The Price of Bounded Preemption

Transcription

1 The Price of Bounded Preemption ABSTRACT Noga Aon * Te-Aviv University Te-Aviv, Israe Princeton University Princeton, New Jersey, USA nogaa@post.tau.ac.i In this paper we provide a tight bound for the price of preemption for scheduing jobs on a singe machine (or mutipe machines). The input consists of a set of jobs to be schedued and of an integer parameter k. Each job has a reease time, deadine, ength (aso caed processing time) and vaue associated with it. The goa is to feasiby schedue a subset of the jobs so that their tota vaue is maxima; whie preemption of a job is permitted, a job may be preempted no more than k times. The price of preemption is the worst possibe (i.e., argest) ratio of the optima non-bounded-preemptive scheduing to the optima k-bounded-preemptive scheduing. Our resuts show that aowing at most k preemptions suffices to guarantee a Θ(min{og k+ n, og k+ P}) fraction of the tota vaue achieved when the number of preemptions is unrestricted (where n is the number of the jobs and P the ratio of the maxima ength to the minima ength), giving us an upper bound for the price; a specific scenario serves to prove the tightness of this bound. We further show that when no preemptions are permitted at a (i.e., k = 0), the price is Θ(min{n, og P}). As part of the proof, we introduce the notion of the Bounded- Degree Ancestor-Independent Sub-Forest (BAS). We investigate the probem of computing the maxima-vaue BAS of a given forest and give a tight bound for the oss factor, which is Θ(og k+ n) as we, where n is the size of the origina forest and k is the bound on the degree of the sub-forest. EYWORDS scheduing jobs; mutipe machines; bounded preemptions; boundeddegree sub-forest ACM Reference Format: Noga Aon, Yossi Azar, and Mark Berin The Price of Bounded Preemption. In Proceedings of 30th ACM Symposium on Paraeism in Agorithms and Architectures (SPAA 8). ACM, New York, NY, USA, 0 pages. *Supported by the ISF and the GIF Supported by: Israe Science Foundation Award 506/6, ICRC Bavatnik Fund 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 author(s) 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. SPAA 8, Juy 6 8, 208, Vienna, Austria 208 Copyright hed by the owner/author(s). Pubication rights icensed to ACM. ACM ISBN /8/07... $ Yossi Azar Te-Aviv University Te-Aviv, Israe azar@tau.ac.i INTRODUCTION. Background Mark Berin Te-Aviv University Te-Aviv, Israe markberin@mai.tau.ac.i We study the bounded preemption rea-time scheduing probem in an off-ine setting, modeed as foows. There is a set J of n jobs avaiabe for scheduing, each of the jobs is associated with a tripet r j,d j,p j, denoting its reease time, deadine and ength (aso caed processing time), respectivey, and with a vaue va(j). In addition, there is an integer k > 0. The jobs can be schedued on a singe machine as we as on mutipe machines (for which we consider the migrative and non-migrative cases aike). A feasibe k-bounded-preemptive schedue is a scheduing of the jobs for execution in a way that at most one job is executed in every moment, the time constraints are not vioated and that no job is preempted (i.e., its execution is stopped) more than k times. The objective is to provide a feasibe (in the sense defined in the ast sentence) schedue for a subset of the jobs of maximum tota vaue. We want to investigate the price of bounding preemption to k, OPT which is defined as PoBP sup (J ) J OPT k (J ), i.e. the amount of vaue we ose by imiting the preemptions to k with respect to the optima -preemptive scheduing. Aso note that Lawer [2] has aready found an optima agorithm for the unbounded case on a singe machine, which in turn can be extended to yied a constant factor approximation for the mutipe identica machines setting; therefore, the price of bounded preemption and the approximation factor for the probem can be reated to each other..2 Motivation The (NP-hard) probem of unbounded preemptive scheduing on mutipe identica machines (denoted P pmtn, r j j ( U j )W j, using the notation of [7]) was first anayzed by Lawer in [2], who found for the singe-machine case an optima soution in pseudopoynomia time, which becomes poynomia for unit-vaue jobs. This (singe-machine) soution can be transformed into an FPTAS using a technique deveoped by Pruhs and Woeginger [25]. This resut can in turn be extended for mutipe (identica) machines in the foowing way. First, one can use the same FPTAS iterativey on the residua jobs set [2], thus obtaining a (2 + ϵ)- approximation for the non-migrative case. Then, this resut can be further extended to the migrative case using the resut of [8], thus obtaining a (6 + ϵ)-approximation. However, in a rea-word setting, preemption comes with a certain price tag (e.g., the sequence of operations required for a context switch), and so one woud want to somehow imit its usage: hence the motivation to study the bounded preemption setting.

