Better Scalable Algorithms for Broadcast Scheduling

1. Better Scalable Algorithms for Broadcast Scheduling RavishankarKrishnaswamy Carnegie Mellon University Joint work with Nikhil Bansal and ViswanathNagarajan (IBM T. J. Watson Research Lab)

2. Outline • Motivation, Problem Definition • Existing Results, Our Results • A Weaker Approximation/ Analysis • Conclusion

3. Motivation: Client-Server System Page A at time 1 Page A at time 3 Page B at time 1 Page C at time 3 Page A at time 2 Page C Page A Page B Page A Clients Server broadcast

4. 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

5. 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

6. A Concrete Example Instance has 3 pages A B C A B Total Response Time: 1 + 2 + 3 + 3 + 3 = 12 A B C A B C A B Total Response Time: 2 + 3 + 1 + 1 + 1 = 8

7. 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)

8. Our Results 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

9. High Level Idea 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

10. 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

11. High Level Idea 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

12. 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

13. Algorithm (with weaker guarantee) • Round Robin: Possible Fix • Round robin over requests! At any time, schedule each outstanding request to the same extent. • Illustration A: 1/3 B: 1/3 C: 1/3 A: 2/4 B: 1/4 C: 1/4 A: 2/5 B: 2/5 C: 1/5 A: 1/4 B: 2/4 C: 1/4 A: 1/3 B: 1/3 C: 1/3 … A B C A B

14. 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.

15. Analysis • 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, Will give us a β-competitive online algorithm 15

16. 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

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

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

19. Analysis Continued • Assume most unfinished requests are completed by OPT • Hope that Φ(t)goes down enough. We make progress on all jobs Each job’s z value goes down by 1/NA(t) Total decrease is NA(t)/2 * 1/NA(t) * 2 Left hand side is non-positive! Speed-Up

20. High Level Idea 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

21. 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) 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.

22. 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? • Thank You