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Better Scalable Algorithms for Broadcast Scheduling

Ravishankar KrishnaswamyCarnegie Mellon University

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

Outline

• Motivation, Problem Definition

• Existing Results, Our Results

• A Weaker Approximation/ Analysis

• Conclusion

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Motivation: Client-Server System

Clients Server

Page A at time 1

Page B at time 1

Page A at time 2Page C at time 3

Page A at time 3

Page APage BPage CPage Abroadcast

Motivation: Formalizing

• Consider a server which has n unit-sized pages– Requests for these pages arrive online, over time– At each time slot, we can broadcast one page• All pending requests for that page are satisfied

• How do we schedule to minimize average response time of requests

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Online Broadcast Scheduling

• Input– A collection of n pages– A request sequence arrives online

• Request r: arrival time a(r), requested page p(r)

• Output– A broadcast of pages, one at a time

• Objective Function– Minimize Average Response Time– Minimize Maximum Response Time– …

This Talk

A Concrete Example

Instance has 3 pages

ABC

A B C

A B

B

A BC

Total Response Time: 1 + 2 + 3 + 3 + 3 = 12

A

Total Response Time: 2 + 3 + 1 + 1 + 1 = 8

Existing 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+є) speed-up, O(1/є2)-competitive[EP09] • (1+є) speed-up, O(1/є11)-competitive [IM10]

(works only for unit-size pages)

Our Results

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A very simple (1+є) speed, O(1/є3)-competitive online algorithm. Can be extended to the setting when the pages have non-uniform sizes

High Level Idea

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Consider “Fractional” Relaxation of Broadcast Scheduling

Get (1+є) speed, O(1/є2) competitive online algorithm

Design an online rounding algorithm, with further O(1/є) loss in obj. function

Fractional Relaxation

• At each time slot, we can broadcast multiple pages, each to extent xpt

– Such that

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

• Minimize

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High Level Idea

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Consider “Fractional” Relaxation of Broadcast Scheduling

Get (1+є) speed, O(1/є2) competitive online algorithm

Design an online rounding algorithm, with further O(1/є) loss in obj. function

Algorithm (with weaker guarantee)

• Round Robin– Known to give online algorithms with good

competitive ratio for other scheduling problems assuming factor of 2 speed-up

– What about broadcast scheduling?– Naïve algorithm is bad• Does not differentiate pages with many outstanding

requests and those with 1 request

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Algorithm (with weaker guarantee)

• Round Robin: Possible Fix– Round robin over requests!

At any time, schedule each outstanding request to the same extent.

• Illustration

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ABC

A B

A: 1/3B: 1/3C: 1/3

A: 2/4B: 1/4C: 1/4

A: 2/5B: 2/5C: 1/5

A: 1/4B: 2/4C: 1/4

A: 1/3B: 1/3C: 1/3

…

Algorithm (with weaker guarantee)

• Round Robin: Possible Fix– Round 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) competitive

• We show that fractionally, it is 2-speed O(1) competitive– Later round it to get integer schedule.

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Analysis

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• Resort to an amortized analysis• Define a potential function Φ(t) which is 0 at t=0 and t=• Show the following:

– At any request arrival,

ΔΦ ≤ 0 – At all other times,

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Will give us a β-competitive online algorithm

For our Problem

• Define

• 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

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Analysis Continued

• New request arrival– It belongs to NA(t) and NO(t)– Does not appear in potential function– No change in value

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Analysis Continued

• Running Condition: Consider [t-1, t)• Opt schedules a page and finishes some requests• These 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 1– Total increase is at most NA(t)

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We’re golden if NO(t) is even a tiny fraction of NA(t)

Analysis Continued

• Assume most unfinished requests are completed by OPT• Hope that Φ(t) goes down enough.

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(i) We make progress on all jobs(ii)Each job’s z value goes down by 1/NA(t)(iii)Total decrease is NA(t)/2 * 1/NA(t) * 2

(iv)Left hand side is non-positive!Speed-Up

High Level Idea

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Consider “Fractional” Relaxation of Broadcast Scheduling

Get (1+є) speed, O(1/є2) competitive online algorithm

Design an online rounding algorithm, with further O(1/є) loss in obj. function

Rounding: One Slide Overview

• Consider the fractional algorithm’s output• Let request r be fractionally completed at time b(r)• Enqueue element <r, b(r) – a(r)>• At any time, choose request with least width and

display corresponding page. Wipe out all outstanding requests for page p(r)

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Suppose 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.

Thank You

Summary + Open Question

• Near-optimal algorithm for broadcast scheduling– Consider “fractional relaxation”– Give good algorithm for fractional problem– Give rounding scheme for integral problem

• But algorithm depends on є– Not fully-scalable – Can we get one such algorithm which works for all є?

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