i1 id
to partition the vectors in V into m sets A ,...,A . The Given as input to VS the vector set V and m partitions, we
1 m
problem is to find a partition to minimize max (cid:107)A (cid:107) constructtheGLBinstanceasfollows.WesetthejobsJ =V.
1≤i≤m i ∞
w(cid:107)Ahe(cid:107)re Aisi t=he(cid:80)injfi∈nAitiypjnoirsmthdeefisunmedoafstthheevmeacxtoimrsuimn Aelei,maenndt Fmoarcheianceh,dpeanrtoittieodnbAyjthaenpdaiitrs(kj,-kth).dTimhuesn,stihoen,cwonestcrounctsetrdumctaa-
i ∞
inthevectorA .Forthecasem≥n,thereisatrivialoptimal chine set M is {(j,k)|j =1,2,...,m and k =1,2,...,d}.
i
solution that assigns vectors to distinct partitions. Therefore, We refer to a machine as a pair of indices so that we can
we only consider the case m<n. map the machine back to its corresponding partition and
For ease of presentation, we give an equivalent integer dimension easily. For a machine t = (j,k) where t ∈ M,
program formulation. Let xij be the indicator variable such we refer to the partition j as t1, and the dimension k as
that xij = 1 if and only if vector pi is assigned to partition t2, i.e., t = (t1,t2). We can see that there are totally d
A . Then machines (t included) corresponding to the same partition as
j
(cid:88) themachinet.Wedenote[t]asthesetofthesemachines,i.e.,
(cid:107)A (cid:107) = max x p
j ∞ ij ik [t]={(j,1),(j,2),...,(j,d)}, where j =t . Among these d
1≤k≤d 1
i
machines,weselectthefirstone(j,1)astheanchormachine,
The vector scheduling problem can be rewritten as denoted by t, such that a vector chooses partition A in VS if
j
and only if the corresponding job chooses t in the new GLB
(cid:88)
minmax x p problem.
ij ik
x j,k The incurred cost c of job i to machine t if i chooses
i ist
subject to machine s is defined as
(cid:88) (VS) 
xij =1, ∀i  pit2 if s=t (1)
j cist = ∞ if s∈[t]∧s(cid:54)=t (2)
xij ∈{0,1}, ∀i,j  0 if s∈/ [t] (3)
B. Generalized Load Balancing where (1)(2) are for the situation where s and t correspond to
the same partition. They force a job to select only the anchor
Thisformulationfirstappearsin[4].Wereformulateitwith
machines. (3) is for the situation where s and t correspond to
slightly different notations. There are a set M of independent
different partitions. In this case, there is no load increase.
machines, and a set J of jobs. If job i is assigned to machine
The resulting GLB instance is defined as VS-GLB:
j, there is non-negative cost c to machine k. The load
ijk
of a machine is defined as its total cost. The problem is to minmax (cid:88)x(cid:48) c
is ist
find an assignment (or schedule) to minimize the makespan, x(cid:48) t
i,s
the maximum load of all the machines. This problem can be
subject to
formally defined as follows. (VS-GLB)
(cid:88)
x(cid:48) =1, ∀i∈J
(cid:88) is
minmax xijcijk s
x k i,j x(cid:48)is ∈{0,1}, ∀i∈J,s∈M
subject to
To avoid the confusion with the general GLB problem, we
(GLB)
(cid:88)
x =1, ∀i∈J intentionally use different notations x(cid:48), s and t. The notation
ij
j i is kept since it is in 1-1 correspondence with the vectors in
x ∈{0,1}, ∀i∈J,j ∈M VS.
ij
As an example, consider the case when d=1. All vectors
where x ∈ {0,1}|J|×|M| is the assignment matrix with in VS have only one element, and there is only one machine
elements x = 1 if and only if job i is assigned to machine in VS-GLB representing a partition in VS. The objective
ij
(cid:80)
j. The two constraints require each job to be assigned to one of VS becomes max x p . On the other hand, the
j i ij i1
machine. objective of VS-GLB is max (cid:80) x(cid:48) c = max (cid:80) x(cid:48) p .
t i it itt t i it i1
Since any machine t corresponds to a distinct partition A ,
j
II. ENCODINGVECTORSCHEDULINGBYGENERALIZED simply changing subscripts shows that the two problems are
LOADBALANCING equivalent. For the case when d > 1, the proof is much
involved, which we delay to Section II-B.