2 The bounded version of the probem was addressed using severa different approaches (e.g., [, 2, 27]; surveyed in [3]). The approach consisting of bounding the number of preemptions per job, which is aso the approach studied in this paper, was first addressed by Abagi-im et a. [], who showed an O()-approximation for the specia cases of unit-vaue and unit-density. Other approaches to the notion of the power of preemption (abeit without regard to specific k) were suggested in [26] (defining it in terms of makespan) and [6] (defining it in terms of cost)..3 Our Resuts Our paper provides both ower and upper bounds for the price of k-bounded preemption in terms of n J (i.e., the number of jobs) and of P max j p j min j p j. First, we expore the probem of the k-bounded Degree Ancestor-Independent Sub-Forest (k-bas) and find: an optima soution using Dynamic Programming; a Θ(og k+ n) bound for the oss factor yieded by the optima soution. We then proceed to bound the price of bounded preemption and achieve the foowing: an upper bound of PoBP k = O(og k+ n), for singe and mutipe machines aike; an upper bound of PoBP k = O(og k+ P), for singe and mutipe machines aike; these above bounds are proven to be tight. We aso show that for the specia case of k = 0, i.e. when no preemptions are aowed for the machine, but an unbounded number of preemptions is aowed for a hypothetica competitor. We estabish PoBP k = Θ(min{n, og P}) for this case..4 Other Reated Work Reca that Lawer s soution for the non-bounded preemption probem on a singe machine [2] can be transformed into an FPTAS, which in turn can be extended to a (2+ϵ)-approximation for mutipe non-migrative machines and to a (6 + ϵ)-approximation for mutipe migrative machines (as described in subsection.2). Aso, Baptiste et a. [5, 6] gave a poynomia agorithm for the specia case of nonbounded preemption on a singe machine with uniform processing time. Reca aso that Abagi-im et a. [] estabished an O()-approximation (and price) for both the unit-vaue and the unit-density sub-probems. Their resuts can be easiy extended, by Cassify-and- Seect, to yied O(og ρ) and O(og σ) approximations (and price) for the genera bounded probem, respectivey, where ρ max j va(j) min j va(j), σ max j (va(j)/p j ) min j (va(j)/p j ). However, their work gave no resut with respect to either n or P, a gap which our work attends to. The on-ine version of the probem with no bound on the preemption was anayzed by [4], whose upper bound for the competitiveness ratio was ater proven to be tight by [3]. The non-preemptive rea-time scheduing probem has been posited and studied aready severa decades ago, and has various favours, severa of which have aso been soved since. Moore devised an O(n og n)-time agorithm [24] soving the simpest version of the probem, maximizing the number of jobs meeting their deadine (equivaent to minimizing the number of ate jobs), where a the jobs are unit-vaue and have the same reease time. Later, Lawer [20] extended this technique for specia sub-cases of the probem of weighted jobs with the same reease time (known to be NP-hard in genera by [9]), and again [22] for specia sub-cases of the probem of unit-vaue jobs with different reease times (known to be NP-hard in genera by [2]). Aso, Lawer and Moore devised [23] an O(n p j )-time (pseudopoynomia) agorithm for maximizing the vaue of jobs meeting the deadine with the same reease time. Bar-Noy et a. [7, 8] showed a constant factor approximation for maximizing the vaue of jobs meeting their deadine with different reease times; other approximations in this setting have been suggested prior to that, particuary [5]. A different approach was used by Bar-Yehuda et a. in [0], who consider the scheduing of jobs aready divided into k intervas which can be schedued ony at specific points in time. Using the oca ratio technique first presented in [9], they achieve an O(k) approximation for the probem itsef and the APX-hard Max-Weight Independent Set (MWIS) probem to which it can be reduced. 2 PRELIMINARIES 2. Statement of Main Probem Let J be a set of n jobs that shoud be schedued for processing on a singe machine (mutipe non-migrating machines), and k > 0 an integer. Each of the jobs j J has a vaue va(j) > 0, a reease time r j, a deadine d j and a ength p j. Definition 2.. (a) A feasibe schedue of a job on a singe machine j J is a set G j of pairwise-disjoint segments such that д G j : д [r j,d j ] and д G j д = p j. (b) A feasibe schedue of a job set J on a singe machine is a set of feasibe schedues of jobs G J = { G j j J }, s.t. j, j 2 J, j j 2 : д G j, д 2 G j2 : д д 2 =. (c) A k-preemptive feasibe schedue of a job set J on a singe machine is a feasibe schedue G J s.t. j J, G j k +, i.e. no job is preempted more than k times. The above definitions can triviay be extended to mutipe nonmigrating machines by making the feasibe schedue for mutipe machines a union of feasibe schedues on singe machine, s.t. no job is schedued twice on different machines. Our objective is, given a set of jobs J, to find J J and a k-preemptive feasibe schedue G J s.t. va(j ) j J va(j) is maxima. 2.2 Usefu Shorthands We wi denote the i-th segment of a job j as д i j = [si j, ti j ]. Aso, we wi define the reation of precedence between any two segments д i j = [s i j, t i j ],д i 2 j2 = [s i 2 j2, t i 2 j 2 ] as foows: д i j д i 2 j2 t i j s i 2 j2. Note, that since a the segments are disjoint, this reation induces a tota order on the segments.

