An Online Scalable Algorithm for Average Flow Time in Broadcast Scheduling

Transcription

1 An Online Scalable Algorithm for Average Flow Time in Broacast Scheuling SUNGJIN IM an BENJAMIN MOSELEY University of Illinois In this paper the online pull-base broacast moel is consiere. In this moel, there are n pages of ata store at a server an requests arrive for pages online. When the server broacasts page p, all outstaning requests for the same page p are simultaneously satisfie. We consier the problem of minimizing average (total) flow time online where all pages are unit-size. For this problem, there has been a ecae-long search for an online algorithm which is scalable, i.e. (1 + )-spee O(1)-competitive for any fixe > 0. In this paper, we give the first analysis of an online scalable algorithm. Categories an Subject Descriptors: F.2.2 [Analysis of Algorithms an Problem Complexity]: Nonnumerical Algorithms an Problems Sequencing an Scheuling General Terms: Algorithms, Theory, Design Aitional Key Wors an Phrases: Broacast, Average Flow Time, Online, Scalable. 1. INTRODUCTION We consier the pull-base broacast scheuling moel. In this moel, there are n pages available at a server an requests arrive for pages over time. The server must satisfy all requests. When the server broacasts a page p, all outstaning requests for the same page p are satisfie simultaneously. This is the main ifference from stanar scheuling settings where the server must process each request separately. The broacast moel is motivate by several applications such as multicast systems an wireless an LAN networks [Wong 1988; Acharya et al. 1995; Aksoy an Franklin 1999; Hall an Täubig 2003]. Besies the practical interest in the moel, broacast scheuling has seen growing interest in algorithmic scheuling literature both in the offline an online settings [Bar-Noy et al. 2002; Aksoy an Franklin 1999; Acharya et al. 1995; Bartal an Muthukrishnan 2000; Hall an Täubig 2003]. Work has also been one in stochastic an queueing theory literature on relate moels [Deb an Serfozo 1973; Deb 1984; Weiss 1979; Weiss an Pliska 1982]. In this paper we concentrate on the online moel with the goal of minimizing the total Authors aresses: S. Im an B. Moseley, Department of Computer Science, University of Illinois, 201 N. Goowin Ave., Urbana, IL 61801; im3@illinois.eu, bmosele2@illinois.eu. B. Moseley was partially supporte by NSF grant CNS S. Im was partially supporte by NSF grants CCF , CNS , an Samsung Fellowship. Permission to make igital/har copy of all or part of this material without fee for personal or classroom use provie that the copies are not mae or istribute for profit or commercial avantage, the ACM copyright/server notice, the title of the publication, an its ate appear, an notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to reistribute to lists requires prior specific permission an/or a fee. c 20YY ACM /20YY/ $5.00 ACM Journal Name, Vol. V, No. N, Month 20YY, Pages 1 0??.

2 2 Im an Moseley (or equivalently average) flow time 1. This is one of the most popular quality of service metrics. The ith request for page p will be enote J p,i. The request J p,i arrives at time a p,i an, in the online moel, this is when the server is first aware of the request. Time is slotte an a single page can be broacaste in a time-slot. Notice that this implies that all pages can be broacast in the same amount of time. Unit (or similar) processing time pages is popular in practice. This moel also captures the algorithmic ifficulty of the problem an this is almost exclusively the moel aresse in previous literature. The total flow i (f p,i a p,i ), where f p,i is the time when time of a given scheule can be written as p J p,i is satisfie. It was shown that without resource augmentation any online eterministic algorithm is Ω(n)-competitive [Kalyanasunaram et al. 2000]. Further, any ranomize online algorithm has a lower boun of Ω( n) on the competitive ratio [Bansal et al. 2005]. Due to these strong lowerbouns we focus on the resource augmentation moel [Kalyanasunaram an Pruhs 2000] where an algorithm A is given s 1 spee an is compare to an optimal offline solution that has 1 spee. We will let A s be the flow time accumulate for an algorithm A when given s spee; sometimes we will allow A s to enote the algorithm itself with s spee if there is no confusion in the context. In the resource augmentation moel, we say that A is s-spee r-competitive if A s ropt for all request sequences, where OPT is an optimal offline solution given 1 spee. The algorithmic ifficulty in broacast scheuling is that two algorithms may have to o ifferent number of broacasts to satisfy the same set of requests. For example, consier the algorithm most-requests-first (MRF) which broacasts the page that has the largest number of unsatisfie requests. This algorithm may seem like the most natural caniate for the problem. However, it was shown that MRF is not O(1)-competitive even when given any fixe extra spee [Kalyanasunaram et al. 2000]. A simple example shows that MRF may repeately broacast the same page, while ignoring requests which eventually accumulate a large amount of flow time. The optimal solution can take avantage of the broacast setting an satisfy the requests MRF was busy working on by a single broacast. This leaves the optimal solution free to work on other requests that are unsatisfie uner MRF s scheule. Further aing to the ifficulty of algorithmic evelopment, previous work has shown that the existence of a O(1)-spee O(1)-competitive online algorithm cannot be prove using stanar techniques. An algorithm A is sai to be locally competitive if the number of requests in A s queue is comparable to the number of requests in the aversary s queue at each time. In [Kalyanasunaram et al. 2000] it was shown that no online algorithm can be locally competitive with an aversary. Local competitiveness has been one of the most popular methos of analysis in stanar scheuling [Kalyanasunaram an Pruhs 2000; Becchetti et al. 2006; Kalyanasunaram et al. 2000]. Even though broacast scheuling has been stuie extensively over the last ecae, the complexity of the problem is yet to be well unerstoo. In the offline setting, minimizing average flow time was first stuie using non-trivial linear programming techniques couple with resource augmentation [Kalyanasunaram et al. 2000; Ganhi et al. 2004; Ganhi et al. 2006]. It was not until later that a complex reuction showe that this problem was in fact NP-Har [Erlebach an Hall 2004]. Recently, a simpler proof of this fact was foun [Chang et al. 2008]. Following this line of work, a (1+)-spee O(1)-approximation 1 Flow time is often referre to response time or wait time