We first create a GLB instance for any VS instance, then
prove their equivalence. At last, we discuss the hardness of Theorem 1. The construction of VS-GLB can be done in
GLB and extend the VS model. polynomial time.
Proof: An instance of VS needs Ω(nd) bits. The con- Set x(cid:48) = x and all others to be 0. We first show that x(cid:48)
is is1
structed VS-GLB instance has n jobs, md machines and is a feasible solution to VS-GLB. Obviously, x(cid:48) is an integer
n(md)2 costs. Since m < n, all three terms are polynomials assignment. We will check that (cid:80) x(cid:48) = 1. Observe that
s is
in n and d. The theorem follows immediately. x(cid:48) = 0 if s (cid:54)= s. We only need to consider m machines
is
ThefollowingtheoremshowsthattheconstructedVS-GLB (1,1),(2,1),...,(m,1). Since x is a feasible solution to VS,
problem is equivalent to its corresponding VS problem. Let T then for any i ∈ V, there exists one and only one partition
be a positive constant. A such that x =1. Our transformation sets x(cid:48) =1 where
j i,j is
s=(j,1). So (cid:80) x(cid:48) =1.
Theorem2. ThereisafeasiblesolutionxtoVSwithobjective s is
Second, we prove that the objective values of the two
valueT ifandonlyifthereisafeasiblesolutionx(cid:48) toVS-GLB
feasible solutions are equal.
with the same objective value T.
(cid:88) (cid:88)
max x(cid:48) c =max x(cid:48) c (7)
This theorem shows that VS and its corresponding VS- is ist is ist
t t
GLB have the same optimal value. In addition, any c- i,s i
s∈[t]
approximation solution to VS-GLB, after transformation, is =max(cid:88)x(cid:48) c (8)
also a c-approximation solution to VS, vice versa. We prove t it itt
i
this theorem in Section II-B. (cid:88)
=max x p
ItisworthmentioningthatVS-GLBisaspecialinstanceof t it1 it2
i
GLB. Since VS-GLB is converted from VS, VS is a special (cid:88)
=max x p ,
instance of GLB, which implies that VS should have approx- ij ik
j,k
imation algorithms at least as good as GLB. Unfortunately, i
on the contrary, the literature shows better approximation where (7) is due to that cist = 0 if s ∈/ [t], and (8) is due to
algorithm for GLB than that for VS. Hence, it is worth our assignment of x(cid:48) that x(cid:48) =0 if s(cid:54)=s.
is
applying algorithms of GLB to VS. “⇐=”Givenx(cid:48) forVS-GLB,constructxforVSasfollows.
Set x = x(cid:48) where s = j. We show that x is a feasible
B. Proof of Equivalence ij is 1
solution to VS. Due to Lemma 1, for any i, there exists one
We first study the properties of feasible solutions to VS- s such that x(cid:48) = 1 and s = s. Therefore, there exists one j
is
GLB in Lemma 1 and Lemma 2, and then prove Theorem 2. such that x =1. On the other hand, there cannot be two js
ij
both with x = 1, otherwise x(cid:48) is not a feasible solution to
Lemma 1. Given a feasible solution x(cid:48) to VS-GLB yielding ij
VS-GLB.
objective value T, for any i∈J, we have
For the objective value, we have
1) ∀s(cid:54)=s,x(cid:48) =0;
is (cid:88) (cid:88)
2) ∃j such that for s=(j,1),x(cid:48) =1. max x p = max x p
is j,k ij ik t=(j,k) it1 it2
s (cid:54)=Pros.of:ThFeonr x1(cid:48)i)s,cissuspp=ose∞x(cid:48)is>=T,1 cfoonrtrasodmicteings wwiitthh i =mtax(cid:88)ix(cid:48)itpit2
max (cid:80) x c =T. i
t i,s is ist (cid:88)
For 2), since (cid:80)sx(cid:48)is = 1, there exists some s such that =mtax x(cid:48)iscist, (9)
x(cid:48) =1. Due to 1), we must have s=s. i,s
is
where (9) is due to Lemma 2. This completes our proof.
