Steps towards Decentralized Deterministic Network Coding

Transcription

1 Steps towards Decentralized Deterministic Network Coding BY Oana Graur Ph.D. Proposal in Electrical Engineering Ph.D Proposal Committee: Prof. Dr.-Ing. Werner Henkel, Dr. Mathias Bode, 1

2 Overview Fundamentals of Graph Theory Introduction to Network Coding Multicast Rate Linear Network Coding Erasures in Network Coding Packet Loss in Real Networks Inferring Link Loss from Path Loss Netscope Measurements Results Ant Colony Optimization (ACO) Finding the Optimum Coding Nodes Hierarchical Network Coding Routing in the Internet Proposed Hierarchical Network Coding Scheme Conclusions and Further Work 1

3 Graph Representation F 0.68 E 1.94 I A(i,j) = { 1, if (i,j) E 0, otherwise. A B D E G H / 6.65 B A C G / 5.2 D 0.93 C A 1.02 B G 5.25 H A = C D E F G H I A D / A C F A F I D E / A B H A G I H E / / / / / Example of a weighted graph Adjacency Matrix Adjacency list representation Spanning Trees 2

4 Traversal of Graphs Depth-First Search (DFS) F 5.2 D 0.93 C E 2.87 A 1.02 B I H G : Initialize Adj(v) for every vertex v in G. 2: i 1 3: T // spanning tree is initialized 4: for each vertex v V(G) do 5: DFI(v) 0 // depth-first index is initialized 6: end for 7: while for some v, DFI(v) = 0 do 8: DFS(v) // call procedure defined below 9: end while 10: 11: procedure DFS(v) 12: DFI(v) i // current node walked in step i 13: i i : for each vertex u Adj(v) do 15: if DFI(u) = 0 then 16: T T {(u,v)}; // add edge (u,v) to the tree 17: DFS(u) // call DFS recursively 18: end if 19: end for 20: Output the spanning tree T. Example of a weighted graph Both Depth-First Search (DFS) and Breadth-First Search (BFS) result in the formation of spanning trees. 3

5 Shortest Paths Dijkstra s Algorithm(G, s) F 5.2 D 0.93 C E A B Dijkstra - first iteration I 6.65 H 0.82 G 1: for each vertex v V(G) do 2: dist(v) // initialize node costs to infinity 3: previous(v) undefined // parents of nodes undefined 4: end for 5: dist(s) 0 // source cost is initialized to zero 6: T // SPT initialized to empty set 7: Q V(G) // initialize queue Q to vertex set 8: while Q do 9: u vertex in Q with smallest dist( ) 10: Q Q \ {u} // dequeue lowest cost vertex 11: if dist(u) = then 12: break 13: 14: end if T T {u} // add vertex to the tree 15: for each neighbor v of u do 16: new dist(v) = dist(u) + w(u, v) // update distances 17: if dist(v) > new dist(v) then 18: dist(v) new dist(v) // keep path with minimum cost 19: previous(v) u 20: reorder Q // reorder the queue of nodes not yet in tree 21: end if 22: end for 23: end while 24: Output the spanning tree T. 4

6 Shortest Paths Dijkstra s Algorithm(G, s) Dijkstra - second iteration 1: for each vertex v V(G) do 2: dist(v) // initialize node costs to infinity 3: previous(v) undefined // parents of nodes undefined 4: end for 5: dist(s) 0 // source cost is initialized to zero 6: T // SPT initialized to empty set 7: Q V(G) // initialize queue Q to vertex set 8: while Q do 9: u vertex in Q with smallest dist( ) 10: Q Q \ {u} // dequeue lowest cost vertex 11: if dist(u) = then 12: break 13: end if 14: T T {u} // add vertex to the tree 15: for each neighbor v of u do 16: new dist(v) = dist(u) + w(u, v) // update distances 17: if dist(v) > new dist(v) then 18: dist(v) new dist(v) // keep path with minimum cost 19: previous(v) u 20: reorder Q // reorder the queue of nodes not yet in tree 21: 22: end if end for 23: end while 24: Output the spanning tree T. 5

