1. June 3, 2015Windows Scheduling Problems for Broadcast System 1 Amotz Bar-Noy, and Richard E. Ladner Presented by Qiaosheng Shi

2. June 3, 2015Windows Scheduling Problems for Broadcast System 2 Outline Windows scheduling problem  The optimal windows scheduling problem  The optimal harmonic windows scheduling problem Perfect schedule and tree representation Asymptotic bounds The greedy algorithm The combination technique Solutions for small h Open problems & my project plan

3. June 3, 2015Windows Scheduling Problems for Broadcast System 3 Outline Windows scheduling problem  The optimal windows scheduling problem  The optimal harmonic windows scheduling problem Perfect schedule and tree representation Asymptotic bounds The greedy algorithm The combination technique Solutions for small h Open problems & my project plan

4. June 3, 2015Windows Scheduling Problems for Broadcast System 4 Windows scheduling problem h slotted channels, and n pages. Each page i has a window size w i. i:1…n  window vector Question: Is there a schedule for the n pages on the h slotted channels  one page each time slot  the gap between two consecutive appearances of page i is no more than w i. The problem is based on the max metric and not the average metric. That is, the next appearance of a page depends only on its previous appearance. But in the average metric, the next appearance of a page depends on all of its previous appearances.

5. June 3, 2015Windows Scheduling Problems for Broadcast System 5 The push systems application The broadcasting environment consists of:  Clients who wish to access information pages from broadcast channels.  Servers who broadcast the information pages on channels  Providers who supply the information pages Window size of each page (quality of service) is determined by the money providers paid to servers. The server is left with the problem: minimize the number of channels (bandwidth) needed to guarantee the quality of service.  The optimal windows scheduling problem

6. June 3, 2015Windows Scheduling Problems for Broadcast System 6  Input: A set W={w 1,w 2,…,w n } of requests for broadcasting. A request with window w i needs to be broadcasted at least once in any window of w i time-slots.  Output: A feasible windows scheduling of W.  Goal: minimize number of channels used H(W). Example: Input: W={2,4,5} Output: one channel 425242425252 … There is at least one transmission of in any window of 5 time-slots 5 There is at least one transmission of in any window of 4 time-slots 4 The Optimal Windows scheduling problem H(W)=1

7. June 3, 2015Windows Scheduling Problems for Broadcast System 7  Medias are broadcast based on customer demand.  A limited number of channels.  The goal: Minimizing clients’ maximal waiting time (delay) with given bandwidth (number of channels).  Assumption  A client that wishes to watch a movie is ‘listening to all the channels’ and is waiting for his movie to start.  Clients have large enough buffer.  Each channel transmits data at the playback rate.  Basic broadcasting schemes  Broadcast popular movies continuously on h channels. The Media-on-Demand application

8. June 3, 2015Windows Scheduling Problems for Broadcast System 8 Staggered broadcasting [Dan96]: Transmit the movie repeatedly on each of the channels. Guaranteed delay: at most 1/h. The Media-on-Demand application Can we do better? Client’s buffer!

9. June 3, 2015Windows Scheduling Problems for Broadcast System 9 Partition the movie into segments (or pages). Early segments (or pages) are transmitted more frequently. The Media-on-Demand application The client can start watching the movie without interruptions. Maximal delay: 1/3. arrive watch & buffer 132 (3 pages) Each time-slot has length 1/3. 0 1/3 2/3 1 4/3 5/3 2 C1:C1: 111111 222333 C2:C2: … … arrive watch & buffer

10. June 3, 2015Windows Scheduling Problems for Broadcast System 10 Why does it work? The 1st page is transmitted in any window of one slot. C1:C1: 111111 222333 C2:C2: … … The 2nd page is transmitted in any window of two slots. The 3rd page is transmitted at least once in any window of three slots. The Media-on-Demand application

11. June 3, 2015Windows Scheduling Problems for Broadcast System 11 The movie is partitioned into n pages, 1,..,n. Necessary and sufficient condition: page i is transmitted at least once in any window of i slots (i- window). The client has page i available on time (from his buffer or from the channels). The maximal delay: one slot = 1/n. Therefore, the goal is to maximize n for given h. The Media-on-Demand application  The optimal harmonic windows scheduling problem

12. June 3, 2015Windows Scheduling Problems for Broadcast System 12  Given h, maximize n such that each i in 1,..,n is scheduled at least once in i time slots. The maximum n is denoted by N(h). The optimal harmonic windows scheduling problem 662222224445553333779988111111111111 … … … C1C1 C2C2 C3C3 111 … C1C1 Examples: h=1, n=1, N(1)=1 C1C1 111111 222333 C2C2 … … h=2, n=3. N(2)=3 h=3, n=9. N(3)=9?

13. June 3, 2015Windows Scheduling Problems for Broadcast System 13 Perfect channel schedule Channel schedule: each page is scheduled on a single channel. A schedule S is called cyclic if it is an infinite concatenation of a finite sequence. Another definition: Matrix schedule

14. June 3, 2015Windows Scheduling Problems for Broadcast System 14 Perfect channel schedule Perfect channel schedule: For page i, there exists a, page i gets one time slot exactly every w i ’ time slots.  the window size of page i in the perfect channel schedule.  Perfect channel schedule is cyclic (least common multiple). Several points:  Avoid busy-waiting: the client actively listen until its movie arrives.  Not optimal for windows scheduling problem  Finding an optimal perfect channel schedule is NP-hard in general.  Only need to record three numbers for one page: channel number, period length and offset.

