Coded Caching for Files with Distinct File Sizes

Transcription

1 Coded Caching for Files with Distinct File Sizes Jinbei Zhang, Xiaojun Lin, Chih-Chun Wang, Xinbing Wang Shanghai Jiao Tong University Purdue University

2 The Importance of Caching Server cache cache cache User 1 User User 3 Data traffic continues to grow at significant rates A major fraction (60-80%) of traffic will be generated by multimedia content, such as video Caching is important for reducing backhaul requirement in serving large volumes of content that multiple users are interested in

3 An Example of Coded Caching [Maddah-Ali and Niesen 14] Server K=3 users Cache size M=1 Broadcast channel User 1 wants A User wants B [1] Fundamental Limits of Caching, M. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, 014. User 3 wants C 3

4 An Example of Coded Caching [Maddah-Ali and Niesen 14] N=3 files: A = (A 1, A, A 3 ) B = (B 1, B, B 3 ) C = (C 1, C, C 3 ) K=3 users Cache size M=1 Broadcast channel Server User 1 wants A User wants B [1] Fundamental Limits of Caching, M. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, 014. User 3 wants C 3

5 An Example of Coded Caching [Maddah-Ali and Niesen 14] N=3 files: A = (A 1, A, A 3 ) B = (B 1, B, B 3 ) C = (C 1, C, C 3 ) Broadcast channel Server Back-hual Requirement: Uncoded Caching K=3 users Cache size M=1 (A 1, B 1, C 1 ) (A 1, B 1, C 1 ) (A 1, B 1, C 1 ) User 1 wants A User wants B [1] Fundamental Limits of Caching, M. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, 014. User 3 wants C 3

6 An Example of Coded Caching [Maddah-Ali and Niesen 14] N=3 files: A = (A 1, A, A 3 ) B = (B 1, B, B 3 ) C = (C 1, C, C 3 ) Back-hual Requirement: Uncoded Caching K=3 users Cache size M=1 (A 1, B 1, C 1 ) Broadcast channel Server (A 1, B 1, C 1 ) A, A 3 B, B 3 C, C 3 (A 1, B 1, C 1 ) User 1 wants A User wants B [1] Fundamental Limits of Caching, M. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, 014. User 3 wants C 3

7 An Example of Coded Caching [Maddah-Ali and Niesen 14] N=3 files: A = (A 1, A, A 3 ) B = (B 1, B, B 3 ) C = (C 1, C, C 3 ) Back-hual Requirement: Uncoded Caching K 1 M N = K=3 users Cache size M=1 (A 1, B 1, C 1 ) Broadcast channel Server (A 1, B 1, C 1 ) A, A 3 B, B 3 C, C 3 (A 1, B 1, C 1 ) User 1 wants A User wants B [1] Fundamental Limits of Caching, M. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, 014. User 3 wants C 3

8 An Example of Coded Caching [Maddah-Ali and Niesen 14] N=3 files: A = (A 1, A, A 3 ) B = (B 1, B, B 3 ) C = (C 1, C, C 3 ) Back-hual Requirement: Uncoded Caching K 1 M N = Coded Caching [1] K=3 users Cache size M=1 (A 1, B 1, C 1 ) Broadcast channel Server User 1 wants A User wants B [1] Fundamental Limits of Caching, M. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, 014. User 3 wants C 3

9 An Example of Coded Caching [Maddah-Ali and Niesen 14] N=3 files: A = (A 1, A, A 3 ) B = (B 1, B, B 3 ) C = (C 1, C, C 3 ) Back-hual Requirement: Uncoded Caching K 1 M N = Coded Caching [1] K=3 users Cache size M=1 Broadcast channel Server (A 1, B 1, C 1 ) (A, B, C ) (A 3, B 3, C 3 ) User 1 wants A User wants B [1] Fundamental Limits of Caching, M. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, 014. User 3 wants C 3

10 An Example of Coded Caching [Maddah-Ali and Niesen 14] N=3 files: A = (A 1, A, A 3 ) B = (B 1, B, B 3 ) C = (C 1, C, C 3 ) Back-hual Requirement: Uncoded Caching K 1 M N = Coded Caching [1] K=3 users Cache size M=1 Broadcast channel Server A B 1 (A 1, B 1, C 1 ) (A, B, C ) (A 3, B 3, C 3 ) A B 1 User 1 wants A User wants B [1] Fundamental Limits of Caching, M. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, 014. User 3 wants C 3