Lemma 2. Given a machine t, a job i, and a feasible
solution x(cid:48) to VS-GLB yielding objective value T, we have
(cid:80)sx(cid:48)iscist =x(cid:48)itpit2. C. Inapproximability for GLB
Proof: Recall that [t]={(t ,1),(t ,2),...,(t ,d)}. We It has been proved that no polynomial time algorithm can
1 1 1
have give c-approximation solution to VS for any c > 1 unless
(cid:88) (cid:88) (cid:88) NP = ZPP [1]. Combining this result with Theorem 2, we
x(cid:48) c = x(cid:48) c + x(cid:48) c
is ist is ist is ist have the following theorem.
s s∈[t] s∈/[t]
(cid:88) Theorem 3. For any constant c > 1, there does not exist a
= x(cid:48) c (4)
is ist polynomial time c-approximation algorithm for GLB, unless
s∈[t]
NP =ZPP.
=x(cid:48) c (5)
it itt
Proof: Since VS-GLB is a special instance of GLB, any
=x(cid:48) p (6)
it it2 c-approximation algorithm for GLB can be used to obtain c-
where (4) is due to (3), (5) is due to Lemma 1, and (6) is due approximation solution to VS-GLB. By Theorem 2, any c-
to (1). approximation solution to VS-GLB is also a c-approximation
With the two lemmas, we can now prove the equivalence. solution to the corresponding VS. Therefore, the approxima-
Proof of Theorem 2: “=⇒” Given a feasible solution x tion algorithm is also a c-approximation algorithm for VS, a
to VS, construct a feasible solution x(cid:48) to VS-GLB as follows. contradiction.
We can obtain a stronger result by relaxing the assumption for the traditional load balancing problem [5], and recently
NP (cid:54)= ZPP to P (cid:54)= NP. (It is a relaxation because extended to the GLB problem [3]. The parameter τ controls
P ⊆ ZPP ⊆ NP.) This can be done by examining the the approximation ratio of the algorithm, as shown in the
inapproximability proof for VS [1]. The inapproximability following lemma.
proof relies on the result that no polynomial time algorithm
Lemma 3 ([5], [3]). Minimizing L norm gives τ l1/τ
can approximate chromatic number to within n1−(cid:15) for any τ ln(2)
approximation ratio to solve GLB where l is the number of
(cid:15) > 0 unless NP = ZPP. Recently, it has been proved
machines.
thatnopolynomialtimealgorithmcanapproximatechromatic
number to within n1−(cid:15) for any (cid:15) > 0 unless P = NP [6]. Setting τ = lnl yields the best approximation ratio elogl.
Thus,wecanchangetheassumptionNP (cid:54)=ZPP toP (cid:54)=NP However, it is unclear whether the computation of L can
lnl
safely. be done in polynomial time. We consider this issue later.
Theorem 4. For any constant c > 1, there does not exist a A. Adapting to VS
polynomial time c-approximation algorithm for GLB, unless
To apply the above algorithm to VS, we can first solve
P =NP.
VS-GLB and transform the solution to VS. This process can
D. Extending to generalized VS be simplified by omitting the transformation between VS and
VS-GLB.
Our construction of VS-GLB and proof can be easily ex-
Recall that the algorithm is to assign vectors one by one.
tendedtoageneralversionofvectorscheduling.Inthecurrent
Consider a vector p in VS. To solve VS-GLB, this vector
VS definition, all machines (partitions) are identical so that i
should choose a machine to minimize the L norm of the
any job (vector) incurs the same vector cost to all machines. τ
resulting load. Due to the construction of VS-GLB, this
The machines can be generalized to be heterogeneous so that
vector can only choose from the anchor machines, otherwise,
each job incurs a different vector cost to different machines.
Formally, job i incurs vector cost p(j) to machine j if i is the resulting Lτ norm would be infinite (definitely not the
i optimal choice). Thus, this is equivalent to picking from the
assigned to machine j. The formulation and transformations
corresponding partitions in VS. After the assignment of any
can be slightly changed as follows. In the integer program
formulationofVS,changetheobjectivetomax (cid:80) x p(j). numberofvectorsthatleadstopartitionsA1,A2,...,Am,the
j,k i ij ik L normoftheloadofmachinesinVS-GLBis,infact,equal
Change p in equation (1) to be p(t1). For Lemma 2, τ
it2 it2 to
change x(cid:48) p to x(cid:48) p(t1). It can be verified that the proof of
Theoremi2t iist2still viatliidt2with minor modifications. The online f(τ)(A1,...,Am)=(cid:0)(cid:107)A1(cid:107)ττ +...+(cid:107)Am(cid:107)ττ(cid:1)1/τ
algorithm adopted later is also valid for this general version
where
of vector scheduling. For simplicity, we mainly focus on the  τ
original VS model. (cid:107)Aj(cid:107)ττ =(cid:88)(cid:88) pik .
