Network coding for online and dynamic networking problems. Tracey Ho Center for the Mathematics of Information California Institute of Technology

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1 Network coding for online and dynamic networking problems Tracey Ho Center for the Mathematics of Information California Institute of Technology 1

2 Work done in collaboration with: Harish Viswanathan: Bell Laboratories Ralf Koetter and Niranjan Ratnakar: Coordinated Science Laboratory, University of Illinois Urbana-Champaign 2

3 Network coding for static single multicast in wired networks Static multicast network connection problem defined on graph (V, E) by given set of sinks T V, source rates λ i, link capacities R ab and link costs C ab (g) Optimal link usage g ab given by optimization with linear constraints [LMHK04]: minimize C ab (g ab ) (a,b) E subject to λ i = λ i = i b:(i,b) E a:(a,β) E f β ib a:(a,i) E f β aβ β T f β ai i V, β T, β i 0 f β ab g ab R ab (a, b) E, β T 3

4 Network coding for static single multicast in wired networks Solution is a union of flows A network code can be constructed over this solution by deterministic or random methods 4

5 Extensions of practical interest static time-varying, packetized, bursty wired wireless single multicast session non-multicast/multiple sessions 5

6 From flows to packets analysis of a dynamic solution static case unit rate source processes unit capacity links coding node dynamic case exogenous source packets coded/forwarded packets formation of a coded packet design of dynamic algorithms? 6

7 c c c c c From wired to wireless flow model must account for wireless multicast advantage: a single wireless link can have multiple destinations, but all receive identical information model each wireless link using a virtual bottleneck node [WuChouKung04] constrain sum of each receiver s flows, via different destination nodes of a wireless link, to be at most the flow on the link [LunMédHoKoe04, HoVis05] 7

8 From wired to wireless (cont d) Minimum energy wireless flow formulation [LunMédHoKoe04]: minimize C ab (g ab ) (a,b) E subject to λ i = f β ib b:(i,b) E a:(a,i) E i V, β T, β i λ i = f β aβ β T i a:(a,β) E f β ab 0 (g ac fac) β 0 (a, b) E, β T c:(a,c) E,(a,c) (a,b) other issues: interference, physical layer cooperation f β ai 8

9 Non-multicast [DouFreZeg05] (1) A scalar linear solution over any ring of characteristic 2 No linear solution for any vector dimension over a finite field with odd characteristic By equating coefficients and manipulating,we have: M i invertible for all i 9

10 Non-multicast [DouFreZeg05] (2) A scalar linear solution over any ring where 2 is a unit No linear solution for any vector dimension over a finite field with characteristic 2 By equating coefficients and manipulating,we have: M i invertible for all i 10

11 Non-multicast [DouFreZeg05] (2) A scalar linear solution over any ring where 2 is a unit No linear solution for any vector dimension over a finite field with characteristic 2 For any value of vector c, there exist values for a,b,d,e such that all six inputs to node 43 are zero 11

12 Non-multicast [DouFreZeg05] (3) 12

13 Multiple multicast/unicast sessions optimal inter-session network coding is an open problem intra-session network coding for multiple multicasts/unicasts improves capacity and optimization complexity over uncoded operation dynamic, bursty settings provides guidance for wireless physical/mac layer choices online network coding across unicast sessions additional flexibility for accommodating more sessions/handling congestion exploits wireless multicast advantage 13

14 Dynamic intra-session network coding for multiple multicast/unicast sessions Problem statement Multi-hop wired/wireless networks with bursty sources, variable channel states, queued packets Multiple multicast/unicast sessions Each session has one or more independent or correlated sources, and one or more sinks Network coding within sessions Problems: network coding, routing, scheduling among packets, power control/scheduling among transmitters, rate control for correlated sources Maximize throughput, minimize cost 14

15 Background back pressure routing introduced for multicommodity unicast problems by [TasEph92,AweLei93] routing and scheduling decisions based on lengths of queues Ui c (t) for each commodity c at each node i supply at sources and removal at sinks create gradients down which packets flow stabilizes any stabilizable set of input rates multicast routing algorithms: predetermined set of multicast trees [SarTas02] a queue for every subset of every session s sinks at each node [AweBriSch02] 15