11 An Example of Coded Caching [Maddah-Ali and Niesen 14] N=3 files: A = (A 1, A, A 3 ) B = (B 1, B, B 3 ) C = (C 1, C, C 3 ) Back-hual Requirement: Uncoded Caching K 1 M N = Coded Caching [1] K=3 users Cache size M=1 Broadcast channel Server A B 1 A 3 C 1 (A 1, B 1, C 1 ) (A, B, C ) (A 3, B 3, C 3 ) A B 1 A 3 C 1 User 1 wants A User wants B [1] Fundamental Limits of Caching, M. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, 014. User 3 wants C 3

12 An Example of Coded Caching [Maddah-Ali and Niesen 14] N=3 files: A = (A 1, A, A 3 ) B = (B 1, B, B 3 ) C = (C 1, C, C 3 ) Back-hual Requirement: Uncoded Caching K 1 M N = Coded Caching [1] K=3 users Cache size M=1 A B 1 A 3 C 1 B 3 C (A 1, B 1, C 1 ) (A, B, C ) (A 3, B 3, C 3 ) A B 1 A 3 C 1 B 3 C User 1 wants A Broadcast channel Server User wants B [1] Fundamental Limits of Caching, M. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, 014. User 3 wants C 3

13 An Example of Coded Caching [Maddah-Ali and Niesen 14] N=3 files: A = (A 1, A, A 3 ) B = (B 1, B, B 3 ) C = (C 1, C, C 3 ) Back-hual Requirement: Uncoded Caching K 1 M N = Coded Caching [1] K 1 M N KM N = 1 K=3 users Cache size M=1 A B 1 A 3 C 1 B 3 C (A 1, B 1, C 1 ) (A, B, C ) (A 3, B 3, C 3 ) A B 1 A 3 C 1 B 3 C User 1 wants A Broadcast channel Server User wants B [1] Fundamental Limits of Caching, M. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, 014. User 3 wants C 3

14 An Example of Coded Caching [Maddah-Ali and Niesen 14] N=3 files: A = (A 1, A, A 3 ) B = (B 1, B, B 3 ) C = (C 1, C, C 3 ) Back-hual Requirement: Uncoded Caching K 1 M N = Coded Caching [1] K 1 M N KM N = 1 K=3 users Cache size M=1 Global caching gain A B 1 A 3 C 1 B 3 C (A 1, B 1, C 1 ) (A, B, C ) (A 3, B 3, C 3 ) A B 1 A 3 C 1 B 3 C User 1 wants A Broadcast channel Server User wants B [1] Fundamental Limits of Caching, M. Maddah-Ali and U. Niesen, IEEE Trans. Inf. Theory, 014. User 3 wants C 3

15 Recent Efforts [Maddah-Ali and Niesen 14] shows that the worst-case transmission rate K 1 M N 1 1+ KM N is at most a constant factor (1x) away from the (information-theoretic) minimum possible Generalized to Decentralized/probabilistic caching schemes [Ali and Niesen `14] Hierarchical caching [Karamchandani et al `14 ] Online caching [Pedarsani et al `13] Expected Rate [Niesen and Ali `14, Ji et al `14, Hachem et al `14] These studies all assume a homogeneous setting where all files are of a same size 4

16 The Question In practice, file sizes are often quite different In heterogeneous settings: Should larger files be cached more aggressively? What is the fundamental limit of the performance of coded caching with distinct file sizes? 5

17 Our Contribution Heterogeneous file-sizes (Roughly) quadratically more content is cached for larger files We show logarithmic-factor bounds While the new achievable schemes are quite intuitive, the corresponding lower bounds are more involved and reveal useful insights 6

18 Network Model: Distinct File Sizes Server with a broadcast channel File 1... Server File N K users: cache size M N files: F = F 1,, F N Request pattern: W i = f i1,, f ik, f ik F Rate for W i is r(w i ) User 1 User... User K Worst-case rate: R = max r(w i ) i 7

