Amotz Bar-Noy, and Richard E. Ladner

1
Windows Scheduling Problems for Broadcast System

• Amotz Bar-Noy, and Richard E. Ladner
• Presented by Qiaosheng Shi

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
Outline

• The greedy algorithm
• The combination technique
• Solutions for small h (2,3,4,5)
• Open problems my project plan

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
  wiltwi.
• The goal decrease the difference wi-wi (wii).

5
The Greedy algorithm

• Basic idea
• Consider the schedule for the pages with smaller
  window size first. (3-gt2 1/6 5-gt4 1/20)
• Insert page i at i-th round, i1,, n.
• At i-th round, find a perfect placement for page
  i such that minimizes the difference wi-wi
  (wii).
• 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
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
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
  havent assigned to pages.
• Let for
• r is the index for minimum
• Let and Ts 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
The Greedy algorithm
h3
(3 mod 1)lt(3 mod 2) dr3
page leaf
(4 mod 2)lt(4 mod 3) dr2
(5 mod 4)lt(5 mod 3) dr1
(1) window label
9
The Greedy algorithm

• For h4

10
Two Possible Modifications

• Try to keep leaves with small window labels open
  as long as possible.
• Split When dr is a composite number, drabc,
  split that node in several steps following an
  increasing order of these prime factors.

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
The Greedy algorithm

• Theorem The greedy algorithm construct a perfect
  lth,ngt 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 wi time
  slots
• There exist perfect channel schedule that cannot
  be embedded in a tree

13
The combination technique

• How to combine schedule together to get new
  schedule for larger number of channels?
• For , let lt h u v gt-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
The combination technique

• Example of Magnification Lemma

lt2,1,3gt schedule
15
The combination technique
Proof
Example lt3,9gt and lt2,3gt gt lt5,39gt
lt3,9gt and lt4,28gt gt lt7,289gt.
16
The combination technique

• Corollary

17
The combination technique

• Theorem for
  any integer . Similar to

• Corollary
  for .

3 divides h
4 divides h
18
Solutions for small h

• For h4

19
Solutions for small h

• Non-perfect lt5,77gt schedule

20
Open Problems

• The harmonic windows scheduling problem
• Is the lt3, 9gt-schedule optimal?
• Algorithm outputs better schedules.

21
Is the lt3, 9gt-schedule optimal?

• If 3 windows a, b, c are prime to each other,
  there is at least one collision in any window of
  abc slots.
• To avoid collision 2, 3, 7, we need at least
  1/42 fraction of a channel.

22
Is the lt3, 9gt-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
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
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 wii.
• All the cyclic channel schedules can be embedded
  in trees.

25
Thanks You !