16 Back pressure multicast with network coding Each node has one queue per sink (independent sources case) or per (source, sink) pair (correlated sources case) local decisions and control: nodes locally determine network coding and packet scheduling for correlated sources, rate control by sinks completely decentralized algorithms for wired networks centralized scheduling and power allocation for interfering wireless transmitters Throughput-optimal given constraint of not coding across sessions 16

17 Back pressure and optimization queue lengths can be interpreted as dual variables (prices) in classical subgradient search methods for multicommodity flow [NeeModRoh05] besides throughput, can optimize cost [Neely05] or achieve fairness [ErySri05] 17

18 Virtual queues Windows of packets that can be coded together Larger windows Handles more burstiness May increase delay Larger code description overhead Packets are replicated in virtual queues for each sink of the session Sink 1 Sink 2 Sink 3 18

19 Case 1: wired network, independent sources Session 1 Sink 1 Sink 2 Active link in this slot In active link in this slot Sink 3 RNC Sink 2 Sink 1 Sink 3 Session 2 Sink 4 RNC Zero fill up to allowed rate if other queue is longer No network coding across sessions Session 1 is chosen as it has a larger sum (over sinks) of positive queue size differences Session 1, Sink 1 queue is not a part of the network code combining as it is less than the corresponding queue at destination of link. 19

20 Algorithm for wired networks, independent sources local decisions based on lengths of virtual queues U cβ i (t) at each node i for each sink β of each session c at each time slot [t, t + T), for each link (a, b), choose one session c ab = arg max c β T c max ( U cβ a ) U cβ b, 0 send at link capacity a random linear combination of data from session c ab queues at a corresponding to all sinks β for which U c ab β a U c ab β b > 0 each sink decodes when it has enough packets by inverting linear combinations 20

21 Case 2: wireless network, independent sources Additional elements: wireless links wireless multicast advantage physical layer cooperation consider generalized links (a, Z) from an originating node a to a set Z of receiving nodes algorithm must choose which queues to update at receiving nodes d Z interference among transmitters; link rates determined by channel states as well as transmit powers algorithm must determine powers/schedules for transmission 21

22 Wireless links 22

23 Network coding on a given wireless link Sink 1 Multiple destination nodes Session 1 Sink 2 Sink 2 Sink 3 RNC Sink 3 Session 2 Sink 1 Sink 4 RNC Zero fill up to allowed rate if other queue is longer No network coding across sessions Session 1 is chosen as it has a larger sum (over sinks) of positive maximum (over destinations of the link) queue size differences For a link (a,z), we denote this sum by 23

24 Dealing with interference transmit scenarios 24

25 Algorithm for wireless, independent sources In each time slot [t, t + T), Scheduling: for each potential link (a, Z), choose one session c az = arg max c β T c max ( ( max Ua cβ b Z ) ) U cβ b, 0 25

26 Algorithm for wireless, independent sources (cont d) Power control: based on channel state S(t), a power allocation P is chosen to maximize µ az (P, S(t))waZ a,z where µ az (P, S(t)) is the link rate and waz = ( ( max U c az a β b Z β T c az max ) ) U c az β b, 0 26

27 Algorithm for wireless, independent sources (cont d) Network coding: for each link (a, Z), a random linear combination ( of data corresponding ) to each sink β for which max b Z U c az a β U c az β b > 0 is sent at the available rate each destination node d Z associates the received information with the virtual( queues corresponding ) to sinks β for which d = arg max b Z U c az a β U c az β b 27

28 Heuristics for choosing transmission scenarios Choices guided by weights w l based on queue lengths, which may also be useful for non-dynamic optimization Form a set of transmission scenarios with transmit powers fixed at maximum (optimal at low SNRs) Build each scenario S starting with a different link Greedily add link l with maximum µ l w l Scenario is completed when potential increase from adding any link is below some threshold Choose the transmission scenario with maximum l S µ lw l Similar to heuristics used in [MukVis04,WuChoZhaJaiZhuKun05] that try to find scenarios maximizing l S µ l 28