3 An Online Scalable Algorithm for Average Flow Time in Broacast Scheuling 3 algorithm was eventually given in [Bansal et al. 2005]. Here, resource augmentation was use even though it is still open if the problem amits an O(1)-approximation. The problem is substantially more ifficult without resource augmentation. No non-trivial analysis was shown without resource augmentation until Bansal et al. gave a O( n)-approximation in [Bansal et al. 2005]. More recently, a O(log 2 n/ log log(n))-approximation was shown in [Bansal et al. 2008]. We note that this result relies on highly non-trivial algorithmic techniques. In the online setting, the strong lowerboun without resource augmentation has le previous work to focus on fining O(1)-spee O(1)-competitive algorithms. The ultimate goal of this line of work is to fin a (1 + )-spee O(1)-competitive algorithms (for any fixe > 0). That is, to show an algorithm that achieves O(1)-competitiveness with the minimum amount of extra resources. For this reason, an online algorithm, which is (1 + )-spee O(1)-competitive, is sai to be scalable. For problems which have strong lowerbouns without resource augmentation, fining a scalable algorithm is the best positive result that can be achieve using worst-case analysis. Previously, there have been two main approaches use to avoi a local argument, however both lines of work o not seem to suggest a way to obtain a scalable algorithm. The first was given by Emons an Pruhs in [Emons an Pruhs 2003]. They showe a nontrivial reuction from the problem of minimizing average flow time in broacast scheuling to a non-clairvoyant scheuling problem. Their reuction takes an algorithm A that is s-spee c-competitive for the non-clairvoyant scheuling problem an creates an algorithm B that is 2s-spee c-competitive for the broacast scheuling problem. Using this reuction, they were able to show an algorithm which is (4 + )-spee O(1)-competitive for minimizing the average flow time in broacast scheuling [Emons 2000; Emons an Pruhs 2003]. More recently, the same authors use this reuction to show another algorithm is (2 + )-spee O(1)-competitive [Emons an Pruhs 2009]. Both of these algorithms can be extene to the case where pages have varying sizes. Notice that a factor of 2 in the spee is lost in the reuction an, therefore, the reuction cannot be use to show a scalable algorithm. The algorithm longest-wait-first (LWF) was first introuce in [Kalyanasunaram et al. 2000]. LWF uses a natural scheuling policy which always scheules the page with the highest flow time. Emons an Pruhs showe that LWF is 6-spee O(1)-competitive using a irect analysis that avoie the use of the reuction [Emons an Pruhs 2005]. In this work, new novel techniques were introuce to avoi a local argument. The techniques presente in the paper were quite complex. In joint work with Chekuri, we were able to simplify these techniques to make the key ieas more transparent. Using this, we were able to show LWF is (4 + )-spee O(1)-competitive [Chekuri et al. 2009a]. In this paper, a generalization of these techniques will be presente which was mae possible by our previous simplification. However, LWF was shown to be n Ω(1) -competitive when given spee less than [Emons an Pruhs 2005]. Results: For the problem of minimizing total flow time in broacast scheuling, we give the first online scalable algorithm LA-W. We prove that LA-W is (1+)-spee O(1/ 11 )- competitive for any 0 < 1, giving a positive answer to a central open problem in the area. Our algorithm LA-W is similar to LWF in that it prioritizes pages with large flow time, however LA-W also gives preference to requests which have arrive recently.

4 4 Im an Moseley Favoring requests which have arrive recently has been shown to be useful in [Emons an Pruhs 2009]. The algorithm LA-W focuses on pages which have requests that arrive recently. This is funamentally ifferent from the algorithm given in [Emons an Pruhs 2009], which focuses on requests that arrive recently without consiering the page they are requesting. Unfortunately, in the broacast setting it is ifficult to categorize which pages have requests that arrive recently, since the arrival of requests can be scattere over time. To counter this, we evelop a novel an robust way to compare the arrival time of requests between two ifferent pages. Overview of the Algorithm: Let F p (t) be the total waiting time of unsatisfie requests for page p at time t an let F max (t) = max p F p (t). LWF scheules a page p such that F p (t) = F max (t). Notice that LWF scheules the page without consiering the number of outstaning requests for the page. Due to this, LWF may broacast a page with a relatively small number of unsatisfie requests which have been waiting to be scheule for a long perio of time. However, a page with a small number of requests oes not accumulate flow time quickly. In some cases, pages which have a large number of unsatisfie requests shoul be broacaste since these requests will rapily accumulate flow time. Using this insight, [Emons an Pruhs 2005] was able to show a lower boun of on the spee LWF require to be O(1)-competitive. Our algorithm LA-W keeps the main spirit of LWF by always broacasting pages with flow time comparable to F max (t) at each time t. However, amongst the pages with flow time comparable to F max (t), LA-W prioritizes pages with requests which have arrive recently. By prioritizing recent requests, we avoi the potentially negative behavior of LWF. This is because a page with requests that arrive recently must have a large number of outstaning requests to have flow time similar to F max. As mentione, we evelop a new way to compare the arrive time of requests for two ifferent pages. Using this technique, we will be able to break up time into intervals an show when requests arrive on these intervals, thus allowing us to etermine how LA-W an the optimal solution must behave on these intervals. The algorithm LA-W broacasts pages with unsatisfie requests that arrive recently to potentially fin pages which have a large number of outstaning requests. The reaer may woner why we chose pages in this manner when we coul simply broacast the page with many outstaning requests. In fact, we have consiere an algorithm which scheules the page with the largest number of outstaning requests amongst the pages with flow time comparable to F max (t). For this algorithm, we have establishe that it is scalable for the problem of minimizing the maximum weighte flow time in broacast scheuling [Chekuri et al. 2009b]. Further, we have preliminary evience that this algorithm is O(1) competitive for average flow time when given more than 2 spee. We however were unable to etermine its performance for average flow time when given less than 2 spee. Other Relate Work: Charikar an Khuller consiere a generalization of average flow time where the goal is to minimize the average flow time for a fraction of the requests [Charikar an Khuller 2006]. Besies work on minimizing the total flow time, other objective functions have been consiere in the broacast moel. In [Bartal an Muthukrishnan 2000; Chang et al. 2008; Chekuri et al. 2009b] a 2-competitive algorithm was given for the problem of minimizing the maximum response time. When each request has a ea-