3 3 BOUNDED-DEGREE ANCESTOR-INDEPENDENT SUB-FOREST 3. Statement of Probem In order to sove the k-bounded preemption probem we first reduce it to the foowing probem of the k-bounded Degree Ancestor- Independent Sub-Forest (k-bas). In this section, we denote by T (u) the sub-tree of a tree T which is rooted in the node u T and by C T (u) the set of the chidren nodes of a tree node u. Aso, the degree of a node u in T is defined to be the number of u s chidren in T, i.e. deg T (u) C T (u). Definition 3.. An Ancestor-Independent Sub-Forest (AISF) of a given forest (tree) T (V, E) is a sub-forest T of T s.t. for any two different connected components C C 2 of T, for any v C,v 2 C 2, v is not a descendant of v 2 with respect to E. Definition 3.2. A k-bounded-degree Ancestor-Independent Sub- Forest (k-bas) of a given forest (tree) T (V, E) is an Ancestor- Independent Sub-Forest T of T, s.t. v V, deg T (v) k. Definition 3.3. An optima k-bas of a given forest (tree) T (V, E) with vaues va : V R + is a k-bas T of T s.t. va(v ) v V va(v) is maximum. Naturay, a k-bas of a tree does not equa the whoe tree in the genera case, but has ess nodes and hence ess tota vaue. We wi formaize this through the notion of the oss factor. Definition 3.4. The oss factor α ALG of an agorithm ALG for the computation of a k-bas of a tree (forest) is the ratio α ALG va(t ) sup va(alg(t )). T Observation 3.5. A max-vaue k-bas of a forest T is the union of the max-vaue k-bass of the trees constituting T. Coroary 3.6. A oss factor α for an agorithm computing a k-bas of a tree impies the same oss factor for the same agorithm appied to a forest. In the foowing sub-sections we wi present an optima agorithm for the computation of a k-bas of a tree and bound its oss factor. By the above reasoning, this oss factor wi hod for forests as we. 3.2 Dynamic Programming Soution Triviay, in order to turn a given tree (of arbitrary degree) into a k-bas, severa of the nodes need to be deeted. Lemma 3.7. Let T (V, E ) be an ancestor-independent sub-forest of a forest (tree) T (V, E). Then v V \ V iff a of its ancestors or a of its descendants are in V \ V. Proof. Assume on the contrary that there is a node v V \ V s.t. both an ancestor of it u and a descendant of it w are V. Since T is a forest, and v is deeted, it foows that u,w beong to two different connected components of T. But w is a descendant of u, contrary to the definition of an AISF. We therefore can divide the nodes of the tree into three groups as foows: () retained nodes, i.e. nodes that are retained in the k-bas (however, some of their descendants might be deeted); (2) pruned-up nodes, i.e. nodes that are deeted with a their ancestors up to the root (in order to fufi Ancestor Independence); (3) pruned-down nodes, i.e. nodes that are deeted with a their descendants. Observation 3.8. For any k-bas: (a) A retained node cannot have pruned-up descendants. (b) A pruned-up node can have descendants of a types. (c) A pruned-down node has ony pruned-down descendants. For a node v, et us denote its aggregate vaue (i.e., the maxima possibe vaue that can be savaged from T (v) after the pruning) as t(v) in case v is retained, and as m(v) in case it is pruned-up (there is no need to introduce a notation for the aggregate vaue of v in case it is pruned-down, since in that case it is discarded). Aso, et us denote as C k (v) the nodes in C(v) with k highest t(v). Observation 3.8 eads us to the foowing formua: t(u) = va(u) + t(v i ) v i C k (u) (3.) m(u) = max(t(v i ),m(v i )) v i C(u) In order to compute the k-bas of a tree rooted in u, we need to decide for each node in this tree whether it is better for it to be retained or pruned-up (it may, of course, so happen that a node is better pruned-down with a its vaue in order to aow for its parent node to be retained). This is done by computing t(u),m(u) using bottom-up traversa and then deciding which of them is better and appying the decision in reverse order for a its descendants. This scheme is, essentiay, Dynamic Programming. For simpicity, the agorithm wi be presented for a tree as an input, but it can be triviay extended to a forest by appying it separatey to each of the trees of which the forest consists. Procedure TM Input: A tree T (V, E) rooted in u. Output: Two numbers, t(u),m(u), being the maxima vaues attained for u being designated retained or removed, respectivey. if u is eaf then 2 t(u) va(u) ; 3 m(u) 0 ; 4 ese 5 foreach v i C(u) do 6 (t(v i ),m(v i )) TM(v i ) ; 7 end foreach 8 t(u) va(u) + t(v i ) ; v i C k (u) 9 m(u) max(t(v i ),m(v i )) ; v i C(u) 0 end if return t(u), m(u); Anaysis. The runtime of the agorithm is triviay O( V ). Since the agorithm impements exacty the formuae presented in equation 3., it is optima, i.e. gives the maxima vaue k-bas possibe for a given tree (forest).

4 3.3 Determining the Loss Factor The anaysis of the quaity of the oss factor directy from the TM agorithm is difficut. We wi therefore ook at a different agorithm, which is easier to anayze and which we wi ca the agorithm of eveed contraction. Using it, we wi prove the foowing theorem: Theorem 3.9. The oss factor of the agorithm TM is at most og k+ n. We wi make use of the simpe observation, that a sub-tree with a degree k can be viewed as a singe arge node, since from the point of view of the choice which node to retain there is no need to ook at each of them individuay, but a of them can be accommodated en boc. Formay: Definition 3.0. A node u of a tree T is k-contractibe iff: either (i) it is a eaf; or (ii) it has no more than k chidren, a which are contractibe as we. Definition 3.. The k-contraction of a k-contractibe node u in a tree T with vaues va : T R + is deeting T (u) from the tree and setting va(u) va(t (u)). Observation 3.2. Let u be a k-contractibe node in a forestt (V, E) with vaues assigned to the nodes. Then the sub-tree T (u) can