29 Minimizing energy Approach of [Neely05] for multiple unicasts extends to multicast with a network coding Goal: minimize energy while stabilizing any stabilizable rates Objective function: E { a V g a ( Z P az (t)) where g a (p) is any convex increasing cost function of the power p used by node a Power control: choose power allocation P to maximize [ µ az (P, S(t))waZ V g a ( ] P az ) a Z Z Parameter V trades off between condition and power optimality } 29

30 Case 3: correlated sources Slepian-Wolf problem [SW73]: separate encoding of correlated sources for a single receiver R 1 X 1. X 2. R 2 achievable rate region: R 1 H(X 1 X 2 ); R 2 H(X 2 X 1 ); R 1 + R 2 H(X 1, X 2 ) generalization to arbitrary networks achieved by random linear network coding for single multicast session with given link capacities [HoMéEffKoe04] 30

31 Case 3: correlated sources Problem: when there are multiple sessions sharing the network, how to choose data rates from correlated sources (which are to some extent interchangeable)? Approach: each node i maintains a virtual queue for each (source, sink) pair (α, β) of each session c each sink β drains the virtual queues of sources α at rates A cαβ out (t) chosen based on lengths of virtual queues at β token queue associated with each virtual queue at sinks keeps track of deficits in received rate caused by source burstiness and link variability reverse of previous algorithms: uses the difference V cαβ i (t) between maximum queue size M and virtual queue length 31

32 Algorithm for correlated sources In each time slot [t, t + T), Outflow rate allocation: Each sink β chooses outflow rates {A cαβ out (t) α S c } to minimize V cαβ β (t + )A cαβ out (t) subject to α S A cαβ out (t) T A cαβ out (t) T α H(S (S c \S )) + ǫ ˆǫ c, S S c λ c α c, α S c. and removes data accordingly from its virtual queues; when queues become empty the deficit is reflected in the token queues 32

33 Algorithm for correlated sources (cont d) Inflow rate control: Each source α adds random linear combinations of exogenous data to its virtual queues at the exogenous data rate, up to a maximum amount of Vα cαβ (t). Network coding, scheduling and power control: analogous to case of independent sources Token queue adjustments: for each virtual queue at the sinks, an amount ( ) Ŵ cαβ (t) = min V V cαβ β (t ), W cαβ (t ) is removed to repay deficit (or, if negative, is added to the deficit) 33

34 Necessary condition (R az ) and {f cαβ abz, gc az, λcαβ } satisfying: Z,a Z f cαβ aiz Z,b Z α S c,b Z ( c f cαβ abz 0 a, b, Z, c, α S c, β T c λ cαβ c, α S c, β T c, i = α f cαβ ibz = λ cαβ c, α S c, β T c, i = β 0 c, α S c, β T c, i / {α, β} f cαβ abz g c az a, Z, c, β T c g c az ) α S λ cαβ (R az ) for some (R az ) Γ λ cαβ λ c α c, α S c, β T c H(S (S c \S )) c, S S c, β T c 34

35 Sufficient condition For some ǫ > 0, (R az ) and {f cαβ abz, gc az, λcαβ } satisfying: Z,a Z f cαβ aiz Z,b Z α S c,b Z ( c f cαβ abz 0 a, b, Z, c, α S c, β T c λ cαβ c, α S c, β T c, i = α f cαβ ibz = λ cαβ c, α S c, β T c, i = β 0 c, α S c, β T c, i / {α, β} f cαβ abz g c az a, Z, c, β T c g c az ) α S λ cαβ (R az ) for some (R az ) Γ λ cαβ λ c α c, α S c, β T c H(S (S c \S )) + ǫ c, S S c, β T c 35

36 Theorem The back-pressure algorithm with V = TBN ˆǫ M = V + Nµ out max, where B = τ max 2 ( 1 N ( E i,c +(µ out max) 2 + (µ in max) 2 ) A cαβ in T, 0 < ˆǫ < ǫ and ) N σ maxµ out maxµ in max is stable and asymptotically achieves the desired multicast rates. 36

