1. On the Placement Delivery Array Design for Coded Caching Scheme
Minquan Cheng
Joint work with: Q. Yan, X. Tang, and Q. Chen

2. Outline
Introduction
Cache systems
Placement delivery array 1

3. Wireless traffic Video-on-demand drives wireless traffic growth
The Available bandwidth is finite
With the development of mobile communication technology, the wireless data transmission is increasing explosively recently. This table is from cisco. Clearly the growth rate is amazing. However the available bandwidth is finite. So this has imposed a tremendous pressure on the data transmission over the core network.
[1] Cisco Visual Networking Index, Global Mobile Data Traffic Forecast Update, , White Paper, 2015.

4. High temporal variability

5. System setting A server stores N files, each of size F packets (N = 3)
Size M N
Files
A server stores N files, each of size F packets (N = 3)
K users, each access a cache of size MF packets (K = 3)

6. Caching System Placement Phase Delivery Phase
parts of each file is partially cached at each user
without the knowledge of user's demands
Delivery Phase
each user proposes a demand, the requested files are
d = (d1, d2, … , dK)
the server sends a signal Xd to users
each user decodes its requested file according to Xd and its cached contents
Xd has the size of RF packets.
A caching system consists of two phases:
1) The first phase is Placement phase during off-peak times: server proactively place some contents in each uesr’s memory. In this phase we do not know what each user will request in the following peak times.
2) The second phase is delivery phase during peak times: each user proposes a random, assume that the requested files are d1,d2,…,dK. Then according to

7. coded caching scheme can be used to further reduce
System setting
In 2014, Ali and Niesen proposed that
coded caching scheme can be used to further reduce
the congestion in the wireless network.
Best paper award of IEEE IT
Cited by 408 times
目前的研究分为两类，一类是不编码缓存，另外一类是编码缓存，2014　Ali和Niesen指出编码缓存要比不编码缓存效果好的多。
[4] M. A. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, 60(5): 2856–2867(2014).

8. A-N scheme (N, K, M, F, t=KM/N)
Placement Phase
N files W1, W2, ...,WN
Each file is split into F= packets, i.e.,
User k caches:
同时他们也给出了一个确定性的编码缓存方案，这个是在放置阶段的方案

9. A-N scheme (N, K, M, F, t=KM/N)
Delivery Phase Assume k requests Wdk
Sever sends
for all such that
这个是在放置阶段的方案

10. A-N scheme (N, K, M, F, t=KM/N)
It is widely used in heterogeneous wireless network , such as D2D, hierarchical network and so on.
Disadvantage: Each file has to be split into F packets, which usually increases exponentially with the number of users K.
Difficult: They can not consider placement phase and delivery phase together.
[5] M. Ji, G. Caire, A. Molisch, IEEE Trans. Inform. Theory, 62(2):   (2016)
[6] N. Karamchandani, U. Niesen, M. A. Maddah-Ali, S. N. Diggavi, IEEE Trans. Inform. Theory ,
62(6):   (2016)
[7] M. A. Maddah-Ali and U. Niesen, IEEE/ACM Trans. Netw. 23(4):   (2015)

11. Placement Delivery Array
Definition: An array on is called PDA if satisfies the following conditions:
̶ Each integer occurs at most once in each row and each column
̶ For any subarray , if (resp ), then it must be of the form
(resp. )

12. Definitions of PDA For an PDA P,
̶ P is said to be a (K,F,Z,S) PDA if each column of P has
non-integer cells
̶ P is a g-regular PDA (g-PDA) if the occurrence of each integer in P is exactly g times

13. Examples K=F=3

14. PDA for caching system
PDA can depict both placement phase and delivery phase together
The problem of finding a proper coded caching scheme can be translated into designing a proper PDA
The rate of the system for any request is S/F

15. PDA: Ali-Niesen A-N scheme corresponds to a regular PDA
Smaller F  Smaller g  Larger R

16. New PDA Theorem For any integers q ≥ 2 and m ≥1, a g- (K,F,Z,S)
PDA is constructed with
K=q(m+1) F=qm Z=qm-1
S=qm+1 ‒ qm g= m+1
such that M/N=1/q and R=q-1

17. Comparison：Ali-Niesen VS New
According to tedious calculations, we find that we have the same delivery rate R, the packet number of our new construction is far less than that AN scheme.

18. Comparison：Example
K=N=6, M=3
RA-N=3/4 VS
RN=1
FA-N=20 VS
FN=4

19. Papers
[1] Q. Yan, M. Cheng, X. Tang and Q. Chen, On the placement delivery array design in
centralized coded caching scheme,已投 IEEE Trans. Inf. Theory, Oct
arXiv: v1. [cs.IT]
[2] M. Cheng, J. Jiang, Q. Yan, X. Tang, H. Cao, Optimal placement delivery arrays,
submitted to IEEE Trans. Inf. Theory, Nov arXiv:  [cs.IT]
[3] M. Cheng, Q. Yan, X. Tang and J. Jiang, Optimal placement delivery arrays with
minimum number of rows, submitted to IEEE Trans. Commun., Mar arXiv:
[cs.IT]
[4] Q. Yan, X. Tang, Q. Chen, M. Cheng, Placement delivery array design through
strong edge coloring of bipartite graphs, CSCIT, Hong Kong.