19 Network Model: Distinct File Sizes Server with a broadcast channel K users: cache size M File 1 1 File Placement... Server File N N files: F = F 1,, F N Request pattern: W i = f i1,, f ik, f ik F Rate for W i is r(w i ) User 1 User... User K Worst-case rate: R = max r(w i ) i 7

20 Network Model: Distinct File Sizes Server with a broadcast channel File 1... Server File N K users: cache size M 1 File Placement N files: F = F 1,, F N Request pattern: W i = f i1,, f ik, f ik F Rate for W i is r(w i ) File Request User 1 User... File Request User K Worst-case rate: R = max r(w i ) i 7

21 Network Model: Distinct File Sizes Server with a broadcast channel K users: cache size M File 1 1 File Placement... Server 3 File N File Transmission N files: F = F 1,, F N Request pattern: W i = f i1,, f ik, f ik F Rate for W i is r(w i ) File Request User 1 User... File Request User K Worst-case rate: R = max r(w i ) i 7

22 Power-of- Simplification All files 8

23 Power-of- Simplification F 1 All files Re-arrangement 8

24 Power-of- Simplification F 1 F 1,N 1 F = F 1,N 0.6F 1 F T = F 1 T 1,N T F 1,N 1 F = F 1,N All files Re-arrangement F T = F 1 T 1,N T 8

25 Power-of- Simplification F 1 F 1,N 1 F = F 1,N 0.6F 1 F T = F 1 T 1,N T F 1,N 1 F = F 1,N All files Re-arrangement F T = F 1 T 1,N T 8

26 Power-of- Simplification F 1,N 1 File 1 1 File Placement... Server 3 File N File Transmission F = F 1,N File Request File Request User 1 User... User K F T = F 1 T 1,N F T = F 1 T 1,N T 9

27 Power-of- Simplification F 1,N 1 File 1 1 File Placement... Server 3 File N File Transmission F = F 1,N File Request File Request User 1 User... User K F T = F 1 T 1,N F T = F 1 T 1,N T T = min{t, log K} Files of type l > T can be virtually neglected Their sizes F 1 /K 9

28 Power-of- Simplification F 1,N 1 File 1 1 File Placement... Server 3 File N File Transmission F = F 1,N File Request File Request User 1 User... User K F T = F 1 T 1,N F T = F 1 T 1,N T T = min{t, log K} Files of type l > T can be virtually neglected Their sizes F 1 /K 9

29 Intuition Towards Quadratic Caching The worst-case rate with uniform file size of 1 [Maddah-Ali & Niesen `14] is given by K 1 M N 1 1+ KM N N M 1 N M, when K >> N M and M<<N Each user caches every bit of each file with probability q = M N Worst-case rate N M = 1 q When the uniform file-size is F, these numbers become: Caching probability q = M NF, Worst-case rate: KF 1 M NF 1 1+ KM NF NF = F, when q << 1 and K >> 1/q M q 10

30 Two Achievable Schemes (UB1 vs UB) Achievable Scheme 1 (UB1) All files are cached with an equal probability q Linearly more content is cached for larger files Cache constraint: T q l=1 N l F l = M with T = min(t, log K) R UB1 O(max l F l q ) 11

31 Two Achievable Schemes (UB1 vs UB) Achievable Scheme 1 (UB1) All files are cached with an equal probability q Linearly more content is cached for larger files Cache constraint: T q l=1 N l F l = M with T = min(t, log K) R UB1 O(max l F l q ) F 1 suffers 11

32 Two Achievable Schemes (UB1 vs UB) Achievable Scheme 1 (UB1) All files are cached with an equal probability q Linearly more content is cached for larger files Cache constraint: T q l=1 N l F l = M with T = min(t, log K) R UB1 O(max l F l q ) F 1 suffers Achievable Scheme (UB) Let the caching probability be q l = F l /c Quadratically more content is cached for larger files Cache constraint: T T l=1 q l N l F l = M l=1 N l F l = cm R UB T l=1 Fl q l + K F T+1 11