15. June 3, 2015Windows Scheduling Problems for Broadcast System 15 Tree representation 1 2 45 3 678 9 66 222222444555 3333779988 111111111111 … … … C1C1 C2C2 C3C3 Page123456789 Channel123223333 Period123446666 Offset000131245 0 1 2 3 4 5 6 7 8 9 10 11 Tree is simple

16. June 3, 2015Windows Scheduling Problems for Broadcast System 16 Tree representation 1 2 45 3 678 9 One ordered tree per channel Leaves represent the pages 1 00 0 1 0 02 1 01 001 Offset The period of each page is the product of the degrees of the nodes on the path from the root to its corresponding leaf.

17. June 3, 2015Windows Scheduling Problems for Broadcast System 17 Tree representation PageABCD Period2666 Offset0135 0 1 2 3 4 5

18. June 3, 2015Windows Scheduling Problems for Broadcast System 18 Tree representation PageABCDEFGH Period6612 4 Offset024101593

19. June 3, 2015Windows Scheduling Problems for Broadcast System 19 If all leaves are distinct in forest, the corresponding schedule is perfect channel schedule. Tree representation However, there exist perfect channel schedule that cannot be embedded in a tree. ABCD1D2D3AD4D5D6D7BAD8D9 D10D11CAD12D13BD14D15AD16D17D18D19D20 … Can we always construct an ordered tree for a perfect channel schedule? Can we always get the perfect channel schedule from an ordered tree? AAA AA BB BC C D1D2D3D4D5D6D7D8D9 D10D11D12D13D14D15D16D17D18D19D20 Degree of root must divide the periods 6, 10, 15

20. June 3, 2015Windows Scheduling Problems for Broadcast System 20 Asymptotic bounds for H(W) Page i requires at least a fraction of a channel Upper bound  It is achieved by constructing a perfect channel schedule. Lower bound  For any window vector Minimum number of channels needed to schedule window vector W, N(h)

21. June 3, 2015Windows Scheduling Problems for Broadcast System 21 Upper bound for H(W) --- simple case Window sizes are all powers of 2. Lemma: there exists a perfect schedule that uses exactly channels. (the first lemma) First upper bound (round the window sizes down to the nearest power of 2): For any window vector W, there exists a perfect schedule that uses no more than channels.

22. June 3, 2015Windows Scheduling Problems for Broadcast System 22 Upper bound for H(W) --- simple case All the window sizes are powers of 2 multiplied by some number u. Lemma: If all the are of the form for some and, then there exists a perfect schedule that uses exactly channels. (the 2nd lemma) Construct an algorithm that for given window vector W creates perfect schedules with about channels.

23. June 3, 2015Windows Scheduling Problems for Broadcast System 23 Upper bound for H(W) --- the algorithm The algorithm use two parameters k and x that are optimized to obtain the best bound. k: the depth of the recursion x: is optimized for each value of k. If k=1, round window size down to closest to get a schedule with at most channels. If k>1, partition the window vector W into x vectors denoted by.  is rounded down to maximal such that, is an odd number and for some u. Then  The set of such that is denoted by

24. June 3, 2015Windows Scheduling Problems for Broadcast System 24 Upper bound for H(W) --- the algorithm channels needed to schedule all windows in Some windows scheduled into non-fully used channels. The set of all these windows is denoted by If x is larger, then is closer to. However, is too big. If x is smaller, then is small. But is too small compared to For each k, find the best value for x.

25. June 3, 2015Windows Scheduling Problems for Broadcast System 25 Upper bound for H(W) --- example Let W= At least 3 channels to schedule the windows in W (2<h(W)<3) k=1: W ’ = k=2, x=3: We get the following 3 vectors 3 44 6 4 5 1 3 5 K=1

26. June 3, 2015Windows Scheduling Problems for Broadcast System 26 Upper bound for H(W) --- major lemma Define r as mapping from the positive integer to the reals by: for k=1 for k>1 The function r is monotonic increasing function whose exist and is approximately 4.6412 For window vector W and positive integers k if then there exists a perfect schedule with number of channels bounded above by

27. June 3, 2015Windows Scheduling Problems for Broadcast System 27 Upper bound for H(W) --- major lemma Theorem: Every window vector W, with h(W)>1, has perfect schedule using number of channels bounded above by, where Theorem: For any window vector W, there exists an algorithm for the optimal windows scheduling problem yielding a solution that is within a factor of of the optimal solution.

28. June 3, 2015Windows Scheduling Problems for Broadcast System 28 Bounds on N(h) Lower bound of H(W):  Upper bound of H(W):  Given h channels, maximize n such that each page i is scheduled at least once in any consecutive i slots

29. June 3, 2015Windows Scheduling Problems for Broadcast System 29 Outline Windows scheduling problem  The optimal windows scheduling problem  The optimal harmonic windows scheduling problem Perfect schedule and tree representation Asymptotic bounds The greedy algorithm The combination technique Solutions for small h Open problems & my project plan