Demand Aware Network ( DAN ) Design Some Results and Open Questions - PowerPoint PPT Presentation

Demand Aware Network ( DAN ) Design Some Results and Open Questions Chen Avin Joint work with Stefan Schmid, Kaushik Mondal, Alexandr Hercules, Andreas Loukas

Motivation • Demand Aware Network Design? • “self-adjust” the networks‘ routing paths (topology) to routing requests • Data Centres? • ProjecTor / Wireless technologies • Skype example? Mirror assembly Reflected beam • Peer-to-Peer Networks Received beam Diffracted beam Towards destination Photodetectors DMDs Lasers Input beam Array of Micromirrors

Outline • Motivation • Problem Settings • Relation to other problems • Lower Bounds • Bounded degree network design • The continuous discrete approach • Future work

Problem Settings 1 2 3 4 5 6 7 2 1 1 1 2 3 1 0 65 13 65 65 65 65 2 1 2 2 0 0 0 0 • Demand distribution, over er V × V . W D 65 65 65 1 1 2 1 3 0 0 0 13 65 65 13 1 2 4 4 0 0 0 0 65 65 65 e the ( ) e 1 3 4 5 0 0 0 0 65 65 65 • Pairwise communication demands 2 3 6 0 0 0 0 0 65 65 3 2 1 3 7 0 0 0 65 65 13 65 7 (a) • Can be represented as directed weighted graph 3 3 1 6 2 2 2 1 • A network N = ( V, E ) 5 5 1 2 5 1 4 • Metric of interest: Expected Path Length 2 3 4 ÿ EPL( D , N ) = E D [d N ( · , · )] = p( u, v ) · d N ( u, v ) ( u,v ) ∈ D g across the host network usually occurs along shortest path ) · d N ( u, v ) - hop distance between u,v in N

Problem Settings 1 2 3 4 5 6 7 2 1 1 1 2 3 1 0 65 13 65 65 65 65 2 1 2 2 0 0 0 0 • Demand distribution, D 65 65 65 1 1 2 1 3 0 0 0 13 65 65 13 1 2 4 4 0 0 0 0 65 65 65 1 3 4 5 0 0 0 0 65 65 65 2 3 6 0 0 0 0 0 • Expected path length 65 65 3 2 1 3 7 0 0 0 65 65 13 65 7 (a) 3 3 ÿ EPL( D , N ) = E D [d N ( · , · )] = p( u, v ) · d N ( u, v ) 1 6 2 2 ( u,v ) ∈ D 2 1 5 • Desired topology family N 5 g across the host network usually occurs along shortest path 1 2 5 1 4 • e.g., bounded degree, trees, sparse, etc. 2 3 4 7 • Optimal Demand Aware Network (DAN) 1 6 N ∗ = arg min N ∈ N EPL( D , N ) 2 5 3 4

Relation to Other Problems 7 3 3 • Minimum Linear Arrangement (MLA) 1 6 2 2 2 1 5 5 1 2 5 1 4 2 3 4

Relation to Other Problems 7 3 3 • Minimum Linear Arrangement (MLA) 1 6 2 2 2 1 5 • Embeddings (guest, host graphs) 5 1 2 5 1 4 • Spanners 2 3 4 • Information Theory / Coding n 1 Entropy : ÿ • H ( X ) = p ( x i ) log 2 p ( x i ) i =1 n Conditional Entropy: ÿ • H ( X | Y ) = p ( y j ) H ( X | Y = y j ) = j =1 Coding - Expected code length •

Lower Bound • For a Δ bounded degree DAN • Theorem N ∗ BND( D , ∆ ) Ø Ω (max( H ∆ ( Y | X ) , H ∆ ( X | Y )) 1 2 3 4 5 6 7 • Proof Idea (using coding): 2 1 1 1 2 3 1 0 65 13 65 65 65 65 2 1 2 2 0 0 0 0 65 65 65 1 1 2 1 3 0 0 0 • Replacing each row with an optimal Δ -ary tree 13 65 65 13 1 2 4 4 0 0 0 0 65 65 65 1 3 4 5 0 0 0 0 65 65 65 2 3 • Same for columns 6 0 0 0 0 0 65 65 3 2 1 3 7 0 0 0 65 65 13 65 • Optimal code length is larger than row Entropy


 
 Bounded Degree DAN • Bounded (e.g., Δ = constant) degree • Theorem: Can design “optimal” network , s.t 
 N EPL( D , N ) ≤ O ( H ( Y | X ) + H ( X | Y )) for, This is asymptotically optimal when ∆ is a • Sparse distributions (weighted, directed) • Local doubling dimension distribution • Possibly dense but uniform and regular