be contracted to a singe (eaf) job without oss of vaue. The contraction itsef is done by a bottom-up marking of the nodes as contractibe. The agorithm is described in the procedure MaxContract, which has the foowing properties: Observation 3.3. After the appication of MaxContract to a tree, a non-eaf nodes have more than k descendants; meaning that the contraction is maxima (i.e., there are no more k-contractibe nodes). Observation 3.4. A contraction (in particuar, the appication of MaxContract) cannot increase the number of eafs. If a node u is not contractibe, the number of eafs of T (u) remains the same. If u is contractibe, then a the eafs of T (u) are repaced by u as a new eaf; but T (u) triviay has at east one eaf. Thus, in either case the number of eafs does not increase. The agorithm LeveedContraction uses MaxContract as a primitive in an iterative fashion: in each iteration it performs a maxima contraction, takes the set of eafs aside and continues to the next one. In the end, the agorithm returns the set of eafs with the maxima vaue. Each of such sets represents a k-bas of the origina tree, as shown in the foowing emma. Observation 3.5. Let S i be the set of eafs S after the contraction at the i-th iteration. Then at the beginning of the i-th iteration a the nodes w j i S i are k-contractibe. { Lemma 3.6. Let S i = wi,w2 i,... w S i i the contraction at the i-th iteration, and et T j } be the set of eafs S after i be the sub-forest of T rooted in s j i in the beginning of that iteration (which is contracted during it). Then the union T i = S i T j i is a k-bas. j= Proof. T i is triviay a sub-forest of T. Thus, in order to prove it is a k-bas of T, we need to prove that its degree is bounded by k and that it is ancestor-independent. Agorithm : Leveed Contraction Procedure MaxContract() Input: A tree T. Output: The tree T after maxima k-contraction. 2 foreach node u of T in bottom-up order do 3 if u is a eaf then mark u as true ; 4 ese if deg(u) k and a chidren of u are marked true then 5 contract chidren of u ; 6 set va(u) va(v i ) ; v i CN (u) 7 mark u as true ; 8 ese mark u as fase ; 9 end foreach 0 end Procedure LeveedContraction() Input: A tree T. Output: A k-bas S of T. 2 S ; 3 whie T do 4 MaxContract(T ) ; 5 S a eafs of T ; 6 S S {S} ; 7 T T \ S ; 8 end whie 9 return argmax va(s) ; S S 20 end Bounded Degree. From observation 3.5 immediatey foows that the degree of each T j i is at most k, and therefore the same hods for their union T i. Ancestor Independence. Assume on the contrary that there exist u T j i,u 2 T j 2 i s.t. u is ancestor of u 2 in the origina T. It therefore means that w j i is ancestor of u 2 as we. However, u 2 was not contracted into w j w j i i ; therefore, on the (singe) path from u 2 to there is a node with a degree higher than k +. This contradicts the fact that w j i is k-contractibe, by observation 3.5. Lemma 3.7. The vaue returned by LeveedContraction is not ess than va(t ) L, where L is the number of iterations. Proof. Let S i be the set of eafs S at the i-th iteration, and S i the set of their parents. Aso, et T i be the k-bas of T corresponding to S i, as expained in emma 3.6. Since the agorithm ony performs contractions, by observation 3.2, no vaue is ost, i.e., va(t ) = Li= va(t i ) = L i= va(s i ). Therefore, the vaue returned by the agorithm is ALG va(t ) L. (3.2) Lemma 3.8. The number of iterations of LeveedContraction is ess than og k+ n.

5 Figure : Rearranging a Lawer Schedue Proof. Due to observation 3.3, S i (k+) S i. After the remova of S i from the tree in the iteration s end, S i becomes the set of the tree s eafs. By observation 3.4, a contraction cannot increase the number of eafs in a tree, and therefore S i+ S i. Combining these, we get that for every iteration i, S i+ S i k + S i = S L (k + ) L i S i where L is the (maximum) number of iterations. Hence foows: L L n S i (k + ) L i = (k + )L (k + ) L k i= i= which in turn impies L og k+ n Combining this resut with equation 3.2 yieds us: ALG va(t ) L va(t ) og k+ n Coroary 3.9. The oss factor of LeveedContraction is not arger than og k+ n. Proof of Theorem 3.9. Since agorithm TM is optima, its oss factor cannot be worse than the oss factor of LeveedContraction. Therefore, it is at most og k+ n, i.e. the agorithm TM (T ) yieds a k-bas T such that va(t ) og k+ n va(t ). Remark. The runtime of the LeveedContraction agorithm is triviay O( V ). 3.4 Tightness of Bound Theorem The oss factor of the TM agorithm for the k-bas probem in the genera case is at east og k+ n, i.e. the oss factor estabished previousy is tight. The proof of this emma is presented in appendix A. 4 THE MAIN PROBLEM 4. Reduction to k-bas Assume we are given an optima soution for the case k =. Observe, that if in a feasibe schedue for a singe machine segments of two jobs A and B intereave, i.e. д a A,дa 2 A G A,д b B,дb 2 B G B s.t. д a A дb B д a 2 A дb 2 B, they can be re-arranged into д a A дa 2 A дb B дb 2 B without affecting the feasibiity. This of course can be done independenty for each machine in the mutipe machine setting (if the machines are non-migrative). It therefore foows that any schedue (in particuar, an optima one) can be rearranged (with no oss of vaue) into a form where a segment of B is between two segments of A iff there are no segments of A between any two of the segments of B. Note, that the described re-arrangement makes the preempts reation aminar. If we now present such a re-arranged schedue G J as a graph T (V, E), where the set of nodes V = J is the set of jobs in the schedue, and the set of edges E = {(u,v) V V v preempts u}, the resuting (directed) graph is therefore a forest, which be wi ca a Schedue Forest. Let T (V, E ) be a max-vaue k-bas of T, computed using the agorithm TM described in subsection 3.2. We wi now describe the way of obtaining a new schedue G J from G ; aso, for the sake of