Multicast Network Coding and Field Sizes

Transcription

1 Multicat Network Coding and Field Size Qifu (Tyler) Sun, Xunrui Yin, Zongpeng Li, and Keping Long Intitute of Advanced Networking Technology and New Service, Univerity of Science and Technology Beijing, Beijing, P R China Department of Computer Science, Univerity of Calgary, Calgary, Canada {qfun, longkeping}@utbeducn, {xunyin, zongpeng}@ucalgaryca Abtract In an acyclic multicat network, it i well known that a linear network coding olution over GF(q) exit when q i ufficiently large In particular, for each prime power q no maller than the number of receiver, a linear olution over GF(q) can be efficiently contructed In thi work, we reveal that a linear olution over a given finite field doe not necearily imply the exitence of a linear olution over all larger finite field Specifically, we prove by contruction that: (i) For every ource dimenion no maller than 3, there i a multicat network linearly olvable over GF(7) but not over GF(8), and there i another multicat network linearly olvable over GF(16) but not over GF(17); (ii) There i a multicat network linearly olvable over GF(5) but not over uch GF(q) that q > 5 i a Merenne prime plu 1, which can be extremely large Index Term Linear network coding, multicat network, field ize, Merenne prime I INTRODUCTION Conider a multicat network, which i a finite directed acyclic multigraph with a unique ource node and a et T of receiver Every edge in the network repreent a noiele tranmiion channel of unit capacity The ource generate ω data ymbol belonging to a fixed ymbol alphabet and will tranmit them to all receiver via the network The maximum flow, which i equal to the number of edge-dijoint path, from to every receiver i aumed to be no maller than ω The network i aid to be olvable if all receiver can recover all ω ource ymbol baed on their repective received data ymbol When the network ha only one receiver, it i olvable by network routing When T > 1, the paradigm of network routing doe not guarantee the network to be olvable The eminal paper [?] introduced the concept of network coding (NC) and proved that the network ha a NC olution over ome infinitely large ymbol alphabet It wa further hown in [1] that linear NC uffice to yield a olution when the ymbol alphabet i algebraically modeled a a ufficiently large finite field, and every intermediate node tranmit a linear combination of it received data ymbol over the ymbol field Since then, there have been extenive tudie on the field ize requirement of a linear olution for a multicat network From an algebraic approach, reference [2] firt howed that a multicat network ha a linear olution over GF(q) a long a the prime power q i larger than ω time the number T of receiver The requirement of q for the exitence of a linear olution over GF(q) i further relaxed by [3] to be larger than T, and uch a olution can be efficiently contructed by the algorithm propoed in [4] Meanwhile, the efficient algorithm in [5] i able to contruct a linear olution over GF(q) when q i no maller than T, and hence thi condition on q i lightly relaxed compared with the one in [3] and [4] Thi efficient algorithm require to initially identify, for each receiver in T, ω edge-dijoint path tarting from the ource and ending at it Denote by η the maximum number of path among the ω T that contain a common edge The parameter η i alway no larger than T By a more elaborate argument, the algorithm in [5] i refined in [6] uch that it can contruct a linear olution over GF(q) a long a q i no maller than η Denote by q min the minimum field ize for the exitence of a linear olution over GF(q min ), and by qmax the maximum field ize for the non-exitence of a linear olution over GF(qmax) (if the network i linearly olvable over every finite field, then qmax i not well defined and we et it to 1 a a convention) The algorithm in [5] implie that T i an upper bound on qmax For the pecial cae that the ource dimenion ω i equal to 2, the upper bound on qmax i reduced to O( T ) in [7] In both cae, the upper bound additionally guarantee the exitence of a linear olution over every