III. ONLINEALGORITHMFORVS k i∈Aj
BasedonTheorem2,wecansolveVSbyitscorresponding Suppose the assignment of vectors p ,p ,...,p leads to
1 2 i−1
VS-GLB. We review the approximation algorithm [3] for partitionsA ,A ,...,A .Letf(τ) beL normoftheresult-
1 2 m i,j τ
GLB, and then modify it to solve VS. ing load if vector p chooses partition A , i.e.,
i j
Given a GLB instance and a positive number τ, the algo-
f(τ) =f(τ)(A ,...,A ∪{p },...,A ).
rithm[3]considersjobsonebyone(inanarbitraryorder)and i,j 1 j i m
assignsthecurrentjobtoamachinetominimizetheL norm2
τ Then,accordingtothealgorithm,vectorp shouldbeassigned
i
oftheresultingloadofallmachines.Specifically,supposejobs
to the partition
arenumberedas1,2,...,n,thesameastheconsideredorder. argminf(τ).
Suppose the load of machine k after jobs 1,2,...,i−1 are j i,j
assigned is li−1. Then job i is assigned to the machine
k TheprocedureisdescribedinAlgorithm1.Foreachincom-
(cid:32) (cid:33)1/τ ing vector, it only needs to execute Lines 5-9.
(cid:88)
argmin (li−1+c )τ . Algorithm 1 with τ =ln(md) is an elog(md) approxima-
k ijk
j tion algorithm to solve the corresponding VS-GLB. Thus, we
k
have the following result due to Theorem 2.
The above optimization problem can be solved by trying each
possiblemachine.Duringtheoptimization,thecomputationof Lemma 4. Algorithm 1 with τ = ln(md) is an elog(md)
the last step of Lτ norm, (·)1/τ, can be omitted. In addition, approximation algorithm to solve VS.
because the algorithm does not require the order of jobs
However, it is unclear whether Algorithm 1 with τ =
and each job is assigned once, it can be implemented in
ln(md) can terminate within polynomial time. The algorithm
an online fashion. This algorithm was originally proposed
requires the computation of xln(md) for some x. First, the
2Lτ normofavectorx=(x1,x2,...,xt)isdefinedas((cid:80)ixτi)1/τ. number ln(md) is irrational, thus cannot be represented by
Algorithm 1: Vector Scheduling Sinceξ ≥lnl,wehavel1/ξ ≤e.Additionally,ξ ≤lnl+1,
(cid:16) (cid:17)
Input: m, the number of partitions; d, the dimension of so 1− lnl ≤ 1 . Therefore, g(cid:48)(ξ) ≤ e 1− lnl ≤
ξ lnl+1 ln(2) ξ
each vector; p1,p2,...,pn, the n vectors to be elog(e). We have
scheduled; τ, the norm lnl+1
Output: A ,...,A , the m partitions elog(e)
1 m g((cid:100)lnl(cid:101))−g(lnl)≤g(lnl+1)−g(lnl)≤
1 begin lnl+1
2 for j from 1 to m do Note that g(lnl)=elog(l). This completes our proof.
3 Aj ←−∅; This theorem holds for general GLB problem, such as the
4 for i from 1 to n do one considered in [3] [5] and [4]. Of course, it holds for VS-
5 if ∃Aj,Aj =∅ then GLBaswell.Tohaveanintuitionontheloss,weplotthetwo
6 Aj ←−Aj(cid:83){pi}; approximation ratios with respect to the number of machines
7 else in Figure 1. We can see that the loss is small.