7 Shortest Paths Dijkstra s Algorithm(G, s) Dijkstra - third iteration 1: for each vertex v V(G) do 2: dist(v) // initialize node costs to infinity 3: previous(v) undefined // parents of nodes undefined 4: end for 5: dist(s) 0 // source cost is initialized to zero 6: T // SPT initialized to empty set 7: Q V(G) // initialize queue Q to vertex set 8: while Q do 9: u vertex in Q with smallest dist( ) 10: Q Q \ {u} // dequeue lowest cost vertex 11: if dist(u) = then 12: break 13: end if 14: T T {u} // add vertex to the tree 15: for each neighbor v of u do 16: new dist(v) = dist(u) + w(u, v) // update distances 17: if dist(v) > new dist(v) then 18: dist(v) new dist(v) // keep path with minimum cost 19: previous(v) u 20: reorder Q // reorder the queue of nodes not yet in tree 21: end if 22: end for 23: end while 24: Output the spanning tree T. 6

8 Shortest Paths Dijkstra s Algorithm(G, s) Dijkstra - fourth iteration 1: for each vertex v V(G) do 2: dist(v) // initialize node costs to infinity 3: previous(v) undefined // parents of nodes undefined 4: end for 5: dist(s) 0 // source cost is initialized to zero 6: T // SPT initialized to empty set 7: Q V(G) // initialize queue Q to vertex set 8: while Q do 9: u vertex in Q with smallest dist( ) 10: Q Q \ {u} // dequeue lowest cost vertex 11: if dist(u) = then 12: break 13: end if 14: T T {u} // add vertex to the tree 15: for each neighbor v of u do 16: new dist(v) = dist(u) + w(u, v) // update distances 17: if dist(v) > new dist(v) then 18: dist(v) new dist(v) // keep path with minimum cost 19: previous(v) u 20: reorder Q // reorder the queue of nodes not yet in tree 21: 22: end if end for 23: end while 24: Output the spanning tree T. 7

9 Minimum Spanning Trees Prim s Algorithm I F 5.2 D E 2.87 A H 1: Initialize V = and T = 2: Designate starting node and include it in V. 3: while V V do 4: Choose an edge e i,j, corresponding to vertices (v i,v j ), with minimum weight, such that v i V and v j (V \ V ). 5: Add vertex v j to V and edge e i,j to T. 6: end while 7: Output the minimum spanning tree, T G C 1.83 B 5.25 ω(t ) = ω(u, v) (u,v) T Prim - first iteration 8

10 Minimum Spanning Trees Prim s Algorithm I F 5.2 D E 2.87 A H 1: Initialize V = and T = 2: Designate starting node and include it in V. 3: while V V do 4: Choose an edge e i,j, corresponding to vertices (v i,v j ), with minimum weight, such that v i V and v j (V \ V ). 5: Add vertex v j to V and edge e i,j to T. 6: end while 7: Output the minimum spanning tree, T G C 1.83 B 5.25 ω(t ) = ω(u, v) (u,v) T Prim - second iteration 9

11 Minimum Spanning Trees Prim s Algorithm F 5.2 D E 2.87 A I 6.65 H : Initialize V = and T = 2: Designate starting node and include it in V. 3: while V V do 4: Choose an edge e i,j, corresponding to vertices (v i,v j ), with minimum weight, such that v i V and v j (V \ V ). 5: Add vertex v j to V and edge e i,j to T. 6: end while 7: Output the minimum spanning tree, T G C B Prim - third iteration ω(t ) = ω(u, v) (u,v) T 10