5 An Online Scalable Algorithm for Average Flow Time in Broacast Scheuling 5 line, constant competitive algorithms were given by [Kim an Chwa 2004; Chan et al. 2004; Zheng et al. 2006; Chrobak et al. 2006] with the the objective of maximizing the number of requests satisfie by their ealines. For the problem of minimizing maximum weighte flow time, a (1 + )-spee O(1)-competitive algorithm was been given by [Chekuri et al. 2009b]. The elay factor is a metric closely relate to weighte flow time. For the problem of minimizing the maximum elay factor a (2 + )-spee O(1)- comeptitive algorithm was given by [Chekuri an Moseley 2009] an this was improve to a (1 + )-spee O(1)-competitive algorihtm in [Chekuri et al. 2009b]. For the problem of minimizing the L k -norms of flow time an the elay factor [Chekuri et al. 2009a] gave O(k)-spee O(k)-competitive algorithms. 2. TIME MODEL AND ALGORITHM We assume that all requests have unit processing time an without loss of generality this is 1. In our time moel we assume that requests arrive only at non-negative integer times. Any scheuling algorithm A with spee s 1 scheules a page every 1/s time-steps starting from time 0. When A broacasts a page p at time t, all alive (unsatisfie) requests for page p which arrive strictly earlier than t are immeiately satisfie by the broacast. If J p,i is a request satisfie by a broacast, it has flow time t a p,i. Note that uner the scheule prouce by the optimal solution with 1-spee, every request has flow time at least 1. On the other han, A with spee s > 1 may finish some requests within a elay less than one. Though it woul seem fair to force A to scheule requests after at least one time step, we o not assume this because our analysis will hol in either case an this assumption improves the reaability of the analysis. Before introucing our algorithm, we state notation that will be use throughout the paper. For any time interval starting at b an ening at e, we let I = e b. For a set of requests R, we will let F (R) be the flow time accumulate for the requests in R by our algorithm. For a page p we will let F p (t) be the total flow time accumulate at time t for unsatisfie requests for page p. We will let F (R, t) be the total flow time accumulate by our algorithm for requests in the set R at time t. Note that if some requests in R arrive after time t then these requests o not contribute to the value of F (R, t). We let F (R) enote the total flow time OPT accumulates for a set of requests R. Fig. 1. R p(t) enotes the alive requests of page p at time t, i.e. the requests of page p which arrive uring [L(p, t), t]. Likewise, R p(τ β p (t)) enotes the requests which arrive uring [L(p, t), τ β p (t)]. We now introuce our algorithm, enote by LA-W for Latest Arrival time with Waiting. We assume that LA-W is given s = 1 + spee where 0 < 1 is a fixe constant. Our algorithm is parameterize by constants c > 1 an β < 1 epening on, later we

6 6 Im an Moseley ( will efine β = 1000 ) 4 an c = ( ). For each page p an time t, let R p (t) enote the set of alive requests for page p at time t. Let L(p, t) be the last time before time t that our algorithm broacaste page p. If there is no such time then L(p, t) = 0. Note that R p (t) is equivalent to the set of the requests for page p which arrive uring [L(p, t), t]. For a page p an time t let τ β p (t) = argmin L(p,t) t t(f (R p (t ), t) (1 β)f p (t)). In other wors, τ β p (t) enotes the earliest time t no later than time t an no earlier than time L(p, t) such that the requests in R p (t ) have total flow time at least (1 β)f p (t) at time t. By this efinition, if R [L(p,t),τ β p (t)] is the set of requests for page p that arrive on the interval [L(p, t), τp β (t)] an R [L(p,t),τ β p (t)) is the set of requests for page p that arrive on the interval [L(p, t), τ β p (t)) then F (R [L(p,t),τ β p (t)], t) (1 β)f p(t) an F (R [L(p,t),τ β p (t)), t) < (1 β)f p(t). See Figure 1. Algorithm: LA-W Let t be a time where our algorithm is not broacasting a page. Let F max (t) = max p F p (t). Broacast one page accoring to Rule 2 every 10 broacasts, an broacast one page accoring to Rule 1 otherwise. Rule 1: broacast the page p = argmax p Q(t)τ β p (t), where Q(t) = {q F q (t) 1 c F max(t)} breaking ties arbitrarily. Rule 2: broacast a page p where F p (t) = F max (t) breaking ties arbitrarily. Our algorithm LA-W broacasts pages mainly accoring to Rule 1 while occasionally broacasting a page accoring to Rule 2. The secon rule uses LWF s scheuling policy which broacasts a page with the highest flow time. The first rule chooses a page p with the latest time τ β p (t) among the pages with flow time close to F max (t). The value of τ β p (t) can be interprete as the latest arrival time of any unsatisfie request for page p after iscounting requests that arrive recently that have small flow time. Since the arrival of requests for the same page p can be scattere over time, we will use τ β p (t) as the representative arrival time of those requests. Notice that if all requests for page p arrive at time t then τ β p (t) = t for any 0 < β 1. We remark that we o not know if Rule 2 is neee for LA-W to be (1 + )-spee O(1)-competitive. Rule 2 will play a crucial role in our analysis, but we o not have a proof that Rule 1 alone performs poorly. 3. ANALYSIS Fig. 2. Events for page p.

7 An Online Scalable Algorithm for Average Flow Time in Broacast Scheuling 7 Let σ be a fixe sequence of requests. OPT enotes a fixe offline optimal solution. We assume LA-W 1+ is always busy scheuling pages for the sequence σ. Otherwise, our arguments can be applie to each maximal time interval where LA-W 1+ is busy. Following the lea of [Emons an Pruhs 2005; Chekuri et al. 2009a], time is partitione into events for each page p. Events for page p are efine by LA-W 1+ s broacasts of page p. Each time LA-W 1+ broacasts a page, an event begins an an event ens. An event E p,x = b p,x, e p,x begins at time b p,x an ens at time e p,x. Here, LA-W 1+ broacasts page p at time b p,x an at time e p,x. These are the xth an (x + 1)st broacasts of page p by LA-W 1+. The (x + 1)st broacast of page p starts a new event E p,x+1 an e p,x = b p,x+1. On the time interval (b p,x, e p,x ) LA-W 1+ oes not broacast page p. The optimal solution can broacast page p zero or more times uring an event E p,x. See Figure 2. For an event E p,x, let R p,x enote the set of requests satisfie by the (x + 1)st broacast of page p. Notice that all requests in R p,x arrive uring E p,x, formally uring [b p,x, e p,x ). Let F p,x = F (R p,x ) be the total flow time LA-W 1+ accumulates for requests in R p,x. Likewise let Fp,x = F (R p,x ) be the flow time OPT accumulates for requests in R p,x. We refer to F p,x as the flow time of E p,x. Similarly to requests, for a set E of events we let F (E) = E F p,x E p,x. Our goal is to show that p x F p,x O(1)OPT. We start by partitioning events into two groups. An event E p,x is calle self-chargeable if F p,x γfp,x where γ 1 is a constant that will be fixe as γ = β later. Let S be the set of all self-chargeable events. The other events are calle non-self-chargeable an are in the set N. By efinition of self-chargeable events, we can easily boun F (S) by OPT. LEMMA 3.1. F (S) γopt. PROOF. F (S) = E p,x S F p,x E p,x S γf p,x γopt. We now concentrate on non-self-chargeable events. Notice that for a non-self-chargeable event E p,x, the optimal solution must broacast page p uring E p,x, formally uring (b p,x, e p,x ). Otherwise, Fp,x F p,x an the event is self-chargeable. We further partition non-self-chargeable events into two classes. Consier a non-self-chargeable event E p,x. Let α an k be constants such that α < 1, k > 1 an βk < 1. We will fix α = an k = 10 ( 1000c + 2) + 1 later. E p,x is in the set N 1 if for some β ρ βk it is the case that at least αs(e p,x τp ρ (e p,x )) self-chargeable events en on the interval [τp ρ (e p,x ), e p,x ). Notice that the time τp ρ (e p,x ) exists because ρ < 1. A non-self-chargeable event not in N 1 is in N 2. The sets N 1 an N 2 are similar to how [Chekuri et al. 2009a] partitions non-self-chargeable events. The events in N 1 can easily be boune by OPT. We o this by bouning F (N 1 ) by the flow time of the self chargeable events ening uring the events in N 1. Knowing that F (S) γopt we will be able to boun F (N 1 ) by OPT. We will formally show F (N 1 ) O( 1 11 )OPT later in Lemma 3.7. The most interesting events are those which are in N 2. Since each event E p,x in N 2 has a relatively small number of self-chargeable events ening uring E p,x, we cannot irectly boun F (N 2 ) by OPT. Instea, we will show that the total flow time of events in N 2 accounts for only a fraction of LA-W 1+ s total flow time, i.e. F (N 2 ) δla-w 1+ for some constant δ < 1 which is inepenent of. In [Chekuri et al. 2009b] an [Emons an Pruhs 2005] spee greater than 3.4 was neee to boun F (N 2 ). Our goal is to ensure 100