8 find j to mi(cid:83)nimize fi(,τj); 20
9 Aj ←−Aj {pi};
15
o
ati
n r
o
polynomial bits to achieve arbitrary resolution. Second, even ati10
m
though we can approximate it by a rational number with xi
o
acceptable resolution, the number xτ˜ may still be irrational, pr
p
whereτ˜istherationalapproximationtoτ.Forexample,when a 5
τ˜ = 1.5, there are lots of values of x such that x1.5 are before rounding
irrational. Though we can still approximate it by a rational after rounding
0
number, it is complicated to theoretically analyze whether 0 20 40 60 80 100
number of machines
the approximation ratio still holds and how the running time
increaseswithrespecttorationalnumberapproximationaccu- Fig. 1. Approximation ratio loss due to rounding. Before rounding, the
racy. This problem has not been addressed in literature. approximationratioisy=elog(x)anditbecomesy=elog(x)+ elog(e)
ln(x)+1
Our solution is to round ln(md) to the next integer afterrounding.
(cid:100)ln(md)(cid:101) and compute the L norm. This guarantees
(cid:100)ln(md)(cid:101)
We have the following corollary due to Theorem 5.
polynomialrunningtime,butcausesthelossofapproximation
ratio. We show in the following that the loss is very small. Corollary 1. With τ = (cid:100)ln(md)(cid:101), Algorithm 1 is an
elog(md)+ elog(e) approximation algorithm to VS, and it
B. Guaranteeing Polynomial Running Time ln(md)+1
runs in polynomial time.
To deal with the irrational number issue, we round ln(md)
tothenextinteger(cid:100)ln(md)(cid:101).Inthefollowing,weanalyzethe Thepolynomialrunningtimecanbeshownbythefollowing
resulting approximation ratio. analysis. We assume τ =(cid:100)ln(md)(cid:101) if not specified. The main
time consuming step is to minimize f(τ) over j for given i.
Theorem 5. Let l be the number of machines. Minimizing i,j
L norm gives elog(l) + elog(e) approximation ratio to We can omit the computation of the outer 1/τ power since
(cid:100)lnl(cid:101) lnl+1 function xy is monotonic for x≥0 and y >0. In computing
solve GLB.
L norm, there is a basic operation, the integer power of a
τ
Proof: This result is obtained from Lemma 3 by per- number, aτ, where a is an element in any vector A . The
j
forming calculus analysis. Let g(x)= x l1/x. Consider the naive approach, which multiplies a iteratively, involves τ −
ln(2)
derivative of g, 1 multiplications. This can be improved by utilizing partial
l1/x (cid:18) lnl(cid:19) multiplication results. For example, computing a8 as ((a2)2)2
g(cid:48)(x)= 1− . onlyneeds3multiplications.Generally,computingaτ requires
ln2 x
(cid:98)log(τ)(cid:99)+ν(τ)−1multiplications,whereν(τ)isthenumber
Forx≥lnl,itholdsthatg(cid:48)(x)≥0sothatthefunctiong(x)is of 1s in the binary representation of τ (Chapter 4.6.3 in [7]).
monotonically increasing. Since ln(l)≤(cid:100)ln(l)(cid:101))≤ln(l)+1, Inthefollowing,weputanupperbound2logτ tothenumber
we have of multiplications needed to compute aτ.
g((cid:100)ln(l)(cid:101))−g(lnl)≤g(lnl+1)−g(lnl). To compute f(τ) for given i and j, it needs d + m − 1
i,j
additions (adding p to A , suppose A is maintained in each
In addition, consider the two points (lnl,g(lnl)) and (lnl+ i j j
iteration) and 2mdlog(τ) multiplications (md numbers, each
1,g(lnl + 1)). Due to Langrange’s mean value theorem in
needstocomputeitsτ power).Tofindtheoptimalj forgiven
calculus, there exists ξ ∈[lnl,lnl+1] such that i, we need to compute f(τ) for all j, and select the optimal
i,j
g(lnl+1)−g(lnl)=g(cid:48)(ξ). one by comparison. This procedure needs m(d + m − 1)
additions, 2dlog(τ)m2 multiplications, and m − 1 compar- Algorithm 2: Sped-up Vector Scheduling with τ =
isons.Insummary,ittakesO(dlog(τ)m2)timeforonevector. (cid:100)ln(md)(cid:101)
For the overall algorithm, it takes O(dlog(τ)nm2) time. Input: m, the number of partitions; d, the dimension of
The computations can be sped up by exploiting the problem each vector; p ,p ,...,p , the n vectors to be
1 2 n
structure. The complexity can be reduced to O(dlog(τ)mn), scheduled
dropping one m factor, as shown in the following. Output: A ,...,A , the m partitions
1 m
C. Computation Speedup 1 begin
2 for j from 1 to m do
Towards VS-GLB, we have the following lemma. Note that
3 Aj ←−∅;
this lemma does not hold for the general GLB problem.