12 Minimum Spanning Trees Prim s Algorithm F 5.2 D E 2.87 A I 6.65 H : Initialize V = and T = 2: Designate starting node and include it in V. 3: while V V do 4: Choose an edge e i,j, corresponding to vertices (v i,v j ), with minimum weight, such that v i V and v j (V \ V ). 5: Add vertex v j to V and edge e i,j to T. 6: end while 7: Output the minimum spanning tree, T G C B Prim - fourth iteration ω(t ) = ω(u, v) (u,v) T 11

13 Minimum Spanning Trees Prim s Algorithm I F 5.2 D E 2.87 A H 1: Initialize V = and T = 2: Designate starting node and include it in V. 3: while V V do 4: Choose an edge e i,j, corresponding to vertices (v i,v j ), with minimum weight, such that v i V and v j (V \ V ). 5: Add vertex v j to V and edge e i,j to T. 6: end while 7: Output the minimum spanning tree, T G C 1.83 B 5.25 ω(t ) = ω(u, v) (u,v) T Prim - fifth iteration 12

14 Minimum Spanning Trees Prim s Algorithm I F 5.2 D E 2.87 A H 1: Initialize V = and T = 2: Designate starting node and include it in V. 3: while V V do 4: Choose an edge e i,j, corresponding to vertices (v i,v j ), with minimum weight, such that v i V and v j (V \ V ). 5: Add vertex v j to V and edge e i,j to T. 6: end while 7: Output the minimum spanning tree, T G C 1.83 B 5.25 ω(t ) = ω(u, v) (u,v) T Prim - sixth iteration 13

15 Minimum Spanning Trees Prim s Algorithm F 5.2 D E 2.87 A I 6.65 H : Initialize V = and T = 2: Designate starting node and include it in V. 3: while V V do 4: Choose an edge e i,j, corresponding to vertices (v i,v j ), with minimum weight, such that v i V and v j (V \ V ). 5: Add vertex v j to V and edge e i,j to T. 6: end while 7: Output the minimum spanning tree, T G C B Prim - seventh iteration ω(t ) = ω(u, v) (u,v) T 14

16 Minimum Spanning Trees Prim s Algorithm F 5.2 D E 2.87 A I 6.65 H : Initialize V = and T = 2: Designate starting node and include it in V. 3: while V V do 4: Choose an edge e i,j, corresponding to vertices (v i,v j ), with minimum weight, such that v i V and v j (V \ V ). 5: Add vertex v j to V and edge e i,j to T. 6: end while 7: Output the minimum spanning tree, T G C B Prim - eigth iteration ω(t ) = ω(u, v) (u,v) T 15

17 Minimum Spanning Trees Prim s Algorithm H 0.82 G F I C 1: Initialize V = and T = 2: Designate starting node and include it in V. 3: while V V do 4: Choose an edge e i,j, corresponding to vertices (v i,v j ), with 5.25 B 0.68 E 1.94 D 0.93 minimum weight, such that v i V and v j (V \ V ). 5: Add vertex v j to V and edge e i,j to T. 6: end while 7: Output the minimum spanning tree, T ω(t ) = ω(u, v) (u,v) T A Prim - Minimum Spanning Tree 16

18 Maximum Flow The Max-Flow/Min-Cut Theorem Network: G = (V,E) Edge capacities: c(u,v) : E R Source: s Sink: t The flow of an edge (u,v), f : E R is subject to: Capacity constraints: f(u,v) < c(u,v) Skew symmetry: f(u,v) = f(v,u) Flow conservation: f(u,v) = 0 unless u = s or u = t. v T 17

19 Maximum Flow The Max-Flow/Min-Cut Theorem Network: G = (V,E) Edge capacities: c(u,v) : E R Source: s Sink: t An s-t cut of the network G = (V,E) is a partition of the vertex set V into two disjoint sets V and V \ V such that one set contains the source s and the other set contains the sink t. Capacity of a cut K(V,V \ V ) = u V,v (V\V ) c(u, v) 18