8 8 Im an Moseley δ < 1 with only (1 + ) spee. Showing this will complete our analysis as follows. Using this, Lemma 3.1 an Lemma 3.7, we have that LA-W 1+ = F (S) + F (N ) = F (S) + F (N 1 ) + F (N 2 ) γopt + O( 1 )OPT + δla-w 11 1+, which simplifies to LA-W 1+ γ+o( 1 11 ) OPT. This will imply the following theorem. 1 δ THEOREM 3.2. For 0 < 1, the algorithm LA-W is (1+)-spee O( 1 11 )-competitive for minimizing average flow time in broacast scheuling with unit size pages. Before continuing, we show some properties of events in N. Say that we set γ 1 β. Then it is not har to show that OPT must broacast page p uring I = [τ β p (e p,x ), e p,x ) for any non-self-chargeable event E p,x. Inee, the requests for page p that arrive uring the interval I have total flow time at least βf p,x in LA-W 1+ s scheule by efinition of τ β. If OPT oes not broacast page p uring I this implies that these requests also have total flow time βf p,x in OPT s scheule. However, then F p,x βf p,x 1 γ F p,x, contraicting the fact that E p,x is non-sef-chargeable. LEMMA 3.3. Suppose that γ 1 β. Then, for any non-self-chargeable event E p,x, the optimal solution must broacast page p uring the interval [τ β p (e p,x ), e p,x ). PROOF. For the sake of contraiction assume the lemma is false. The event E p,x is nonself-chargeable therefore the optimal solution must broacast page p at some time uring (b p,x, τ β p (e p,x )). Let t be the latest broacasting time of page p by the optimal solution uring (b p,x, τ β p (e p,x )). Let S [bp,x,t] an S (t,ep,x) be the set of requests for page p which arrive uring [b p,x, t] an (t, e p,x ), respectively. We know that F (S [bp,x,t]) < (1 β)f p,x by efinition of τ β p (e p,x ) an t < τ β p (e p,x ). Thus F (S (t,ep,x)) = F (R p,x \ S [bp,x,t]) > βf p,x. Since the optimal solution oes not broacast page p uring (t, e p,x ), it follows that F p,x F (S (t,ep,x)) > βf p,x 1 γ F p,x, which is a contraiction to E p,x being a non-self-chargeable event. Now say that we set γ β. Using similar ieas as in Lemma 3.3, we will be 2 able to show that [τp β (e p,x ), e p,x ) This will be use to ensure that the intervals 2 consiere in our remaining arguments are sufficiently long. LEMMA 3.4. Suppose γ β. Then, for any non-self-chargeable event E p,x, [τ β p (e p,x ), e p,x ] PROOF. For the sake of contraiction, assume that there exists a non-self-chargeable event E p,x such that [τp β (e p,x ), e p,x ] < Let S be the set of requests for page p 2 which arrive on the interval [τp β (e p,x ), e p,x ). By efinition of τp β (e p,x ) it must be the case that F (S) > βf p,x. We now want to boun the number of requests in S. Since each request in S can accumulate flow time at most [τp β (e p,x ), e p,x ] < 10000, we have that 2 F (S) < S 10000, thus βf 2 p,x < S Hence we have that S > βf p,x. The optimal solution must accumulate at least S flow time for the requests in S, therefore Fp,x S > βf p,x 1 γ F p,x. This is a contraiction to E p,x being non-selfchargeable. To boun F (N 1 ) an F (N 2 ), we nee the following two lemmas. For any event E p,x, the first lemma will be use to boun the flow time accumulate for page p at ifferent times uring E p,x. This will help us to compare the flow time of E p,x to the flow time of

9 An Online Scalable Algorithm for Average Flow Time in Broacast Scheuling 9 events ening uring E p,x. The proof of this lemma follows easily by efinition of flow time. LEMMA 3.5. For any event E p,x, let R R p,x. Let t be such that b p,x t < e p,x. Suppose that all requests in R arrive no later than time t. Then for any 0 η < 1, F (R, t+η(e p,x t)) ηf (R ). Further, if F (R ) υf p,x, then F (R, t+η(e p,x t)) ηυf p,x. PROOF. F (R, t+η(e p,x t)) = J p,i R (t+η(e p,x t) a p,i ) = J p,i R ((1 η)t+ ηe p,x a p,i ) J p,i R ((1 η)a p,i+ηe p,x a p,i ) = η J p,i R (e p,x a p,i ) = ηf (R ). The inequality hols, since any request J p,i in R arrives no later than time t. The next lemma gives a global charging scheme built on Hall s theorem, which is a generalization of the techniques use in [Emons an Pruhs 2005; Chekuri et al. 2009a].This lemma shows how to charge the flow time of some events to the total flow time LA-W 1+ accumulates. The proof is technical an we efer its proof to Section 3.4. LEMMA 3.6. Let A be a set of events. Let µ, κ > 0 be some constants. Let λ 1 be an integer. For each event E p,x A, suppose there exists an interval I p,x an a set of events B p,x such that The optimal solution broacasts page p at least λ times uring the interval I p,x. Further, I p,x is isjoint with I p,x for any E p,x A s.t. x x. B p,x µ I p,x an E q,y B p,x only if e q,y I p,x an F q,y κf p,x. Let B = (p,x):e p,x A B p,x an = min Ep,x A I p,x. Then, F (A) ( 2 λκµ ( 2 λκµ +1 )( )LA-W )( )F (B) This lemma can be interprete as follows. For a set of events A N 2, we charge the flow time of each event E p,x A to some events ening uring I p,x. In our analysis, I p,x will always be a subinterval of E p,x ; thus for any fixe page p, {I p,x E p,x A} are isjoint. If the following conitions hol for each event E p,x A, then F (A) 2 LA-W 1+. (1) There are at least λ broacasts by OPT of page p uring I p,x. (2) We can fin a sufficiently large fraction of events ening uring I p,x, enote by µ, such that each of these events have flow time at least κf p,x. (3) I p,x is sufficiently long for all E p,x A. The boun we get on F (A) improves by either fining many broacasts of page p by OPT uring I p,x or by fining sufficiently many events with very large flow time ening uring I p,x. Using the global charging scheme given in Lemma 3.6, we can boun the flow time of events in N 1 by OPT. This is not too ifficult an follows easily by combining the analysis given in [Chekuri et al. 2009a], the efinition of τ an Lemma 3.6. The proof is similar to that given in [Chekuri et al. 2009a]. LEMMA 3.7. F (N 1 ) O( 1 11 )OPT. PROOF. We apply Lemma 3.6 using the notation given in the lemma. Let A be the set of all N 1 events. Consier any event E p,x A. Let I p,x = [τp ρ (e p,x ), e p,x ) for some fixe β ρ β( 10 ( 1000c + 2) + 1) such that at least αs(e p,x τ ρ (e p,x )) self-chargeable events en on I p,x. Note that ρ exists by efinition of N 1 events. By Lemma 3.3, the 2 A B shoul rea as A < ξb for some constant ξ < 1.