simpicity, we wi ca the chid nodes of a node (job) j in T the sub-jobs of j. The set of jobs J in the new schedue corresponds directy to V. For each retained job j V, its segments between two consecutive sub-jobs remain as is. If, however, a sub-job of j is removed (pruned-down see observation 3.8), the segments of j surrounding it are merged to the eft. This eft-merge may affect severa segments at once if severa consecutive sub-jobs are removed. Lemma 4.. The schedue G J obtained as described from an origina unbounded-preemption schedue G J is a feasibe k-bounded schedue as defined in definition 2.. Proof. Feasibiity of Individua Job Schedues. The feasibiity of the individua job schedues G j G J foows from the feasibiity of the origina G j G J, as the segments of the jobs are either retained as is or merged to the eft, i.e., earier than the deadine, but nevertheess not earier than the reease time. Aso, since segments are removed ony as part of the remova of a job as a whoe, the ength constraint is kept as we. Feasibiity of the Schedue Overa. The unchanged segments do not intersect each other by feasibiity of the origina G J. The merged segments repace sub-jobs which are pruned-down, i.e. removed with a their descendants and therefore occupy a pace which woud otherwise be empty. Preemption Bound. The segments of any job j are schedued so that between any two of them there is a sub-job of j (because of the eft-merge). Since by the definition of a k-bas each j J has no more than k sub-jobs, the number of segments of each job j is not more than k +. Remark. In the mutipe non-migrative machines setting the reduction can be appied to each machine separatey. The forests obtained per machine are simpy merged to a singe forest. Since migration can be eiminated by using 6 times more machines [8], if we keep the number of machines constant, we retain at east the 6 -th part of the optima soution after this eimination and prior to appying the

6 reduction. Thus a the prices achieved beow using this reduction stay the same in terms of O notation. Remark. The described reduction is not bidirectiona, i.e. the optima soution for the k-bas probem is not necessariy optima for the corresponding k-bounded Preemptive Schedue probem. This due to the probem of adequatey transating a case where a singe job A is preempted by a string of successive jobs. In this case A is preempted ony once, however if each of these other jobs is in turn preempted, say, k + times, we are forced to represent it in the tree as if A were preempted more than once (modeing it as an infinitey sma segment of A between each of the jobs preempting it in succession), which may force us to forgo some of these jobs, which of course is not necessary in the schedue probem and thus the reverse reduction might yied a non-optima soution. However, the reduction does provide an upper bound for the oss factor. 4.2 The Price of Bounded Preemption as Function of the Number of the Jobs Theorem 4.2. The price of imiting preemption to k is at most og k+ n. Proof. Lemma 4. describes the transformation of an origina optima unbounded preemptive schedue G J into a k-bounded preemption schedue G J. Triviay, the tota vaue of the new set of jobs J is equa to the tota vaue of the V of the max-vaue k-bas T, which in turn is optima (see subsection 3.2) and obeys theorem 3.9. Thus, PoBP k sup J OPT (J) OPT k (J) va(j) va(j ) = va(v ) va(v ) og k+ n The foowing theorem estabishes the tightness of this bound; the proof is presented in appendix B. Theorem 4.3. PoBP k = Θ(og k+ n). 4.3 The Price of Bounded Preemption as Function of the Length-Ratio In order to anayze the Price of Bounded Preemption as a function of the ength ratio P, we wi first introduce the notion of reative axity of jobs. Definition 4.4. The reative axity of a job j J is the ratio λ j d j r j p j. The maxima reative axity in an instance of the probem is denoted λ max. We wi divide the set of jobs J into two by their reative axity (i.e., J = { j J λ j k + }, J 2 = { j J λ j > k + } ) and sove each case separatey. We show that in both cases the price bound is og k+ P, and therefore it is O(og k+ P) for the overa probem. In other words, we prove the foowing: Theorem 4.5. The price of imiting preemption to k is O(og k+ P) Strict Jobs. Let us define the window of a schedued job j J in, denoted w(j), as the time difference between its reease time and its deadine. Triviay, j J,w(j) = p j λ j. Note that the window of a job necessariy incudes the window of a its descendants in the origina tree. Thus, if we ook back at the proof of emma 3.8 and define W i min j Si w(j), we have, by observation 3.3, that for every iteration i, W i+ (k + ) W i p max λ max W L (k + ) L i W i where L is the (maximum) number of iterations. Hence foows: p max λ max (k + ) L W 0 (k + ) L p min L og k+ (P λ max ) By emma 3.7, this in turn impies: Lemma 4.6. For a tree T created from a schedue satisfying λ max k +, the k-bas T returned by agorithm Leveed- Contraction satisfies va(t ) og k+ P va(t ) Lax Jobs. The treatment of the ax jobs is sighty different and requires a different agorithm. It aso requires the estabishment of severa auxiiary emmas presented beow. Lemma 4.7. Let S be a set of intervas. There exists a subset S S such that each point in I S I is covered by at east one and at most two different intervas in S. Proof. Assume WLOG I S I is a singe segment (otherwise appy the proof separatey for each segment and a intervas contained in that segment). Let I 0 be the interva whose eft endpoint is the eftmost among a intervas in S. We start with S I 0 and iterativey add intervas to S. Let I i be the interva whose right endpoint is the rightmost among a intervas in S that intersect an interva in S. Add I i to S, that is S S {I i }. Repeat this procedure unti I S I = I S I. Coroary 4.8. If we number the intervas in S by the order of their