20 Maximum Flow The Max-Flow/Min-Cut Theorem Network: G = (V,E) Edge capacities: c(u,v) : E R Source: s Sink: t The maximum flow through a network is equal to the minimum value among all possible s t cuts [Elias, Shannon]. Flow of an arbitrary cut F(G) = u V,v (V\V ) f(u,v) u (V\V ),v V f(u,v) 19

21 Maximum Flow Ford-Fulkerson Algorithm residual capacity cf (u,v) = c(u,v) f(u,v) augmented path Designate G, source s, sink t, and edge capacities c(u,v). f(u,v) 0 for all edges (u,v) while there is an augmenting path P from s to t do find c f (P) = min{c f (u,v) : (u,v) P} for each edge (u,v) P do f(u,v) f(u,v) + c f (P) f(v,u) f(v,u) c f (P) end for end while Output the maximum flow f(s). 20

22 Maximum Flow Ford-Fulkerson Algorithm A B C D E F A B C D E F A B C D E F A B C D E F A B C D E F A B C D E F Iteration 2 Iteration 1 Iteration 3 Iteration 4 Iteration 5 Iteration 6 21

23 Random Graph Theory Erdős-Rényi Model First model: En,N set of all graphs with n vertices and N edges ( ( ) n ) A graph is chosen at random with uniform probability 1/ 2. N Second model: En,p - connect every possible pair of nodes with probability p A graph results with pn(n 1)/2 edges distributed randomly. Degree distribution P(k) = As n, ( ) n 1 p k (1 p) n 1 k k P(k) = e α λ k, for λ = (n 1)p. k! 22

24 Network Modeling and Barabasi-Albert (BA) Model Scale free model for simulation of node degree distribution growth preferential attachment Starting with n0 vertices, at each time increment a new node is connected to the an existing node i with probability Degree distribution Π(k i ) = k i k j j P(k) = 2m 1/β k γ for γ=3, β=0.5 Clustering coefficient C i = 2E i k i (k i 1) Scale-free network generated by the B-A model 23

25 Introduction to Network Coding The max flow given by the Max-Flow/Min-Cut Theorem is hardly achieved in practice (store-and-forward). The maximum flow through a network is achievable through the use of network coding [Ahlswede, Cai, and Li]. System of equations not linearly dependent. A butterfly network - simple network coding 24

26 Multicast Rate Single-Source Multicast Directed acyclic network G = (V,E) with unit edge capacities Transmitted vector x = [x 1 x 2 x ω], x F w Condition for sink ti to receive x maxflow(t i ) ω The value of maxflow for sink ti is given by the Ford-Fulkerson algorithm. The multicast rate for a network with multiple sinks is the minimum maxflow rate among all sinks t i. A butterfly network - simple network coding h = maxflow(t i ) 25

27 Linear Network Coding Local Encoding Kernel [Yeung, Li, Cai] Let F be a finite field and ω a positive integer. An ω-dimensional F-valued linear network code on an acyclic communication network consists of a scalar k d,e, called the local encoding kernel, for every adjacent pair (d,e). Meanwhile, the local encoding kernel at node u can be written as the In(u) Out(u) matrix K u = [k d,e ], d In(u), e Out(u). maps data from incoming edge d to outgoing edge e, at a node Local encoding kernels (in black) and global encoding kernels (in blue) 26

28 Linear Network Coding Global Encoding Kernel [Yeung, Li, Cai] Let F be a finite field and ω a positive integer. An ω-dimensional F-valued linear network code on an acyclic communication network consists of a scalar k d,e in the network as well as ω-dimensional column vector f e, known as the global encoding kernel, for every channel e such that: fe = k d,e f d, where e Out(u). d In(u) The vectors fe for the ω imaginary channels e In(s) form the natural basis of the vector space F ω. maps the source message x to the output on edge e Local encoding kernels (in black) and global encoding kernels (in blue) 27