GF(q) with q larger than the bound On the other hand, reference [8] and [9] independently contructed a cla of multicat network with q min lower bounded by Θ( T ) On any network in thi cla, a linear olution exit over every GF(q) with q no maller than q min, that i, qmax < q min Thee reult indicate that in many cae, ( ) A multicat network that i linearly olvable over a given finite field GF(q) i alo linearly olvable over every larger finite field To the bet of our knowledge, there i no explicit proof or diproof of the above claim for a general multicat network and for the cae q < T ; furthermore, all known multicat network tudied in the network coding literature atify q max < q min Although it ha been hown in [10] that the (4, 2)-combination network depicted in Fig 1 ha a nonlinear olution over a ternary ymbol alphabet but ha neither a linear nor a nonlinear olution over any ymbol alphabet of ize 6, it doe not hed light on diproving the claim ( ) becaue thi combination network ha a linear olution over every GF(q) with q 3 (See, for example, [11]) Moreover, in the cae ω = 2, a revealed in [9] and [7], there exit a linear olution over GF(q) if and only if there exit a (q 1)-vertex coloring in an appropriately defined aociated graph Since a q-vertex coloring in a graph alway guarantee a q -vertex coloring with

2 q > q in the ame graph, a multicat network with ω = 2 i linearly olvable over every GF(q ) with q q min Thi evidence eemed to add more upport for the correctne of the claim ( ) for an arbitrary multicat network Fig 1 The (4, 2)-combination network ha a unique ource with ω = 2 and 6 receiver at the bottom It ha a nonlinear olution over a ternary ymbol alphabet but no olution over any ymbol alphabet of ize 6 It ha a linear olution over every GF(q) with q 3 In the preent paper, we hall how by contructive proof that the claim ( ) i not alway true In particular, we how that there i a multicat network that i linearly olvable over GF(q min ) but not over GF(q max) for each of: (i) q min = 5, q max = 8; (ii) q min = 7, q max = 8; (iii) q min = 16, q max = 17; we how that for any poitive integer d with le than 17,425,170 (bae-10) digit, there i a multicat network with q min = 5 wherea q max > d The inight of our reult i that not only the field ize but alo the order of the proper multiplicative ubgroup in the ymbol field affect the linear olvability over the finite field A we hall ee, if a finite field doe not contain a large enough proper multiplicative ubgroup, or the complement of a large multiplicative ubgroup in the finite field i not large enough, it i poible to contruct a multicat network that i not linearly olvable over thi finite field but linearly olvable over a maller finite field In comparion, the characteritic of the ymbol field doe not appear a important in deigning example here a in the one in [12] which how the nonexitence of a linear olution for a general multi-ource multicat network The claical olvable network that i not linearly olvable, propoed in [12], make ue of two type of ubnetwork: one i linearly olvable only over a field with even characteritic wherea the other i linearly olvable only over a field with odd characteritic Conequently, the propoed network a a whole i not linearly olvable over any field, even or odd Our reult bring about a new facet on the connection between the ymbol field tructure and network coding olvability problem The remainder of thi paper i organized a follow Section II preent the fundamental reult that there exit multicat network uch that q min < qmax Section III dicue how large the gap qmax q min can be Section IV conclude the paper and lit ome intereting problem along thi new reearch thread in network coding theory Due to pace limit, all detailed proof of lemma in thi paper are omitted and they can be found in the technical report [13] II FUNDAMENTAL RESULTS Convention A multicat network i a finite directed acyclic multigraph with a unique ource node and a et T of receiver On a multicat network, for every node v, denote by In(v) and Out(v), repectively, the et of it incoming and outgoing edge Without lo of