eftmost coordinate and divide them into two disjoint subsets S E, S O by the parity of this numbering, then each of these subsets is a set of pairwise-disjoint intervas. Proof. Assume on the contrary WLOG that S E does not satisfy the above. Then k s.t. I 2k I 2k+2. By construction, I 2k I 2k+ and I 2k+2 I 2k+. However, since I 2k+ is contiguous, this necessary impies the existence of a point beonging to a three at once, in contradiction of the previous emma. The foowing emma was proven in [4]: Lemma 4.9. Given a sequence {a,..., a n }, a non-increasing nonnegative sequence {b,...,b n } and two sets X,Y {,..., n}, et X i = X {,..., i} and Y i = Y {,..., i}. Then i n : a j > α a j = a j b j > α a j b j j X i j Y i j X j Y Let us now return to the main probem. Reca that we assume a the jobs are ax, specificay j J, λ j > k +. We wi prove the foowing: Lemma 4.0. For a given set of jobs J excusivey of jobs for which λ j k +, the agorithm LSA_CS yieds a feasibe k-preemptive schedue G Jin s.t. va(j in ) 6 og k+ P va(j). In order to prove this theorem, we wi present the Leftmost- Schedue Agorithm (LSA) suggested by [], with the difference that the jobs wi be sorted by their density σ j va(j) p j rather than by vaue. In the foowing anaysis we wi use the same terminoogy

7 Agorithm 2: The Leftmost Schedue Agorithm Procedure LSA_CS() Input: A set of ax jobs J (i.e., satisfying j J, λ j k + ). Output: A feasibe schedue for jobs J in J, s.t. va(j in ) va(j OPT) 6 og k+ P. 2 Separate a jobs in J into casses J c, c oд k+ P by ength, s.t. c oд k+ P, 3 j J c, (k + ) c p j (k + ) c ; 4 foreach c og k+ P do 5 J c,in, G Jc,in LSA(J c ) ; 6 end foreach 7 return J c,in, G Jc,in, s.t. va(j c,in) is maxima among the J c,in ; 8 end 9 Procedure LSA() Input: A set of ax jobs J (satisfying P k + ). Output: A feasibe schedue for jobs J in J (s.t. va(j in ) 6 va(j OPT)). 0 Sort J in descending order of the jobs density ; foreach j J do 2 Let S be the set of the eftmost k + ide segments in [r j,d j ] ; 3 repeat 4 if j fits into the segments in S then 5 Schedue j in members of S in the eftmost possibe way ; 6 break; 7 ese 8 Remove shortest segment from S and repace it with the next ide segment in [r j,d j ] ; 9 end if 20 unti a ide segments are exhausted; 2 end foreach 22 end of busy-segment (a maxima subset д of the timeine s.t. j J : д G j : д д ) and ide-segment (a maxima subset д of the timeine s.t. j J : д G j : д д = ). Aso, for the sake of brevity, we introduce the notation J out J \ J in for the set of the jobs not schedued. Lemma 4.. The ength of each busy-segment in the schedue created by LSA is at east the ength of the shortest job. Proof is omitted and wi be presented in the fu paper. Now et j be a discarded job. If it was discarded, it means that aready at the moment of its discarding in the span of time [r j,d j ] which is at east p j (k + ) ong there coud be found no k + ide segments of tota cumuative ength p j. Let us denote the the number of the ide segments in [r j,d j ] as f. We wi ook at two distinct cases. Either (i) f k + ; or (ii) f > k +. In case (i), since the tota ength of the busy segments and ide segments in [r j,d j ] is equa to d j r j p j (k + ), and the tota ength of (a) the ide segments is ess than p j, it necessariy means that the tota ength of the busy segments is more than p j k, and so in the worst case we have that va(j ) k+ k va(j). In case (ii), f > k +, yet j coud not be schedued; this impies that the sum of the k + ongest ide segments in [r j,d j ] is ess than p j. Therefore, if we denote the tota cumuative ength of a ide segments in [r j,d j ] as L ide, it hods that L ide f k+ p j. On the other hand, the number of busy segments in [r j,d j ] is triviay at east f, making their tota ength L busy f (by emma 4.). It hods that: L ide f k + p j L busy f L busy L busy + L ide = L ide L busy + L ide L busy f f + 2P k+ p j k + = 2P k + k + 2P + k +, meaning that the busy segments constitute at east the k+ 2P+k+ -th part of the span [r j,d j ]. We have thus proved the foowing emma: Lemma 4.2. For a given set of jobs J excusivey of jobs for which λ j k +, agorithm LSA yieds a feasibe k-preemptive schedue G Jin s.t. j J out, the timespan [r j,d j ] is at east b 0 -oaded, where b 0 = k+ 2P+k+. Remark. Since in LSA we operate under cassify and seect, in each cass P(J c ) k +, and therefore b 0 3. Now et us assume WLOG that a the union of the rejected jobs timespans U = j J out [r j,d j ] is a singe contiguous time segment. Appying to U the construction described in emmas 4.7, 4.8, we get two disjoint sets of pairwise disjoint timespans U E,U O, in which in turn correspond to two disjoint sets of jobs J E, J O. Let us denote the timespans set with the arger tota ength U, and the corresponding job set J. By emma 4.2, the timespan of every job in J out, and in particuar of every job j in J J out, is at east b 0 -oaded by jobs from J in. Since the timespans of jobs in J are disjoint, we can safey caim that p j U 2 U = 2 λ j p j 2 p j. b j J OPT j J 0 j J in Note that since we made no specia assumptions, this inequaity appies to every contiguous subset of U invoving at east two timespans (and by inearity, to every union of disjoint pairs of timespans) and to the corresponding subsets of J OPT, J in. (For subsets of U consisting of a singe timespan the stricter emma 4.2 hods.) Let us assume an order on J by the jobs densities from the argest to the smaest; since agorithm LSA considers the jobs in the same order, and so the corresponding members of J in are aways considered prior to the corresponding members of J OPT, we can caim that this inequaity appies to a the prefixes of J OPT, J in by this order. Now we wi utiize emma 4.9 to achieve: OPT (J) = σ j p j 2 σ j p j = 2 va(j in ) b j J 0 b OPT j J 0 in Note that since LSA operates on casses of jobs separatey, this inequaity hods for every cass J c.