33 Two Achievable Schemes (UB1 vs UB) Achievable Scheme 1 (UB1) All files are cached with an equal probability q Linearly more content is cached for larger files Cache constraint: T q l=1 N l F l = M with T = min(t, log K) R UB1 O(max l F l q ) F 1 suffers Achievable Scheme (UB) Let the caching probability be q l = F l /c Quadratically more content is cached for larger files Cache constraint: T T l=1 q l N l F l = M l=1 N l F l = cm R UB T Fl q l + K F T+1 l=1 Type T # of types * rate for each type 11

34 Two Achievable Schemes (UB1 vs UB) Achievable Scheme 1 (UB1) R UB1 O( F T 1 l=1 4M Nl F l Consider T = log K, N l+1 = 4N l Achievable Scheme (UB) ) R UB ( T + 1) l=1 T N l F l M 1

35 log(rate) Two Achievable Schemes (UB1 vs UB) Achievable Scheme 1 (UB1) R UB1 O( F T 1 l=1 4M Nl F l Achievable Scheme (UB) ) R UB ( T + 1) T l=1 N l F l M Consider T = log K, N l+1 = 4N l R UB1 K N 1F 1 8M R UB ( T + 1) T N 1F 1 M UB1 UB1 K/(log K) UB UB Number of types 1

36 log(rate) Two Achievable Schemes (UB1 vs UB) Achievable Scheme 1 (UB1) R UB1 O( F T 1 l=1 4M Nl F l Achievable Scheme (UB) ) R UB ( T + 1) T l=1 N l F l M Consider T = log K, N l+1 = 4N l R UB1 K N 1F 1 8M R UB ( T + 1) T N 1F 1 M Critical to cache quadratically more content for larger files! UB1 UB1 K/(log K) UB UB Number of types 1

37 Lower Bound: Uniform File-size NF/(M) users 1 NF/(M) M/F Request patterns N Files In order to serve all request patterns, we must have [Maddah-Ali & Niesen `14] M F R F + NF M M NF R F NF 4M = F 4q 13

38 Lower Bound: First Try L users l Φ N l L Request patterns l Φ N l Files of types in Φ In order to serve all request patterns, we must have l Φ N l R F + L M N L l F l Maximizing over L and Φ l Φ R l Φ N l F l F max Φ 4M l Φ N l (LB1) 14

39 log(rate) LB1 vs UB Lower Bound 1 (LB1) Achievable Bound (UB) Consider N l+1 = 4N l R LB1 N 1F 1 M R UB = ( T + 1) l=1 T N l F l M UB1 UB (log K) LB1 Number of types 15

40 log(rate) LB1 vs UB Lower Bound 1 (LB1) Achievable Bound (UB) Consider N l+1 = 4N l R LB1 N 1F 1 M R UB = ( T + 1) l=1 T N l F l M UB1 LB1 fails to account for heterogeneous caching probabilities! UB (log K) LB1 Number of types 15

41 log(rate) An Improved Lower Bound (LB) Proposition 1: Under Assumption 1: M F 1 and N lf l M are integers for all 1 l T Assumption : We must have T l=1 N l F l M K R F (LB) T Nl F l l=1 4M UB1 UB log K Compared with UB: R UB = ( T + 1) l=1 T N l F l M Number of types LB LB1 16

42 Intuition for LB (Two types: F = F 1 /) R T N F l F l l=1 4M =N 1F 1 4M +N F 4M (LB) s 1 = N 1F 1 M users (U 1 ) s = N F M users (U ) 17

43 Intuition for LB (Two types: F = F 1 /) R T N F l F l l=1 4M =N 1F 1 4M +N F 4M (LB) s 1 = N 1F 1 M users (U 1 ) s = N F M users (U ) patterns N 1 files of type 1 (F 1 ) 17

44 Intuition for LB (Two types: F = F 1 /) R T N F l F l l=1 4M =N 1F 1 4M +N F 4M (LB) s 1 = N 1F 1 M users (U 1 ) s = N F M users (U ) patterns N 1 files of type 1 (F 1 ) R F N 1F 1 4M 17

45 Intuition for LB (Two types: F = F 1 /) R T N F l F l l=1 4M =N 1F 1 4M +N F 4M (LB) s 1 = N 1F 1 M users (U 1 ) s = N F M users (U ) patterns N 1 files of type 1 (F 1 ) F 1 17