generality (WLOG), aume that Out() = ω (otherwie a new ource can be created, connected to the old ource with ω edge) For an arbitrary et N of non-ource node, denote by maxf low(n) the maximum number of edge-dijoint path tarting from and ending at node in N Each node t in the et T of receiver ha maxflow(t) = ω A linear network code (LNC) over GF(q) i an aignment of a coding coefficient k d,e GF(q) to every pair (d, e) of edge uch that k d,e = 0 when (d, e) i not an adjacent pair, that i, when there i not a node v uch that d In(v) and e Out(v) The LNC uniquely determine a coding vector f e, which i an ω-dim column vector, for each edge e in the network uch that: {f e, e Out()} form the natural bai of GF(q) ω f e = d In(v) k d,ef d when e Out(v) for ome v WLOG, we aume throughout thi paper that all LNC on a given multicat network have coding coefficient k d,e = 1 for all thoe adjacent pair (d, e) where d i the unique incoming edge to ome node A multicat network i aid to be linearly olvable over GF(q) if there i an LNC over GF(q) uch that for each receiver t T, the ω In(t) matrix [f e ] e In(t) over GF(q) i full rank Such an LNC i called a linear olution over GF(q) for the multicat network Denote by q min the minimum field ize for the exitence of a linear olution over GF(q min ), and by qmax the maximum field ize for the nonexitence of a linear olution over GF(qmax) Specific to a finite field GF(q), let GF(q) repreent the multiplicative group of nonzero element in GF(q) When q T, the efficient algorithm in [5] can be adopted to contruct a linear olution over GF(q) In the cae ω = 2, it ha been hown in [9], [7] that every linear olution over GF(q) can induce a (q 1)-vertex coloring in an appropriately aociated graph of the multicat network and vice vera Since every (q 1)-vertex coloring in a graph can alo be regarded a a (q 1)-vertex coloring for q > q, it in turn induce a linear olution over GF(q ) when q i a prime power Thu, every linear olution over a finite field can induce a linear olution over a larger finite field All thee are tempting fact for one to conjecture that when ω > 2 and q < T, a linear olution over a finite field might alo imply the exitence of a linear olution over a larger field The central theme of the preent paper i to refute thi conjecture in everal apect Theorem 1 A multicat network with ω 3 that i linearly olvable over a finite field i not necearily linearly olvable over all larger finite field

3 Proof: When ω = 3, thi theorem i a direct conequence of the lemma below Aume ω > 3 Expand the network N depicted in Fig 2(a) to a new multicat network N a follow Create ω 3 new node each of which ha an incoming edge emanating from the ource and an outgoing edge entering every receiver In N, every receiver ha the maximum flow from the ource equal to ω Conider an LNC over a given GF(q) By the topology of N, the coding vector for every edge that i originally in N i a linear combination of the coding vector for thoe edge in Out() that are alo in N Since the network N i linearly olvable over GF(q) with every q 7 except for q = 8 by the lemma below, o i the network N Lemma 2 Conider the multicat network depicted in Fig 2(a) Denote by e i, 1 i 9 the unique incoming edge to node n i For every et N of 3 grey node with maxflow(n) = 3, there i a receiver connected from it Thi network i linearly olvable over every finite field GF(q) with q q min = 7 except for q = q max = 8 In the network in Fig 2(a), oberve that every receiver i connected with three node n i, n j, n k where either i/3, j/3, k/3 are ditinct (uch a {, n 4, n 7 }) or two among i/3, j/3, k/3 are ame (uch a {, n 2, n 4 }) One can then check an LNC over GF(7) with [f e1 f e9 ] = qualifie to be a linear olution for the network Note that all nonzero entrie in thi matrix belong to a proper ubgroup of order 3 in the multiplicative group GF(7) The key reaon that reult in the network not linearly olvable over GF(8) i that there i not a proper ubgroup of order no maller than 3 in the multiplicative group GF(8) In general, it can be hown that the network in Fig 2(a) i linearly olvable over a finite field