8 Figure 2: Lower bound for k=0 Proof of Lemma 4.0. Reca that LSA_CS divides the ax jobs into og k+ P casses J c, for each of which the above inequaity hods with b 0 3. Therefore, for each cass J c the appication of the interna function LSA wi yied J c,in s.t. c, va(j c,in ) 6 va(j c,opt ). Since necessariy c [og k+ P] s.t. va(j c,opt) og k+ P va(j OPT), it foows that va(j c,in) 6 va(j c,opt) 6 og k+ P va(j OPT). Thus, cassifying a the ax jobs as described, running LSA separatey for each of the casses and taking the best resut wi yied us J in s.t. va(j in ) 6 og k+ P va(j OPT) Overa Price of Preemption. To concude the proof, we combine the resuts of emmas 4.6 and 4.0 into a singe agorithm. Agorithm 3: k-preemptioncombined Input: A pair J, G J, where J is a set of jobs, and G J is a feasibe -preemptive schedue for them. Output: A feasibe k-preemptive schedue G J for J J, s.t. va(j ) O( og k+ P ) va(j OPT). J, G J a jobs j and schedues G j for which λ j k + ; 2 J 2, G J2 a jobs j and schedues G j for which λ j k + ; 3 J LeveedContraction(J ) ; 4 J 2 LSA_CS(J 2) ; 5 return max(j, J 2 ) ; Anaysis. Since the union of J, J 2 yieds back a the jobs J, one of them has at east haf the tota vaue of J; simiary, the optima unbounded preemptive scheduing for one of them is at east haf of the optima one for the whoe. Since both agorithms yied the O(og k+ P)-th fraction of the input s optima unbounded preemption s vaue, the tota vaue of at east one of J, J 2 is a O(og k+ P)-th fraction of va(j OPT ) itsef. Choosing the better resut ensures this factor further. Remark. Like in subsection 4., we can eiminate migration and retain a east a 6 of the optima job set prior to appying the agorithm LSA_CS. Thus, the price of the migrative setting is sti O(og k+ P). From the above we can concude the proof of theorem 4.5. Proof of Theorem 4.5. OPT (J) PoBP k sup J OPT k (J) 2 max (OPT (J ), OPT (J 2 )) ) = O(og k+ P) max (va(j ), va(j 2 ) This bound is tight, as we wi prove in appendix B: Theorem 4.3. The Price of Bounded Preemption for k preemptions is Θ(og k+ P) Extending to Mutipe Machines. The resut for strict jobs can be triviay extended to mutipe non-migrative machines by appying the reduction from subsection 4. and agorithm TM for each machine separatey, which yieds the same bound on the price. As for the ax jobs, the LSA_CS agorithm can be extended to mutipe non-migrative machines in the foowing way. In each iteration i, the i-th machine is assigned the schedue given by LSA_CS for the jobs remaining (formay, J i = J \ i k= J k ). By a known resut from [2], this adds at most to the price, thus preserving the O(og k+ P) ratio (price). This overa resut for non-migrative machines, being constructive, immediatey impies an approximation with the same O(og k+ P) factor. This approximation can be further extended to the migrative machines setting using the resuts of [8], which adds a mutipicative factor to the quaity of the approximation. This approximation (reative to J,OPT ) immediatey impies a O(og k+ P) price for the migrative setting as we (since necessariy J k,opt J,OPT and therefore the price is smaer than the approximation factor). 5 SPECIAL CASE: k = 0 Note that the expressions og k+ n, og k+ P are defined ony for k. To compete our work, et us now ook at the specia case where k = 0, i.e. we are not aowed to preempt at a whereas the optima agorithm can preempt as much as is required. Let us ook at a specific exampe of a series of unit-vaue jobs as presented in figure 2. Since the jobs engths form a geometrica progression with q = 2, it is cear that even with one preemption per job aowed, a of the jobs can be accommodated. On the other hand, if no preemptions are aowed, one can ony accommodate a singe job. Therefore the price of bounding preemption in this case is n and aso, due to the nature of the exampe, og P; these are the ower bounds for the price in case k = 0. These ower bounds can be triviay extended to mutipe machines by simpy mutipying the setting by the number of machines (aong a third axis, so to speak); the bound of og P remains the same, whereas the bound of n becomes O(n) for any fixed number of machines. The n ower bound for the price is triviay tight, since we can aways choose from a set of jobs the one with the maxima vaue and thus achieve a feasibe schedue; however, the tightness of the og P approximation ratio shoud be examined further. Let us utiize the agorithm LSA_CS, from subsection 4.3, adjusting it for the case k = 0, i.e. mandating scheduing to be made soey en

9 Figure 3: k-bas ower bound boc and cassifying the jobs s.t. J c, P(J c ) 2. Lemma 4. sti hods in this setting. Let us now ook at the set J out J \ J ; specificay, et us ook at j J out at the moment of its rejection. Again, since the agorithm schedues the jobs in descending order of density, it suffices to compare the tota engths of the busy and ide segments. If j is rejected, this means that the tota ength of the ide segments in [r j,d j ] is ess than p j, i.e.l I P. On the other hand, triviay L B, since if there were no busy segments in the span, it woud be possibe to schedue j. Thus b 0 = } L I P L B L B L B + L I = L I L B P + L I + P. L B By the same technique as in subsection 4.3 this yieds us va(j in ) +P va(j OPT) for each cass. Since (in this case) we cassify the jobs so that P 2, by the same reasoning as before, taking the maximavaue J c,in yieds us in the overa va(j in ) 3 