29 Linear Network Coding F u out = Fu in Ku F uin = [f d1 f d2 f d In(u) ] F u out = [fe 1 fe 2 fe Out(u) ] d In(u) e Out(u) k d1 e k 1 d1 e k 2 d1 e Out(u) k d2 e k 1 d2 e k 2 d2 e Out(u) [f e1 f e2 f e Out(u) ] = [f d1 f d2 f d In(u) ] k d In(u) e k 1 d In(u) e k 2 d In(u) e Out(u) 28

30 Linear Network Coding For a certain node u, the symbol leaving the outgoing edge e is given by x f e = x d In(u) k d,e f d = d In(u) k d,e (x f d ) The vector of received symbols y = [y1 y 2 y ω] can be written as y = xg, where column j of G holds the global encoding kernel of the jth incoming link of sink t i. Decoding at sink ti is possible if G is nonsingular. There is also Random Linear Network Coding (RLNC), where all intermediate nodes perform coding and the encoding kernels are chosen at random. Local encoding kernels (in black) and global encoding kernels (in blue) 29

31 Erasures in Network Coding congested links (buffer overflows, etc.) packet loss Binary Erasure Channel (BEC) P set of all edges of a certain path L pi erasure probability of link e i, 1 i P overall erasure probability of path L is given by P P L = 1 (1 p i ) i=1 If GEK poorly chosen, performance is significantly decreased Local encoding kernels (in black) and global encoding kernels (in blue) 30

32 Packet Loss in Real Networks H1 R5 H4 limited accesibility to intermediate nodes end-to-end measurements possible (path measurements) e1 R1 e2 R6 R4 Netscope [Nguyen, Ghita] network tomography infer link loss from path loss measurements construct a probability distribution function of the link losses H2 e3 R2 R3 H3 Random network example 31

33 Inferring Link Loss from Path Loss - Netscope Network: G = (V,E) Edge capacities: c(u,v) : E R Number vertices: n v = V Number edges: n e = E Set of paths: P Number of paths: n p = P End-to-end measurements: y = [y 1 y 2...y np ] T Link transmission rate estimates: x = [x 1 x 2...x ne ] T Routing matrix: R, size n p n e y = Rx y y 2 y 3 = y x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 H1 H2 e1 R5 R4 R1 e2 R6 R2 R3 e3 Random network example H3 H4 Routing matrix is always rank deficient [Ghita, Nguyen]. 32

34 Inferring Link Loss from Path Loss - Netscope Network: G = (V,E) Edge capacities: c(u,v) : E R Number vertices: n v = V Number edges: n e = E Set of paths: P Number of paths: n p = P End-to-end measurements: y = [y 1 y 2...y np ] T H1 R5 H4 Link loss estimates: x = [x 1 x 2...x ne ] T Routing matrix: R, size n p n e e1 R1 R4 y = Rx e2 R6 Not all links are lossy, e.g., backbone links. Packet loss of a subset of the links (k) can be approximated by zero. H2 e3 R2 R3 H3 k opt = E rank(r) Random network example Nguyen et al. argue that the links with zero transmission loss can be identified by first computing their variance. The more congested a link is, the higher the variance of its loss rate over time. 33

35 Inferring Link Loss from Path Loss - Netscope Practical aspects and assumptions H1 R5 H4 not all links are distinguishable virtual links paths meet, split and meet again route fluttering route fluttering highly site-dependent (2% 40%) route fluttering attributed to load balancing topology varies over time (only 2/3 of routes persist for more than one day) Reduced routing matrix R, size n p n c, n c < n e. H2 e1 R1 e3 e2 R2 R6 R4 R3 H3 Random network example 34

