Embed Size (px)
Citation preview
23/4/18 Windows Scheduling Problems for Broadcast System
1
Windows Scheduling Problems for Broadcast
System
Amotz Bar-Noy, and Richard E. Ladner
Presented by Qiaosheng Shi
23/4/18 Windows Scheduling Problems for Broadcast System
2
Review
Windows scheduling problem The optimal windows scheduling problem, H(W). The optimal harmonic windows scheduling proble
m, 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)
23/4/18 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
23/4/18 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 wi’<=wi. The goal: decrease the difference wi-wi’ (wi=i).
23/4/18 Windows Scheduling Problems for Broadcast System
5
The Greedy algorithm
Basic idea Consider the schedule for the pages with smaller wi
ndow 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 su
ch that minimizes the difference wi-wi’ (wi=i). In order to keep track of placements for pages, we r
epresent 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.
23/4/18 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.
23/4/18 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 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)
kxxx ...21
)mod( jj xim kj 1
jm
rr x
idrx
1rd
rd
rrdx
rm
23/4/18 Windows Scheduling Problems for Broadcast System
8
The Greedy algorithm
(1)(1) (1)
1
2
4 5
3
6 7 8 9
1
2 (2)3 (3) (3)
54
1
2 3 (3) (3)
h=3
page leaf
(1)window label
(3 mod 1)<(3 mod 2)
dr=3
(4)
(4 mod 2)<(4 mod 3)
dr=2
(5 mod 4)<(5 mod 3)
dr=1
23/4/18 Windows Scheduling Problems for Broadcast System
9
The Greedy algorithm
For h=4
23/4/18 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 dr is a composite number, dr=a*b*c…, split that node in several steps following an increasing order of these prime factors.
dr=12(xr)
12xr (12xr) … … …
(12xr)
… …
12xr (12xr) (12xr)
(4xr)
(2xr)
23/4/18 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
23/4/18 Windows Scheduling Problems for Broadcast System
12
The Greedy algorithm
Theorem: The greedy algorithm construct a perfect <h,n> 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 s
lots There exist perfect channel schedule that cannot b
e embedded in a tree
23/4/18 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 < h; u; v >-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 .
viu
vu
)1)1(,,(,, vlluhvuh1l
ju )( jul 1)( jul 1)( ljul
… … …… …
uvj 0
23/4/18 Windows Scheduling Problems for Broadcast System
14
The combination technique
Example of Magnification Lemma
<2,1,3> 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 … …
<2,10,39> schedule
10l
23/4/18 Windows Scheduling Problems for Broadcast System
15
The combination technique
111,,1,,,1, 21212211 nnhhnhnh
111,,,, 21212211 nnhhnhnh
Combination theorem:
Proof:
111,1,,1, 211222 nnnhnh)1( 1 nl :
Example: <3,9> and <2,3> => <5,39>; <3,9> and <4,28> => <7,289>.
23/4/18 Windows Scheduling Problems for Broadcast System
16
The combination technique
Corollary
12,1, nhnh
12,11)1(2,1,,1,1 nhnhnh
12, hh
Apply this corollary h-1 times starting with <1,1> 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.
12, hh
23/4/18 Windows Scheduling Problems for Broadcast System
17
The combination technique
Theorem: for any integer . Similar to
11,, lnlknk 1l12,1,1 hh
Corollary: for . 11,, h
k nhnk lkh
110,9,3 3 h
h
129,28,4 4 h
h
3 divides h
4 divides h
23/4/18 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]=3/40Page 27,28: [27;27;26]=3/80
23/4/18 Windows Scheduling Problems for Broadcast System
19
Solutions for small h
Non-perfect <5,77> schedule
23/4/18 Windows Scheduling Problems for Broadcast System
20
Open Problems
The harmonic windows scheduling problem Is the <3, 9>-schedule optimal? Algorithm outputs better schedules.
23/4/18 Windows Scheduling Problems for Broadcast System
21
Is the <3, 9>-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 … … 42
1
7
1
6
1
… … … 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 … …… … 3 7 _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 … …
7 742
1
3
1
14
1
… … 2 7 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ … …… … … 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 _ _ 3 … …
7 7 742
1
2
1
21
1
23/4/18 Windows Scheduling Problems for Broadcast System
22
Is the <3, 9>-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.
23/4/18 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)
23/4/18 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 wi=i.
All the cyclic channel schedules can be embedded in trees.
1p 2p lp… … …
… …
23/4/18 Windows Scheduling Problems for Broadcast System
25
Thanks You !