GF(q) if and only if there exit α i, β i, δ i GF(q), 1 i 3 ubject to α i α j, β i β j, δ i δ j, 1 i < j 3, {δ 1, δ 2, δ 3 } GF(q) \{ α i β j : 1 i, j 3} A detailed technical proof for Lemma 2 can be found in [13] On a olvable multi-ource multicat network, it i wellknown that linear network coding (with linearity in term of a more general algebraic tructure) i not ufficient to yield a olution The claical example in [12] to how thi conit of two ubnetwork, one i only linearly olvable over a field with even characteritic, wherea the other i only linearly olvable over a field with odd characteritic More generally, a procedure i introduced in [14] to contruct a (matroidal) multiource multicat network baed on a matroid uch that the contructed network i linearly olvable over GF(q) if and only if the matroid i repreentable over GF(q) Thi connection i powerful for deigning a number of non-linearly olvable network from a variety of intereting matroid tructure (See the Appendix in [15] for example,) and the network in [12] n 3 n 2 n 7 n 8 (a) n 9 n 4 n 5 n 6 n 5 1 n 20 Fig 2 For both multicat network, the ource dimenion ω i 3 There are totally 9 grey node in (a) and 20 grey node in (b) For every et N of 3 grey node that ha maxflow(n) = 3, there i a receiver connected from it, which i omitted in the depiction for implicity The network (a) i linearly olvable over every finite field GF(q) with q q min = 7 except for q = qmax = 8 The network in (b) i linearly olvable over every finite field GF(q) with q q min = 16 except for q = qmax = 17 i an intance under thi contruction However, the example of (ingle-ource) multicat network preented in thi paper cannot be etablihed by the procedure introduced in [14] Moreover, a a reult of Lemma 2 and the next lemma, the role of the characteritic of a finite field in the example deigned in the preent paper i not a important a in the example in [12] Lemma 3 Conider the multicat network depicted in Fig 2(b) There are in total 20 grey node For every et N of 3 grey node with maxf low(n) = 3, there i a receiver connected from it Thi network i linearly olvable over every finite field GF(q) with q q min = 16 except for q = q max = 17 A detailed proof of thi lemma can be found in [13] Similar to the cae in Fig 2(a), it can be hown that the network in Fig 2(b) i linearly olvable over a finite field GF(q) if and only if there i an aignment of {α i, β i } 1 i 5, {δ i } 1 i 10 from GF(q) ubject to and α i α j, β i β j, δ k δ l, (b) 1 i < j 5, 1 k < l 10, {δ 1,, δ 10 } GF(q) \{ α i β j : 1 i, j 5} For example, when q = 16, we can et α i = β i = ξ 3i for all 1 i 5, where ξ i a primitive element in GF(16), and et δ 1,, δ 10 to be the 10 element in GF(16) \{ξ 3i : 1 i 5} Such an aignment atifie the above condition and hence the network i linearly olvable over GF(16) The inight for the network to be linearly olvable over GF(16) rather than GF(17) i that there i uch a ubgroup in the multiplicative group GF(16) but not in GF(17) that (i) the ubgroup ha order no maller than 5; (ii) the complement of the ubgroup in the multiplicative group contain at leat 10 element Corollary 4 Given a multicat network with q min < q max, q min can be of either even or odd characteritic n 6 0

4 III GAP BETWEEN q min AND q max To the bet of our knowledge, all known multicat network tudied in the network coding literature have the property that q max < q min The reult in the lat ection reveal that it i poible for q min < q max However, both example illutrating thi fact have q max = q min + 1 A natural quetion next i how far away can q max be from q min The main reult in thi ection i to how that for ome multicat network, the difference q max q min can be extremely large Conider the Swirl Network 1 with ource dimenion ω 3 depicted in Fig 3 For every et N of ω grey node with maxf low(n) = ω, there i a non-depicted receiver connected from it Correponding to each node n i of in-degree 2, where 1 i ω, let d i, e i denote the two outgoing edge from it We next derive the neceary and ufficient condition on q for the Swirl Network to be linearly olvable over GF(q) n n 2 Fig 3 The Swirl