36 Inferring Link Loss from Path Loss - Netscope Estimate of the successful transmission rate on path P i : ˆφ i Transmission rate of link e k for the packets traveling along path P i : ˆφ i,ek Loss rate of P i : 1-φ i, φ i = E[ˆφ i ] Transmission rate of link e k : φ ek = E[ˆφ ek ] y i = log ˆφ i x k = log ˆφ ek Assumptions: ˆφ i,ek = ˆφ ek random variables xk are independent Covariance matrix of x: H1 H2 e1 R1 e3 R5 e2 R2 R6 R4 R3 H3 H4 Γ x = diag([v 1,v 2,...,v nc ]) Σ = RΓ xr T Random network example 35

37 Inferring Link Loss from Path Loss - Netscope Covariance matrix of x: y = Rx H1 e1 R1 R5 R4 H4 Γ x = diag([v 1,v 2,...,v nc ]) e2 R6 Σ = RΓ xr T σ 2 y 1 cov(y 1,y 2 )... cov(y 1,y np ) cov(y 2,y 1 ) σ 2 y 2... cov(y 1,y np ) Σ = cov(y np,y 1 ) cov(y np,y 2 )... σ 2 y np H2 R2 R3 e3 Random network example H3 36

38 Inferring Link Loss from Path Loss - Netscope Let A be an augmented matrix of dimension n p(n p +1)/2 n c whose rows consist of the component-wise product of each pair of different rows from R. The rows of A are arranged as follows: A ((i 1) np+(j i)+1) = R i R j for all 1 i j n p. E.g., R = A = R 1 R 2 R = R = ( ) R 1 R 2 R np. R np A = R 1 R 1 R 1 R 2. R 1 R np R 2 R 2 R 2 R 3. R 2 R np. R np R np 37

39 The equations Σ = RΓ xr T = Rdiag(v)R T are equivalent to the equations Σ = Av, where Σ is a vector of length n p(n p + 1)/2 n c and Σ ((i 1) np+(j i)+1) =Σ i,j for all 1 i j n p [Ghita, Nguyen]. Proof: Σ = R 1 R 2. R np v v v v nc ( R T ) 1 RT 2 RT np R 11 v 1 R 12 v 2 R 1nc v nc R 11 R 21 R np1 R 21 v 1 R 22 v 2 R 2nc v nc R 12 R 22 R np2 Σ = R np1 v 1 R np2 v 2 R npnc v nc R 1nc R 2nc R npnc (R 1 R 1 )v (R 1 R 2 )v (R 1 R np )v (R 2 R 1 )v (R 2 R 2 )v (R 2 R np )v Σ = (R np R 1 )v (R np R 2 )v (R np R np )v 38

40 Inferring Link Loss from Path Loss - Netscope Netscope [Ghita, Nguyen] Input: Reduced routing matrix R and m + 1 snapshots Learning the link variances: Step 1: Compute an estimate of Σ, given the m snapshots, as in Eq. (??). Step 2: Solve Σ = Av with the first m 1 snapshots to find v k for all links e k E. Inferring link loss rates: Step 1: Sort v k in increasing order. Step 2: Initialize R = R. while R is not of full column rank do Step 3: Remove R 1 from R. Step 4: Solve y = R x for snapshot m. end while Approximate φ ek 1 for all links e k whose columns were removed from R. END 39

41 PlanetLab Measurements Internet testbed for research 226 end hosts, both beacons and probing destinations ping, traceroute Number of active paths P = IP aliasing resolution (active probing Ally) IP ID [RFC 791] counter and TTL counter comparison Number of intermediate routers reduced from 3049 to 1920 No. of IP aliases in each router 10 3 Router with 251 IP aliases Unique router index [n] IP aliases The router with the highest numbers of aliases (251) in the PlanetLab Europe network was identified as being part of NORDUnet (Nordic Infrastructure for Research & Education network). 40

42 PlanetLab Measurements 10 0 Ratio out of the total no. of edges Success probability of transmission [%] Derived link loss (erasure) distribution based on path loss measurements, employing network tomography techniques (Netscope). 41