10 10 Im an Moseley optimal solution must broacast page p uring I p,x. Due to this we set λ = 1. Since I p,x by Lemma 3.4, we have = min 2 Ep,x A I p,x Let B p,x be the self-chargeable events ening uring I p,x = [τp ρ (e p,x ) + α 2 (e p,x τp ρ (e p,x )), e p,x ). Note that there are at most αs 2 I p,x events ening uring I p,x \ I p,x, because the algorithm broacasts a page every 1 s time steps uring I p,x \ I p,x. Therefore there exist at least αs I p,x αs 2 I p,x αs 2 I p,x αs 4 I p,x self-chargeable events ening uring I p,x. Hence, B p,x αs 4 I p,x an we can set µ = αs 4. Let E q,y B p,x. By Lemma 3.5 an the efinition of τp ρ (e p,x ) we know that at anytime t I p,x it is the case that F p,x (t) α 2 (1 ρ)f p,x. Since our algorithm chose to broacast page q at time e p,x I p,x over page p, we have F q,y α 2c (1 ρ)f p,x. Therefore we can set κ = α 2c (1 ρ). In sum, by Lemma 3.6, F (N 1 ) 2 λκµ c F (S) = ( α 2 s )( 1 1 ρ ) (γopt) = O( )OPT. 11 We now focus on bouning the flow time of events in N 2. To exploit Lemma 3.6, N 2 is partitione into three isjoint sets T 1, T 2 an T 3. To iscuss the high level interpretation of the sets T 1, T 2 an T 3 we fix an event E p,x N 2 an page p an rop the subscript p, x. For the event E we will consier ifferent subintervals of E efine by τ. Let I i 10 β( = [τ i+1) (e), e) for i N. Notice that I i is a subinterval of I i+1 for all i. We will concentrate on the intervals I i for ifferent values of i. Concentrating on these intervals will allow us to break up the event E so that we can better unerstan when the requests for page p arrive uring E an how the optimal solution an LA-W 1+ behave uring E. The event E will be in the set T 1 if for some i it is the case that page p is not in the queue Q for a sufficiently large number of broacasts by LA-W 1+ uring I i. By efinition of Q, if p is not in Q(t) then there exists another page q such that F q (t) > cf p (t). Rule 2 of LA-W broacasts a page with the highest flow time every 10 broacasts. Using this, we will be able to fin sufficiently many events ening uring E with flow time much larger than the flow time of event E. Then Lemma 3.6 can be use to show that F (T 1 ) LA-W 1+. Intuitively, the requests in T 1 cannot account for most of LA-W 1+ s flow time since there exists other events with flow time much larger than those in T 1. If the event E is not in the set T 1 an if the length of I i+1 is sufficiently longer than the length of I i for many ifferent values of i then the event E will be in the set T 2. For such an event E, the requests for page p that arrive uring E will be groupe accoring to when they arrive. We will show that each of these groups contributes to a substantial amount of event E s flow time. Knowing that E is non-self-chargeable, we will show that OPT must perform a unique broacast of page p for each of these groups uring E. This allows us to show that F (T 2 ) LA-W 1+ using Lemma 3.6. Intuitively, since the optimal solution has to perform a lot of broacasts for each event in T 2, there cannot be many events in T 2. Therefore the events in T 2 o not account for a large portion of LA-W 1+ s flow time. Finally T 3 will consist of all events in N 2 that are not in T 1 or T 2. Using the efinitions of T 1, T 2 an τ we will be able to show that no events can be in T 3 an this will complete our analysis. Showing that T 3 = is the most ifficult part of the analysis an this is where Rule 1 an resource augmentation plays a crucial role. We now formally efine

11 An Online Scalable Algorithm for Average Flow Time in Broacast Scheuling 11 the sets T 1, T 2 an T 3. For simplicity of notation, let τp,x β,i = τ E p,x is in 10 β( i+1) p (e p,x ). A N 2 event T 1 if an only if for some 0 i 1000c + 2 the page p is not in Q for at least s β,i 10 [τp,x, e p,x ) broacasts by our algorithm on the interval [τp,x, β,i e p,x ). T 2 if an only if E p,x / T 1 an for all 0 i 1000c, τp,x β,i τp,x β,i+1 10 (e p,x τp,x) β,i T 3 otherwise. We note that if β an c are chosen such that β( 10 ( 1000c + 2) + 1) < 1, then the time τp,x β,i must exist for all 0 i 1000c +2. The rest of the paper is organize as follows. In Section 3.1 we will show that F (T 1 ) LA-W 1+. Then in Section 3.2 we will show that F (T 2 ) LA-W 1+. Finally we will show that T 3 = in Section 3.3. Before continuing, we fix our constants, so that our arguments can be verifie. As alreay mentione, we let β = ( 1000 )4, c = 10000, γ = β, α = an k = ( 1000c + 2) + 1. Note that τp,x β, 1000c +2 = τp βk (e p,x ) for any page p by efinition of k an τp,x. β,i Recall that our algorithm is parameterize by β an c. Here we have chosen c an β so that the analysis is reaable an easy to verify an not to optimize the analysis. 3.1 Bouning T 1 events. In this section we boun F (T 1 ). By efinition of T 1, for each event E p,x T 1 the page p is not in Q for at least s 10 [t p,x, e p,x ) broacasts by LA-W 1+ on the interval [t p,x, e p,x ) where t p,x = τp,x β,i for some fixe 0 i 1000c + 2. Recall that our goal is to show that there are many events ening uring E p,x with flow time much larger than F p,x. After fining these events, we will charge F p,x to these events. We begin by actually fining such events in the next lemma. LEMMA 3.8. For an event E p,x T 1 there exist at least ( 2 s 205 ) [t p,x, e p,x ) events ening on the interval [t p,x, e p,x ) with flow time at least c 20 (1 βk)f p,x. PROOF. Let S [bp,x,t p,x] be the requests for page p which arrive uring [b p,x, t p,x ]. By the efinition of t p,x an τ, we have F (S [bp,x,t p,x]) (1 β( 10 i + 1))F p,x (1 βk)f p,x. Let I = [t p,x + 20 (e p,x t p,x ), e p,x ). For any time t I, by Lemma 3.5, F (S [bp,x,t p,x], t) 20 (1 βk)f p,x. (1) By efinition of T 1, there are at least s 10 (e p,x t p,x ) broacasts by our algorithm on the interval [t p,x, e p,x ) where page p is not in Q. At most s 20 (e p,x t p,x ) of these broacasts en on the interval [t p,x, t p,x + 20 (e p,x t p,x )). Therefore, there are at least s 10 (e p,x t p,x ) s 20 (e p,x t p,x ) s 20 (e p,x t p,x ) broacasts by our algorithm on the interval I where page p is not in Q when these broacasts were scheule. Now consier a time t I where page p is not in Q(t). By efinition of Q, at time t there must exist some page q such that F q (t) cf p (t). Our algorithm scheules the page with the largest flow time every broacasts accoring to Rule 2. Therefore, within broacasts some page q where F q (t) cf p (t) is broacaste by the algorithm. Using this an (1), there exists an event E q,y with flow time at least F q,y > c 20 (1 βk)f p,x such that