4 µj ←−0; // vector Aj
Lemma 5. For any j1,j2, it holds that fi(,τj1) > fi(,τj2) if and 5 δj ←−0 ; // scalar (cid:107)Aj(cid:107)ττ
only if 6 for i from 1 to n do
(cid:13)(cid:13)(cid:13)Aj1 ∪{pi}(cid:13)(cid:13)(cid:13)ττ −(cid:13)(cid:13)Aj1(cid:13)(cid:13)ττ >(cid:13)(cid:13)(cid:13)Aj2 ∪{pi}(cid:13)(cid:13)(cid:13)ττ −(cid:13)(cid:13)Aj2(cid:13)(cid:13)ττ 78 if ∃AAjj,←A−j =Aj∅(cid:83)th{epni};
Proof:Adding(cid:107)A (cid:107)τ+...+(cid:107)A (cid:107)τ tobothsidesproves 9 µj ←−pi;
the lemma. 1 τ m τ 10 δj ←−(cid:107)pi(cid:107)ττ;
11 else
Algorithm 2 shows the final design. For each partition A ,
the algorithm maintains two variables, the vector A (µ ijn 12 jmin ←−1; // partition index
the algorithm) and its norm (cid:107)A (cid:107)τ (δ in the algorijthm)j. If 13 µmin ←−µ1+p1;
thereisnoemptypartition,thenejacτhincjomingvectorsearches 14 δmin ←−(cid:107)µmin(cid:107)ττ;
(cid:13) (cid:13)τ 15 for j from 2 to m do
o(cid:13)(cid:13)vAejr(cid:13)(cid:13)aττll(Lpainrteistio1n2s-2t4o).finAdstLheemjmtoa 5misnhiomwizs,et(cid:13)(cid:13)hAisjis∪e{qpuii}v(cid:13)(cid:13)aτlen−t 1167 µδ˜˜←←−−(cid:107)µµ˜j(cid:107)+ττ;pi; // vector addition
to minimize fi(,τj). 18 if δmin−δjmin >δ˜−δj then
For the running time, consider a new vector that cannot 19 jmin ←−j;
findanemptypartition.Therearemdadditions(Lines13,16), 20 µmin ←−µ˜;
2mdlog(τ) multiplications (Lines 14,17), 2(m−1) subtrac- 21 δmin ←−δ˜;
tionsandm−1comparisons(Line18).Thedominatingfactor
is mdlog(τ). This is for one vector. For all n vectors, the 22 µjmin ←−µmin;
running time is O(mndlog(τ)), compared to O(m2ndlogτ) 23 δjmin ←−δmin; (cid:83)
before speedup. Substituting τ = (cid:100)ln(md)(cid:101) into the formula 24 Ajmin ←−Ajmin {pi};
yields O(nmdlnln(md)) running time, polynomial in the
input length (note m < n). This analysis, together with
Corollary 1 and Lemma 5, gives the following theorem.
Theorem 6. Algorithm 2 is an elog(md)+ elog(e) approx- we design a polynomial time algorithm for vector scheduling
ln(md)+1 by using an existing algorithm for GLB and modifying it to
imation algorithm to VS. It runs in O(nmdlnln(md)) time.
guarantee polynomial running time. The resulting algorithm
It should be noted that we treat multiplications as basic has two advantages. It can be performed in an online fashion,
operations in the above running time analysis. The running and it provides better approximation bound to solve VS than
time will be different if we further consider the complexity existingofflinepolynomialtimealgorithm.Notethatweround
of computing multiplications. Multiplying two n-bit integers ln(md) to an integer to guarantee polynomial running time,
takes time O(n1.59) for a recursive algorithm (Chapter 5.5 which causes a small approximation ratio loss. It is unclear
in [8]). Applying such analysis to Algorithm 2, however, whether computing L norm directly, without rounding,
ln(md)
requires the consideration of the length of the binary repre- can be done in polynomial time.
sentation of each numeric value in the vectors, which may
be complicated. Nevertheless, it is clear that multiplications REFERENCES
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