Sparse Distributions • Proof idea i i Optimal bounded degree tree

Sparse Distributions • Proof idea i i Optimal bounded degree tree Problem i j Solution

Sparse Distributions • Proof idea i i Optimal bounded degree tree Problem j i i j Solution

Doubling Dimensions Dist. • Local Doubling Dimension distribution 2-hops balls can be covered by 1-hop balls

Doubling Dimensions Dist. • Local Doubling Dimension distribution 2-hops balls can be covered by 1-hop balls • Can be a dense graph

Continuous-Discrete Design 1 2 3 4 5 6 7 2 1 1 1 2 3 1 0 • Greedy routing 65 13 65 65 65 65 2 1 2 2 0 0 0 0 65 65 65 1 1 2 1 3 0 0 0 13 65 65 13 1 2 4 4 0 0 0 0 65 65 65 1 3 4 5 0 0 0 0 65 65 65 2 3 6 0 0 0 0 0 65 65 3 2 1 3 7 0 0 0 65 65 13 65 (a)

Continuous-Discrete Design 1 2 3 4 5 6 7 2 1 1 1 2 3 1 0 • Greedy routing 65 13 65 65 65 65 2 1 2 2 0 0 0 0 65 65 65 1 1 2 1 3 0 0 0 13 65 65 13 1 2 4 4 0 0 0 0 65 65 65 1 3 4 5 0 0 0 0 65 65 65 2 3 6 0 0 0 0 0 𝑐𝑏𝑑𝑙(𝑦) 65 65 𝑚𝑓𝑔𝑢(𝑦) 𝑠𝑗𝑕ℎ𝑢(𝑦) 3 2 1 3 7 0 0 0 65 65 13 65 𝑦 𝑦 + 1 2𝑦 mod 1 0 𝑦 1 2 2 (a) x 1 = 0 x 1 = 0 u 1 x 2 = F(u 1 ) x 2 = 0.1 u 2 u 6 0.8 = x 6 right( s 4 ) x 3 = 0.25 s(x i ) = [x i , x i+1 ) u 5 0.7 = x 5 u x i+1 F(u i ) = left( s 4 ) 3 s 4 cs 4 x i = F(u i-1 ) x 4 = 0.45 u 4 cs(i) = [cw(i), cw(i)+2 -l(u i ) )

Continuous-Discrete Design 1 2 3 4 5 6 7 2 1 1 1 2 3 1 0 • Greedy routing 65 13 65 65 65 65 2 1 2 2 0 0 0 0 65 65 65 1 1 2 1 3 0 0 0 13 65 65 13 1 2 4 4 0 0 0 0 65 65 65 1 3 4 5 0 0 0 0 65 65 65 2 3 6 0 0 0 0 0 65 65 3 2 1 3 7 0 0 0 65 65 13 65 ( x ) (a) F 0 0 000 1 100 1 001 ( ) 0 010 0 F x } i ( ) ( ) p x F x 0 1 i i 1 0 ( ) F x � 1 i 101 1 1 0 011 110 0 111 1 x x x x x � 1 1 0 1 i i Shannon-Fano-Elias Coding De-Bruijn Graph

Continuous-Discrete Design 1 2 3 4 5 6 7 2 1 1 1 2 3 1 0 • Greedy routing 65 13 65 65 65 65 2 1 2 2 0 0 0 0 65 65 65 1 1 2 1 3 0 0 0 13 65 65 13 1 2 4 4 0 0 0 0 • Theorem: 65 65 65 1 3 4 5 0 0 0 0 65 65 65 2 3 6 0 0 0 0 0 • Linear size 65 65 3 2 1 3 7 0 0 0 65 65 13 65 (a) • Fair (please explain) • Robust to failures • Expected path length: EPL ( R , G, A ) < min { H ( p s ) , H ( p d ) } + 2 .

Future Work / Discussion • New “Graph Entropy” measure for networks • Online algorithms - Amortize analysis • Splay-nets example • Distributed algorithms? • Practical use ???

Thank you avin@cse.bgu.ac.il See papers: 
 • Demand-Aware Network Designs of Bounded Degree . Chen Avin, Kaushik Mondal, and Stefan Schmid.. ArXiv Technical Report, May 2017. https://arxiv.org/ abs/1705.06024 • Towards Communication-Aware Robust Topologies . Chen Avin, Alexandr Hercules, Andreas Loukas, and Stefan Schmid. 
 https://arxiv.org/abs/1705.07163 • SplayNet: Towards Locally Self-Adjusting Networks . Stefan Schmid, Chen Avin, Christian Scheideler, Michael Borokhovich, Bernhard Haeupler, and Zvi Lotker. IEEE/ACM Transactions on Networking (ToN). 
 http://ieeexplore.ieee.org/document/7066977/