37 Proof outline Compare with a randomized policy that depends on the (unknown) flow solution At t = (kt), token queue adjustments for both policies: for each virtual queue at the sinks, an amount ( ) Ŵ cαβ (t) = min V V cαβ β (t ), W cαβ (t ) is removed to repay token queue deficit (or, if negative, is added to the deficit), i.e. V cαβ β (t + ) = V cαβ (t ) + Ŵ cαβ (t) β W cαβ (t + ) = W cαβ (t ) Ŵ cαβ (t) 37

38 Proof outline Outflow rate allocation: back pressure policy: each sink β chooses {A cαβ out (t) α S c } to minimize V cαβ β (t + )A cαβ out (t) subject to α S A cαβ out (t) T A cαβ out (t) T α H(S (S c \S )) + ǫ ˆǫ c, S S c λ c α c, α S c randomized policy: A cαβ out (t) = T(λ cαβ ˆǫ) 38

39 Inflow at sources: Proof outline back pressure policy: A cαβ in (t) = Tλc α randomized policy: A cαβ in (t) = Tλcαβ add a corresponding amount of data to source queues up to maximum capacity Scheduling and power control: back pressure policy: choose so as to maximize µ az (P, S(t))waZ a,z where µ az (P, S(t)) is the link rate and waz = ( ( max U c az a β b Z β T c az max ) ) U c az β b, 0 39

40 randomized policy: choose so as to achieve average rates power allocation: for each S, choose allocation from a finite set according to a set of probabilities for each link (a, Z), one session c is chosen randomly with g c az probability c gc az each of its sinks β is independently chosen with probability α,b fcαβ abz g c az for each chosen sink, one (source, destination node) pair (α S c, b Z) is chosen with probability f cαβ abz α,b fcαβ abz 40

41 Proof outline Network coding: send a random linear combination of data corresponding to the chosen (session, source, sink, destination)-tuples (c, α, β, d) on each link (a, Z) at its instantaneous rate (back pressure policy) c or at a fraction gc az R az of its instantaneous rate (randomized policy) up to maximum capacity of destination queue 41

42 Evolution of queues in interval (t,t + T): V cαβ α (t + T) max V α cαβ (t) A cαβ in (t) T a,z aαz (t), 0 µ cαβ +T b,z V cαβ i (t + T) max µ cαβ αbz (t) V cαβ i (t) T a,z aiz (t), 0 µ cαβ +T b,z µ cαβ ibz (t) i / {α, β} 42

43 Evolution of sink queues in interval (t,t + T): V cαβ β ((t + T) ) max V cαβ β (t + ) T a,z aβz (t), 0 µ cαβ +T b,z µ cαβ βbz (t) + Âcαβ out (t) W cαβ ((t + T) ) = W cαβ (t + ) + A cαβ out (t) Âcαβ out (t) 43

44 Lyapunov functions For each (c, α S c, β T c ), define the Lyapunov function L cαβ (V cαβ, W cαβ ) = i (V cαβ i ) 2 + 2V W cαβ and let L cαβ (t) = L cαβ (V cαβ (t), W cαβ (t)). 44

45 Evolution of Lyapunov functions At t = kt, L cαβ (t + ) L cαβ (t ) = Ŵ cαβ (t) (Ŵ cαβ (t) + 2V cαβ β (t ) 2V ) 0 45

46 Evolution of Lyapunov functions (cont d) For (t, t + T), E{L cαβ ((t + T) ) L cαβ (t + ) V (t + ), W(t + )} c,α S c,β T c 2T 2 BN 2T V cαβ i (t)e µ cαβ aiz c,α S c,β T c i a,z b,z { } { }] +Vα cαβ A cαβ in (t)e V cαβ β (t + A cαβ out )E T T µ cαβ ibz = D(V (t + ), W(t + )) where B = τ max 2 ( 5 2N c,α (λ c α) 2 + 4σ maxµ in maxµ out max N + (µ out max) 2 + (µ in max) 2 ) 46

