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

Windows 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)

Windows 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

Windows 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).

Windows 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.

Windows 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.

Windows 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 )

Windows Scheduling Problems for Broadcast System 8 The Greedy algorithm (1) (2) 3 (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

Windows Scheduling Problems for Broadcast System 9 The Greedy algorithm For h=4

Windows 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 )

Windows 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

Windows 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

Windows 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. … … … …

Windows Scheduling Problems for Broadcast System 14 The combination technique Example of Magnification Lemma schedule … … … … 10 … … … … 19 … … 20 … … … … 39 … … schedule

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

Windows 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.

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

Windows 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

Windows Scheduling Problems for Broadcast System 19 Solutions for small h Non-perfect schedule

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

Windows 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 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ … … … … … 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 … … 77 7

Windows 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.

Windows 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)

Windows 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. … … … …

Windows Scheduling Problems for Broadcast System 25 Thanks You !