og P va(j OPT). By the same ogic as in 4.3.4, this resut remains the same for the mutipe machines setting. Thus the upper bound for the price in case k = 0 is estabished to be O(og P), and in conjunction with the previous, it means that this bound is tight. REFERENCES [] S. Abagi-im, B. Schieber, H. Shachnai, and T. Tamir. Rea-time k-bounded preemptive scheduing. In Proc. of the 8th Workshop on Agorithm Engineering and Experiments (ALENEX), pages 27 37, 206. [2] B. Awerbuch, Y. Azar, A. Fiat, S. Leonardi, and A. Rosén. On-ine competitive agorithms for ca admission in optica networks. Agorithmica, 3():29 43, 200. [3] Y. Azar and O. Gion. Scheduing with deadines and buffer management with processing requirements. Agorithmica, 78(4): , 207. [4] Y. Azar and O. Regev. Combinatoria agorithms for the unspittabe fow probem. Agorithmica, 44():49 66, [5] P. Baptiste. Poynomia time agorithms for minimizing the weighted number of ate jobs on a singe machine with equa processing times. 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Worst-case response time anaysis of rea-time tasks under fixed-priority scheduing with deferred preemption revisited. In Proc. 9th Euromicro Conf. ReaTime Syst. (ECRTS), pages , [3] G. C. Buttazzo, M. Bertogna, and G. Yao. Limited preemptive scheduing for reatime systems. a survey. IEEE Transactions on Industria Informatics, 9():3 5, 203. [4] R. Canetti and S. Irani. Bounding the power of preemption in randomized scheduing. SIAM J. Comput., 27(4):993 05, 998. [5] S. Dauzère-Pérès. Minimizing ate jobs in the genera one machine scheduing probem. European Journa of Operationa Research, 8():34 42, 995. [6] L. Epstein, A. Levin, A. J. Soper, and V. A. Strusevich. Power of preemption for minimizing tota competion time on uniform parae machines. SIAM J. Discrete Math., 3():0 23, 207. [7] R. Graham, E. L. Lawer, J. Lenstra, and A. R. an. Optimization and approximation in deterministic sequencing and scheduing: A survey. Arm. Discr. Math., 5:75 90, 979. [8] B. ayanasundaram and. Pruhs. 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The nodes of the tree are arranged in og n + eves, numbered from 0 (the topmost) to L og n (the owermost). In each eve i, the number of nodes is i and the vaue of each of them i. Every node has exacty chidren. is taken to be an arbitrary integer number stricty arger than k (ater, we wi assign it a specific vaue); in particuar, since k, this impies >.

10 Figure 4: Genera ower bound: exampe for = 3 Observation A.. The tota vaue of a the nodes in a eve of the tree is ; thus the tota vaue of a the nodes in the tree is L +. Lemma A.2. For each node v on eve i, agorithm TM yieds t(v) = i L i ( ) k j L (i+), m(v) = i ( ) k j. Note that this impies that for every node v at eve i, t(v) > m(v). Proof. By induction on the appication of the agorithm TM (from eve L up to 0). Base: For each eaf v (i = L), t(v) = L, m(v) = 0. Hypothesis: For a node (job) v at eve i, t(v) = i L i ( ) k j L (i+), m(v) = i Step: Let u be a job at eve i. Then: t(u) = va(u) + t(v i ) = i + k i L i v i CN k (u) m(u) = max(t(v i ),m(v i )) = i L i v i CN (u) ( k ( k ( ) k j. ) j = i L i+ ) j = i L i ( ) k j ( ) k j Coroary A.3. The tota vaue of the k-bas returned by TM on the exampe is ess than k. Proof. Denote the root node v 0. The tota vaue of the k-bas is L ( ) k j ALG = max{t(v 0 ),m(v 0 )} = t(v 0 ) = < k = k. Proof of Theorem We have estabished in subsection 3.2 that the TM agorithm yieds the optima resut. If we take the exampe and set = 2k, we get: B } OPT = og k+ n + OPT k < 2k k 2k = 2 OPT OPT k = Ω(og k+ n) LOWER BOUND FOR PRICE IN k-bounded PREEMPTION PROBLEM In this appendix we estabish the ower bound for the price of k- bounded preemption by providing a specific exampe, as shown in figure 4. A the caims in this section pertain soey to said exampe. The exampe is constructed as foows: A the jobs are arranged in L + eves numbered from 0 to L, where L og 3 2 P. Leve contains jobs, numbered from 0 to. The m-th job at the -th eve is denoted j m. The vaue of a job j m is. The ength of a job j m is p () = P (3 2 ). The reative axity of a the jobs is λ = + 3. For a job j m, the jobs {j + m m m (m + ) } are caed the chid jobs of j m. This term can be triviay extended to introduce the notion of a descendant job. The reease times are given by the recursive formua r (+,m ) = r (,m) + (m m + ) p() p(+), where m m (m + ) and r (0,0) = 0. The deadine times are given by d (,m) = r (,m) + p () λ. From the above construction one can deduce the foowing (proofs are omitted and wi be presented in the fu version of the paper): Lemma B.. For every job j m, a singe preemption aows the scheduing of at most one of j m s chid jobs. Therefore, in a feasibe soution, a job is schedued with at most k chid jobs. Lemma B.2. OPT = L + > og 3 2 P = og P og (3 2 ) > 3 og P ( ) OPT k = L ( ) L+ i k k = < = i=0 k k k Proof of Theorem 4.3. Let us choose = 2k; then: OPT > 3 og 2k P } OPT k < 2k k 2k = 2 PoBP k > Ω(og k+ P) Proof of Theorem 4.3. We use the same exampe. Then L n = = L+ < L 3 2 L og n < L, =0 and if we, as in the previous paragraph, choose = 2k, we get } OPT > og 2k n OPT k < 2k k 2k = 2 PoBP k > Ω(og k+ n) Remark. For the mutipe machine setting, this exampe can be mutipied aong a third axis, effectivey forcing to sove the same basic exampe for each machine separatey. The Ω(og k+ P) bound wi hod as is; the Ω(og k+ n) wi hod for any fixed number of machines.