12 12 Im an Moseley e q,y [t, t + 1 s 10 ). Using Lemma 3.4 to ensure the interval [t p,x, e p,x ) is sufficiently long, we conclue that there exist at least ( s 20 (e p,x t p,x ) / 10 ) ( 2 s 205 ) [t p,x, e p,x ) events ening uring I with flow time at least c 20 (1 βk)f p,x. We can now easily boun F (T 1 ) by LA-W 1+ using lemmas 3.6, 3.8, 3.3 an 3.4. LEMMA 3.9. F (T 1 ) < LA-W 1+. PROOF. We apply Lemma 3.6 using the notation given in the lemma. Consier any E p,x T 1. Let I p,x = [t p,x, e p,x ). We know that the optimal solution must broacast page p at least once on the interval [t p,x, e p,x ) by Lemma 3.3, since [τp β (e p,x ), e p,x ) is a subinterval of [t p,x, e p,x ). So we can set λ = 1. By Lemma 3.8 we have that for any event E p,x T 1 there exist at least 2 s 205 [t p,x, e p,x ) events ening on the interval [t p,x, e p,x ) of flow time at least c 20 (1 βk)f p,x. If we let the set B p,x consist of these events, we can set µ = 2 s c 205 an κ = 20 (1 βk). Using Lemma 3.4 we know that I p,x 10000/ 2 an therefore = min Ep,x A I p,x 10000/ 2. Thus we have F (T 1 ) κµλ LA-W = ( 50(1 + ) )( 1 1 βk ) + 1 LA-W 1+ < LA-W Bouning T 2 events. Fig. 3. For any event E p,x in T 2, OPT must broacast page p uring [t 1, t 3 ). In this section, we boun F (T 2 ). Recall that our goal is to show that for any event E p,x T 2, the optimal solution must broacast page p many times uring E p,x. To fin these broacasts by the optimal solution, we break up each event E p,x T 2 into the time intervals [τp,x β,i+2, τp,x). β,i By efinition of τ, we know that the requests for page p that arrive uring [τp,x β,i+2, τp,x β,i+1 ] account for a substantial portion of the flow time of event E p,x. Knowing this an that the length of [τp,x β,i+1, τp,x) β,i is sufficiently long by efinition of events in T 2, we will be able to show that the optimal solution must broacast page p uring [τp,x β,i+2, τp,x). β,i Otherwise, these requests wait for a sufficiently long time to be scheule by OPT an, therefore, OPT must accumulate flow time at least 1 γ F p,x for these requests. This contraicts the fact that events in T 2 are non-self-chargeable. LEMMA Let E p,x be an event in T 2. For any integer i s.t. 0 i 1000c, the optimal solution must broacast page p uring the interval [τp,x β,i+2, τp,x). β,i PROOF. For any fixe integer i such that 0 i 1000c, let t 1 = τp,x β,i+2, t 2 = τp,x β,i+1, an t 3 = τp,x. β,i Note that t 3 t 2 10 (e p,x t 3 ) an t 1 < t 2 < t 3, since E p,x T 2. See Figure 3. Let S [t1,e p,x), S (t2,e p,x) an S [t1,t 2] be the set of requests for page p which arrive

13 An Online Scalable Algorithm for Average Flow Time in Broacast Scheuling 13 on the intervals [t 1, e p,x ), (t 2, e p,x ) an [t 1, t 2 ], respectively. By efinition of t 1 an t 2, we have that F (S [t1,e p,x)) > β( 10 (i + 2) + 1)F p,x an F (S (t2,e p,x)) β( 10 (i + 1) + 1)F p,x. Thus we have, F (S [t1,t 2]) = F (S [t1,e p,x)) F (S (t2,e p,x)) > 10 βf p,x. (2) With the fact t 3 t 2 10 (e p,x t 3 ), the fact that the requests in S [t1,t 2] arrive by time t 2, an (2), by applying Lemma 3.5 we have F (S [t1,t 2], t 3 ) ( 10 )(10 )βf p,x = βf p,x. (3) For the sake of contraiction, suppose that the optimal solution oes not broacast page p on the interval [t 1, t 3 ). Then F p,x F (S [t1,t 2], t 3 ) βf p,x 1 γ F p,x. (4) This is a contraiction to E p,x being non-self-chargeable. COROLLARY For each event E p,x T 2, the optimal solution broacasts page p at least 500c times uring the interval [τ βk p,x, e p,x ). At this point, we have shown the most interesting property of events in T 2 an we are almost reay to boun F (T 2 ). Before bouning F (T 2 ), we first fin events to charge to. For each event E p,x T 2, we want to charge F p,x to some events ening uring [τp,x, βk e p,x ) because we know OPT broacasts page p many times uring this interval. Knowing that LA-W always broacasts the page with flow time close to the highest flow time, we can easily fin events ening uring [τp,x, βk e p,x ) with sufficiently large flow time. LEMMA Consier any event E p,x T 2. Let I p,x = [τ βk p,x, e p,x ). There exist at least (1 + ) I p,x events ening uring I p,x with flow time at least 1 2c (1 βk)f p,x. PROOF. Let I p,x = [τp,x βk (e p,x τp,x), βk e p,x ). Note that there are at least (1 + ) 1 2 I p,x (1+) I p,x events ening uring I p,x; the inequality is ue to Lemma 3.4 to ensure I p,x is sufficiently long. Let E q,y be an event such that e q,y I p,x. We now show that F q,y 1 2c (1 βk)f p,x. By Lemma 3.5 an the efinition of τp,x βk we have F p (e q,y ) 1 2 (1 βk)f p,x. Since our algorithm chose page q over page p at time t, accoring to either Rule 1 or Rule 2, F q,y 1 c F p(e q,y ). Hence we conclue that F q,y 1 2c (1 βk)f p,x. Finally we boun the flow time of T 2 events by charging an event E p,x T 2 to the events we foun in Lemma Notice that the events we are charging to can have flow time less that F p,x, but we counter this by fining many broacasts of page p by OPT uring E p,x. LEMMA For 0 < 1, F (T 2 ) < LA-W 1+. PROOF. We apply Lemma 3.6. Let E p,x T 2 an I p,x = [τp,x, βk e p,x ). By Corollary 3.11 we can set λ = 500c. By letting B p,x be the set of events foun for E p,x in Lemma 3.12, we can set κ = c (1 βk) an µ = 100 (1 + ). Using Lemma 3.4 we know that I p,x 10000/ 2 an therefore = min Ep,x A I p,x 10000/ 2. The esire result follows by simple calculation.