47 Comparing the policies Token queue adjustment at t = kt identical for both policies For (t, t + T), D = K 2T + c,α,β = K 2T + c,α,β i,c Vα cαβ (t)e a,b,z V cαβ i α S c,β T c c,β Vα cαβ (t)e { } A cαβ in T { E µ cαβ abz { A cαβ in T } (t)e } ( c,α,β V cαβ a c,α,β a,z V cαβ β (t)e µ cαβ aiz b,z µ cαβ ibz { } A cαβ out T ) (t) V cαβ b (t) V cαβ β (t)e { } A cαβ out T 47

48 Comparing the policies The three terms of the preceding expression involve disjoint sets of policy-dependent variables The back-pressure policy maximizes each of them subject to constraints which are also satisfied by the randomized policy Thus, D backpressure (V (t + ), W(t + )) D randomized (V (t + ), W(t + )) 48

49 Evolution over one time step For the randomized policy, E{µ cαβ abz (t)} = fcαβ abz a, b, Z, c, α S c, β T c A cαβ in = Tλ cαβ c, α S c, β T c A cαβ out = T ˆλ cαβ = T(λ cαβ ˆǫ) c, α S c, β T c independently of V (t), W(t) Thus, E { L cαβ ((t T) ) L cαβ (t ) } D backpressure D randomized 2T 2 BN 2Tˆǫ c V cαβ β α S c,β T c (t) 49

50 Stability of back pressure policy If W cαβ (t + ) = 0 for all (c, α, β), L cαβ (t ) NM 2 t (t, t + T) If W cαβ (t + ) > 0 for some (c, α, β), then V cαβ β (t + ) = V, and E { L cαβ ((t T) ) L cαβ (t ) } 2T 2 BN 2TˆǫV 0 for V = TBN ˆǫ 50

51 Coding across unicast sessions wired 51

52 Coding across unicast sessions wired 52

53 Coding across unicast sessions wired 53

54 Coding across unicast sessions wired 54

55 Coding across unicast sessions wireless 55

56 Coding across unicast sessions wireless 56

57 Coding across unicast sessions wireless 57

58 Coding across sessions advantages in capacity, online incremental solutions how to find the minimum cost way to add a connection? additional flow requirements depend on which existing flow(s), if any, are coded with new flow use butterfly structure as a basic building block 58

59 re m e d y re q u e s t re m e d y re q u e s t Wired butterfly structure 59

60 Linear constraints for adding a new connection x e (i): usage of link e by flow i p e (m n, u): poison on flow n originating from merger with flow m at node u q e (m n, u): associated remedy request r e (m n, u): associated remedy r e (m n, u) is a positive real flow; p e (m n, u) and q e (m n, u) are negative virtual flows completing loop in opposite direction to real flows 60

61 Linear constraints for adding a new connection Src 1 x(1) = 1, q(1 2) = 1 r(1 2) = 1 Rcv 2 x(2) = 1, p(1 2) = 1 61

62 Linear constraints for adding a new connection New flow of size r n from source s n to sink t n x e (n) 0 v / {s n, t n } x e (n) = r n v = t n e: d(e)=v e: o(e)=v r n v = s n Conservation of flow in loops (p e (m n, u) + q e ( ) + r e ( )) = e: d(e)=v e: o(e)=v (p e ( ) + q e ( ) + r e ( )) Virtual flows are associated with corresponding real flows x e (n) + (p e (m n, u) + q e (n m, u)) 0 u m 62

63 Linear constraints for adding a new connection p e (m n, u) generated at u and diminishes going downstream p e (m n, u) p e (m n, u) e: d(e)=u p e (m n, v) e: o(e)=u e: d(e)=v e: o(e)=v p e (m n, v) v u q e (m n, u) generated at u and diminishes going upstream q e (m n, u) q e (m n, u) e: d(e)=u q e (m n, v) e: o(e)=u e: d(e)=v e: o(e)=v q e (m n, v) v u 63

64 Linear constraints for adding a new connection Poison generated symmetrically p e (m n, o(e)) = p e (n m, o(e)) Link capacity constraint n max(p e (m n, u), p e (n m, u)) + x e (i) u m + u i=1 (r e (m n, u) + r e (n m, u)) z e m 64