46 Intuition for LB (Two types: F = F 1 /) R T N F l F l l=1 4M =N 1F 1 4M +N F 4M (LB) s 1 = N 1F 1 M users (U 1 ) s = N F M users (U ) patterns N 1 files of type 1 (F 1 ) F 1 patterns N 1 files of type 1 (F 1 ) 17

47 Intuition for LB (Two types: F = F 1 /) R T N F l F l l=1 4M =N 1F 1 4M +N F 4M (LB) s 1 = N 1F 1 M users (U 1 ) s = N F M users (U ) patterns N 1 files of type 1 (F 1 ) F 1 patterns N 1 files of type 1 (F 1 ) F 17

48 Intuition for LB (Two types: F = F 1 /) R T N F l F l l=1 4M =N 1F 1 4M +N F 4M (LB) s 1 = N 1F 1 M users (U 1 ) s = N F M users (U ) patterns patterns N 1 files of type 1 (F 1 ) N 1 files of type 1 (F 1 ) F 1 N files of type (F ) F 17

49 Proof: (Two types: F = F 1 /) s 1 = N 1F 1, s M = N F M H M 1 = 0.5H F 1 = N 1F 1 H M = 0.5H F = N F H R D1, M 1 = H(R D1, M 1 F 1 ) + I R D1, M 1 ; F 1 18

50 Proof: (Two types: F = F 1 /) s 1 = N 1F 1, s M = N F M H M 1 = 0.5H F 1 = N 1F 1 H M = 0.5H F = N F H R D1, M 1 = H(R D1, M 1 F 1 ) + I R D1, M 1 ; F 1 0.5H(F 1 ) H(F 1 ) 18

51 Proof: (Two types: F = F 1 /) s 1 = N 1F 1, s M = N F M H M 1 = 0.5H F 1 = N 1F 1 H M = 0.5H F = N F H R D1, M 1 = H(R D1, M 1 F 1 ) + I R D1, M 1 ; F 1 0.5H(F 1 ) H(F 1 ) H R D1 H(R D1, M 1 F 1 ) + 0.5H(F 1 ) 18

52 Proof: (Two types: F = F 1 /) s 1 = N 1F 1, s M = N F M H M 1 = 0.5H F 1 = N 1F 1 H M = 0.5H F = N F H R D1, M 1 = H(R D1, M 1 F 1 ) + I R D1, M 1 ; F 1 0.5H(F 1 ) H(F 1 ) H R D1 H(R D1, M 1 F 1 ) + 0.5H(F 1 ) H R D H(R D, M 1 F 1 ) + 0.5H(F 1 ) 18

53 Proof: (Two types: F = F 1 /) s 1 = N 1F 1, s M = N F M H M 1 = 0.5H F 1 = N 1F 1 H M = 0.5H F = N F H R D1, M 1 = H(R D1, M 1 F 1 ) + I R D1, M 1 ; F 1 0.5H(F 1 ) H(F 1 ) H R D1 H(R D1, M 1 F 1 ) + 0.5H(F 1 ) H R D H(R D, M 1 F 1 ) + 0.5H(F 1 ) H(M ) H R D1 + H R D H F 0.5H(F ) +H(F 1 ) 18

54 Proof: (T>=3) 4N 1 s 1 R F 4I R D1 ; F N 1F 1 I R D1 R D ; F F 1 N F + I R D1 R D R D3 R D4 ; F 3 F 1, F N 3 F 3 19

55 Without Assumptions 1 & Intuition Memory sharing argument Assume there exists a lower bound R File Size R R A user s storage Effective Cache T 1 M = M N l (F l R) M l=1 Group 1 F l R Cache F l R portion Group (can be empty) F l < R, F l > M Each user requests 1 file Group 3 F l M Using previous results T 1 T 3 T T 4 File Type 0

56 Comparisons With Assumptions 1 & 1

57 Comparisons With Assumptions 1 & General Result (without Assumptions 1 and ): R UB (3log K + )R LB 1

58 Conclusion and Discussions Heterogeneous file-sizes Quadratically more content is cached for larger files We show logarithmic-factor gap While the new achievable schemes are quite intuitive, the corresponding lower bounds more involved and reveal useful insights Future Work: A tighter lower-bound? New techniques in the achievable scheme? Remove the logarithmic factor Sharper insights into the general lower bound

59 Thank you!