43 PlanetLab Measurements B-A Theoretical Model PlanetLab Measurements Probability Node degree 42

44 Ant Colony Optimization for Finding Shortest Path Ant Colony Optimization (ACO) [Dorigo] Ants walk through the network, looking for food sources. Once a food source is found, ants return to the colony, on the same path, laying down a pheromone trail. Each ant has a memory of the current path. At every node, an ant chooses one among the adjacent links based on previous pheromone levels of the links. Pheromone concentration will be more significant on shorter paths. Trail intensity diffuses over time. Ants path ACO is preferable since it does not require a centralized view of the topology. 43

45 Ant Colony Optimization for Finding Shortest Path Ant Colony Optimization (ACO) [Dorigo] Network: G = (V,E) Number of vertices: n = V Edge mappings: w : E R Pheromone level of edge e i,j : τ i,j (t) Number of ants: m Transition probability τ ij α (t)η β ij p ij (t) = τ is α(t)ηβ is sallowed Pheromone update τ ij (t + n) = ρτ ij (t) + τ ij, τ i,j = m k=1 τ k i,j, Ants path { Q τi,j k, if ant k deposited pheromones on e = L i,j in time t k 0, otherwise. P k L k = 1 φ ei i=1 44

46 ACO Convergence Prob. of detection within tolerance level % tolerance 5% tolerance perfect fit, 0% tolerance 0 n 0,5 n 1 n 1,5 n 2 n 2,5 n 3 n No. of iterations (n=size of network) α = 1, β = 5, ρ = 0.99, n = m =

47 Finding Disjoint Paths Select a source node and sink nodes Sinks First disjoint path Second disjoint path Sink 16 Sink 17 Sink 18 Sink 19 46

48 Finding Disjoint Paths Select a source node and sink nodes Sinks First disjoint path Second disjoint path Sink 16 Sink 17 Sink 18 Sink 19 47

49 Finding Disjoint Paths Select a source node and sink nodes Sinks First disjoint path Second disjoint path Sink 16 Sink 17 Sink 18 Sink 19 48

50 Finding Disjoint Paths Get multicast rate mi for every pair (source,sink i ). Overall multicast rate is min(m i ). For all sinks, find min(mi ) disjoint paths Sinks First disjoint path Second disjoint path Sink 16 Sink 17 Sink 18 Sink 19 P 1 :

51 Finding Disjoint Paths Get multicast rate mi for every pair (source,sink i ). Overall multicast rate is min(m i ). For all sinks, find min(mi ) disjoint paths Sinks First disjoint path Second disjoint path Sink 16 P 1 : P 2 : Sink 17 Sink 18 Sink 19 50

52 Finding Disjoint Paths Get multicast rate mi for every pair (source,sink i ). Overall multicast rate is min(m i ). For all sinks, find min(mi ) disjoint paths Sinks First disjoint path Second disjoint path Sink 16 P 1 : P 2 : Sink 17 P 3 : Sink 18 Sink 19 51

53 Finding Disjoint Paths Get multicast rate mi for every pair (source,sink i ). Overall multicast rate is min(m i ). For all sinks, find min(mi ) disjoint paths Sinks First disjoint path Second disjoint path Sink 16 P 1 : P 2 : Sink 17 P 3 : P 4 : Sink 18 Sink 19 52

54 Finding Disjoint Paths Get multicast rate mi for every pair (source,sink i ). Overall multicast rate is min(m i ). For all sinks, find min(mi ) disjoint paths Sinks First disjoint path Second disjoint path Sink 16 P 1 : P 2 : Sink 17 P 3 : P 4 : Sink 18 P 5 : Sink 19 53

55 Finding Disjoint Paths Get multicast rate mi for every pair (source,sink i ). Overall multicast rate is min(m i ). For all sinks, find min(mi ) disjoint paths Sinks First disjoint path Second disjoint path Sink 16 P 1 : P 2 : Sink 17 P 3 : P 4 : Sink 18 P 5 : P 6 : Sink 19 54

