1. 2015-6-3Windows Scheduling Problems for Broadcast System 1 Amotz Bar-Noy, and Richard E. Ladner Presented by Qiaosheng Shi

2. 2015-6-3Windows Scheduling Problems for Broadcast System 2 Review Windows scheduling problem  The optimal windows scheduling problem, H(W).  The optimal harmonic windows scheduling problem, N(h). Perfect schedule and tree representation  If all leaves are distinct in forest, the corresponding schedule is perfect channel schedule.  However, there exist perfect channel schedule that cannot be embedded in a tree. Asymptotic bounds for H(W) and N(h)

3. 2015-6-3Windows Scheduling Problems for Broadcast System 3 Outline The greedy algorithm The combination technique Solutions for small h (=2,3,4,5) Open problems & my project plan

4. 2015-6-3Windows Scheduling Problems for Broadcast System 4 The Greedy algorithm For harmonic windows scheduling problem  Can be generalized to the general windows scheduling problems. Several points  Perfect channel schedule (NP-hard)  Tree representation  To avoid collisions, we have to decrease the window size of some pages (temporally)  In perfect channel schedule, each page has w i ’<=w i.  The goal: decrease the difference w i -w i ’ (w i =i).

5. 2015-6-3Windows Scheduling Problems for Broadcast System 5 The Greedy algorithm Basic idea  Consider the schedule for the pages with smaller window size first. (3->2: 1/6; 5->4: 1/20)  Insert page i at i-th round, i=1,…, n.  At i-th round, find a perfect placement for page i such that minimizes the difference w i -w i ’ (w i =i).  In order to keep track of placements for pages, we represent each channel by a tree, where pages are assigned only to some leaf of the trees.  Terminate when there is no place for page i.

6. 2015-6-3Windows Scheduling Problems for Broadcast System 6 The Greedy algorithm Two labels: page and window. Open tree: there is some leaves not assigned to pages. Close tree: all leaves are assigned to pages. Initially, all the trees are open trees with one window leaf whose value is 1. Insert one page at a time and terminate when all trees are closed.  Terminate when there is no place for current page.

7. 2015-6-3Windows Scheduling Problems for Broadcast System 7 The Greedy algorithm The way to find the placement for page i:  is the ordered list of the labels of all the leaves in the forest that haven ’ t assigned to pages. Let for r is the index for minimum Let and T s be the tree that contains  If, then assign that leaf to page i. ( replacement )  Otherwise, add children to that leaf. The first child is labeled with page label i and the rest are labeled with the window label ( split )

8. 2015-6-3Windows Scheduling Problems for Broadcast System 8 The Greedy algorithm (1) 1 2 45 3 678 9 1 2 (2) 3 (3) 5 4 1 2 3 h=3 page leaf (1) window label (3 mod 1)<(3 mod 2) d r =3 (4) (4 mod 2)<(4 mod 3) d r =2 (5 mod 4)<(5 mod 3) d r =1

9. 2015-6-3Windows Scheduling Problems for Broadcast System 9 The Greedy algorithm For h=4

10. 2015-6-3Windows Scheduling Problems for Broadcast System 10 Two Possible Modifications Try to keep leaves with small window labels open as long as possible. Split: When d r is a composite number, d r =a*b*c…, split that node in several steps following an increasing order of these prime factors. d r =12 (x r ) 12x r (12x r ) … … … (12x r ) … 12x r (12x r ) (4x r ) (2x r )

11. 2015-6-3Windows Scheduling Problems for Broadcast System 11 Two Possible Modifications In the second modification, the algorithm sometimes prefers to assigning the new label i to a large window label on the expense of not minimizing i-i ’. On this way, it leaves smaller window labels for possibly better split operations It was not the case that one version always outperforms the other versions