14 14 Im an Moseley 3.3 There are no events in T 3. Fig. 4. For an event E p,x in T 3, uring [t 1, e p,x) OPT must make a unique broacast for most events which en uring [t 3, e p,x). In this section we show T 3 =. For the sake of contraiction suppose that T 3 is non-empty. Fix an event E p,x T 3. For some fixe 0 i 1000c we have that τp,x β,i τp,x β,i+1 < 10 (e p,x τp,x) β,i because E p,x / T 2. Let t 1 = τp,x β,i+1 an t 2 = τp,x. β,i Let t 3 = t (e p,x t 2 ). Let E be all the non-self-chargeable events ening uring [t 3, e p,x ) which were scheule by Rule 1 when page p was in Q. Our goal is to show that OPT must make a unique broacast for each event in E on the interval [t 1, e p,x ). Then it will be shown that E > [t 1, e p,x ) + 1 by showing E (1 + ) [t 3, e p,x ) > [t 1, e p,x ) + 1. Since OPT has 1 spee, this will show that OPT cannot complete these broacasts on the interval [t 1, e p,x ). This contraiction will imply that T 3 =. See Figure 4. Recall that by Lemma 3.3, for any E q,y E, the optimal solution must broacast page q on the interval [τq β (e q,y ), e q,y ) because E q,y is non-self-chargeable. Further, note that such broacasts are unique to E q,y, i.e. not containe in E q,y for any y y because E q,y an E q,y are isjoint by efinition. For any E q,y E, if we show that τq β (e q,y ) [t 1, e p,x ) then we will know that OPT performs these broacasts on [t 1, e p,x ). This is where Rule 1 will play a crucial role in our analysis. We will first show that τp β (t) t 1 for all times t [t 3, e p,x ). By efinition, if page q was scheule by Rule 1 an page p was in Q(t) then τp β (t) τq β (t). Hence, for any E q,y E we will have that t 1 τp β (e q,y ) τq β (e q,y ) an OPT broacasts page q on [t 1, e p,x ). LEMMA For the event E p,x T 3, at any time t [t 3, e p,x ), τ β p (t) t 1. PROOF. For the sake of contraiction assume that τp β (t) < t 1. Let t = τp β (t). Note that t < t 1 t 2 < t < e p,x. Let S [t1,e p,x), S (t2,e p,x) an S [t1,t 2] be the set of requests which arrive for page p on the intervals [t 1, e p,x ), (t 2, e p,x ), an [t 1, t 2 ], respectively. By efinition of t 1 an t 2, we have F (S [t1,e p,x)) > β( 10 (i + 1) + 1)F p,x an F (S (t2,e p,x)) β( 10 i + 1)F p,x. Hence, F (S [t1,t 2]) = F (S [t1,e p,x)) F (S (t2,e p,x)) > 10 βf p,x. (5) By the efinition of t = τ β p (t), we have F (S (t,t 2], t) F (S (t,t], t) βf p (t) βf p,x. Since t t (e p,x t 2 ), by Lemma 3.5, 9 F (S (t,t 2], e p,x ) F (S (t,t 2], t). Thus we have, F (S (t,t 2]) = F (S (t,t 2], e p,x ) 9 F (S (t,t 2], t) 9 βf p,x. (6) Knowing that F (S (t,t 2]) F (S [t1,t 2]), this is a contraiction to (5).

15 An Online Scalable Algorithm for Average Flow Time in Broacast Scheuling 15 Finally we are reay to show that T 3 =. This lemma follows by using the previous lemma an counting the number of broacasts the optimal solution must o on the interval [t 1, e p,x ). It is in the next lemma that we rely strongly on resource augmentation. LEMMA It must be the case that T 3 =. PROOF. Recall that E is the set of all the non-self-chargeable events ening uring [t 3, e p,x ) which were scheule by Rule 1 when page p was in Q. We first show E > s( )(e p,x t 2 ) by a simple counting argument. We know that at least s(1 9 )(e p,x t 2 ) events en uring [t 3, e p,x ) by efinition of t 3 an t 2. Among these events we know that at most αs(e p,x t 2 ) events are self-chargeable, since E p,x N 2 ; at most s(e p,x t 2 ) / s 900 (e p,x t 2 ) broacasts are scheule by Rule 2, since our algorithm performs accoring to Rule 2 every 10 broacasts (the inequality is ue to Lemma 3.4); an at most s 10 (e p,x t 2 ) events were scheule when p is not in the queue Q, since E p,x / T 1. By subtracting these numbers from the number of events ening uring I p,x an knowing that (e p,x t 2 ) by Lemma 3.4, we have 2 ( )(1 + )(e p,x t 2 ) E. (7) Knowing that t 2 t 1 < 10 (e p,x t 2 ), we have [t 1, e p,x ] < ( )(e p,x t 2 ). (8) As iscusse previously, Lemma 3.14 implies that OPT must make a unique broacast for each event in E uring [t 1, e p,x ). Since the optimal solution has 1 spee, with Lemma 3.4, it must be the case that E [t 1, e p,x ) + 1 ( ) [t 1, e p,x ). (9) By combining (7), (8), an (9), we have that ( )(1 + ) < ( )(1 + any 0 < 1 this is not true, so we obtain a contraiction ). For By lemmas 3.9, 3.13 an 3.15 we have that F (N 2 ) LA-W 1+. The proof of Theorem 3.2 follows easily by combining this an lemmas 3.1 an 3.7. To complete the analysis it only remains to prove Lemma 3.6. This is one in the next section. 3.4 Proof of Lemma 3.6 Here we prove Lemma 3.6. The proof of this lemma relies on a generalization of Hall s theorem. This generalization of Hall s theorem was implicitly use in [Emons an Pruhs 2005], an later formalize in [Chekuri et al. 2009a]. DEFINITION [Chekuri et al. 2009a][g-covering] Let G = (X Y, E) be a bipartite graph whose two parts are X an Y, respectively. Let l : E [0, 1]. We say {l u,v } is a g-covering if u X, v Y l u,v = 1 an v Y, u X l u,v g. The following lemma follows easily from either Hall s Theorem or the Max-Flow Min- Cut Theorem. LEMMA [Chekuri et al. 2009a][Fractional Hall s theorem] Let G = (V = X Y, E) be a bipartite graph whose two parts are X an Y, respectively. For a subset S of