network of ource dimenion ω 3 ha a receiver, which i not depicted, connected from every et N of ω grey node that ha maxflow(n) = ω Correponding to each node n i of in-degree 2, 1 i ω, let d i, e i denote the two outgoing edge from it The lemma below can be hown baed on the pecial topology of the Swirl network Refer to [13] for a detailed proof Lemma 5 The Swirl network i linearly olvable over a given GF(q) if and only if there i an LNC over GF(q) with coding vector for {d i, e i } 1 i ω precribed by n 4 [f d1 f e1 f d2 f e2 f dω f eω ] α = α , α ω 1 δ 1 δ 2 1 The name i due to both the hape of the network and the cloe connection to a matroid tructure referred to a the free wirl (See Chapter 14 in [15]) n 3 (1) where and α 1,, α ω 1, δ 1, δ 2 GF(q), δ 1 δ 2 and α j 1 for all 1 j ω 1, {δ 1, δ 2 } GF(q) \{( 1) ω γ 1 γ 2 γ ω 1 : (2) γ i {1, α j } for all 1 j ω 1} (3) Example Conider the Swirl network with ω = 6, and an L- NC over GF(5) with the coding vector for {d 1, e 1, d 6, e 6 } precribed by = [f d1 f e1 f d6 f e6 ] Apparently condition (2) i atified by thi LNC Since {1, 4} i a ubgroup of GF(5), it i cloed under multiplication by element in it Moreover, {2, 3} = GF(5) \{( 1) 6 1, ( 1) 6 4}, condition (3) i alo atified Thu, thi LNC over GF(5) qualifie a a linear olution On the other hand, ince GF(8) doe not have a proper ubgroup other than {1}, it i not difficult to check that for arbitrary α 1,, α 5 GF(8) \{1}, the et {γ 1 γ 2 γ 5 : γ i {1, α i } for all 1 i 5} contain at leat 6 element Thu, there are not enough ditinct element in GF(8) to aign for α 1,, α 5 and δ 1, δ 2 ubject to condition (2) and (3) Hence, the network i not linearly olvable over GF(8) The argument in the example above can imply be generalized to derive the linear olvability of the Swirl network with general ource dimenion ω over a given GF(q) Firt, aume that the order of the multiplicative group GF(q) i not prime Thi implie q 5 Then there i a ubgroup G in GF(q) with G 2, and GF(q) \{( 1) ω g : g G} contain no le than 2 element Let a be an element in G not equal to 1 and b, c be two ditinct element in GF(q) \{( 1) ω g : g G} We can aign α i = a for all 1 i ω 1 and δ 1 = b, δ 2 = c Such an aignment obey condition (2) and (3) Hence, the network i linearly olvable over GF(q) Next, aume that the order of GF(q) i a prime, in which cae q can alway be written in the form of 2 p for ome prime p We hall how that for any 1 a 1, a 2,, a n 2 p 2, the et S n = {b 1 + +b n mod 2 p 1 : b i {0, a i }} contain at leat min{n+1, 2 p 1} element Thi i obviouly true when n = 1 Aume that it i true for n = m and conider the cae n = m + 1 Note that S n = S m {an + b mod 2 p 1 : b S m } If S m 2 p 1, then o i S n Aume that 2 p 1 > S m m + 1 Since a n 0 and 2 p 1 i a prime, there i at leat one element in {a n + b mod 2 p 1 : b S m } that i not in S m Thu, S n S m + 1 n + 1 A a conequence, we conclude that for any aignment of α 1 = ξ a 1,, α ω 1 = ξ a ω 1, where ξ i a primitive element in GF(2 p ) and 1 a 1,, a ω 1

5 2 p 2, the et {γ 1 γ ω 1 : γ i {1, ξ a i }} contain at leat ω element In order to further uccefully aign δ 1, δ 2 ubject to (2) and (3), the ize of GF(2 p ) ha to be at leat ω + 3 We have proved the following theorem Theorem 6 The Swirl network with ω 3 ha q min = 5 It i not linearly olvable over thoe GF(2 p ) when 2 p ω + 2 and 2 p 1 i a prime Recall that a prime integer in the form 2 p 1 i known to be a Merenne prime Then q max for the Swirl network with ω 3 i equal to one plu the larget Merenne prime no larger than ω + 1 While whether there are infinitely many Merenne prime i till an open problem, the 48 th known Merenne prime (which i alo the larget known prime) found under the GIMPS project (See [16]) ha a length of 17,425,170 digit under bae 10 Thu, when ω i ufficiently large, the difference q max q min for the Swirl network i o enormou a to having ten of million of digit Corollary 7 If there are infinitely many Merenne prime, then there are infinitely many multicat network with q max > q min, and moreover, the difference q max q min can tend to infinity IV SUMMARY In an acyclic multicat network, if