56 Finding Disjoint Paths Get multicast rate mi for every pair (source,sink i ). Overall multicast rate is min(m i ). For all sinks, find min(mi ) disjoint paths Sinks First disjoint path Second disjoint path Sink 16 P 1 : P 2 : Sink 17 P 3 : P 4 : Sink 18 P 5 : P 6 : Sink 19 P 7 :

57 Finding Disjoint Paths Get multicast rate mi for every pair (source,sink i ). Overall multicast rate is min(m i ). For all sinks, find min(mi ) disjoint paths Sinks First disjoint path Second disjoint path Sink 16 P 1 : P 2 : Sink 17 P 3 : P 4 : Sink 18 P 5 : P 6 : Sink 19 P 7 : P 8 :

58 Finding Coding Nodes Obtain coding nodes from the intersection of paths corresponding to different sinks Sinks First disjoint path Second disjoint path Sink 16 P 1 : P 2 : Sink 17 P 3 : P 4 : Sink 18 P 5 : P 6 : Sink 19 P 7 : P 8 :

59 Finding Coding Nodes Obtain coding nodes from the intersection of paths corresponding to different sinks Sinks First disjoint path Second disjoint path Sink 16 P 1 : P 2 : Sink 17 P 3 : P 4 : Sink 18 P 5 : P 6 : Sink 19 P 7 : P 8 :

60 Finding Coding Nodes a 1 b 2 3 a+b a b a+b 5 a+b Obtain coding nodes from the intersection of paths corresponding to different sinks. 8 a 6 a+b a+b b 11 b a a 16 b a+b a+b a b a b 19 Sinks First disjoint path Second disjoint path Sink 16 P 1 : P 2 : Sink 17 P 3 : P 4 : Sink 18 P 5 : P 6 : Sink 19 P 7 : P 8 :

61 Finding Coding Nodes a 1 b 2 3 a+b a b a+b 5 a+b Ensure that at all times each edge passes either coded or not coded messages, not both at the same time, otherwise flip disjoint paths. 8 a 6 a+b a+b b 11 b a a a+b a+b b 16 b a b a

62 Finding Coding Nodes 1 a 1 a 2 3 b b 4 b a a b a+b a b b a+b a+b (a) (b) Counter example GEK not uniquely specified yi = xg i Gi needs to be invertible for every sink t i 61

63 Hierarchical Network Structure 1 D a b 3 C E F log. node I a P a+b T b B 4.9 A G H exit Hierarchical network structure Autonomous Systems (AS) set of routers under a single technical administration with a clearly defined routing policy [RFC 1771]. Routing protocols IGP (OSPF), EGP (BGP) OSPF builds Dijkstra trees not network coding aware 4400 ASs identified and 9 % of the flow between ASs accounts for 90 % of the bytes transmitted 62

64 Hierarchical Network Structure B I G C F E H A E D C A E H A I B G F C I H G F D B 0.5 D (a) (b) (c) Dijkstra trees vs. minimum spanning tree 63

65 Conclusions B-A network model inferred link loss from packet loss pdf of link losses single-source multicast Ant Colony Optimization to determine coding nodes solve the problem of GEK not uniquely determined by the coding links choice for certain topologies hierarchical network coding scheme 64

66 Further Work dependence of network coding on other link parameters solve the problem of GEK not uniquely determined by the coding links choice for certain topologies multiple-source multicast optimization of the ACO parameters effect of the fluctuation speed of the code on performance effect of topological changes (non-stationarity) of the network effect of the fluctuation speed of the code on performance effect of topological changes (non-stationarity) of the network hierarchical network coding scheme extend current research on achieving a hierarchical routing protocol that is network coding aware study the distortion of the degree distribution for LT-codes, given the LT symbols are subsequently coded within the network 65

67 Thank you! 66