12. 2015-6-3Windows Scheduling Problems for Broadcast System 12 The Greedy algorithm Theorem: The greedy algorithm construct a perfect schedule for some value n. Problem  No analytical bounds  Perfect channel schedule each page is scheduled on a single channel each page is periodic: one exactly every w i ’ time slots  There exist perfect channel schedule that cannot be embedded in a tree

13. 2015-6-3Windows Scheduling Problems for Broadcast System 13 The combination technique How to combine schedule together to get new schedule for larger number of channels? For, let -schedule be a schedule of the pages u,.., v on h channels such that page i appears at least once in any consecutive i slots for. Magnification Lemma:, for any integer. … … … …

14. 2015-6-3Windows Scheduling Problems for Broadcast System 14 The combination technique Example of Magnification Lemma schedule 1 1 1 1 1 1 … … 2 3 2 3 2 3 … … 10 … 19 10 … 19 10 … 19 10 … 19 … … 20 … 29 30 … 39 20 … 29 30 … 39 … … schedule

15. 2015-6-3Windows Scheduling Problems for Broadcast System 15 The combination technique Combination theorem: Proof: : Example: and => ; and =>.

16. 2015-6-3Windows Scheduling Problems for Broadcast System 16 The combination technique Corollary Apply this corollary h-1 times starting with schedule we have the well-known schedule. A better asymptotic result than the schedule may be obtained by taking other known schedule on h>1 channels and applying the combination theorem.

17. 2015-6-3Windows Scheduling Problems for Broadcast System 17 The combination technique Theorem: for any integer. Similar to Corollary: for. 3 divides h 4 divides h

18. 2015-6-3Windows Scheduling Problems for Broadcast System 18 Solutions for small h For h=4 Page 7: [7;7;6]=3/20 Page 14: [14;13;13]=3/40 Page 27,28: [27;27;26]=3/80

19. 2015-6-3Windows Scheduling Problems for Broadcast System 19 Solutions for small h Non-perfect schedule

20. 2015-6-3Windows Scheduling Problems for Broadcast System 20 Open Problems The harmonic windows scheduling problem  Is the -schedule optimal?  Algorithm outputs better schedules.

21. 2015-6-3Windows Scheduling Problems for Broadcast System 21 Is the -schedule optimal? If 3 windows a, b, c are prime to each other, there is at least one collision in any window of a*b*c slots. To avoid collision {2, 3, 7}, we need at least 1/42 fraction of a channel. 7 … … … 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 … … … … 7 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 … … … … … 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 … … … … 3 7 _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 … … 77 … … 2 7 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ … … … … … 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 … … 77 7

22. 2015-6-3Windows Scheduling Problems for Broadcast System 22 Is the -schedule optimal? Other collisions: {2, 3, 5}, {2, 5, 7}, {2, 5, 9}, {2, 7, 9}, {3, 4, 5}, {3, 4, 7}, {3, 5, 7}, {3, 5, 8}, {3, 7, 8}, {3, 7, 10}, {4, 5, 7}, {4, 5, 9}, {4, 7, 9}, {5, 6, 7}, {5, 7, 8}, {5, 7, 9}, {5, 8, 9}, {7, 8, 9}, {7, 9, 10}. These collisions are not independent of each other.

23. 2015-6-3Windows Scheduling Problems for Broadcast System 23 Rough idea of my project Two constraints for perfect channel schedule:  each page in one channel  a fixed window size for each page Our constraints:  each page in one channel  Schedule is cyclic Tree representation of cyclic channel schedule  One ordered tree per channel  Leaves represent pages. But the leaves are not distinct  Same way to compute period length and offset of leaves (not pages) (Cyclic channel schedule)

24. 2015-6-3Windows Scheduling Problems for Broadcast System 24 Rough idea of my project New constraints:  All the leaves for page i should have same period length.  The gap of offsets between two consecutive leaves for page i is less than w i =i. All the cyclic channel schedules can be embedded in trees. … … … …

25. 2015-6-3Windows Scheduling Problems for Broadcast System 25 Thanks You !