there i a linear olution over GF(q), could we claim that there i a linear olution over every GF(q ) with q q? It would be tempting to anwer it poitively becaue by the reult in [5], the claim i correct when q i no maller than the number of receiver and moreover, a a conequence of the reult in [7], the poitive anwer i affirmed for the pecial cae that the ource dimenion of the network i equal to 2 In the preent paper, however, we how the negative anwer for general cae by contructing everal clae of multicat network with different emphai Thee network are the firt one dicovered in the network coding literature with the property that q max, the maximinum field ize for the nonexitence of a linear olution over GF(q max), i larger than q min, the minimum field ize for the exitence of a linear olution over GF(q min ) The inight of variou exemplifying network etablihed in the preent paper i that not only the field ize of GF(q), but alo the order of the proper multiplicative ubgroup of GF(q) affect the network linear olvability over GF(q) The reult in thi paper bring about a new thread on the fundamental tudy of linear network coding We end thi paper by propoing a number of open problem: For a multicat network, what i the mallet prime power q larger than qmax (uch that the network i linearly olvable over all GF(q ) with q q)? Can the gap qmax q min > 0 tend to infinity? Are there infinitely many prime power pair (q, q ) with q < q uch that each (q, q ) correpond to (q min, qmax) of ome multicat network? If a multicat network i linearly olvable over uch a GF(q) that GF(q) doe not contain any proper multiplicative ubgroup other than {1}, i it linearly olvable over all larger finite field than GF(q)? If a multicat network i linearly olvable over GF(2), i it linearly olvable over all finite field? If a multicat network i linearly olvable over both GF(2) and GF(3), i it linearly olvable over all finite field? ACKNOWLEDGMENT The author would like to thank Sidharth Jaggi for helping identify and motivate the problem tudied in thi work REFERENCES [1] S-Y R Li, R W Yeung, and N Cai, Linear network coding, IEEE Tran Inf Theory, vol 49, no 2, pp , Feb, 2003 [2] R Koetter and M Médard, An algebraic approach to network coding, IEEE/ACM Tran Netw, vol 11, No 5, pp , Oct, 2003 [3] T Ho, D Karger, M Médard, Network coding from a network flow perpective, Int Symp Information Theory, Yokohama, Japan, Jun 2003 [4] N J A Harvey, D R Karger, and K Murota, Determinitic network coding by matrix completion, Annual ACM-SIAM Sympoium on Dicrete Algorithm, 2005 [5] S Jaggi, P Sander, P A Chou, M Effro, S Egner, K Jain, and L Tolhuizen, Polynomial time algorithm for multicat network code contruction, IEEE Tran Inf Theory, vol 51, no 6, pp 1973C1982, Jun 2005 [6] R W Yeung, A unification of network coding and routing, Information Theory and Application Workhop (ITA), Feb, 2012 [7] C Fragouli and E Soljanin, Information flow decompoition for network coding, IEEE Tran Inf Theory, vol 52, no 3, pp , Mar 2006 [8] M Feder, D Ron, and A Tavory, Bound on linear code for network multicat, Electronic Colloquium on Computational Complexity (ECCC), Rep 33, pp 1C9, 2003 [9] A Raala Lehman and E Lehman, Complexity claification of network information flow problem, 14th Annual Sympoium on Dicrete Algorithm (SODA), New Orlean, LA, Jan 2004 [10] R Dougherty, C Freiling, and K Zeger, Linearity and olvability in multicat network, IEEE Tran Inf Theory, vol 50, no 10, pp , Oct 2004 [11] S-Y R Li, Q T Sun, and Z Shao, Linear network coding: theory and algorithm, Proc IEEE, vol 99, no 3, pp , Mar, 2011 [12] R Dougherty, C Freiling, and K Zeger, Inufficiency of linear coding in network information flow, IEEE Tran Inf Theory, vol 51, no 8, pp , Aug 2005 [13] Q T Sun, X Yin, Z Li, and K Long, Multicat network coding and field ize, arxiv: [14] R Dougherty, C Freiling, and K Zeger, Network, matroid, and non- Shannon information inequalitie, IEEE Tran Inf Theory, vol 53, no 6, pp , Jun 2007 [15] J G Oxley, Matroid Theory, New York: Oxford Univ Pre, 2nd edition, 2006 [16] Great Internet Merenne Prime Search (GIMPS) Project,