65 Wireless multicast advantage 65

66 Linear constraints for adding a new connection wireless multicast advantage Poison generation Link capacity constraints ( max u m b Z p ubz (m n, u) = b Z p ubz (n m, u) p abz (n m, u) p abz (m n, u), b Z b Z + [ ( max x az (m), ) r abz (m n, u) m u b ( + max x az (n), )] r abz (n m, u) u b ) z abz 66

67 Competitive online algorithms we extend the approach of Awerbuch et al. [AAP93] on throughput-competitive online routing to network coding sequentially consider unicast session requests β j, j = 1, 2,...,k, each specifying source s j, destination d j, duration, bandwidth and an associated reward goal is to maximize total reward (e.g. throughput) online algorithm either allocates sufficient capacity to accommodate request or rejects request; no rerouting Evaluate online algorithm by competitive ratio: supremum, over all possible input sequences, of the ratio of the optimal offline reward to the online reward 67

68 m links in network Notation and assumptions reward ρ j of j th request bounded within range [1, R] consider single period and unit bandwidth requests for simplicity capacity u e of each link e bounded within range [log µ, P], where P 1 and µ = 2mPR 68

69 Online routing approach of Awerbuch et al. Online strategy: assign link costs exponential in fractional usage accept a connection only if worthwhile compared to costs Competitive ratio of O(log n) achieved, where n is the number of nodes 69

70 Incorporating network coding allow up to two connections to share capacity by being coded together associate with each edge e a cost, after considering j 1 requests, based on fraction λ e,x (j) of its capacity already assigned to x = 1, 2 or more flows: where 0 α < 1 c e (j) = 1 m µλ e,1(j)+αλ e,2 (j)(1 1 ue λ e,1(j)) + 70

71 Online strategy algorithm accommodates request β j iff it can find a solution such that (a e,1 + αa e,2 ) c e (j) ρ j u e e a e,1 + αa e,2 u e a e,2 u e λ e,1 (j) λ e,2 (j) e & 1 log µ e where a e,x is the capacity of e used in adding β j that becomes shared by x = 1, 2 connections 71

72 Finding a low-cost solution for a connection by linear programming based on the equations described earlier, or by heuristics, e.g. searching for remedy paths within limited local areas, greedy Dijkstra-like approach 72

73 Performance A (2 log µ + 3) competitive ratio is obtained as long as each connection added uses a solution costing no more than the lowest cost routing-only solution This is order-optimal [AAP93] 73

74 Enforcing capacity constraints When λ e,1 (j) > 1 1 u e, using the assumption that log µ u e P, we have 1 c e (j) > µ1 u e mu e µ 2mP = R, so capacity will not be exceeded log µ 74

75 Lower bounding online performance If request β j is admitted, (c e (j + 1) c e (j)) e e e c e (j) ρ j log µ ( ) µ a e,1 +αa e,2 ue 1 ( ) ae,1 + αa e,2 c e (j) log µ Summing over set A of all accepted requests, ρ j log µ ( c e (k + 1) 1 ) m j A e = e where k is the total number of requests. c e (k + 1) 1, u e 75

76 Upper bounding off-line performance If request j is admitted by the off-line algorithm but not the online algorithm, ρ j < e P j c e (j) u e e P j c e (k + 1) u e where P j is the path from s j to t j that forms part of the set of links used by the off-line algorithm to accommodate β j Total reward from the set Q of these requests is at most j Q e P j c e (k + 1) u e = e: e P j, j Q c e (k + 1) j Q: e P j 1 u e 2 e c e (k + 1) 76

77 Competitive ratio Off-line reward is at most j Qρ j + j Aρ j 2 e c e (k + 1) + j A ρ j 2 + (2 log µ + 1) j Aρ j Competitive ratio is 2 log µ

78 Conclusions and further work Dynamic back pressure-based algorithms for intra-session multicast with independent or correlated sources Online algorithm for sequentially phasing in multiple unicast connections Further work: fully decentralized dynamic algorithms for wireless evaluation and comparisons on typical or random graphs randomized online network coding algorithms 78