Click here to load reader

Better Scalable Algorithms for Broadcast Scheduling

  • View
    23

  • Download
    0

Embed Size (px)

DESCRIPTION

Better Scalable Algorithms for Broadcast Scheduling. Ravishankar Krishnaswamy Carnegie Mellon University Joint work with Nikhil Bansal and Viswanath Nagarajan (IBM T. J. Watson Research Lab). Outline. Motivation, Problem Definition Existing Results, Our Results - PowerPoint PPT Presentation

Text of Better Scalable Algorithms for Broadcast Scheduling

Online and Stochastic Survivable Network Design

1Better Scalable Algorithms for Broadcast SchedulingRavishankar KrishnaswamyCarnegie Mellon University

Joint work with Nikhil Bansal and Viswanath Nagarajan (IBM T. J. Watson Research Lab)

1OutlineMotivation, Problem Definition

Existing Results, Our Results

A Weaker Approximation/ Analysis

Conclusion2Motivation: Client-Server System

ClientsServer

Page A at time 1Page B at time 1Page A at time 2Page C at time 3Page A at time 3Page APage BPage CPage Abroadcast

Motivation: FormalizingConsider a server which has n unit-sized pagesRequests for these pages arrive online, over timeAt each time slot, we can broadcast one pageAll pending requests for that page are satisfied

How do we schedule to minimize average response time of requests

4Online Broadcast SchedulingInputA collection of n pagesA request sequence arrives onlineRequest r: arrival time a(r), requested page p(r)

OutputA broadcast of pages, one at a time

Objective FunctionMinimize Average Response TimeMinimize Maximum Response TimeThis TalkA Concrete ExampleInstance has 3 pagesABCABCA

BBABCTotal Response Time: 1 + 2 + 3 + 3 + 3 = 12ATotal Response Time: 2 + 3 + 1 + 1 + 1 = 8Existing Results (Average Response Time) In the offline setting O(log2n)-approximation algorithm [BCS06]

In the online setting very strong lower bounds if no speed-up (2+) speed-up, O(1/2)-competitive[EP09] (1+) speed-up, O(1/11)-competitive[IM10] (works only for unit-size pages)

Our Results8A very simple (1+) speed, O(1/3)-competitive online algorithm. Can be extended to the setting when the pages have non-uniform sizesHigh Level Idea9Consider Fractional Relaxation of Broadcast SchedulingGet (1+) speed, O(1/2) competitive online algorithmDesign an online rounding algorithm, with further O(1/) loss in obj. functionFractional RelaxationAt each time slot, we can broadcast multiple pages, each to extent xptSuch that

A request r is satisfied at the first time b(r) when

Minimize 10

High Level Idea11Consider Fractional Relaxation of Broadcast SchedulingGet (1+) speed, O(1/2) competitive online algorithmDesign an online rounding algorithm, with further O(1/) loss in obj. functionAlgorithm (with weaker guarantee)Round RobinKnown to give online algorithms with good competitive ratio for other scheduling problems assuming factor of 2 speed-upWhat about broadcast scheduling?Nave algorithm is badDoes not differentiate pages with many outstanding requests and those with 1 request12Algorithm (with weaker guarantee)Round Robin: Possible FixRound robin over requests!At any time, schedule each outstanding request to the same extent.Illustration13ABCA

BA: 1/3B: 1/3C: 1/3A: 2/4B: 1/4C: 1/4A: 2/5B: 2/5C: 1/5A: 1/4B: 2/4C: 1/4A: 1/3B: 1/3C: 1/3 Algorithm (with weaker guarantee)Round Robin: Possible FixRound robin over requests!At any time, schedule each outstanding request to the same extent.Can we show anything for this algorithm?Edmonds and Pruhs showed it is 4-speed O(1) competitiveWe show that fractionally, it is 2-speed O(1) competitiveLater round it to get integer schedule.14Analysis15Resort to an amortized analysisDefine a potential function (t) which is 0 at t=0 and t=Show the following:At any request arrival, 0 At all other times,

15

Will give us a -competitive online algorithmFor our ProblemDefine

rank(r) is sorted order of requests w.r.t arrival times (most recent has highest rank)

z(r,t) is the amount of time the online algorithm will dedicate towards request r, in the future, i.e. after time t

16

Analysis ContinuedNew request arrivalIt belongs to NA(t) and NO(t)Does not appear in potential functionNo change in value

17

Analysis ContinuedRunning Condition: Consider [t-1, t)Opt schedules a page and finishes some requestsThese terms will now appear in the potential function.How much increase will it cause?The sum of the z(r,t) over all these requests is at most 1Total increase is at most NA(t)

18

Were golden if NO(t) is even a tiny fraction of NA(t)Analysis ContinuedAssume most unfinished requests are completed by OPTHope that (t) goes down enough.

19

We make progress on all jobsEach jobs z value goes down by 1/NA(t)Total decrease is NA(t)/2 * 1/NA(t) * 2Left hand side is non-positive!Speed-UpHigh Level Idea20Consider Fractional Relaxation of Broadcast SchedulingGet (1+) speed, O(1/2) competitive online algorithmDesign an online rounding algorithm, with further O(1/) loss in obj. functionRounding: One Slide OverviewConsider the fractional algorithms outputLet request r be fractionally completed at time b(r)Enqueue element At any time, choose request with least width and display corresponding page. Wipe out all outstanding requests for page p(r)21Suppose a request was forced to wait for too much time.Then many other requests for different pages all having smaller width. Too much mass packed fractionally. A contradiction.21Thank YouSummary + Open QuestionNear-optimal algorithm for broadcast schedulingConsider fractional relaxationGive good algorithm for fractional problemGive rounding scheme for integral problemBut algorithm depends on Not fully-scalable Can we get one such algorithm which works for all ?

22

Search related