16 16 Im an Moseley X, let N G (S) = {v Y uv E, u S}, be the neighborhoo of S. For every S X, if N G (S) 1 g S, then there exists a g-covering for X. Now we are reay to prove Lemma 3.6. Proof of [Lemma 3.6] We start by creating a bipartite graph G = (X Y, E). There is one vertex u p,x X for each event E p,x A an there is a vertex v q,y Y for each event in E q,y B. Let u p,x X an v q,y Y. There is an ege connecting u p,x an v q,y if an only if E q,y B p,x. For any set Z X, let I(Z) be the set of intervals corresponing to events in Z, i.e. I(Z) = { I p,x u p,x Z}. We let I(Z) enote the union of intervals in I(Z). We enote the sum of length of maximal subintervals in I(Z) by I(Z). We will now show that G has a (( 2 ))-covering for X. Consier any fixe set Z X. We know the optimal solution must perform λ unique broacasts for each event in Z uring I(Z) an these broacasts can only occur at integral times by efinition of OPT. We observe that each maximal interval I in I(Z) contains at most I + 1 = I +1 I I +1 I integers. Thus it follows that I(Z) contains at most λµ )( I(Z) integers. Therefore we have λ Z + 1 I(Z). (10) From now on, for simplicity, we assume that I(Z) is one continuous interval; otherwise our argument can be applie to each maximal subinterval in I(Z). Let I I(Z) be such that for any two intervals I p,x, I q,y I it is the case that I p,x is not completely containe in I q,y, an also I = I(Z). By efinition, I = I(Z). (11) We orer all intervals in I in the increasing orer of starting points. We pick intervals from I one by one an label them by the orer they are picke; the i th selecte interval is enote by I i. Starting with I 1, we pick I i+1 so that I i+1 the least overlaps with I i ; here we will say I i+1 overlaps with I i even when I i+1 starts exactly where I i ens. Let I o an I even be the o inexe an even inexe intervals, respectively. WLOG, assume that I o I even. Since I o an I even is a partition of I, we know that I o + I even I. Thus we have I o 1 2 I. (12) Let N G (Z) be the neighborhoo of Z. We now show that N G (Z) µ I o. Note that u p,x, corresponing to I p,x in I o, has at least µ I p,x neighbors. Also note that all intervals in I o are isjoint by construction of I o. Hence, by summing up all neighbors of vertices corresponing to intervals in I o, we have N G (Z) µ I o. (13) From (10), (11), (12) an (13), We have N G (Z) ( λµ 2 )( covering using Lemma Let l be such a covering ) Z an G has a (( λµ )( ))- F (A) = u p,x X F p,x

17 An Online Scalable Algorithm for Average Flow Time in Broacast Scheuling 17 = u p,xv q,y E u p,xv q,y E l up,xv q,y F p,x l up,xv q,y F q,y κ ( 2 κλµ )( + 1 ) v q,y Y [By efinition of the covering] [By F q,y κf p,x ] F q,y [Change orer of the summation an l is a (( 2 +1 λµ )( ))-covering] = ( 2 κλµ )( + 1 )F (B) [Since Y is the set of vertices corresponing to events in B] ( 2 κλµ )( + 1 )LA-W 1+ [Since B is a subset of all events] 4. CONCLUSION In this paper we have given the first (1 + )-spee O(1)-competitive algorithm for the objective of minimizing the total flow time in broacast scheuling with unit size pages. Recently, Bansal et al. gave another scalable algorithm with better competitive ratio which works for varying size pages [Bansal et al. 2009]. It is also important to note that the algorithm LA-W is parameterize by an so is the algorithm given in [Bansal et al. 2009]. It woul be interesting to show a (1 + )-spee O(1)-competitive algorithm which scales with without knowlege of. Acknowlegements: We thank Chanra Chekuri for many helpful iscussions an his encouragement. We also thank the anonymous referees for their valuable suggestions. REFERENCES ACHARYA, S., FRANKLIN, M., AND ZDONIK, S Dissemination-base ata elivery using broacast isks. Personal Communications, IEEE [see also IEEE Wireless Communications] 2, 6 (Dec), AKSOY, D. AND FRANKLIN, M. J rxw: A scheuling approach for large-scale on-eman ata broacast. IEEE/ACM Trans. Netw. 7, 6, BANSAL, N., CHARIKAR, M., KHANNA, S., AND NAOR, J. S Approximating the average response time in broacast scheuling. In SODA 05: Proceeings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms BANSAL, N., COPPERSMITH, D., AND SVIRIDENKO, M Improve approximation algorithms for broacast scheuling. SIAM J. Comput. 38, 3, BANSAL, N., KRISHNASWAMY, R., AND NAGARAJAN, V Better scalable algorithms for broacast scheuling. Tech. Rep. CMU-CS , Carnegie Mellon University, Pittsburgh, PA. BAR-NOY, A., BHATIA, R., NAOR, J. S., AND SCHIEBER, B Minimizing service an operation costs of perioic scheuling. Math. Oper. Res. 27, 3, BARTAL, Y. AND MUTHUKRISHNAN, S Minimizing maximum response time in scheuling broacasts. In SODA 00: Proceeings of the eleventh annual ACM-SIAM symposium on Discrete algorithms BECCHETTI, L., LEONARDI, S., MARCHETTI-SPACCAMELA, A., AND PRUHS, K Online weighte flow time an ealine scheuling. J. Discrete Algorithms 4, 3, CHAN, W.-T., LAM, T. W., TING, H.-F., AND WONG, P. W. H New results on on-eman broacasting with ealine via job scheuling with cancellation. In COCOON, K.-Y. Chwa an J. I. Munro, Es. Lecture Notes in Computer Science, vol CHANG, J., ERLEBACH, T., GAILIS, R., AND KHULLER, S Broacast scheuling: algorithms an complexity. In SODA 08: Proceeings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms. Society for Inustrial an Applie Mathematics,